Research ArticleEvolutionary Game Analysis of BIM Adoption amongStakeholders in PPP Projects

Chong Jia Ruixue Zhang and Dan Wang

School of Business Administration Liaoning Technical University Huludao 125105 China

Correspondence should be addressed to Chong Jia 826108193qqcom

Received 18 March 2021 Revised 24 July 2021 Accepted 7 August 2021 Published 18 August 2021

Academic Editor Daniele Salvati

Copyright copy 2021 Chong Jia et al +is is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

With the development of building information technology Building Information Modeling (BIM) has become an important wayto effectively solve the cross-organization information collaboration of Public-Private Partnership (PPP) projects and how topromote the adoption of BIM in PPP projects has become a realistic problem to be solved urgently +is study discusses theadoption of BIM among stakeholders in PPP projects based on prospect theory and evolutionary game theory A tripartiteevolutionary game model including governments social capitals and contractors is established +e behavioral evolutionmechanism of each stakeholder on BIM adoption is explored by analyzing the evolutionary equilibrium and the key influencingfactors of equilibrium strategy are analyzed by using numerical simulation+e results demonstrate that first the degree of the costto all stakeholders involved in the adoption of BIM as well as the punishment for governmentsrsquo passive promotion of BIM thepunishment for social capitalsrsquo passive adoption of BIM and the reward for contractorsrsquo active application of BIM are the keyfactors affecting evolutionary stability Second according to prospect theory the main stakeholders usually make decisionsthrough subjective judgment and perceived value which ultimately lead to deviation in their behaviors +e deviations will hinderthe establishment of ESS point (1 1 1) and make the system difficult to converge to the optimal state Finally from the perspectiveof governments social capitals and contractors countermeasures and management implications are put forward to effectivelypromote the adoption of BIM in PPP projects

1 Introduction

Public-Private Partnership (PPP) can not only save the costsfor the government but also inject the private sector effi-ciency in the government sector domain [1] It has been animportant way to reshape the infrastructure constructionindustry in China [2 3] Different from general constructionprojects PPP projects always involve a complex network ofpublic and private partners (designers contractors andoperators) beginning in the planning stage of the projects[4 5] with each of these partners having different prioritiesand leading to conflicting objectives [6] During the longcontract period problems such as information asymmetryand poor communication often occur [7ndash10] How to realizethe effective cross-organizational collaboration betweenstakeholders of PPP projects has become one of the im-portant factors affecting the performance and even thesuccess or failure of PPP projects [11]

With the development of informatization in construc-tion industry Building Information Modeling (BIM) hasbeen widely used in the whole process of constructionprojects and reshaping the production mode of the con-struction industry [12] BIM is not a simple 3D modelingprogram Its biggest feature is that it can carry a largeamount of information of architectural models and can beapplied in multiple fields +e advantages of BIM whichconsidered to be the best way to solve the problem ofcross-organizational collaboration in construction projectscoincide well with the needs of PPP project life cyclemanagement [13] +e key issue to promote the adoption ofBIM in PPP projects is the behavior evolutionary among themajor stakeholders such as government social capital andcontractor It can be seen that they usually have the commonvision to achieve the project objectives but not the same inthe collaborative adoption of BIM [14] Some scholars haveresearched the adoption of BIM in construction industry

HindawiComplexityVolume 2021 Article ID 5553785 14 pageshttpsdoiorg10115520215553785

based on evolutionary game theory +ey mainly build theevolutionary game model of BIM adoption between two orthree main stakeholders and analyzed the evolutionarymechanism or influencing factors [15ndash18] But in existingliterature there is little research on BIM adoption in PPPprojects Chong Jia et al build an evolutionary game modelof BIM adoption between government and social capital inPPP projects and analyzed the stable equilibrium strategiesand implementation conditions of both parties [19] +enthe authors find that the existing research on behaviorevolutionary of BIM adoption is all based on the expectedutility theory Prospect theory put forward by Dan-ielmiddotKahneman and AmosmiddotTversky pointed out that the be-havior characteristics of the individual under the riskconditions are not consistent with the basic principle of theexpected utility theory [20] Individual behavior is not onlycompletely following the utility maximization but also willbe affected by a variety of psychological factors In PPPprojects governments social capitals and contractors showthe characteristics of bounded rationality under the influ-enced and interfered by uncertain factors such as envi-ronment policy and information [21]+e decision-makingof them is on the basis of their own perceived profit and lossso the expected utility theory which has been widely ac-cepted for a long time is not suitable for the situation whenpeoplersquos psychological perception is uncertain [22] +eapplication of prospect theory can effectively solve the aboveproblems the evolutionary game model among stakeholdersof PPP projects for the adoption of BIM can be build basedon bounded rationality and psychological perception valueand the influence of individual psychological perception canbe considered in the evolutionary game process +e fol-lowing research questions will be answered in this study

(i) What is the behavior evolution law of BIM adoptionamong stakeholders in PPP projects

(ii) What are the influencing factors of behaviorstrategy selection for the adoption of BIM in PPPprojects

(iii) How to promote the adoption of BIM in PPPprojects

+e research results will help to clarify the behaviorevolution mechanism and the influencing factors of theevolutionary stabilization strategy (ESS) for adopting BIMamong stakeholders in PPP projects so as to improve thecross-organization information collaboration efficiency ofPPP projects and promote the informatization of the con-struction industry +e application of prospect theoryconsidering the psychological perception value will makeevolutionary game more in line with reality +is study isstructured as follows First we review the relevant literatureand point out our research opportunities Second we buildthe tripartite evolutionary game model for adopting BIMand analyze the stability of stakeholders of PPP projectsFinally we analyze the key influencing factors of behaviorstrategy by numerical simulation and put forward somemanagement implications on the adoption of BIM in PPPprojects

2 Related Literature

21 Adoption of BIM inPPPProjects Many scholars at homeand abroad have studied the feasibility and applicability ofBIM adoption in PPP projects Porwal and Hewage pro-posed a new procurement framework for public facilities andservices based on BIM and proved the feasibility of thismethod in public construction projects by example analysis[23] Love et al put forward that BIM can be used as a way tomeasure the life cycle performance of PPP projects andprovide decision-makers with the key decision informationacross the whole life cycle [24] Yin et al proposed a set ofVfM quantitative evaluation and financial affordabilityevaluation system based on BIM data which effectivelydemonstrates the feasibility of PPP projects in different stagesof the whole life cycle [25] Wang et al explored the appli-cation of BIM in the delicacy management of PPP projectsand demonstrated its feasibility [26] Chen andChen analyzedthe risk types of PPP projectsrsquo whole life cycle and constructedthe risk management model of PPP projects based on BIM[27] Jinhua Li and Zengzhao Zhang established the operationand supervision platform of PPP projects based on BIM andexplored the application effect of BIM in the whole life cycleinformation management of PPP projects Luo and Zhan [28]and Zhou and Li [29] discussed the specific applicationcontent of BIM in the whole process of project

22 EvolutionaryGame)eory Evolutionary game theory isan application of the mathematical framework of gametheory to the dynamics of animal conflicts [30 31] Differentfrom game theory evolutionary game theory which is basedon the assumption of bounded rationality not only considersthe limited rationality of individuals but also dynamicallyadjusts their strategies through learning and imitation Itboth well describes the dynamic evolutionary process amongthe core stakeholders and better explains the formationprocess of equilibrium [32 33] +e two most importantconcepts in evolutionary game theory are evolutional sta-bility strategy (ESS) and replicator dynamics [34]

Evolutionary game theory provides a wide range ofresearch tools its application ranges from evolutionarybiology extends to various fields such as natural science andsocial science and is especially widely used in economics+eresearch in this area applies evolutionary game theory infollowing main contexts (1) the evolution of social customstraditions and norms Peyton Young applied evolutionarygame theory to analyze the relationship between the for-mation of social norms and social welfare [35] Fudenbergand Harris applied evolutionary game theory to analyze thesocial learning process [36] (2)+e changes of the economicsystem and the evolution of market environment Bester andGuth applied the evolutionary game theory to study theexistence and evolutionary stability of human altruism ineconomic activities [37] +e evolution of market survivaland market credit are both studied based on the theory[38 39] (3) +e research of enterprise organization tech-nology innovation and strategy Freidman and Fung appliedevolutionary game to analyze the evolution of the

2 Complexity

organization mode of firms with and without trade based onAmerican and Japanese firms [40] Wang and Meng appliedthe evolutionary game theory to study the evolution of thecooperative competition mechanism of supply chain en-terprises [41] Following the development of the theory ithas been widely applied to a wider range of fields For in-stance Gao and Sheng analyzed evolutionary game theory inpower market [42] Yan Zhang and Mohsen Guizani de-scribed how to employ the theory in infrastructure-basedwireless networks and multihop networks to reduce powerconsumption while improving system capacity decreasingpacket loss and enhancing network resilience [43] Lu usedthe theory to analyze the strategy selection behavior amongpollution-producing enterprises environmental protectiondepartments and pollution-producing enterprises [44]

In construction industry evolutionary game theory hasbeen widely used to analyze the behavioral evolutionmechanism of BIM adoption For example Qi Wang andTeng Wang constructed a tripartite evolutionary game modelon BIM application among governments enterprises andconsumers which provided a reference for BIM applicationand promotion in Chinese construction industry [15]Qiaoling Yin constructed evolutionary game models betweengovernment and enterprises as well as owner and contractorson BIM diffusion in prefabricated buildings explored the keyinfluencing factors and put forward corresponding sugges-tions [16] Tang et al used evolutionary game theory toanalyze behaviors evolution of BIM application in integratedfacility management organization from six scenarios [17]Jiaren Song used evolutionary game theory to build a dynamicmodel of the owner replication group and discussed theevolution strategy of incentives and developers for socialstability and finally put forward relevant suggestions toprovide the beneficial reference for promoting BIM [18]

23 Prospect )eory Prospect theory is the integration ofpsychological research into economic research and reveals thedecision-making mechanism under uncertain conditions [45]Different from expected utility theory prospect theory findsthrough a series of experiments that peoplersquos decision choicedepends on the gap between results and prospects rather thanthe results themselves When people make decisions they havea preset reference point in mind and then they measurewhether each outcome is above or below this reference pointFor the return results above the reference point people tend toshow risk aversion and preference for certain small returns Forloss-type outcomes below the reference point people show riskappetite again hoping for good luck to avoid losses [46]

Initially finance is the field of economics whereprospect theory has been most actively applied [47] +eresearch in this area applies prospect theory in three maincontexts (1) the cross-section of average returns where thegoal is to understand why some financial assets have higheraverage returns than others Benartzi and +aler appliedprospect theory to explain the famous equity premiumpuzzle the fact that the average return of the US stockmarket has historically exceeded the average return oftreasury bills by a much greater margin than predicted by

traditional consumption-based models of asset prices [48](2) +e aggregate stock markets Barberis and Huangstudied asset prices in a one-period economy populated byinvestors who derive prospect theory utility from the changein the value of their portfolios over the course of the period[49] (3) +e trading of financial assets over time Frazzinifound through the empirical that both individual investorsand mutual fund managers have a greater propensity to sellstocks that have risen in value since purchase rather thanstocks that have fallen in value [50] At present the appli-cation of prospect theory has been quite extensive includinginsurance marketing consumer behavior and other aspectsFor example Hu and Scott argued that prospect theory offersa way of understanding why annuities are unpopular In theirframework people think of an annuity as a risky gamblewhose payoff unknown at the moment of retirement is thepresent value of the payouts to be received from the annuitybefore death minus the amount initially paid for the annuity[51] Koszegi and Rabin proposed a way of incorporating theideas in prospect theory into a dynamic model of con-sumption choice +e model builds on the authorsrsquo earlieridea that expectations are an important reference point [52]Vamvakas et al proposed a novel dynamic spectrum man-agement scheme for 5G nonorthogonal multiple access(NOMA) wireless networks based on prospect theory whereusers are offered the option to transmit via licensed andunlicensed bands [53]

In construction industry many scholars have introducedprospect theory into evolutionary games and used percep-tual expectation function instead of the expected utilityfunction so as to better analyze behavior evolutionaryproblems He et al studied the economic interest gamebetween the three stakeholders of government social capitaland public in major engineering projects and analyzes itsbehavior +e three-way evolutionary game model based onprospect theory and the perceptual benefit matrix is con-structed [22] Zhou and Li established and analyzed thegame revenue perception matrix which introduces the ad-ministrative supervision mechanism of higher-level gov-ernment departments into PPP projects based on prospecttheory and the assumption of bounded rationality and theresearch considered that it is necessary to consider thirdparties from outside the regulatory structure to achieveeffective regulation [54] Zhang et al constructed the ex-pected utility model of PPP investorsrsquo investment decisionbased on prospect theory and considering the imperfectrational state of investors in the investment decision and theresearch considered prospect theory is more suitable tostudy the decision behavior of PPP project investors com-pared with the traditional expected utility theory [55]

3 Evolutionary Game Model Based onProspect Theory

Evolutionary game theory which has been widely used forbehavioral strategy evolution mechanism analysis is applied toestablish the behavioral evolutionary game model of BIMadoption Based on the above background evolutionary gamebased on expected utility does not consider the characteristics

Complexity 3

of psychological perception factors of PPP projectsrsquo stake-holders Obviously under the condition of bounded rationalityindividual psychological perception often has a key influenceon behavioral decision-making [46] So this study replaces theexpected utility of the traditional evolutionary game with thevalue function v(Δωi) of prospect theory and replaces theactual probability with the weight function π(pi) so as toobtain the perceived value of the total expected utility of thestrategy which can be expressed as

V 1113944i

π pi( 1113857v Δωi( 1113857 (1)

+erein the value function v(Δωi)

(Δωi)α Δωi ge 0

minusλ(minusΔωi)β Δωi lt 0

1113896 and the weight function π(pi) Pc

[Pc + (1 minus P)c]1c In the formula Δωi is the differencebetween the actual income and the reference point when thebehavior subject adopts strategy i α β(0lt αlt 1 0lt βlt 1)

are the risk attitude coefficients which represent the degreeof decline in the editor of the game subjectrsquos perceived valueof profit and loss+e higher the value the greater the degreeof marginal decline+e loss avoidance coefficient is denotedby λ (λge 1) and the greater the value the more sensitive thegame subject to the loss+e actual occurrence probability ofstrategy i is denoted by pi and the decision weight is denotedby π(pi)(π(0) 0 π(1) 1) When p is very small therewill be π(pi)＞p and when p is very large there will beπ(pi)＜p +at is low probability events are usually over-estimated in prospect theory while high probability eventsare usually underestimated +e construction of the mul-tiagent evolutionary game model based on prospect theorycan fully reflect the psychological perceived value of eachstakeholder and is more conducive to clarifying the evo-lutionary game mechanism and influencing factors

31 Basic Assumptions and Evolutionary Game Strategies

311 Basic Assumptions

Assumption 1 According to the above analysis socialcapitals are the important subjects of BIM adoption in PPPprojects contractors are the implementation subject of thepractical application of BIM and governments are thedriving subject of BIM adoption +erefore the behaviorstrategies of governments are set as positive promotionpassive promotion the behavioral strategies of socialcapitals are set as positive adoption passive adoption andthe behavioral strategies of contractors are set as positiveapplication passive application

Assumption 2 +e probability of ldquopositive promotionrdquo is setas x(0lexle 1) and the probability of ldquopassive promotionrdquo isequal to 1 minus x +e probability of ldquopositive adoptionrdquo is set asy(0leyle 1) and the probability of ldquopassive adoptionrdquo isequal to 1minusy +e probability of ldquopositive applicationrdquo is setas z(0le zle 1) and the probability of ldquopassive applicationrdquo isequal to 1 minus z

Assumption 3 Governments social capitals and contrac-tors will generate benefits and expenses when adoptingdifferent behavior strategies +e profit and loss indexes ofeach subject are given in Table 1

312 Game Strategiesrsquo Perceived Value of Profit and LossBased on the above assumptions the profit and loss matrixof the perceived value among governments social capitalsand contractors is constructed as given in Table 2

32 Construction and Solution of the Evolutionary GameModel

321 Replicator Dynamics Equation and Equilibrium PointAnalysis of Governments According to Table 2 we can get theperceived expected value Eg1 of governments taking strategy ofldquopositive promotionrdquo and the perceived expected value Eg2 ofgovernments taking strategy of ldquopassive promotionrdquo

Eg1 yz(Rg1 minus Cg minus Sp) + y(1 minus z)(Rg1 minus Cg minus Sp + Lc) + z(1 minus y)(Rg2 minus Cg minus Sp + Lp)

+(1 minus y(1 minus z)(minusCg + Lp + Lc minus Sp))

Eg2 yz(Rg1 minus αLg1) + y(1 minus z)(Rg2 minus αLg1) + z(1 minus y)(1 minus z)(minusLg minus αLg1)

(2)

+en the average perceived expected value of govern-ments is as follows

Eg xEg1 +(1 minus x)Eg2 (3)

+erefore the replication dynamic equation of gov-ernments is as follows

F(x) dx

dt x(Eg1 minus Eg) x(1 minus x)(Eg1 minus Eg2)

x(1 minus x)(minusCg minus Sp + Lp + Lc + Lg + αLg1 minus zLc minus yLp minus yLg)

(4)

4 Complexity

+e first derivative of x is equal to

d(F(x))

dx (1 minus 2x)(minusCg minus Sp + Lp)

+ Lc + Lg + αLg1 minus zLc minus yLp minus yLg)

(5)

When F(x) 0 and d(F(x))dxlang0 governmentsrsquostrategy of ldquopositive promotionrdquo will be in a stable state

+erefore there will be F(x) equiv 0 when z minusCgminus

ySp + Lp + Lc + Lg minus yLp minus αLg1Lc and all the values of xare in a stable state When zlang minus Cg minus ySp + Lp+

Lc + Lg minus yLp minus αLg1Lc there will be d(F(x))dx|x1lang0and x 1 is the ESS of the system When zrang minus Cg minus ySp+

Lp + Lc + Lg minus yLp minus αLg1Lc there will be d(F(x))dx|x0lang0 and x 0 is the ESS of the system +e phase

diagram of governmentsrsquo behavioral strategy evolution isshown in Figure 1+e slant z minusCg minus ySp + Lp + Lc + Lg minus

yLp minus αLg1Lc divides the three-dimensional spaceV (x y z| 0lexle 1 0leyle 1 0le zle 1) into parts of V1and V2 In part V1 governments gradually evolved to x 0and finally chose the strategy of ldquopassive promotionrdquo In partV2 governments gradually evolved to x 1 and finallychose the strategy of ldquopositive promotionrdquo

322 Replicator Dynamics Equation and Equilibrium PointAnalysis of Social Capitals According to Table 2 we can getthe perceived expected value Ep1 of social capitals takingstrategy of rdquopositive adoptionrdquo and the perceived expectedvalue Ep2 of social capitals taking the strategy of ldquopassiveadoptionrdquo

Table 1 +e profit and loss indexes of stakeholders

Index Explanation of indexRg1 +e benefit brought to governments when social capitals adopt BIM activelyRg2 +e benefit brought to governments when social capitals adopt BIM passivelyCg +e cost of governmentsrsquo promotion of BIMSp +e subsidy provided by governments when social capitals adopt BIM activelyLp +e punishment imposed by governments when social capitals adopt BIM passivelyLc +e punishment imposed by governments when contractors apply BIM passively

Lg+e adverse effects on the development of construction industry when governments promote BIM passively and social capitals

adopt BIM passivelyLg1 +e punishment imposed by the superior supervision authority when governments promote BIM passivelyα +e probability of the superior supervision authority finding that governments promote BIM passivelyRp +e benefit brought to social capitals when contractors applying BIM activelyCp1 +e cost of social capitalsrsquo positive adoption of BIMCp2 +e cost of social capitalsrsquo passive adoption of BIMLp1 +e adverse effects on social capitals when contractors apply BIM passivelySc +e reward provided by social capitals when contractors apply BIM activelyRc +e benefit obtained by contractors apply BIM activelyCc +e cost incurred by the contractors apply BIM actively

Lc1 +e compensation given by contractors undertaking passive application of BIM to social capitals undertaking positive adoption ofBIM when governments promote BIM actively

Lc2 +e compensation given by contractors undertaking passive application of BIM to social capitals undertaking positive adoption ofBIM when governments promote BIM passively

Table 2 +e matrix of the perceived value of profit and loss among stakeholders

Participants of the evolutionary gameGovernments

Positive promotion(x)

Passive promotion(1 minus x)

Social capitals

Positive adoption(y)

Contractors

Positive application(z)

Rg1 minus Cg minus Sp Rg1 minus αLg1Rp minus Cp1 minus Sc + Sp Rp minus Cp1 minus Sc

Rc minus Cc + Sc RcminusCc+ Sc

Passive application(1 minus z)

Rg1 minus Cg minus Sp + Lc Rg1 minus αLg1Rp minus Cp1 + Sp + Lc1 minus Lp1 Rp minus Cp1 + Lc2 minus Lp1

minusLc minus Lc1 minus Lc2

Passive adoption(1 minus y)

Positive application(z)

Rg2 minus Cg minus Sp + Lp Rg2 minus Lg minus αLg1Rp minus Cp2 + Sp minus Lp minus Sc Rp minus Cp2 minus Sc

Rc minus Cc + Sc Rc minus Cc + Sc

Passive application(1 minus z)

minusCg + Lp + Lc minus Sp minusLg minus αLg1RpminusCp2+ Spminus Lpminus Lp1 RpminusCp2minus Lp1

minusLc 0

Complexity 5

Ep1 xz(Rp minus Cp minus Sc + Sp) + z(1 minus x)(Rp minus Cp1 minus Sc) + x(1 minus z)(Rp minus Cp1 + Sp

+ Lc1 minus Lp1) +(1 minus z)(1 minus x)(Rp1 minus Cp1 + Lc2 minus Lp1)

Ep2 xz(Rp1 minus Cp2 minus Lp minus Sc + Sp) + z(1 minus x)(Rp minus Cp2 minus Sc) + x(1 minus z)

middot (Rp minus Cp2 minus Lp + Sp minus Lp1) +(1 minus z)(1 minus x)(Rp minus Cp2 minus Lp)

(6)

+en the average perceived expected value of socialcapitals is as follows

Ep yEp1 +(1 minus y)Ep2 (7)

+erefore the replication dynamic equation of socialcapitals is as follows

F(y) dy

dt y(Ep1 minus Ep) y(1 minus y)(Ep1 minus Ep2)

y(1 minus y)(Cp2 minus Cp1 + Lc2 + xLc1 minus xLc2

minus zLc2 + xLp minus xzLc1 + xzLc2)

(8)

+e first derivative of y is equal to

d(F(y))

dt (1 minus 2y)(Cp2 minus Cp1 + Lc2 + xLc1 minus xLc2

minus zLc2 + xLp minus xzLc1 + xzLc2)

(9)

When F(y) 0 and d(F(y))dylang0 social capitalsrsquostrategy of ldquopositive adoptionrdquo will be in a stable state+erefore there will be F(y) equiv 0 when z Cp2 minus Cp1 + Lc2

+xLc1 minus xLc2 + xLpLc2 + xLc1 minus xLc2 and all the values ofy are in a stable state When zlangCp2 minus Cp1 + Lc2 + xLc1 minus

xLc2+ xLpLc2 + xLc1 minus xLc2 there will be d(F(y))dy|y1lang0 and y 1 is the ESS of the systemWhen zrangCp2 minus Cp1 +

Lc2 + xLc1minus xLc2 + xLpLc2 + xLc1 minus xLc2 there will bed(F(y))dy|y0lang0 and y 0 is the ESS of the system +ephase diagram of social capitalsrsquo behavioral strategy evo-lution is shown in Figure 2 +e surface z Cp2 minus Cp1 +

Lc2 + xLc1 minus xLc2 + xLpLc2 + xLc1 minus xLc2 divides thethree-dimensional space V (x y z| 0lexle 1 0leyle1 0le zle 1) into parts of V1 and V2 In part V1 socialcapitals gradually evolved to y 1 and finally chose thestrategy of ldquopositive adoptionrdquo In part V2 social capitalsgradually evolved to y 0 and finally chose the strategy ofldquopassive adoptionrdquo

323 Replicator Dynamics Equation and Equilibrium PointAnalysis of Contractors According to Table 2 we can get theperceived expected valueEc1 of contractors taking strategy ofldquopositive applicationrdquo and the perceived expected value Ec2of contractors taking strategy of ldquopassive applicationrdquo

Ec1 xy(Rc minus Cc + Sc) + y(1 minus x)(Rc minus Cc + Sc) +(1 minus y)(1 minus x)(Rc minus Cc + Sc)

Rc minus Cc + Sc

Ec2 xy(minusLc minus Lc1) + y(1 minus x)(minusLc2) + x(1 minus y)(minusLc) +(1 minus y)(1 minus x)(0) xyLc2 minus xLc minus yLc minus xyLc

(10)

z

1

1

x

V2

V1

1 y

Figure 1 Phase diagram of governmentsrsquo behavioral strategy evolution

6 Complexity

+en the average perceived expected value of contrac-tors is as follows

Ec zEc1 +(1 minus z)Ec2 (11)

+erefore the replication dynamic equation of con-tractors is as follows

F(z) dz

dt z(Ec1 minus Ec) z(1 minus z)(Ec1 minus Ec2)

z(1 minus z)(Rc minus Cc + Sc + xLc + yLc2 + xyLc1 minus xyLc2)

(12)

+e first derivative of z is equal to

d(F(z))

dz (1 minus 2z)(Rc minus Cc + Sc + xLc + yLc2 + xyLc1 minus xyLc2)

(13)

When F(z) 0 and d(F(z))dzlang0 contractorsrsquo strategyof ldquopositive applicationrdquo will be in a stable state +ereforethere will be F(z) equiv 0 when y Rc minus Cc + Sc + xLcx(Lc2 minus Lc1) minus Lc2 and all the values of z are in a stablestate When ylangRc minus Cc + Sc + xLcx(Lc2 minus Lc1) minus Lc2there will be d(F(z))dz|z1lang0 and z 1 is the ESS of thesystem When yrangRc minus Cc + Sc + xLcx(Lc2 minus Lc1) minus Lc2there will be d(F(z))dz|z0lang0 and z 0 is the ESS of thesystem +e phase diagram of contractorsrsquo behavioralstrategy evolution is shown in Figure 3 +e surface y

Rc minus Cc + Sc + xLcx(Lc2 minus Lc1) minus Lc2 divides the three-dimensional space V (x y z| 0lexle 1 0le yle 1 0lezle 1) into parts of V1 and V2 In part V1 contractorsgradually evolved to z 1 and finally chose the strategy ofldquopositive applicationrdquo In part V2 contractors graduallyevolved to z 0 and finally chose the strategy of ldquopassiveapplicationrdquo

33 Evolutionary Stability Analysis +e Jacobin matrix ofthe system is represented by J

J

J1 J2 J3

J4 J5 J6

J7 J8 J9

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

zf(x)

zx

zf(x)

zy

zf(x)

zz

zf(y)

zx

zf(y)

zy

zf(y)

zz

zf(z)

zx

zf(z)

zy

zf(z)

zz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(14)

Among them

zf(x)

zx (1 minus 2x)(Cg2 minus Cg1 + Lg minus Sp + Lp + Lc minus zLc minus yLp minus zLg)

zf(y)

zy (1 minus 2y)(Cp2 minus Cp1 + Lc2 + xLc1 minus xLc2 minus zLc2 + xLp minus xzLc1 + xzLc2)

zf(z)

zz (1 minus 2z)(Rc minus Cc + Sc + xLc + yLc2 + xyLc1 minus xyLc2)

(15)

+e local stability of equilibrium points is given inTable 3

According to the indirect method of Lyapunov theequilibrium point is asymptotically stable if all its eigen-values of the Jacobian matrix [56] have negative real parts InTable 3 it can be seen that if we want the system evolving tothe ESS point (1 1 1) the following conditions must be metCg + Splt αLg1 Cp1ltCp2 + Lp and CcltRc + Sc

However it should be noted that in the actual PPPprojects the main stakeholders cannot make the best use ofthe existing information in the decision-making because of

their limited rationality +ey usually make decisionsthrough subjective judgment which ultimately leads to de-viation in their behaviors +e deviations will hinder theestablishment of ESS point (1 1 1) and make the systemdifficult to converge to the optimal state +e reasons areanalyzed based on the prospect theory as followstake governments for example the condition that gov-ernments promoting BIM actively is Cg + Splt αLg1 Basedon the prospect theory individuals always overestimate thelow probability loss and underestimate the high probabilitygain namely individuals tend to choose risk aversion in the

z

1

1

x

V2

V11 y

Figure 2 Phase diagram of social capitalsrsquo behavioral strategyevolution

Complexity 7

Table 3 +e eigenvalues of each equilibrium point

Equilibrium pointEigenvalues of Jacobi matrix

λ1 λ2 λ3E(0 0 0) minusCg minus Sp + Lg + Lp + αLg1 Cp2 minus Cp1 + Lc2 Rc minus Cc + ScE(1 0 0) Cg + Sp minus Lg minus Lp minus Lc minus αLg1 Cp2 minus Cp1 + Lc1 + Lp Rc minus Cc + Sc + LcE(0 1 0) minusCg minus Sp + Lc + αLg1 Cp2 minus Cp1 minus Lc2 Rc minus Cc + Sc + Lc2E(0 0 1) minusCg minus Sp + Lc + αLg1 Cp2 minus Cp1 minusRc + Cc + ScE(1 1 0) Cg + Sp minus Lc minus αLg1 minusCp2 minus Cp1 minus Lc1 minus Lp Rc minus Cc + Sc + Lc + Lc1E(1 0 1) Cg + Sp minus Lp minus Lg minus αLg1 Cp2 minus Cp1 + Lp minusRc + Cc minus Sc minus LcE(0 1 1) minusCg minus Sp + αLg1 minusCp2 + Cp1 minusRc + Cc minus Sc minus Lc2E(1 1 1) Cg + Sp minus αLg1 minusCp2 minus Cp1 + Lp minusRc + Cc minus Sc minus Lc minus Lc1

z

1

1

x

V2

V1

1 y

Figure 3 Phase diagram of contractorsrsquo behavioral strategy evolution

111

0908070605

z

108

06y04 05 06 07 08

x

09 1

Cg=20Cg=15Cg=10

(a)

11

1

09

08

07

06

05

z

Cg=20Cg=15Cg=10

1 095 09 085 08 075 07 065 06 055 05x

(b)

Figure 4 Influence of Cg change on the evolutionary game process

8 Complexity

face of gains but tend to prefer risk preference when facinglosses [57] In the practice of PPP projects governmentsoften think that their behavior will not be easily detected bythe superior supervision authority due to the commonexistence of fluke mentality +erefore governments tend tounderestimate the probability α and the value of punishmentLg1 by the superior supervision authority which means thatthe right part of the conditional formula Cg + Splt αLg1 willbe underestimated in practice On the other hand whengovernments choose the active strategy it will definitely paythe promotion cost while choosing the passive strategy willonly have a certain probability of punitive losses Accordingto the prospect theory governments will be more inclined torisk preference that is they prefer to take risks and choosethe passive strategy rather than to bear the deterministic

expenditure +erefore the above reasons will cause certaindifficulties for the establishment of the formulaCg + Splt αLg1

4 Numerical Simulation and Analysis

In order to better analyze the behavior evolution mechanismamong governments social capitals and contractors in BIMadoption this study uses MATLAB R2018a to simulate thetrend and influencing factors of the evolutionary game +evalues of parameters are randomly assigned under the as-sumption which do not represent the real benefits ofstakeholders in PPP projects

According to the above analysis if the system convergesto the ESS point (1 1 1) the following conditions need to be

111

0908070605

z

108

06y04 04

0608

x

1

a=05a=08a=1

(a)

a=05a=08a=1

11

1

09

08

07

06

05

z

11 1 09 08 07 06 05x

(b)

Figure 5 Influence of α change on the evolutionary game process

Lg1=80Lg1=100Lg1=120

111

0908070605

z

108

06y04 04

0608

x

1

(a)

Lg1=80Lg1=100Lg1=120

11

1

09

08

07

06

05

z

11 1 09 08 07 06 05x

(b)

Figure 6 Influence of Lg1 change on the evolutionary game process

Complexity 9

satisfied Cg + Splt αLg1 Cp1ltCp2 + Lp and CcltRc + Sc+e initial values of each parameter are set to x 05 y

05 α 05Cg 20 Sp 10 Lgl 80Cp1 30Cp2

20 Sp 10 Lg1 80Cp1 30Cp2 20 Lp 20Cc

30Rc 40 Sc 10 Lg 10 Lc 10 Lc1 10 LC2 5

41 Influence of Cg1 and Cg2 Changes on the EvolutionaryGame Process Keep the other parameters unchanged andthe influence on the evolution trend of the system is ob-served by changing the values of Cg α and Lg1 as shown inFigures 4(a) 5(a) and 6(a) In Figure 4(b) we can see thatthe convergence rate of x to the ESS point (1 1 1) isaccelerated as the value of Cg goes down +e simulationresults show that reducing the cost of governmentsrsquo strategyof ldquopositive promotionrdquo is helpful to enhance the willingness

of governments to promote the adoption of BIM in PPPprojects In Figures 5(b) and 6(b) we can see that theconvergence rate of x to the ESS point (1 1 1) is acceleratedas the value of α and Lg1 go bigger +e simulation resultsshow that increasing the supervision probability and theseverity of punishment will help increase the willingness ofgovernments to promote BIM application in PPP projects

42 Influence of Cp1 and Lp Changes on the EvolutionaryGame Process Keep the other parameters unchanged andthe influence on the evolution trend of the system is ob-served by changing the values of Cp1 and Lp as shown inFigures 7(a) and 8(a) In Figure 7(b) we can see that theconvergence rate of y to the ESS point (1 1 1) is acceleratedas the value of Cp1 goes down +e simulation results show

111

0908070605

z

108

06y04 05 06 07 08

x

091

Cp1=30Cp1=20Cp1=10

(a)

Cp1=30Cp1=20Cp1=10

11

1

09

08

07

06

05

z

11 1 09 08 07 06 05y

(b)

Figure 7 Influence of Cp1 change on the evolutionary game process

111

0908070605

z

108

06y04 05

06 07 x08 09 1

Lp=20Lp=30Lp=40

(a)

Lp=20Lp=30Lp=40

11

1

09

08

07

06

05

z

11 1 09 08 07 06 05y

(b)

Figure 8 Influence of Lp change on the evolutionary game process

10 Complexity

that reducing the cost of social capitalsrsquo strategy of ldquopositiveadoptionrdquo is helpful to enhance the willingness of socialcapitals to adopt BIM in PPP projects In Figure 8(b) we cansee that the convergence rate of y to the ESS point (1 1 1) isaccelerated as the value of Lp goes bigger +e simulationresults show that increasing the punishment of social cap-italsrsquo strategy of ldquopassive adoptionrdquo is helpful to enhance thewillingness of social capitals to adopt BIM in PPP projects

43 Influence of Cc and Sc Changes on the Evolutionary GameProcess Keep the other parameters unchanged and theinfluence on the evolution trend of the system is observed bychanging the values of Cc and Sc as shown in Figures 9(a)and 10(a) In Figure 9(b) we can see that the convergence

rate of z to the ESS point (1 1 1) is accelerated as the value ofCc goes down+e simulation results show that reducing thecost of contractorsrsquo strategy of ldquopositive applicationrdquo ishelpful to enhance the willingness of contractors to applyBIM in PPP projects In Figure 10(b) we can see that theconvergence rate of z to the ESS point (1 1 1) is acceleratedas the value of Sc goes bigger +e simulation results showthat increasing the punishment of contractorsrsquo strategy ofldquopassive applicationrdquo is helpful to enhance the willingness ofcontractors to apply BIM in PPP projects

5 Conclusions and Management Implications

Under the assumption of bounded rationality this studyconstructs a tripartite evolutionary game model including

111

0908070605

z

108

06y04 05 06 07 08

x

091

Cc=30Cc=20Cc=10

(a)

Cc=30Cc=20Cc=10

11

1

09

08

07

06

05

z

1 09095 08085 07075 06065 05055y

(b)

Figure 9 Influence of Cc change on the evolutionary game process

111

0908070605

z

108

06y04 05 06 07 x

08 091

Sc=20Sc=30Sc=40

(a)

Sc=20Sc=30Sc=40

11

1

09

08

07

06

05

z

1 09095 08085 07075 06065 05055y

(b)

Figure 10 Influence of Sc change on the evolutionary game process

Complexity 11

governments social capitals and contractors +e satisfac-tion conditions that the system converging to the ESS point(1 1 1) are analyzed based on prospect theory and the mainfactors influencing stakeholdersrsquo behavior decision-makingare simulated by MATLAB +e research results will help toclarify the behavior evolution mechanism and the influ-encing factors of the ESS for adopting BIM among stake-holders in PPP projects so as to improve the cross-organization information collaboration efficiency of PPPprojects and promote the informatization of the construc-tion industry +e primary conclusions and managementimplications are as follows

First the cost of BIM promotion by governments and thepunishment from the superior supervision authority whengovernments choose the passive strategy are the key factorsaffecting governmentsrsquo behavior choice When governmentsactively promote the cost reduction of BIM or punishedmore by the superior supervision authority governmentswill choose the active promotion strategy and the behaviorgradually converges to the ESS point (1 1 1)

Second the cost of active adoption of BIM and thepunishment of passive adoption of BIM are the key factorsaffecting the choice of social capitalsrsquo behavior When thecost of active adoption of BIM is reduced or the penalty ofpassive adoption of BIM is increased the intention of socialcapitals to choose active adoption of BIM strategy is in-creased and the behavior of social capitals gradually con-verges to the ESS point (1 1 1)

Finally the cost of active application of BIM and thereward of active application of BIM are the key factorsinfluencing the choice of contractorsrsquo behavior When thecost of the active application of BIM technology is reducedor the penalty of the negative application of BIM technologyis increased the intention of the contractor to choose thestrategy of active application of BIM technology is increasedand the contractorrsquos behavior gradually converges to the ESSpoint (1 1 1)

+e adoption of BIM in PPP projects is a major reform ofthe traditional building mode whichrequires all stakeholders to change the traditionalthinking and actively cooperate In order to promoteBIM adoption among stakeholders of PPP projects thefollowing management implications are proposedImprove the supervision efficiency of superior supervisionauthority on governmentsrsquo promotion of BIM and furtherenrich the supervision means with the help of the media andthe public Increase the punishment for the governmentsrsquopassive promotion of BIM so as to improve the perceivedvalue of governments for regulatory punishment

Strengthen the punishment for the passive adoption ofBIM by social capitals and the reward for the active ap-plication of BIM by contractors Improve the cognition ofsocial capitals and contractors on BIM and other advancedinformation technology through publicity and training andgradually transform the system promotion into habitualadoption

Under the PPP mode the adoption of BIM can bepromoted by the governmentrsquos supervision but the gov-ernmentrsquos behavior also needs supervision +e public is the

users and beneficiaries of PPP projectsrsquo goods and the publicshould also have the right to know and to supervise theimplementation of the projects +erefore in the next stagethe authors will further research the role of public super-vision in the adoption of BIM in PPP projects in order toexplore the evolutionary mechanism and influencing factorsunder more stakeholders

Data Availability

Some or all data models or code generated or used tosupport the findings of this study are available from thecorresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interest

Acknowledgments

+is work was supported by the National Natural ScienceFoundation of China (NSFC) (71801119) in part by theSocial Science Planning Foundation of Liaoning(L18BGL020) and in part by the Basic Scientific ResearchProjects in Institutions of Higher Learning of LiaoningEducation Department (LJ2017QW009)

References

[1] N Carbonara and R Pellegrino ldquoRevenue guarantee inpublic-private partnerships a win-win modelrdquo ConstructionManagement amp Economics vol 36 no 10 pp 584ndash598 2018

[2] L Cheng ldquoDiscussion on the application of PPPmodel in newinfrastructure constructionrdquo Open Journal of Social Sciencesvol 7 no 9 pp 283ndash288 2019

[3] D Wu and S Q Wang ldquoResearch development and trend ofpublic private partnership in Chinardquo Journal of EngineeringManagement vol 28 no 6 pp 75ndash80 2014

[4] K Leidel and H W Alfen Lifecycle-Oriented Risk Manage-ment for PPP-Projects in Public Real Estate pp 587ndash604 EresLos Angeles CA USA 2009

[5] J Yuan X Li Y Ke W Xu Z Xu and M SkibnewskildquoDeveloping a building information modelingndashbased per-formance management system for publicndashprivate partner-shipsrdquo Engineering Construction and ArchitecturalManagement vol 27 no 8 pp 1727ndash1762 2020

[6] S Jayasuriya G Zhang and R J Yang ldquoExploring the impactof stakeholder management strategies on managing issues inPPP projectsrdquo International Journal of Construction Man-agement vol 20 no 6 pp 666ndash678 2020

[7] C X Jiang ldquoCauses and implications of the failure intransportation infrastructure public privatepartnershipsmdashmdasha study of 25 typical PPP casesrdquo Journal ofBeijing Jiaotong University vol 15 no 3 pp 50ndash58 2016

[8] R Osei-Kyei and A P C Chan ldquoReview of studies on thecritical success factors for public-private partnership(PPP) projects from 1990 to 2013rdquo International Journal ofProject Management vol 33 no 6 pp 1335ndash1346 2015

[9] B Zhang Y Wang X Liu et al ldquo+e configuration of PPPfailuremdashmdasha crisp-set qualitative comparative analysisrdquoJournal of Public Administration vol 12 no 04pp 65ndash81+191 2019

12 Complexity

[10] N Costantino and R Pellegrino ldquoPublicndashprivate partnershipimplementation cross-case analysisrdquo Proceedings of the In-stitution of Civil Engineers-management Procurement andLaw vol 168 no 4 pp 164ndash176 2015

[11] P Zhang K Zhan and Y Zhou ldquoGame model of three-partyshared risk allocation for PPP projectsmdashmdashbased on theperspective of incomplete informationrdquo IOP Conference Se-ries Earth and Environmental Science vol 295 no 4pp 42ndash49 2019

[12] N W Young S A Jones and H M Bernstein BuildingInformation Modeling Transforming Design and Constructionto Achieve Greater Industry Productivity McGraw HillConstruction New York NY USA 2008

[13] N D Aziz A H Nawawi and N R M Ariff ldquoBuildinginformation modelling (BIM) in facilities management op-portunities to be considered by facility managersrdquo Procedia-Social and Behavioral Sciences vol 234 pp 353ndash362 2016

[14] W Zou M Kumaraswamy and J Chung ldquoIdentifying thecritical success factors for relationship management in PPPprojectsrdquo International Journal of Project Managementvol 32 no 2 pp 265ndash274 2014

[15] QWang and TWang ldquoResearch on promotion of BIM basedon game analysis of stakeholdersrdquo Sichuan BuildingMaterialsvol 41 no 6 p 272 + 278 2015

[16] Q L Yin Evolutionary Game Analysis on BIM TechnologyDiffusion of Prefabricated Construction Qingdao Universityof Technology Qingdao China 2019

[17] H X Tang J M Cao H S Xu and C F Han ldquoAnalysis onBIM application cooperation behaviors in integrated facilitymanagement organization based on evolutionary game the-oryrdquo Industrial Engineering amp Management vol 25 no 4pp 1ndash8 2020

[18] J R Song ldquoResearch on incentive and countermeasures ofBIM application based on game theoryrdquo Value Engineeringvol 36 no 33 pp 43-44 2017

[19] C Jia Z H Wang and L Zhao ldquoEvolutionary game analysisand simulation for cooperative application of BIM technologyin PPPrdquo Journal of Liaoning Technical University vol 22no 4 pp 270ndash277 2020

[20] K A Tversky ldquoProspect theory an analysis of decision underriskrdquo Econometrica vol 47 no 2 pp 263ndash291 1979

[21] R L Gao and Q Bao ldquoSelection of government supervisionmode during the operational period in PPP projects based onevolutionary game theoryrdquo Operations Research and Man-agement Science vol 28 no 4 pp 155ndash162 2019

[22] S K He G W Liang and J B Meng ldquoMulti-subjectsbenefit game and behavior evolution mechanism of majorengineering based on prospect theoryrdquo Science andTechnology Management Research vol 40 no 5 pp 207ndash214 2020

[23] A Porwal and K N Hewage ldquoBuilding information modeling(BIM) partnering framework for public construction proj-ectsrdquo Automation in Construction vol 31 no 3 pp 204ndash2142013

[24] P E D Love J Liu J Matthews C-P Sing and J SmithldquoFuture proofing PPPs life-cycle performance measurementand building information modellingrdquo Automation in Con-struction vol 56 pp 26ndash35 2015

[25] Y L Yin Q J Liu and X Wang ldquoResearch on the whole lifecycle supervision system of public projectsmdashmdashbased on thecollaboration of VfM evaluation and BIM technologyrdquo ProjectManagement Technology vol 14 no 07 pp 17ndash20 2016

[26] S Q Wang R Tiong S Ting et al ldquoEvaluation and man-agement of political risks in Chinarsquos BOTprojectsrdquo Journal of

Construction Engineering and Management vol 126 no 3pp 242ndash250 2000

[27] Y M Chen and H F Chen ldquo+e risk management of PPPproject life cycle based on BIM technologyrdquo Project Man-agement Technology vol 15 no 5 pp 51ndash55 2017

[28] G Luo and X X Zhan ldquoResearch on the application of BIM inPPP project whole-life cyclerdquo Technology and EconomicGuide vol 26 no 5 p 168 2018

[29] B L Zhou and C L Lin ldquo+e application of BIM in PPPproject whole-life cyclerdquo Engineering Economy vol 29 no 3pp 51ndash54 2019

[30] J M Smith Evolution and the )eory of Games CambridgeUniversity Press Cambridge UK 1982

[31] J W Weibull Evolutionary Game )eory MIT PressCambridge MA USA 1997

[32] K Binmore ldquoModeling rational players part Irdquo Economicsand Philosophy vol 3 no 2 pp 179ndash214 1987

[33] K Binmore ldquoModeling rational players part IIrdquo Economicsand Philosophy vol 4 1988

[34] P D Taylor and L B Jonker ldquoEvolutionarily stable strategiesand game dynamicsrdquoMathematical Biosciences vol 40 no 1-2 pp 145ndash156 1978

[35] P Young ldquoConventional contractsrdquo )e Review of EconomicStudies vol 65 no 4 pp 773ndash792 1998

[36] D Fudenberg and C Harris ldquoEvolutionary dynamics withaggregate shocksrdquo Journal of Economic )eory vol 57 no 2pp 420ndash441 1992

[37] H Bester W Guth and W Guth ldquoIs altruism evolutionarilystablerdquo Journal of Economic Behavior amp Organization vol 34no 2 pp 193ndash209 1998

[38] K E Boulding R R Nelson and S G Winter ldquoAn evolu-tionary theory of economic changrdquo American Journal ofAgricultural Economics vol 66 no 4 p 535 1984

[39] S Y Xie Economic Game )eory Fudan University PressShanghai China 2002

[40] D Friedman and K C Fung ldquoInternational trade and theinternal organization of firms an evolutionary approachrdquoJournal of International Economics vol 41 no 1-2 pp 113ndash137 1996

[41] Y P Wang and W D Meng ldquoEvolutionary game analysis onco-competition mechanism of supply chainrdquo Journal of In-dustrial Engineering and Engineering Management vol 18no 2 pp 96ndash98 2004

[42] J Gao and Z H Shen ldquoElementary groping for evolutionarygame theory and its application in electricity marketrdquo Au-tomation of Electric Power Systems vol 27 no 18 pp 18ndash212003

[43] Y Zhang and M Guizani Game )eory for Wireless Com-munications and Networking CRC Press Boca Raton FLUSA 2011

[44] F Y Lu ldquoEvolutionary game analysis on environmentalpollution problemrdquo Systems Engineering-)eory amp Practicevol 27 no 9 pp 148ndash152 2007

[45] S L Dreisbach ldquoImpact of psychological research on moraltheory prospect theory and the doctrine of doing andallowingrdquo Dissertations amp )eses-Gradworks University ofCalifornia Santa Cruz CA USA 2012

[46] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Econometrica vol 47 no 2 1979

[47] N C Barberis ldquo+irty years of prospect theory in economicsa review and assessmentrdquo )e Journal of Economic Perspec-tives vol 27 no 1 pp 173ndash196 2013

Complexity 13

[48] S Benartzi and R H +aler ldquoMyopic loss aversion and theequity premium puzzlerdquo Quarterly Journal of Economicsvol 110 no 1 pp 73ndash92 1995

[49] N Barberis and M Huang ldquoStocks as lotteries the impli-cations of probability weighting for security pricesrdquo )eAmerican Economic Review vol 98 no 5 pp 2066ndash21002008

[50] A Frazzini ldquo+e disposition effect and underreaction tonewsrdquo )e Journal of Finance vol 61 no 4 pp 2017ndash20462006

[51] W Y Hu and J S Scott ldquoBehavioral obstacles in the annuitymarketrdquo Financial Analysts Journal vol 63 no 6 pp 71ndash822007

[52] B Koszegi andM Rabin ldquoReference-dependent consumptionplansrdquo )e American Economic Review vol 99 no 3pp 909ndash936 2009

[53] P Vamvakas E E Tsiropoulou and S Papavassiliou ldquoDy-namic spectrum management in 5G wireless networks a real-life modeling approachrdquo in Proceedings of the IEEE INFO-COM 2019-IEEE Conference on Computer CommunicationsIEEE Paris France April 2019

[54] Y N Zhou and J C Liu ldquoStudy on supervision strategy of pppproject considering the participation of higher-level gov-ernmentrdquo Chinese Journal of Management Science vol 1ndash162021 httpsdoiorg1016381jcnkiissn1003-207x20200801

[55] H Q Zhang J C Wang and H L Tao ldquoResearch the de-cision-making behavior of investors based on prospect theoryin the ppp projectrdquo Soft Science vol 32 no 8 pp 129ndash1332018

[56] D Friedman ldquoEvolutionary games in economicsrdquo Econo-metrica vol 59 no 3 pp 637ndash666 1991

[57] Z B Zhao and Q P Man ldquoEvolutionary game analysis of riskmanagement behavior of major infrastructure projects basedon projects based on prospect theoryrdquo Journal of SystemsManagement vol 27 no 1 pp 112ndash120 2018

14 Complexity

organization mode of firms with and without trade based onAmerican and Japanese firms [40] Wang and Meng appliedthe evolutionary game theory to study the evolution of thecooperative competition mechanism of supply chain en-terprises [41] Following the development of the theory ithas been widely applied to a wider range of fields For in-stance Gao and Sheng analyzed evolutionary game theory inpower market [42] Yan Zhang and Mohsen Guizani de-scribed how to employ the theory in infrastructure-basedwireless networks and multihop networks to reduce powerconsumption while improving system capacity decreasingpacket loss and enhancing network resilience [43] Lu usedthe theory to analyze the strategy selection behavior amongpollution-producing enterprises environmental protectiondepartments and pollution-producing enterprises [44]

In construction industry evolutionary game theory hasbeen widely used to analyze the behavioral evolutionmechanism of BIM adoption For example Qi Wang andTeng Wang constructed a tripartite evolutionary game modelon BIM application among governments enterprises andconsumers which provided a reference for BIM applicationand promotion in Chinese construction industry [15]Qiaoling Yin constructed evolutionary game models betweengovernment and enterprises as well as owner and contractorson BIM diffusion in prefabricated buildings explored the keyinfluencing factors and put forward corresponding sugges-tions [16] Tang et al used evolutionary game theory toanalyze behaviors evolution of BIM application in integratedfacility management organization from six scenarios [17]Jiaren Song used evolutionary game theory to build a dynamicmodel of the owner replication group and discussed theevolution strategy of incentives and developers for socialstability and finally put forward relevant suggestions toprovide the beneficial reference for promoting BIM [18]

23 Prospect )eory Prospect theory is the integration ofpsychological research into economic research and reveals thedecision-making mechanism under uncertain conditions [45]Different from expected utility theory prospect theory findsthrough a series of experiments that peoplersquos decision choicedepends on the gap between results and prospects rather thanthe results themselves When people make decisions they havea preset reference point in mind and then they measurewhether each outcome is above or below this reference pointFor the return results above the reference point people tend toshow risk aversion and preference for certain small returns Forloss-type outcomes below the reference point people show riskappetite again hoping for good luck to avoid losses [46]

Initially finance is the field of economics whereprospect theory has been most actively applied [47] +eresearch in this area applies prospect theory in three maincontexts (1) the cross-section of average returns where thegoal is to understand why some financial assets have higheraverage returns than others Benartzi and +aler appliedprospect theory to explain the famous equity premiumpuzzle the fact that the average return of the US stockmarket has historically exceeded the average return oftreasury bills by a much greater margin than predicted by

traditional consumption-based models of asset prices [48](2) +e aggregate stock markets Barberis and Huangstudied asset prices in a one-period economy populated byinvestors who derive prospect theory utility from the changein the value of their portfolios over the course of the period[49] (3) +e trading of financial assets over time Frazzinifound through the empirical that both individual investorsand mutual fund managers have a greater propensity to sellstocks that have risen in value since purchase rather thanstocks that have fallen in value [50] At present the appli-cation of prospect theory has been quite extensive includinginsurance marketing consumer behavior and other aspectsFor example Hu and Scott argued that prospect theory offersa way of understanding why annuities are unpopular In theirframework people think of an annuity as a risky gamblewhose payoff unknown at the moment of retirement is thepresent value of the payouts to be received from the annuitybefore death minus the amount initially paid for the annuity[51] Koszegi and Rabin proposed a way of incorporating theideas in prospect theory into a dynamic model of con-sumption choice +e model builds on the authorsrsquo earlieridea that expectations are an important reference point [52]Vamvakas et al proposed a novel dynamic spectrum man-agement scheme for 5G nonorthogonal multiple access(NOMA) wireless networks based on prospect theory whereusers are offered the option to transmit via licensed andunlicensed bands [53]

In construction industry many scholars have introducedprospect theory into evolutionary games and used percep-tual expectation function instead of the expected utilityfunction so as to better analyze behavior evolutionaryproblems He et al studied the economic interest gamebetween the three stakeholders of government social capitaland public in major engineering projects and analyzes itsbehavior +e three-way evolutionary game model based onprospect theory and the perceptual benefit matrix is con-structed [22] Zhou and Li established and analyzed thegame revenue perception matrix which introduces the ad-ministrative supervision mechanism of higher-level gov-ernment departments into PPP projects based on prospecttheory and the assumption of bounded rationality and theresearch considered that it is necessary to consider thirdparties from outside the regulatory structure to achieveeffective regulation [54] Zhang et al constructed the ex-pected utility model of PPP investorsrsquo investment decisionbased on prospect theory and considering the imperfectrational state of investors in the investment decision and theresearch considered prospect theory is more suitable tostudy the decision behavior of PPP project investors com-pared with the traditional expected utility theory [55]

3 Evolutionary Game Model Based onProspect Theory

Evolutionary game theory which has been widely used forbehavioral strategy evolution mechanism analysis is applied toestablish the behavioral evolutionary game model of BIMadoption Based on the above background evolutionary gamebased on expected utility does not consider the characteristics

Complexity 3

of psychological perception factors of PPP projectsrsquo stake-holders Obviously under the condition of bounded rationalityindividual psychological perception often has a key influenceon behavioral decision-making [46] So this study replaces theexpected utility of the traditional evolutionary game with thevalue function v(Δωi) of prospect theory and replaces theactual probability with the weight function π(pi) so as toobtain the perceived value of the total expected utility of thestrategy which can be expressed as

V 1113944i

π pi( 1113857v Δωi( 1113857 (1)

+erein the value function v(Δωi)

(Δωi)α Δωi ge 0

minusλ(minusΔωi)β Δωi lt 0

1113896 and the weight function π(pi) Pc

[Pc + (1 minus P)c]1c In the formula Δωi is the differencebetween the actual income and the reference point when thebehavior subject adopts strategy i α β(0lt αlt 1 0lt βlt 1)

are the risk attitude coefficients which represent the degreeof decline in the editor of the game subjectrsquos perceived valueof profit and loss+e higher the value the greater the degreeof marginal decline+e loss avoidance coefficient is denotedby λ (λge 1) and the greater the value the more sensitive thegame subject to the loss+e actual occurrence probability ofstrategy i is denoted by pi and the decision weight is denotedby π(pi)(π(0) 0 π(1) 1) When p is very small therewill be π(pi)＞p and when p is very large there will beπ(pi)＜p +at is low probability events are usually over-estimated in prospect theory while high probability eventsare usually underestimated +e construction of the mul-tiagent evolutionary game model based on prospect theorycan fully reflect the psychological perceived value of eachstakeholder and is more conducive to clarifying the evo-lutionary game mechanism and influencing factors

31 Basic Assumptions and Evolutionary Game Strategies

311 Basic Assumptions

Assumption 1 According to the above analysis socialcapitals are the important subjects of BIM adoption in PPPprojects contractors are the implementation subject of thepractical application of BIM and governments are thedriving subject of BIM adoption +erefore the behaviorstrategies of governments are set as positive promotionpassive promotion the behavioral strategies of socialcapitals are set as positive adoption passive adoption andthe behavioral strategies of contractors are set as positiveapplication passive application

Assumption 2 +e probability of ldquopositive promotionrdquo is setas x(0lexle 1) and the probability of ldquopassive promotionrdquo isequal to 1 minus x +e probability of ldquopositive adoptionrdquo is set asy(0leyle 1) and the probability of ldquopassive adoptionrdquo isequal to 1minusy +e probability of ldquopositive applicationrdquo is setas z(0le zle 1) and the probability of ldquopassive applicationrdquo isequal to 1 minus z

Assumption 3 Governments social capitals and contrac-tors will generate benefits and expenses when adoptingdifferent behavior strategies +e profit and loss indexes ofeach subject are given in Table 1

312 Game Strategiesrsquo Perceived Value of Profit and LossBased on the above assumptions the profit and loss matrixof the perceived value among governments social capitalsand contractors is constructed as given in Table 2

32 Construction and Solution of the Evolutionary GameModel

321 Replicator Dynamics Equation and Equilibrium PointAnalysis of Governments According to Table 2 we can get theperceived expected value Eg1 of governments taking strategy ofldquopositive promotionrdquo and the perceived expected value Eg2 ofgovernments taking strategy of ldquopassive promotionrdquo

Eg1 yz(Rg1 minus Cg minus Sp) + y(1 minus z)(Rg1 minus Cg minus Sp + Lc) + z(1 minus y)(Rg2 minus Cg minus Sp + Lp)

+(1 minus y(1 minus z)(minusCg + Lp + Lc minus Sp))

Eg2 yz(Rg1 minus αLg1) + y(1 minus z)(Rg2 minus αLg1) + z(1 minus y)(1 minus z)(minusLg minus αLg1)

(2)

+en the average perceived expected value of govern-ments is as follows

Eg xEg1 +(1 minus x)Eg2 (3)

+erefore the replication dynamic equation of gov-ernments is as follows

F(x) dx

dt x(Eg1 minus Eg) x(1 minus x)(Eg1 minus Eg2)

x(1 minus x)(minusCg minus Sp + Lp + Lc + Lg + αLg1 minus zLc minus yLp minus yLg)

(4)

4 Complexity

+e first derivative of x is equal to

d(F(x))

dx (1 minus 2x)(minusCg minus Sp + Lp)

+ Lc + Lg + αLg1 minus zLc minus yLp minus yLg)

(5)

When F(x) 0 and d(F(x))dxlang0 governmentsrsquostrategy of ldquopositive promotionrdquo will be in a stable state

+erefore there will be F(x) equiv 0 when z minusCgminus

ySp + Lp + Lc + Lg minus yLp minus αLg1Lc and all the values of xare in a stable state When zlang minus Cg minus ySp + Lp+

Lc + Lg minus yLp minus αLg1Lc there will be d(F(x))dx|x1lang0and x 1 is the ESS of the system When zrang minus Cg minus ySp+

Lp + Lc + Lg minus yLp minus αLg1Lc there will be d(F(x))dx|x0lang0 and x 0 is the ESS of the system +e phase

diagram of governmentsrsquo behavioral strategy evolution isshown in Figure 1+e slant z minusCg minus ySp + Lp + Lc + Lg minus

yLp minus αLg1Lc divides the three-dimensional spaceV (x y z| 0lexle 1 0leyle 1 0le zle 1) into parts of V1and V2 In part V1 governments gradually evolved to x 0and finally chose the strategy of ldquopassive promotionrdquo In partV2 governments gradually evolved to x 1 and finallychose the strategy of ldquopositive promotionrdquo

322 Replicator Dynamics Equation and Equilibrium PointAnalysis of Social Capitals According to Table 2 we can getthe perceived expected value Ep1 of social capitals takingstrategy of rdquopositive adoptionrdquo and the perceived expectedvalue Ep2 of social capitals taking the strategy of ldquopassiveadoptionrdquo

Table 1 +e profit and loss indexes of stakeholders

Index Explanation of indexRg1 +e benefit brought to governments when social capitals adopt BIM activelyRg2 +e benefit brought to governments when social capitals adopt BIM passivelyCg +e cost of governmentsrsquo promotion of BIMSp +e subsidy provided by governments when social capitals adopt BIM activelyLp +e punishment imposed by governments when social capitals adopt BIM passivelyLc +e punishment imposed by governments when contractors apply BIM passively

Lg+e adverse effects on the development of construction industry when governments promote BIM passively and social capitals

adopt BIM passivelyLg1 +e punishment imposed by the superior supervision authority when governments promote BIM passivelyα +e probability of the superior supervision authority finding that governments promote BIM passivelyRp +e benefit brought to social capitals when contractors applying BIM activelyCp1 +e cost of social capitalsrsquo positive adoption of BIMCp2 +e cost of social capitalsrsquo passive adoption of BIMLp1 +e adverse effects on social capitals when contractors apply BIM passivelySc +e reward provided by social capitals when contractors apply BIM activelyRc +e benefit obtained by contractors apply BIM activelyCc +e cost incurred by the contractors apply BIM actively

Lc1 +e compensation given by contractors undertaking passive application of BIM to social capitals undertaking positive adoption ofBIM when governments promote BIM actively

Lc2 +e compensation given by contractors undertaking passive application of BIM to social capitals undertaking positive adoption ofBIM when governments promote BIM passively

Table 2 +e matrix of the perceived value of profit and loss among stakeholders

Participants of the evolutionary gameGovernments

Positive promotion(x)

Passive promotion(1 minus x)

Social capitals

Positive adoption(y)

Contractors

Positive application(z)

Rg1 minus Cg minus Sp Rg1 minus αLg1Rp minus Cp1 minus Sc + Sp Rp minus Cp1 minus Sc

Rc minus Cc + Sc RcminusCc+ Sc

Passive application(1 minus z)

Rg1 minus Cg minus Sp + Lc Rg1 minus αLg1Rp minus Cp1 + Sp + Lc1 minus Lp1 Rp minus Cp1 + Lc2 minus Lp1

minusLc minus Lc1 minus Lc2

Passive adoption(1 minus y)

Positive application(z)

Rg2 minus Cg minus Sp + Lp Rg2 minus Lg minus αLg1Rp minus Cp2 + Sp minus Lp minus Sc Rp minus Cp2 minus Sc

Rc minus Cc + Sc Rc minus Cc + Sc

Passive application(1 minus z)

minusCg + Lp + Lc minus Sp minusLg minus αLg1RpminusCp2+ Spminus Lpminus Lp1 RpminusCp2minus Lp1

minusLc 0

Complexity 5

Ep1 xz(Rp minus Cp minus Sc + Sp) + z(1 minus x)(Rp minus Cp1 minus Sc) + x(1 minus z)(Rp minus Cp1 + Sp

+ Lc1 minus Lp1) +(1 minus z)(1 minus x)(Rp1 minus Cp1 + Lc2 minus Lp1)

Ep2 xz(Rp1 minus Cp2 minus Lp minus Sc + Sp) + z(1 minus x)(Rp minus Cp2 minus Sc) + x(1 minus z)

middot (Rp minus Cp2 minus Lp + Sp minus Lp1) +(1 minus z)(1 minus x)(Rp minus Cp2 minus Lp)

(6)

+en the average perceived expected value of socialcapitals is as follows

Ep yEp1 +(1 minus y)Ep2 (7)

+erefore the replication dynamic equation of socialcapitals is as follows

F(y) dy

dt y(Ep1 minus Ep) y(1 minus y)(Ep1 minus Ep2)

y(1 minus y)(Cp2 minus Cp1 + Lc2 + xLc1 minus xLc2

minus zLc2 + xLp minus xzLc1 + xzLc2)

(8)

+e first derivative of y is equal to

d(F(y))

dt (1 minus 2y)(Cp2 minus Cp1 + Lc2 + xLc1 minus xLc2

minus zLc2 + xLp minus xzLc1 + xzLc2)

(9)

When F(y) 0 and d(F(y))dylang0 social capitalsrsquostrategy of ldquopositive adoptionrdquo will be in a stable state+erefore there will be F(y) equiv 0 when z Cp2 minus Cp1 + Lc2

+xLc1 minus xLc2 + xLpLc2 + xLc1 minus xLc2 and all the values ofy are in a stable state When zlangCp2 minus Cp1 + Lc2 + xLc1 minus

xLc2+ xLpLc2 + xLc1 minus xLc2 there will be d(F(y))dy|y1lang0 and y 1 is the ESS of the systemWhen zrangCp2 minus Cp1 +

Lc2 + xLc1minus xLc2 + xLpLc2 + xLc1 minus xLc2 there will bed(F(y))dy|y0lang0 and y 0 is the ESS of the system +ephase diagram of social capitalsrsquo behavioral strategy evo-lution is shown in Figure 2 +e surface z Cp2 minus Cp1 +

Lc2 + xLc1 minus xLc2 + xLpLc2 + xLc1 minus xLc2 divides thethree-dimensional space V (x y z| 0lexle 1 0leyle1 0le zle 1) into parts of V1 and V2 In part V1 socialcapitals gradually evolved to y 1 and finally chose thestrategy of ldquopositive adoptionrdquo In part V2 social capitalsgradually evolved to y 0 and finally chose the strategy ofldquopassive adoptionrdquo

323 Replicator Dynamics Equation and Equilibrium PointAnalysis of Contractors According to Table 2 we can get theperceived expected valueEc1 of contractors taking strategy ofldquopositive applicationrdquo and the perceived expected value Ec2of contractors taking strategy of ldquopassive applicationrdquo

Ec1 xy(Rc minus Cc + Sc) + y(1 minus x)(Rc minus Cc + Sc) +(1 minus y)(1 minus x)(Rc minus Cc + Sc)

Rc minus Cc + Sc

Ec2 xy(minusLc minus Lc1) + y(1 minus x)(minusLc2) + x(1 minus y)(minusLc) +(1 minus y)(1 minus x)(0) xyLc2 minus xLc minus yLc minus xyLc

(10)

z

1

1

x

V2

V1

1 y

Figure 1 Phase diagram of governmentsrsquo behavioral strategy evolution

6 Complexity

+en the average perceived expected value of contrac-tors is as follows

Ec zEc1 +(1 minus z)Ec2 (11)

+erefore the replication dynamic equation of con-tractors is as follows

F(z) dz

dt z(Ec1 minus Ec) z(1 minus z)(Ec1 minus Ec2)

z(1 minus z)(Rc minus Cc + Sc + xLc + yLc2 + xyLc1 minus xyLc2)

(12)

+e first derivative of z is equal to

d(F(z))

dz (1 minus 2z)(Rc minus Cc + Sc + xLc + yLc2 + xyLc1 minus xyLc2)

(13)

When F(z) 0 and d(F(z))dzlang0 contractorsrsquo strategyof ldquopositive applicationrdquo will be in a stable state +ereforethere will be F(z) equiv 0 when y Rc minus Cc + Sc + xLcx(Lc2 minus Lc1) minus Lc2 and all the values of z are in a stablestate When ylangRc minus Cc + Sc + xLcx(Lc2 minus Lc1) minus Lc2there will be d(F(z))dz|z1lang0 and z 1 is the ESS of thesystem When yrangRc minus Cc + Sc + xLcx(Lc2 minus Lc1) minus Lc2there will be d(F(z))dz|z0lang0 and z 0 is the ESS of thesystem +e phase diagram of contractorsrsquo behavioralstrategy evolution is shown in Figure 3 +e surface y

Rc minus Cc + Sc + xLcx(Lc2 minus Lc1) minus Lc2 divides the three-dimensional space V (x y z| 0lexle 1 0le yle 1 0lezle 1) into parts of V1 and V2 In part V1 contractorsgradually evolved to z 1 and finally chose the strategy ofldquopositive applicationrdquo In part V2 contractors graduallyevolved to z 0 and finally chose the strategy of ldquopassiveapplicationrdquo

33 Evolutionary Stability Analysis +e Jacobin matrix ofthe system is represented by J

J

J1 J2 J3

J4 J5 J6

J7 J8 J9

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

zf(x)

zx

zf(x)

zy

zf(x)

zz

zf(y)

zx

zf(y)

zy

zf(y)

zz

zf(z)

zx

zf(z)

zy

zf(z)

zz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(14)

Among them

zf(x)

zx (1 minus 2x)(Cg2 minus Cg1 + Lg minus Sp + Lp + Lc minus zLc minus yLp minus zLg)

zf(y)

zy (1 minus 2y)(Cp2 minus Cp1 + Lc2 + xLc1 minus xLc2 minus zLc2 + xLp minus xzLc1 + xzLc2)

zf(z)

zz (1 minus 2z)(Rc minus Cc + Sc + xLc + yLc2 + xyLc1 minus xyLc2)

(15)

+e local stability of equilibrium points is given inTable 3

According to the indirect method of Lyapunov theequilibrium point is asymptotically stable if all its eigen-values of the Jacobian matrix [56] have negative real parts InTable 3 it can be seen that if we want the system evolving tothe ESS point (1 1 1) the following conditions must be metCg + Splt αLg1 Cp1ltCp2 + Lp and CcltRc + Sc

However it should be noted that in the actual PPPprojects the main stakeholders cannot make the best use ofthe existing information in the decision-making because of

their limited rationality +ey usually make decisionsthrough subjective judgment which ultimately leads to de-viation in their behaviors +e deviations will hinder theestablishment of ESS point (1 1 1) and make the systemdifficult to converge to the optimal state +e reasons areanalyzed based on the prospect theory as followstake governments for example the condition that gov-ernments promoting BIM actively is Cg + Splt αLg1 Basedon the prospect theory individuals always overestimate thelow probability loss and underestimate the high probabilitygain namely individuals tend to choose risk aversion in the

z

1

1

x

V2

V11 y

Figure 2 Phase diagram of social capitalsrsquo behavioral strategyevolution

Complexity 7

Table 3 +e eigenvalues of each equilibrium point

Equilibrium pointEigenvalues of Jacobi matrix

λ1 λ2 λ3E(0 0 0) minusCg minus Sp + Lg + Lp + αLg1 Cp2 minus Cp1 + Lc2 Rc minus Cc + ScE(1 0 0) Cg + Sp minus Lg minus Lp minus Lc minus αLg1 Cp2 minus Cp1 + Lc1 + Lp Rc minus Cc + Sc + LcE(0 1 0) minusCg minus Sp + Lc + αLg1 Cp2 minus Cp1 minus Lc2 Rc minus Cc + Sc + Lc2E(0 0 1) minusCg minus Sp + Lc + αLg1 Cp2 minus Cp1 minusRc + Cc + ScE(1 1 0) Cg + Sp minus Lc minus αLg1 minusCp2 minus Cp1 minus Lc1 minus Lp Rc minus Cc + Sc + Lc + Lc1E(1 0 1) Cg + Sp minus Lp minus Lg minus αLg1 Cp2 minus Cp1 + Lp minusRc + Cc minus Sc minus LcE(0 1 1) minusCg minus Sp + αLg1 minusCp2 + Cp1 minusRc + Cc minus Sc minus Lc2E(1 1 1) Cg + Sp minus αLg1 minusCp2 minus Cp1 + Lp minusRc + Cc minus Sc minus Lc minus Lc1

z

1

1

x

V2

V1

1 y

Figure 3 Phase diagram of contractorsrsquo behavioral strategy evolution

111

0908070605

z

108

06y04 05 06 07 08

x

09 1

Cg=20Cg=15Cg=10

(a)

11

1

09

08

07

06

05

z

Cg=20Cg=15Cg=10

1 095 09 085 08 075 07 065 06 055 05x

(b)

Figure 4 Influence of Cg change on the evolutionary game process

8 Complexity

face of gains but tend to prefer risk preference when facinglosses [57] In the practice of PPP projects governmentsoften think that their behavior will not be easily detected bythe superior supervision authority due to the commonexistence of fluke mentality +erefore governments tend tounderestimate the probability α and the value of punishmentLg1 by the superior supervision authority which means thatthe right part of the conditional formula Cg + Splt αLg1 willbe underestimated in practice On the other hand whengovernments choose the active strategy it will definitely paythe promotion cost while choosing the passive strategy willonly have a certain probability of punitive losses Accordingto the prospect theory governments will be more inclined torisk preference that is they prefer to take risks and choosethe passive strategy rather than to bear the deterministic

expenditure +erefore the above reasons will cause certaindifficulties for the establishment of the formulaCg + Splt αLg1

4 Numerical Simulation and Analysis

In order to better analyze the behavior evolution mechanismamong governments social capitals and contractors in BIMadoption this study uses MATLAB R2018a to simulate thetrend and influencing factors of the evolutionary game +evalues of parameters are randomly assigned under the as-sumption which do not represent the real benefits ofstakeholders in PPP projects

According to the above analysis if the system convergesto the ESS point (1 1 1) the following conditions need to be

111

0908070605

z

108

06y04 04

0608

x

1

a=05a=08a=1

(a)

a=05a=08a=1

11

1

09

08

07

06

05

z

11 1 09 08 07 06 05x

(b)

Figure 5 Influence of α change on the evolutionary game process

Lg1=80Lg1=100Lg1=120

111

0908070605

z

108

06y04 04

0608

x

1

(a)

Lg1=80Lg1=100Lg1=120

11

1

09

08

07

06

05

z

11 1 09 08 07 06 05x

(b)

Figure 6 Influence of Lg1 change on the evolutionary game process

Complexity 9

satisfied Cg + Splt αLg1 Cp1ltCp2 + Lp and CcltRc + Sc+e initial values of each parameter are set to x 05 y

05 α 05Cg 20 Sp 10 Lgl 80Cp1 30Cp2

20 Sp 10 Lg1 80Cp1 30Cp2 20 Lp 20Cc

30Rc 40 Sc 10 Lg 10 Lc 10 Lc1 10 LC2 5

41 Influence of Cg1 and Cg2 Changes on the EvolutionaryGame Process Keep the other parameters unchanged andthe influence on the evolution trend of the system is ob-served by changing the values of Cg α and Lg1 as shown inFigures 4(a) 5(a) and 6(a) In Figure 4(b) we can see thatthe convergence rate of x to the ESS point (1 1 1) isaccelerated as the value of Cg goes down +e simulationresults show that reducing the cost of governmentsrsquo strategyof ldquopositive promotionrdquo is helpful to enhance the willingness

of governments to promote the adoption of BIM in PPPprojects In Figures 5(b) and 6(b) we can see that theconvergence rate of x to the ESS point (1 1 1) is acceleratedas the value of α and Lg1 go bigger +e simulation resultsshow that increasing the supervision probability and theseverity of punishment will help increase the willingness ofgovernments to promote BIM application in PPP projects

42 Influence of Cp1 and Lp Changes on the EvolutionaryGame Process Keep the other parameters unchanged andthe influence on the evolution trend of the system is ob-served by changing the values of Cp1 and Lp as shown inFigures 7(a) and 8(a) In Figure 7(b) we can see that theconvergence rate of y to the ESS point (1 1 1) is acceleratedas the value of Cp1 goes down +e simulation results show

111

0908070605

z

108

06y04 05 06 07 08

x

091

Cp1=30Cp1=20Cp1=10

(a)

Cp1=30Cp1=20Cp1=10

11

1

09

08

07

06

05

z

11 1 09 08 07 06 05y

(b)

Figure 7 Influence of Cp1 change on the evolutionary game process

111

0908070605

z

108

06y04 05

06 07 x08 09 1

Lp=20Lp=30Lp=40

(a)

Lp=20Lp=30Lp=40

11

1

09

08

07

06

05

z

11 1 09 08 07 06 05y

(b)

Figure 8 Influence of Lp change on the evolutionary game process

10 Complexity

that reducing the cost of social capitalsrsquo strategy of ldquopositiveadoptionrdquo is helpful to enhance the willingness of socialcapitals to adopt BIM in PPP projects In Figure 8(b) we cansee that the convergence rate of y to the ESS point (1 1 1) isaccelerated as the value of Lp goes bigger +e simulationresults show that increasing the punishment of social cap-italsrsquo strategy of ldquopassive adoptionrdquo is helpful to enhance thewillingness of social capitals to adopt BIM in PPP projects

43 Influence of Cc and Sc Changes on the Evolutionary GameProcess Keep the other parameters unchanged and theinfluence on the evolution trend of the system is observed bychanging the values of Cc and Sc as shown in Figures 9(a)and 10(a) In Figure 9(b) we can see that the convergence

rate of z to the ESS point (1 1 1) is accelerated as the value ofCc goes down+e simulation results show that reducing thecost of contractorsrsquo strategy of ldquopositive applicationrdquo ishelpful to enhance the willingness of contractors to applyBIM in PPP projects In Figure 10(b) we can see that theconvergence rate of z to the ESS point (1 1 1) is acceleratedas the value of Sc goes bigger +e simulation results showthat increasing the punishment of contractorsrsquo strategy ofldquopassive applicationrdquo is helpful to enhance the willingness ofcontractors to apply BIM in PPP projects

5 Conclusions and Management Implications

Under the assumption of bounded rationality this studyconstructs a tripartite evolutionary game model including

111

0908070605

z

108

06y04 05 06 07 08

x

091

Cc=30Cc=20Cc=10

(a)

Cc=30Cc=20Cc=10

11

1

09

08

07

06

05

z

1 09095 08085 07075 06065 05055y

(b)

Figure 9 Influence of Cc change on the evolutionary game process

111

0908070605

z

108

06y04 05 06 07 x

08 091

Sc=20Sc=30Sc=40

(a)

Sc=20Sc=30Sc=40

11

1

09

08

07

06

05

z

1 09095 08085 07075 06065 05055y

(b)

Figure 10 Influence of Sc change on the evolutionary game process

Complexity 11

governments social capitals and contractors +e satisfac-tion conditions that the system converging to the ESS point(1 1 1) are analyzed based on prospect theory and the mainfactors influencing stakeholdersrsquo behavior decision-makingare simulated by MATLAB +e research results will help toclarify the behavior evolution mechanism and the influ-encing factors of the ESS for adopting BIM among stake-holders in PPP projects so as to improve the cross-organization information collaboration efficiency of PPPprojects and promote the informatization of the construc-tion industry +e primary conclusions and managementimplications are as follows

First the cost of BIM promotion by governments and thepunishment from the superior supervision authority whengovernments choose the passive strategy are the key factorsaffecting governmentsrsquo behavior choice When governmentsactively promote the cost reduction of BIM or punishedmore by the superior supervision authority governmentswill choose the active promotion strategy and the behaviorgradually converges to the ESS point (1 1 1)

Second the cost of active adoption of BIM and thepunishment of passive adoption of BIM are the key factorsaffecting the choice of social capitalsrsquo behavior When thecost of active adoption of BIM is reduced or the penalty ofpassive adoption of BIM is increased the intention of socialcapitals to choose active adoption of BIM strategy is in-creased and the behavior of social capitals gradually con-verges to the ESS point (1 1 1)

Finally the cost of active application of BIM and thereward of active application of BIM are the key factorsinfluencing the choice of contractorsrsquo behavior When thecost of the active application of BIM technology is reducedor the penalty of the negative application of BIM technologyis increased the intention of the contractor to choose thestrategy of active application of BIM technology is increasedand the contractorrsquos behavior gradually converges to the ESSpoint (1 1 1)

+e adoption of BIM in PPP projects is a major reform ofthe traditional building mode whichrequires all stakeholders to change the traditionalthinking and actively cooperate In order to promoteBIM adoption among stakeholders of PPP projects thefollowing management implications are proposedImprove the supervision efficiency of superior supervisionauthority on governmentsrsquo promotion of BIM and furtherenrich the supervision means with the help of the media andthe public Increase the punishment for the governmentsrsquopassive promotion of BIM so as to improve the perceivedvalue of governments for regulatory punishment

Strengthen the punishment for the passive adoption ofBIM by social capitals and the reward for the active ap-plication of BIM by contractors Improve the cognition ofsocial capitals and contractors on BIM and other advancedinformation technology through publicity and training andgradually transform the system promotion into habitualadoption

Under the PPP mode the adoption of BIM can bepromoted by the governmentrsquos supervision but the gov-ernmentrsquos behavior also needs supervision +e public is the

users and beneficiaries of PPP projectsrsquo goods and the publicshould also have the right to know and to supervise theimplementation of the projects +erefore in the next stagethe authors will further research the role of public super-vision in the adoption of BIM in PPP projects in order toexplore the evolutionary mechanism and influencing factorsunder more stakeholders

Data Availability

Some or all data models or code generated or used tosupport the findings of this study are available from thecorresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interest

Acknowledgments

+is work was supported by the National Natural ScienceFoundation of China (NSFC) (71801119) in part by theSocial Science Planning Foundation of Liaoning(L18BGL020) and in part by the Basic Scientific ResearchProjects in Institutions of Higher Learning of LiaoningEducation Department (LJ2017QW009)

References

[1] N Carbonara and R Pellegrino ldquoRevenue guarantee inpublic-private partnerships a win-win modelrdquo ConstructionManagement amp Economics vol 36 no 10 pp 584ndash598 2018

[2] L Cheng ldquoDiscussion on the application of PPPmodel in newinfrastructure constructionrdquo Open Journal of Social Sciencesvol 7 no 9 pp 283ndash288 2019

[3] D Wu and S Q Wang ldquoResearch development and trend ofpublic private partnership in Chinardquo Journal of EngineeringManagement vol 28 no 6 pp 75ndash80 2014

[4] K Leidel and H W Alfen Lifecycle-Oriented Risk Manage-ment for PPP-Projects in Public Real Estate pp 587ndash604 EresLos Angeles CA USA 2009

[5] J Yuan X Li Y Ke W Xu Z Xu and M SkibnewskildquoDeveloping a building information modelingndashbased per-formance management system for publicndashprivate partner-shipsrdquo Engineering Construction and ArchitecturalManagement vol 27 no 8 pp 1727ndash1762 2020

[6] S Jayasuriya G Zhang and R J Yang ldquoExploring the impactof stakeholder management strategies on managing issues inPPP projectsrdquo International Journal of Construction Man-agement vol 20 no 6 pp 666ndash678 2020

[7] C X Jiang ldquoCauses and implications of the failure intransportation infrastructure public privatepartnershipsmdashmdasha study of 25 typical PPP casesrdquo Journal ofBeijing Jiaotong University vol 15 no 3 pp 50ndash58 2016

[8] R Osei-Kyei and A P C Chan ldquoReview of studies on thecritical success factors for public-private partnership(PPP) projects from 1990 to 2013rdquo International Journal ofProject Management vol 33 no 6 pp 1335ndash1346 2015

[9] B Zhang Y Wang X Liu et al ldquo+e configuration of PPPfailuremdashmdasha crisp-set qualitative comparative analysisrdquoJournal of Public Administration vol 12 no 04pp 65ndash81+191 2019

12 Complexity

[10] N Costantino and R Pellegrino ldquoPublicndashprivate partnershipimplementation cross-case analysisrdquo Proceedings of the In-stitution of Civil Engineers-management Procurement andLaw vol 168 no 4 pp 164ndash176 2015

[11] P Zhang K Zhan and Y Zhou ldquoGame model of three-partyshared risk allocation for PPP projectsmdashmdashbased on theperspective of incomplete informationrdquo IOP Conference Se-ries Earth and Environmental Science vol 295 no 4pp 42ndash49 2019

[12] N W Young S A Jones and H M Bernstein BuildingInformation Modeling Transforming Design and Constructionto Achieve Greater Industry Productivity McGraw HillConstruction New York NY USA 2008

[13] N D Aziz A H Nawawi and N R M Ariff ldquoBuildinginformation modelling (BIM) in facilities management op-portunities to be considered by facility managersrdquo Procedia-Social and Behavioral Sciences vol 234 pp 353ndash362 2016

[14] W Zou M Kumaraswamy and J Chung ldquoIdentifying thecritical success factors for relationship management in PPPprojectsrdquo International Journal of Project Managementvol 32 no 2 pp 265ndash274 2014

[15] QWang and TWang ldquoResearch on promotion of BIM basedon game analysis of stakeholdersrdquo Sichuan BuildingMaterialsvol 41 no 6 p 272 + 278 2015

[16] Q L Yin Evolutionary Game Analysis on BIM TechnologyDiffusion of Prefabricated Construction Qingdao Universityof Technology Qingdao China 2019

[17] H X Tang J M Cao H S Xu and C F Han ldquoAnalysis onBIM application cooperation behaviors in integrated facilitymanagement organization based on evolutionary game the-oryrdquo Industrial Engineering amp Management vol 25 no 4pp 1ndash8 2020

[18] J R Song ldquoResearch on incentive and countermeasures ofBIM application based on game theoryrdquo Value Engineeringvol 36 no 33 pp 43-44 2017

[19] C Jia Z H Wang and L Zhao ldquoEvolutionary game analysisand simulation for cooperative application of BIM technologyin PPPrdquo Journal of Liaoning Technical University vol 22no 4 pp 270ndash277 2020

[20] K A Tversky ldquoProspect theory an analysis of decision underriskrdquo Econometrica vol 47 no 2 pp 263ndash291 1979

[21] R L Gao and Q Bao ldquoSelection of government supervisionmode during the operational period in PPP projects based onevolutionary game theoryrdquo Operations Research and Man-agement Science vol 28 no 4 pp 155ndash162 2019

[22] S K He G W Liang and J B Meng ldquoMulti-subjectsbenefit game and behavior evolution mechanism of majorengineering based on prospect theoryrdquo Science andTechnology Management Research vol 40 no 5 pp 207ndash214 2020

[23] A Porwal and K N Hewage ldquoBuilding information modeling(BIM) partnering framework for public construction proj-ectsrdquo Automation in Construction vol 31 no 3 pp 204ndash2142013

[24] P E D Love J Liu J Matthews C-P Sing and J SmithldquoFuture proofing PPPs life-cycle performance measurementand building information modellingrdquo Automation in Con-struction vol 56 pp 26ndash35 2015

[25] Y L Yin Q J Liu and X Wang ldquoResearch on the whole lifecycle supervision system of public projectsmdashmdashbased on thecollaboration of VfM evaluation and BIM technologyrdquo ProjectManagement Technology vol 14 no 07 pp 17ndash20 2016

[26] S Q Wang R Tiong S Ting et al ldquoEvaluation and man-agement of political risks in Chinarsquos BOTprojectsrdquo Journal of

Construction Engineering and Management vol 126 no 3pp 242ndash250 2000

[27] Y M Chen and H F Chen ldquo+e risk management of PPPproject life cycle based on BIM technologyrdquo Project Man-agement Technology vol 15 no 5 pp 51ndash55 2017

[28] G Luo and X X Zhan ldquoResearch on the application of BIM inPPP project whole-life cyclerdquo Technology and EconomicGuide vol 26 no 5 p 168 2018

[29] B L Zhou and C L Lin ldquo+e application of BIM in PPPproject whole-life cyclerdquo Engineering Economy vol 29 no 3pp 51ndash54 2019

[30] J M Smith Evolution and the )eory of Games CambridgeUniversity Press Cambridge UK 1982

[31] J W Weibull Evolutionary Game )eory MIT PressCambridge MA USA 1997

[32] K Binmore ldquoModeling rational players part Irdquo Economicsand Philosophy vol 3 no 2 pp 179ndash214 1987

[33] K Binmore ldquoModeling rational players part IIrdquo Economicsand Philosophy vol 4 1988

[34] P D Taylor and L B Jonker ldquoEvolutionarily stable strategiesand game dynamicsrdquoMathematical Biosciences vol 40 no 1-2 pp 145ndash156 1978

[35] P Young ldquoConventional contractsrdquo )e Review of EconomicStudies vol 65 no 4 pp 773ndash792 1998

[36] D Fudenberg and C Harris ldquoEvolutionary dynamics withaggregate shocksrdquo Journal of Economic )eory vol 57 no 2pp 420ndash441 1992

[37] H Bester W Guth and W Guth ldquoIs altruism evolutionarilystablerdquo Journal of Economic Behavior amp Organization vol 34no 2 pp 193ndash209 1998

[38] K E Boulding R R Nelson and S G Winter ldquoAn evolu-tionary theory of economic changrdquo American Journal ofAgricultural Economics vol 66 no 4 p 535 1984

[39] S Y Xie Economic Game )eory Fudan University PressShanghai China 2002

[40] D Friedman and K C Fung ldquoInternational trade and theinternal organization of firms an evolutionary approachrdquoJournal of International Economics vol 41 no 1-2 pp 113ndash137 1996

[41] Y P Wang and W D Meng ldquoEvolutionary game analysis onco-competition mechanism of supply chainrdquo Journal of In-dustrial Engineering and Engineering Management vol 18no 2 pp 96ndash98 2004

[42] J Gao and Z H Shen ldquoElementary groping for evolutionarygame theory and its application in electricity marketrdquo Au-tomation of Electric Power Systems vol 27 no 18 pp 18ndash212003

[43] Y Zhang and M Guizani Game )eory for Wireless Com-munications and Networking CRC Press Boca Raton FLUSA 2011

[44] F Y Lu ldquoEvolutionary game analysis on environmentalpollution problemrdquo Systems Engineering-)eory amp Practicevol 27 no 9 pp 148ndash152 2007

[45] S L Dreisbach ldquoImpact of psychological research on moraltheory prospect theory and the doctrine of doing andallowingrdquo Dissertations amp )eses-Gradworks University ofCalifornia Santa Cruz CA USA 2012

[46] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Econometrica vol 47 no 2 1979

[47] N C Barberis ldquo+irty years of prospect theory in economicsa review and assessmentrdquo )e Journal of Economic Perspec-tives vol 27 no 1 pp 173ndash196 2013

Complexity 13

[48] S Benartzi and R H +aler ldquoMyopic loss aversion and theequity premium puzzlerdquo Quarterly Journal of Economicsvol 110 no 1 pp 73ndash92 1995

[49] N Barberis and M Huang ldquoStocks as lotteries the impli-cations of probability weighting for security pricesrdquo )eAmerican Economic Review vol 98 no 5 pp 2066ndash21002008

[50] A Frazzini ldquo+e disposition effect and underreaction tonewsrdquo )e Journal of Finance vol 61 no 4 pp 2017ndash20462006

[51] W Y Hu and J S Scott ldquoBehavioral obstacles in the annuitymarketrdquo Financial Analysts Journal vol 63 no 6 pp 71ndash822007

[52] B Koszegi andM Rabin ldquoReference-dependent consumptionplansrdquo )e American Economic Review vol 99 no 3pp 909ndash936 2009

[53] P Vamvakas E E Tsiropoulou and S Papavassiliou ldquoDy-namic spectrum management in 5G wireless networks a real-life modeling approachrdquo in Proceedings of the IEEE INFO-COM 2019-IEEE Conference on Computer CommunicationsIEEE Paris France April 2019

[54] Y N Zhou and J C Liu ldquoStudy on supervision strategy of pppproject considering the participation of higher-level gov-ernmentrdquo Chinese Journal of Management Science vol 1ndash162021 httpsdoiorg1016381jcnkiissn1003-207x20200801

[55] H Q Zhang J C Wang and H L Tao ldquoResearch the de-cision-making behavior of investors based on prospect theoryin the ppp projectrdquo Soft Science vol 32 no 8 pp 129ndash1332018

[56] D Friedman ldquoEvolutionary games in economicsrdquo Econo-metrica vol 59 no 3 pp 637ndash666 1991

[57] Z B Zhao and Q P Man ldquoEvolutionary game analysis of riskmanagement behavior of major infrastructure projects basedon projects based on prospect theoryrdquo Journal of SystemsManagement vol 27 no 1 pp 112ndash120 2018

14 Complexity

of psychological perception factors of PPP projectsrsquo stake-holders Obviously under the condition of bounded rationalityindividual psychological perception often has a key influenceon behavioral decision-making [46] So this study replaces theexpected utility of the traditional evolutionary game with thevalue function v(Δωi) of prospect theory and replaces theactual probability with the weight function π(pi) so as toobtain the perceived value of the total expected utility of thestrategy which can be expressed as

V 1113944i

π pi( 1113857v Δωi( 1113857 (1)

+erein the value function v(Δωi)

(Δωi)α Δωi ge 0

minusλ(minusΔωi)β Δωi lt 0

1113896 and the weight function π(pi) Pc

[Pc + (1 minus P)c]1c In the formula Δωi is the differencebetween the actual income and the reference point when thebehavior subject adopts strategy i α β(0lt αlt 1 0lt βlt 1)

are the risk attitude coefficients which represent the degreeof decline in the editor of the game subjectrsquos perceived valueof profit and loss+e higher the value the greater the degreeof marginal decline+e loss avoidance coefficient is denotedby λ (λge 1) and the greater the value the more sensitive thegame subject to the loss+e actual occurrence probability ofstrategy i is denoted by pi and the decision weight is denotedby π(pi)(π(0) 0 π(1) 1) When p is very small therewill be π(pi)＞p and when p is very large there will beπ(pi)＜p +at is low probability events are usually over-estimated in prospect theory while high probability eventsare usually underestimated +e construction of the mul-tiagent evolutionary game model based on prospect theorycan fully reflect the psychological perceived value of eachstakeholder and is more conducive to clarifying the evo-lutionary game mechanism and influencing factors

31 Basic Assumptions and Evolutionary Game Strategies

311 Basic Assumptions

Assumption 1 According to the above analysis socialcapitals are the important subjects of BIM adoption in PPPprojects contractors are the implementation subject of thepractical application of BIM and governments are thedriving subject of BIM adoption +erefore the behaviorstrategies of governments are set as positive promotionpassive promotion the behavioral strategies of socialcapitals are set as positive adoption passive adoption andthe behavioral strategies of contractors are set as positiveapplication passive application

Assumption 2 +e probability of ldquopositive promotionrdquo is setas x(0lexle 1) and the probability of ldquopassive promotionrdquo isequal to 1 minus x +e probability of ldquopositive adoptionrdquo is set asy(0leyle 1) and the probability of ldquopassive adoptionrdquo isequal to 1minusy +e probability of ldquopositive applicationrdquo is setas z(0le zle 1) and the probability of ldquopassive applicationrdquo isequal to 1 minus z

Assumption 3 Governments social capitals and contrac-tors will generate benefits and expenses when adoptingdifferent behavior strategies +e profit and loss indexes ofeach subject are given in Table 1

312 Game Strategiesrsquo Perceived Value of Profit and LossBased on the above assumptions the profit and loss matrixof the perceived value among governments social capitalsand contractors is constructed as given in Table 2

32 Construction and Solution of the Evolutionary GameModel

321 Replicator Dynamics Equation and Equilibrium PointAnalysis of Governments According to Table 2 we can get theperceived expected value Eg1 of governments taking strategy ofldquopositive promotionrdquo and the perceived expected value Eg2 ofgovernments taking strategy of ldquopassive promotionrdquo

Eg1 yz(Rg1 minus Cg minus Sp) + y(1 minus z)(Rg1 minus Cg minus Sp + Lc) + z(1 minus y)(Rg2 minus Cg minus Sp + Lp)

+(1 minus y(1 minus z)(minusCg + Lp + Lc minus Sp))

Eg2 yz(Rg1 minus αLg1) + y(1 minus z)(Rg2 minus αLg1) + z(1 minus y)(1 minus z)(minusLg minus αLg1)

(2)

+en the average perceived expected value of govern-ments is as follows

Eg xEg1 +(1 minus x)Eg2 (3)

+erefore the replication dynamic equation of gov-ernments is as follows

F(x) dx

dt x(Eg1 minus Eg) x(1 minus x)(Eg1 minus Eg2)

x(1 minus x)(minusCg minus Sp + Lp + Lc + Lg + αLg1 minus zLc minus yLp minus yLg)

(4)

4 Complexity

+e first derivative of x is equal to

d(F(x))

dx (1 minus 2x)(minusCg minus Sp + Lp)

+ Lc + Lg + αLg1 minus zLc minus yLp minus yLg)

(5)

When F(x) 0 and d(F(x))dxlang0 governmentsrsquostrategy of ldquopositive promotionrdquo will be in a stable state

+erefore there will be F(x) equiv 0 when z minusCgminus

ySp + Lp + Lc + Lg minus yLp minus αLg1Lc and all the values of xare in a stable state When zlang minus Cg minus ySp + Lp+

Lc + Lg minus yLp minus αLg1Lc there will be d(F(x))dx|x1lang0and x 1 is the ESS of the system When zrang minus Cg minus ySp+

Lp + Lc + Lg minus yLp minus αLg1Lc there will be d(F(x))dx|x0lang0 and x 0 is the ESS of the system +e phase

diagram of governmentsrsquo behavioral strategy evolution isshown in Figure 1+e slant z minusCg minus ySp + Lp + Lc + Lg minus

yLp minus αLg1Lc divides the three-dimensional spaceV (x y z| 0lexle 1 0leyle 1 0le zle 1) into parts of V1and V2 In part V1 governments gradually evolved to x 0and finally chose the strategy of ldquopassive promotionrdquo In partV2 governments gradually evolved to x 1 and finallychose the strategy of ldquopositive promotionrdquo

322 Replicator Dynamics Equation and Equilibrium PointAnalysis of Social Capitals According to Table 2 we can getthe perceived expected value Ep1 of social capitals takingstrategy of rdquopositive adoptionrdquo and the perceived expectedvalue Ep2 of social capitals taking the strategy of ldquopassiveadoptionrdquo

Table 1 +e profit and loss indexes of stakeholders

Index Explanation of indexRg1 +e benefit brought to governments when social capitals adopt BIM activelyRg2 +e benefit brought to governments when social capitals adopt BIM passivelyCg +e cost of governmentsrsquo promotion of BIMSp +e subsidy provided by governments when social capitals adopt BIM activelyLp +e punishment imposed by governments when social capitals adopt BIM passivelyLc +e punishment imposed by governments when contractors apply BIM passively

Lg+e adverse effects on the development of construction industry when governments promote BIM passively and social capitals

adopt BIM passivelyLg1 +e punishment imposed by the superior supervision authority when governments promote BIM passivelyα +e probability of the superior supervision authority finding that governments promote BIM passivelyRp +e benefit brought to social capitals when contractors applying BIM activelyCp1 +e cost of social capitalsrsquo positive adoption of BIMCp2 +e cost of social capitalsrsquo passive adoption of BIMLp1 +e adverse effects on social capitals when contractors apply BIM passivelySc +e reward provided by social capitals when contractors apply BIM activelyRc +e benefit obtained by contractors apply BIM activelyCc +e cost incurred by the contractors apply BIM actively

Lc1 +e compensation given by contractors undertaking passive application of BIM to social capitals undertaking positive adoption ofBIM when governments promote BIM actively

Lc2 +e compensation given by contractors undertaking passive application of BIM to social capitals undertaking positive adoption ofBIM when governments promote BIM passively

Table 2 +e matrix of the perceived value of profit and loss among stakeholders

Participants of the evolutionary gameGovernments

Positive promotion(x)

Passive promotion(1 minus x)

Social capitals

Positive adoption(y)

Contractors

Positive application(z)

Rg1 minus Cg minus Sp Rg1 minus αLg1Rp minus Cp1 minus Sc + Sp Rp minus Cp1 minus Sc

Rc minus Cc + Sc RcminusCc+ Sc

Passive application(1 minus z)

Rg1 minus Cg minus Sp + Lc Rg1 minus αLg1Rp minus Cp1 + Sp + Lc1 minus Lp1 Rp minus Cp1 + Lc2 minus Lp1

minusLc minus Lc1 minus Lc2

Passive adoption(1 minus y)

Positive application(z)

Rg2 minus Cg minus Sp + Lp Rg2 minus Lg minus αLg1Rp minus Cp2 + Sp minus Lp minus Sc Rp minus Cp2 minus Sc

Rc minus Cc + Sc Rc minus Cc + Sc

Passive application(1 minus z)

minusCg + Lp + Lc minus Sp minusLg minus αLg1RpminusCp2+ Spminus Lpminus Lp1 RpminusCp2minus Lp1

minusLc 0

Complexity 5

Ep1 xz(Rp minus Cp minus Sc + Sp) + z(1 minus x)(Rp minus Cp1 minus Sc) + x(1 minus z)(Rp minus Cp1 + Sp

+ Lc1 minus Lp1) +(1 minus z)(1 minus x)(Rp1 minus Cp1 + Lc2 minus Lp1)

Ep2 xz(Rp1 minus Cp2 minus Lp minus Sc + Sp) + z(1 minus x)(Rp minus Cp2 minus Sc) + x(1 minus z)

middot (Rp minus Cp2 minus Lp + Sp minus Lp1) +(1 minus z)(1 minus x)(Rp minus Cp2 minus Lp)

(6)

+en the average perceived expected value of socialcapitals is as follows

Ep yEp1 +(1 minus y)Ep2 (7)

+erefore the replication dynamic equation of socialcapitals is as follows

F(y) dy

dt y(Ep1 minus Ep) y(1 minus y)(Ep1 minus Ep2)

y(1 minus y)(Cp2 minus Cp1 + Lc2 + xLc1 minus xLc2

minus zLc2 + xLp minus xzLc1 + xzLc2)

(8)

+e first derivative of y is equal to

d(F(y))

dt (1 minus 2y)(Cp2 minus Cp1 + Lc2 + xLc1 minus xLc2

minus zLc2 + xLp minus xzLc1 + xzLc2)

(9)

When F(y) 0 and d(F(y))dylang0 social capitalsrsquostrategy of ldquopositive adoptionrdquo will be in a stable state+erefore there will be F(y) equiv 0 when z Cp2 minus Cp1 + Lc2

+xLc1 minus xLc2 + xLpLc2 + xLc1 minus xLc2 and all the values ofy are in a stable state When zlangCp2 minus Cp1 + Lc2 + xLc1 minus

xLc2+ xLpLc2 + xLc1 minus xLc2 there will be d(F(y))dy|y1lang0 and y 1 is the ESS of the systemWhen zrangCp2 minus Cp1 +

Lc2 + xLc1minus xLc2 + xLpLc2 + xLc1 minus xLc2 there will bed(F(y))dy|y0lang0 and y 0 is the ESS of the system +ephase diagram of social capitalsrsquo behavioral strategy evo-lution is shown in Figure 2 +e surface z Cp2 minus Cp1 +

Lc2 + xLc1 minus xLc2 + xLpLc2 + xLc1 minus xLc2 divides thethree-dimensional space V (x y z| 0lexle 1 0leyle1 0le zle 1) into parts of V1 and V2 In part V1 socialcapitals gradually evolved to y 1 and finally chose thestrategy of ldquopositive adoptionrdquo In part V2 social capitalsgradually evolved to y 0 and finally chose the strategy ofldquopassive adoptionrdquo

323 Replicator Dynamics Equation and Equilibrium PointAnalysis of Contractors According to Table 2 we can get theperceived expected valueEc1 of contractors taking strategy ofldquopositive applicationrdquo and the perceived expected value Ec2of contractors taking strategy of ldquopassive applicationrdquo

Ec1 xy(Rc minus Cc + Sc) + y(1 minus x)(Rc minus Cc + Sc) +(1 minus y)(1 minus x)(Rc minus Cc + Sc)

Rc minus Cc + Sc

Ec2 xy(minusLc minus Lc1) + y(1 minus x)(minusLc2) + x(1 minus y)(minusLc) +(1 minus y)(1 minus x)(0) xyLc2 minus xLc minus yLc minus xyLc

(10)

z

1

1

x

V2

V1

1 y

Figure 1 Phase diagram of governmentsrsquo behavioral strategy evolution

6 Complexity

+en the average perceived expected value of contrac-tors is as follows

Ec zEc1 +(1 minus z)Ec2 (11)

+erefore the replication dynamic equation of con-tractors is as follows

F(z) dz

dt z(Ec1 minus Ec) z(1 minus z)(Ec1 minus Ec2)

z(1 minus z)(Rc minus Cc + Sc + xLc + yLc2 + xyLc1 minus xyLc2)

(12)

+e first derivative of z is equal to

d(F(z))

dz (1 minus 2z)(Rc minus Cc + Sc + xLc + yLc2 + xyLc1 minus xyLc2)

(13)

When F(z) 0 and d(F(z))dzlang0 contractorsrsquo strategyof ldquopositive applicationrdquo will be in a stable state +ereforethere will be F(z) equiv 0 when y Rc minus Cc + Sc + xLcx(Lc2 minus Lc1) minus Lc2 and all the values of z are in a stablestate When ylangRc minus Cc + Sc + xLcx(Lc2 minus Lc1) minus Lc2there will be d(F(z))dz|z1lang0 and z 1 is the ESS of thesystem When yrangRc minus Cc + Sc + xLcx(Lc2 minus Lc1) minus Lc2there will be d(F(z))dz|z0lang0 and z 0 is the ESS of thesystem +e phase diagram of contractorsrsquo behavioralstrategy evolution is shown in Figure 3 +e surface y

Rc minus Cc + Sc + xLcx(Lc2 minus Lc1) minus Lc2 divides the three-dimensional space V (x y z| 0lexle 1 0le yle 1 0lezle 1) into parts of V1 and V2 In part V1 contractorsgradually evolved to z 1 and finally chose the strategy ofldquopositive applicationrdquo In part V2 contractors graduallyevolved to z 0 and finally chose the strategy of ldquopassiveapplicationrdquo

33 Evolutionary Stability Analysis +e Jacobin matrix ofthe system is represented by J

J

J1 J2 J3

J4 J5 J6

J7 J8 J9

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

zf(x)

zx

zf(x)

zy

zf(x)

zz

zf(y)

zx

zf(y)

zy

zf(y)

zz

zf(z)

zx

zf(z)

zy

zf(z)

zz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(14)

Among them

zf(x)

zx (1 minus 2x)(Cg2 minus Cg1 + Lg minus Sp + Lp + Lc minus zLc minus yLp minus zLg)

zf(y)

zy (1 minus 2y)(Cp2 minus Cp1 + Lc2 + xLc1 minus xLc2 minus zLc2 + xLp minus xzLc1 + xzLc2)

zf(z)

zz (1 minus 2z)(Rc minus Cc + Sc + xLc + yLc2 + xyLc1 minus xyLc2)

(15)

+e local stability of equilibrium points is given inTable 3

According to the indirect method of Lyapunov theequilibrium point is asymptotically stable if all its eigen-values of the Jacobian matrix [56] have negative real parts InTable 3 it can be seen that if we want the system evolving tothe ESS point (1 1 1) the following conditions must be metCg + Splt αLg1 Cp1ltCp2 + Lp and CcltRc + Sc

However it should be noted that in the actual PPPprojects the main stakeholders cannot make the best use ofthe existing information in the decision-making because of

their limited rationality +ey usually make decisionsthrough subjective judgment which ultimately leads to de-viation in their behaviors +e deviations will hinder theestablishment of ESS point (1 1 1) and make the systemdifficult to converge to the optimal state +e reasons areanalyzed based on the prospect theory as followstake governments for example the condition that gov-ernments promoting BIM actively is Cg + Splt αLg1 Basedon the prospect theory individuals always overestimate thelow probability loss and underestimate the high probabilitygain namely individuals tend to choose risk aversion in the

z

1

1

x

V2

V11 y

Figure 2 Phase diagram of social capitalsrsquo behavioral strategyevolution

Complexity 7

Table 3 +e eigenvalues of each equilibrium point

Equilibrium pointEigenvalues of Jacobi matrix

λ1 λ2 λ3E(0 0 0) minusCg minus Sp + Lg + Lp + αLg1 Cp2 minus Cp1 + Lc2 Rc minus Cc + ScE(1 0 0) Cg + Sp minus Lg minus Lp minus Lc minus αLg1 Cp2 minus Cp1 + Lc1 + Lp Rc minus Cc + Sc + LcE(0 1 0) minusCg minus Sp + Lc + αLg1 Cp2 minus Cp1 minus Lc2 Rc minus Cc + Sc + Lc2E(0 0 1) minusCg minus Sp + Lc + αLg1 Cp2 minus Cp1 minusRc + Cc + ScE(1 1 0) Cg + Sp minus Lc minus αLg1 minusCp2 minus Cp1 minus Lc1 minus Lp Rc minus Cc + Sc + Lc + Lc1E(1 0 1) Cg + Sp minus Lp minus Lg minus αLg1 Cp2 minus Cp1 + Lp minusRc + Cc minus Sc minus LcE(0 1 1) minusCg minus Sp + αLg1 minusCp2 + Cp1 minusRc + Cc minus Sc minus Lc2E(1 1 1) Cg + Sp minus αLg1 minusCp2 minus Cp1 + Lp minusRc + Cc minus Sc minus Lc minus Lc1

z

1

1

x

V2

V1

1 y

Figure 3 Phase diagram of contractorsrsquo behavioral strategy evolution

111

0908070605

z

108

06y04 05 06 07 08

x

09 1

Cg=20Cg=15Cg=10

(a)

11

1

09

08

07

06

05

z

Cg=20Cg=15Cg=10

1 095 09 085 08 075 07 065 06 055 05x

(b)

Figure 4 Influence of Cg change on the evolutionary game process

8 Complexity

face of gains but tend to prefer risk preference when facinglosses [57] In the practice of PPP projects governmentsoften think that their behavior will not be easily detected bythe superior supervision authority due to the commonexistence of fluke mentality +erefore governments tend tounderestimate the probability α and the value of punishmentLg1 by the superior supervision authority which means thatthe right part of the conditional formula Cg + Splt αLg1 willbe underestimated in practice On the other hand whengovernments choose the active strategy it will definitely paythe promotion cost while choosing the passive strategy willonly have a certain probability of punitive losses Accordingto the prospect theory governments will be more inclined torisk preference that is they prefer to take risks and choosethe passive strategy rather than to bear the deterministic

expenditure +erefore the above reasons will cause certaindifficulties for the establishment of the formulaCg + Splt αLg1

4 Numerical Simulation and Analysis

In order to better analyze the behavior evolution mechanismamong governments social capitals and contractors in BIMadoption this study uses MATLAB R2018a to simulate thetrend and influencing factors of the evolutionary game +evalues of parameters are randomly assigned under the as-sumption which do not represent the real benefits ofstakeholders in PPP projects

According to the above analysis if the system convergesto the ESS point (1 1 1) the following conditions need to be

111

0908070605

z

108

06y04 04

0608

x

1

a=05a=08a=1

(a)

a=05a=08a=1

11

1

09

08

07

06

05

z

11 1 09 08 07 06 05x

(b)

Figure 5 Influence of α change on the evolutionary game process

Lg1=80Lg1=100Lg1=120

111

0908070605

z

108

06y04 04

0608

x

1

(a)

Lg1=80Lg1=100Lg1=120

11

1

09

08

07

06

05

z

11 1 09 08 07 06 05x

(b)

Figure 6 Influence of Lg1 change on the evolutionary game process

Complexity 9

satisfied Cg + Splt αLg1 Cp1ltCp2 + Lp and CcltRc + Sc+e initial values of each parameter are set to x 05 y

05 α 05Cg 20 Sp 10 Lgl 80Cp1 30Cp2

20 Sp 10 Lg1 80Cp1 30Cp2 20 Lp 20Cc

30Rc 40 Sc 10 Lg 10 Lc 10 Lc1 10 LC2 5

41 Influence of Cg1 and Cg2 Changes on the EvolutionaryGame Process Keep the other parameters unchanged andthe influence on the evolution trend of the system is ob-served by changing the values of Cg α and Lg1 as shown inFigures 4(a) 5(a) and 6(a) In Figure 4(b) we can see thatthe convergence rate of x to the ESS point (1 1 1) isaccelerated as the value of Cg goes down +e simulationresults show that reducing the cost of governmentsrsquo strategyof ldquopositive promotionrdquo is helpful to enhance the willingness

of governments to promote the adoption of BIM in PPPprojects In Figures 5(b) and 6(b) we can see that theconvergence rate of x to the ESS point (1 1 1) is acceleratedas the value of α and Lg1 go bigger +e simulation resultsshow that increasing the supervision probability and theseverity of punishment will help increase the willingness ofgovernments to promote BIM application in PPP projects

42 Influence of Cp1 and Lp Changes on the EvolutionaryGame Process Keep the other parameters unchanged andthe influence on the evolution trend of the system is ob-served by changing the values of Cp1 and Lp as shown inFigures 7(a) and 8(a) In Figure 7(b) we can see that theconvergence rate of y to the ESS point (1 1 1) is acceleratedas the value of Cp1 goes down +e simulation results show

111

0908070605

z

108

06y04 05 06 07 08

x

091

Cp1=30Cp1=20Cp1=10

(a)

Cp1=30Cp1=20Cp1=10

11

1

09

08

07

06

05

z

11 1 09 08 07 06 05y

(b)

Figure 7 Influence of Cp1 change on the evolutionary game process

111

0908070605

z

108

06y04 05

06 07 x08 09 1

Lp=20Lp=30Lp=40

(a)

Lp=20Lp=30Lp=40

11

1

09

08

07

06

05

z

11 1 09 08 07 06 05y

(b)

Figure 8 Influence of Lp change on the evolutionary game process

10 Complexity

that reducing the cost of social capitalsrsquo strategy of ldquopositiveadoptionrdquo is helpful to enhance the willingness of socialcapitals to adopt BIM in PPP projects In Figure 8(b) we cansee that the convergence rate of y to the ESS point (1 1 1) isaccelerated as the value of Lp goes bigger +e simulationresults show that increasing the punishment of social cap-italsrsquo strategy of ldquopassive adoptionrdquo is helpful to enhance thewillingness of social capitals to adopt BIM in PPP projects

43 Influence of Cc and Sc Changes on the Evolutionary GameProcess Keep the other parameters unchanged and theinfluence on the evolution trend of the system is observed bychanging the values of Cc and Sc as shown in Figures 9(a)and 10(a) In Figure 9(b) we can see that the convergence

rate of z to the ESS point (1 1 1) is accelerated as the value ofCc goes down+e simulation results show that reducing thecost of contractorsrsquo strategy of ldquopositive applicationrdquo ishelpful to enhance the willingness of contractors to applyBIM in PPP projects In Figure 10(b) we can see that theconvergence rate of z to the ESS point (1 1 1) is acceleratedas the value of Sc goes bigger +e simulation results showthat increasing the punishment of contractorsrsquo strategy ofldquopassive applicationrdquo is helpful to enhance the willingness ofcontractors to apply BIM in PPP projects

5 Conclusions and Management Implications

Under the assumption of bounded rationality this studyconstructs a tripartite evolutionary game model including

111

0908070605

z

108

06y04 05 06 07 08

x

091

Cc=30Cc=20Cc=10

(a)

Cc=30Cc=20Cc=10

11

1

09

08

07

06

05

z

1 09095 08085 07075 06065 05055y

(b)

Figure 9 Influence of Cc change on the evolutionary game process

111

0908070605

z

108

06y04 05 06 07 x

08 091

Sc=20Sc=30Sc=40

(a)

Sc=20Sc=30Sc=40

11

1

09

08

07

06

05

z

1 09095 08085 07075 06065 05055y

(b)

Figure 10 Influence of Sc change on the evolutionary game process

Complexity 11

governments social capitals and contractors +e satisfac-tion conditions that the system converging to the ESS point(1 1 1) are analyzed based on prospect theory and the mainfactors influencing stakeholdersrsquo behavior decision-makingare simulated by MATLAB +e research results will help toclarify the behavior evolution mechanism and the influ-encing factors of the ESS for adopting BIM among stake-holders in PPP projects so as to improve the cross-organization information collaboration efficiency of PPPprojects and promote the informatization of the construc-tion industry +e primary conclusions and managementimplications are as follows

First the cost of BIM promotion by governments and thepunishment from the superior supervision authority whengovernments choose the passive strategy are the key factorsaffecting governmentsrsquo behavior choice When governmentsactively promote the cost reduction of BIM or punishedmore by the superior supervision authority governmentswill choose the active promotion strategy and the behaviorgradually converges to the ESS point (1 1 1)

Second the cost of active adoption of BIM and thepunishment of passive adoption of BIM are the key factorsaffecting the choice of social capitalsrsquo behavior When thecost of active adoption of BIM is reduced or the penalty ofpassive adoption of BIM is increased the intention of socialcapitals to choose active adoption of BIM strategy is in-creased and the behavior of social capitals gradually con-verges to the ESS point (1 1 1)

Finally the cost of active application of BIM and thereward of active application of BIM are the key factorsinfluencing the choice of contractorsrsquo behavior When thecost of the active application of BIM technology is reducedor the penalty of the negative application of BIM technologyis increased the intention of the contractor to choose thestrategy of active application of BIM technology is increasedand the contractorrsquos behavior gradually converges to the ESSpoint (1 1 1)

+e adoption of BIM in PPP projects is a major reform ofthe traditional building mode whichrequires all stakeholders to change the traditionalthinking and actively cooperate In order to promoteBIM adoption among stakeholders of PPP projects thefollowing management implications are proposedImprove the supervision efficiency of superior supervisionauthority on governmentsrsquo promotion of BIM and furtherenrich the supervision means with the help of the media andthe public Increase the punishment for the governmentsrsquopassive promotion of BIM so as to improve the perceivedvalue of governments for regulatory punishment

Strengthen the punishment for the passive adoption ofBIM by social capitals and the reward for the active ap-plication of BIM by contractors Improve the cognition ofsocial capitals and contractors on BIM and other advancedinformation technology through publicity and training andgradually transform the system promotion into habitualadoption

Under the PPP mode the adoption of BIM can bepromoted by the governmentrsquos supervision but the gov-ernmentrsquos behavior also needs supervision +e public is the

users and beneficiaries of PPP projectsrsquo goods and the publicshould also have the right to know and to supervise theimplementation of the projects +erefore in the next stagethe authors will further research the role of public super-vision in the adoption of BIM in PPP projects in order toexplore the evolutionary mechanism and influencing factorsunder more stakeholders

Data Availability

Some or all data models or code generated or used tosupport the findings of this study are available from thecorresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interest

Acknowledgments

+is work was supported by the National Natural ScienceFoundation of China (NSFC) (71801119) in part by theSocial Science Planning Foundation of Liaoning(L18BGL020) and in part by the Basic Scientific ResearchProjects in Institutions of Higher Learning of LiaoningEducation Department (LJ2017QW009)

References

[1] N Carbonara and R Pellegrino ldquoRevenue guarantee inpublic-private partnerships a win-win modelrdquo ConstructionManagement amp Economics vol 36 no 10 pp 584ndash598 2018

[2] L Cheng ldquoDiscussion on the application of PPPmodel in newinfrastructure constructionrdquo Open Journal of Social Sciencesvol 7 no 9 pp 283ndash288 2019

[3] D Wu and S Q Wang ldquoResearch development and trend ofpublic private partnership in Chinardquo Journal of EngineeringManagement vol 28 no 6 pp 75ndash80 2014

[4] K Leidel and H W Alfen Lifecycle-Oriented Risk Manage-ment for PPP-Projects in Public Real Estate pp 587ndash604 EresLos Angeles CA USA 2009

[5] J Yuan X Li Y Ke W Xu Z Xu and M SkibnewskildquoDeveloping a building information modelingndashbased per-formance management system for publicndashprivate partner-shipsrdquo Engineering Construction and ArchitecturalManagement vol 27 no 8 pp 1727ndash1762 2020

[6] S Jayasuriya G Zhang and R J Yang ldquoExploring the impactof stakeholder management strategies on managing issues inPPP projectsrdquo International Journal of Construction Man-agement vol 20 no 6 pp 666ndash678 2020

[7] C X Jiang ldquoCauses and implications of the failure intransportation infrastructure public privatepartnershipsmdashmdasha study of 25 typical PPP casesrdquo Journal ofBeijing Jiaotong University vol 15 no 3 pp 50ndash58 2016

[8] R Osei-Kyei and A P C Chan ldquoReview of studies on thecritical success factors for public-private partnership(PPP) projects from 1990 to 2013rdquo International Journal ofProject Management vol 33 no 6 pp 1335ndash1346 2015

[9] B Zhang Y Wang X Liu et al ldquo+e configuration of PPPfailuremdashmdasha crisp-set qualitative comparative analysisrdquoJournal of Public Administration vol 12 no 04pp 65ndash81+191 2019

12 Complexity

[10] N Costantino and R Pellegrino ldquoPublicndashprivate partnershipimplementation cross-case analysisrdquo Proceedings of the In-stitution of Civil Engineers-management Procurement andLaw vol 168 no 4 pp 164ndash176 2015

[11] P Zhang K Zhan and Y Zhou ldquoGame model of three-partyshared risk allocation for PPP projectsmdashmdashbased on theperspective of incomplete informationrdquo IOP Conference Se-ries Earth and Environmental Science vol 295 no 4pp 42ndash49 2019

[12] N W Young S A Jones and H M Bernstein BuildingInformation Modeling Transforming Design and Constructionto Achieve Greater Industry Productivity McGraw HillConstruction New York NY USA 2008

[13] N D Aziz A H Nawawi and N R M Ariff ldquoBuildinginformation modelling (BIM) in facilities management op-portunities to be considered by facility managersrdquo Procedia-Social and Behavioral Sciences vol 234 pp 353ndash362 2016

[14] W Zou M Kumaraswamy and J Chung ldquoIdentifying thecritical success factors for relationship management in PPPprojectsrdquo International Journal of Project Managementvol 32 no 2 pp 265ndash274 2014

[15] QWang and TWang ldquoResearch on promotion of BIM basedon game analysis of stakeholdersrdquo Sichuan BuildingMaterialsvol 41 no 6 p 272 + 278 2015

[16] Q L Yin Evolutionary Game Analysis on BIM TechnologyDiffusion of Prefabricated Construction Qingdao Universityof Technology Qingdao China 2019

[17] H X Tang J M Cao H S Xu and C F Han ldquoAnalysis onBIM application cooperation behaviors in integrated facilitymanagement organization based on evolutionary game the-oryrdquo Industrial Engineering amp Management vol 25 no 4pp 1ndash8 2020

[18] J R Song ldquoResearch on incentive and countermeasures ofBIM application based on game theoryrdquo Value Engineeringvol 36 no 33 pp 43-44 2017

[19] C Jia Z H Wang and L Zhao ldquoEvolutionary game analysisand simulation for cooperative application of BIM technologyin PPPrdquo Journal of Liaoning Technical University vol 22no 4 pp 270ndash277 2020

[20] K A Tversky ldquoProspect theory an analysis of decision underriskrdquo Econometrica vol 47 no 2 pp 263ndash291 1979

[21] R L Gao and Q Bao ldquoSelection of government supervisionmode during the operational period in PPP projects based onevolutionary game theoryrdquo Operations Research and Man-agement Science vol 28 no 4 pp 155ndash162 2019

[22] S K He G W Liang and J B Meng ldquoMulti-subjectsbenefit game and behavior evolution mechanism of majorengineering based on prospect theoryrdquo Science andTechnology Management Research vol 40 no 5 pp 207ndash214 2020

[23] A Porwal and K N Hewage ldquoBuilding information modeling(BIM) partnering framework for public construction proj-ectsrdquo Automation in Construction vol 31 no 3 pp 204ndash2142013

[24] P E D Love J Liu J Matthews C-P Sing and J SmithldquoFuture proofing PPPs life-cycle performance measurementand building information modellingrdquo Automation in Con-struction vol 56 pp 26ndash35 2015

[25] Y L Yin Q J Liu and X Wang ldquoResearch on the whole lifecycle supervision system of public projectsmdashmdashbased on thecollaboration of VfM evaluation and BIM technologyrdquo ProjectManagement Technology vol 14 no 07 pp 17ndash20 2016

[26] S Q Wang R Tiong S Ting et al ldquoEvaluation and man-agement of political risks in Chinarsquos BOTprojectsrdquo Journal of

Construction Engineering and Management vol 126 no 3pp 242ndash250 2000

[27] Y M Chen and H F Chen ldquo+e risk management of PPPproject life cycle based on BIM technologyrdquo Project Man-agement Technology vol 15 no 5 pp 51ndash55 2017

[28] G Luo and X X Zhan ldquoResearch on the application of BIM inPPP project whole-life cyclerdquo Technology and EconomicGuide vol 26 no 5 p 168 2018

[29] B L Zhou and C L Lin ldquo+e application of BIM in PPPproject whole-life cyclerdquo Engineering Economy vol 29 no 3pp 51ndash54 2019

[30] J M Smith Evolution and the )eory of Games CambridgeUniversity Press Cambridge UK 1982

[31] J W Weibull Evolutionary Game )eory MIT PressCambridge MA USA 1997

[32] K Binmore ldquoModeling rational players part Irdquo Economicsand Philosophy vol 3 no 2 pp 179ndash214 1987

[33] K Binmore ldquoModeling rational players part IIrdquo Economicsand Philosophy vol 4 1988

[34] P D Taylor and L B Jonker ldquoEvolutionarily stable strategiesand game dynamicsrdquoMathematical Biosciences vol 40 no 1-2 pp 145ndash156 1978

[35] P Young ldquoConventional contractsrdquo )e Review of EconomicStudies vol 65 no 4 pp 773ndash792 1998

[36] D Fudenberg and C Harris ldquoEvolutionary dynamics withaggregate shocksrdquo Journal of Economic )eory vol 57 no 2pp 420ndash441 1992

[37] H Bester W Guth and W Guth ldquoIs altruism evolutionarilystablerdquo Journal of Economic Behavior amp Organization vol 34no 2 pp 193ndash209 1998

[38] K E Boulding R R Nelson and S G Winter ldquoAn evolu-tionary theory of economic changrdquo American Journal ofAgricultural Economics vol 66 no 4 p 535 1984

[39] S Y Xie Economic Game )eory Fudan University PressShanghai China 2002

[40] D Friedman and K C Fung ldquoInternational trade and theinternal organization of firms an evolutionary approachrdquoJournal of International Economics vol 41 no 1-2 pp 113ndash137 1996

[41] Y P Wang and W D Meng ldquoEvolutionary game analysis onco-competition mechanism of supply chainrdquo Journal of In-dustrial Engineering and Engineering Management vol 18no 2 pp 96ndash98 2004

[42] J Gao and Z H Shen ldquoElementary groping for evolutionarygame theory and its application in electricity marketrdquo Au-tomation of Electric Power Systems vol 27 no 18 pp 18ndash212003

[43] Y Zhang and M Guizani Game )eory for Wireless Com-munications and Networking CRC Press Boca Raton FLUSA 2011

[44] F Y Lu ldquoEvolutionary game analysis on environmentalpollution problemrdquo Systems Engineering-)eory amp Practicevol 27 no 9 pp 148ndash152 2007

[45] S L Dreisbach ldquoImpact of psychological research on moraltheory prospect theory and the doctrine of doing andallowingrdquo Dissertations amp )eses-Gradworks University ofCalifornia Santa Cruz CA USA 2012

[46] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Econometrica vol 47 no 2 1979

[47] N C Barberis ldquo+irty years of prospect theory in economicsa review and assessmentrdquo )e Journal of Economic Perspec-tives vol 27 no 1 pp 173ndash196 2013

Complexity 13

[48] S Benartzi and R H +aler ldquoMyopic loss aversion and theequity premium puzzlerdquo Quarterly Journal of Economicsvol 110 no 1 pp 73ndash92 1995

[49] N Barberis and M Huang ldquoStocks as lotteries the impli-cations of probability weighting for security pricesrdquo )eAmerican Economic Review vol 98 no 5 pp 2066ndash21002008

[50] A Frazzini ldquo+e disposition effect and underreaction tonewsrdquo )e Journal of Finance vol 61 no 4 pp 2017ndash20462006

[51] W Y Hu and J S Scott ldquoBehavioral obstacles in the annuitymarketrdquo Financial Analysts Journal vol 63 no 6 pp 71ndash822007

[52] B Koszegi andM Rabin ldquoReference-dependent consumptionplansrdquo )e American Economic Review vol 99 no 3pp 909ndash936 2009

[53] P Vamvakas E E Tsiropoulou and S Papavassiliou ldquoDy-namic spectrum management in 5G wireless networks a real-life modeling approachrdquo in Proceedings of the IEEE INFO-COM 2019-IEEE Conference on Computer CommunicationsIEEE Paris France April 2019

[54] Y N Zhou and J C Liu ldquoStudy on supervision strategy of pppproject considering the participation of higher-level gov-ernmentrdquo Chinese Journal of Management Science vol 1ndash162021 httpsdoiorg1016381jcnkiissn1003-207x20200801

[55] H Q Zhang J C Wang and H L Tao ldquoResearch the de-cision-making behavior of investors based on prospect theoryin the ppp projectrdquo Soft Science vol 32 no 8 pp 129ndash1332018

[56] D Friedman ldquoEvolutionary games in economicsrdquo Econo-metrica vol 59 no 3 pp 637ndash666 1991

[57] Z B Zhao and Q P Man ldquoEvolutionary game analysis of riskmanagement behavior of major infrastructure projects basedon projects based on prospect theoryrdquo Journal of SystemsManagement vol 27 no 1 pp 112ndash120 2018

14 Complexity

+e first derivative of x is equal to

d(F(x))

dx (1 minus 2x)(minusCg minus Sp + Lp)

+ Lc + Lg + αLg1 minus zLc minus yLp minus yLg)

(5)

When F(x) 0 and d(F(x))dxlang0 governmentsrsquostrategy of ldquopositive promotionrdquo will be in a stable state

+erefore there will be F(x) equiv 0 when z minusCgminus

ySp + Lp + Lc + Lg minus yLp minus αLg1Lc and all the values of xare in a stable state When zlang minus Cg minus ySp + Lp+

Lc + Lg minus yLp minus αLg1Lc there will be d(F(x))dx|x1lang0and x 1 is the ESS of the system When zrang minus Cg minus ySp+

Lp + Lc + Lg minus yLp minus αLg1Lc there will be d(F(x))dx|x0lang0 and x 0 is the ESS of the system +e phase

diagram of governmentsrsquo behavioral strategy evolution isshown in Figure 1+e slant z minusCg minus ySp + Lp + Lc + Lg minus

yLp minus αLg1Lc divides the three-dimensional spaceV (x y z| 0lexle 1 0leyle 1 0le zle 1) into parts of V1and V2 In part V1 governments gradually evolved to x 0and finally chose the strategy of ldquopassive promotionrdquo In partV2 governments gradually evolved to x 1 and finallychose the strategy of ldquopositive promotionrdquo

322 Replicator Dynamics Equation and Equilibrium PointAnalysis of Social Capitals According to Table 2 we can getthe perceived expected value Ep1 of social capitals takingstrategy of rdquopositive adoptionrdquo and the perceived expectedvalue Ep2 of social capitals taking the strategy of ldquopassiveadoptionrdquo

Table 1 +e profit and loss indexes of stakeholders

Index Explanation of indexRg1 +e benefit brought to governments when social capitals adopt BIM activelyRg2 +e benefit brought to governments when social capitals adopt BIM passivelyCg +e cost of governmentsrsquo promotion of BIMSp +e subsidy provided by governments when social capitals adopt BIM activelyLp +e punishment imposed by governments when social capitals adopt BIM passivelyLc +e punishment imposed by governments when contractors apply BIM passively

Lg+e adverse effects on the development of construction industry when governments promote BIM passively and social capitals

adopt BIM passivelyLg1 +e punishment imposed by the superior supervision authority when governments promote BIM passivelyα +e probability of the superior supervision authority finding that governments promote BIM passivelyRp +e benefit brought to social capitals when contractors applying BIM activelyCp1 +e cost of social capitalsrsquo positive adoption of BIMCp2 +e cost of social capitalsrsquo passive adoption of BIMLp1 +e adverse effects on social capitals when contractors apply BIM passivelySc +e reward provided by social capitals when contractors apply BIM activelyRc +e benefit obtained by contractors apply BIM activelyCc +e cost incurred by the contractors apply BIM actively

Lc1 +e compensation given by contractors undertaking passive application of BIM to social capitals undertaking positive adoption ofBIM when governments promote BIM actively

Lc2 +e compensation given by contractors undertaking passive application of BIM to social capitals undertaking positive adoption ofBIM when governments promote BIM passively

Table 2 +e matrix of the perceived value of profit and loss among stakeholders

Participants of the evolutionary gameGovernments

Positive promotion(x)

Passive promotion(1 minus x)

Social capitals

Positive adoption(y)

Contractors

Positive application(z)

Rg1 minus Cg minus Sp Rg1 minus αLg1Rp minus Cp1 minus Sc + Sp Rp minus Cp1 minus Sc

Rc minus Cc + Sc RcminusCc+ Sc

Passive application(1 minus z)

Rg1 minus Cg minus Sp + Lc Rg1 minus αLg1Rp minus Cp1 + Sp + Lc1 minus Lp1 Rp minus Cp1 + Lc2 minus Lp1

minusLc minus Lc1 minus Lc2

Passive adoption(1 minus y)

Positive application(z)

Rg2 minus Cg minus Sp + Lp Rg2 minus Lg minus αLg1Rp minus Cp2 + Sp minus Lp minus Sc Rp minus Cp2 minus Sc

Rc minus Cc + Sc Rc minus Cc + Sc

Passive application(1 minus z)

minusCg + Lp + Lc minus Sp minusLg minus αLg1RpminusCp2+ Spminus Lpminus Lp1 RpminusCp2minus Lp1

minusLc 0

Complexity 5

Ep1 xz(Rp minus Cp minus Sc + Sp) + z(1 minus x)(Rp minus Cp1 minus Sc) + x(1 minus z)(Rp minus Cp1 + Sp

+ Lc1 minus Lp1) +(1 minus z)(1 minus x)(Rp1 minus Cp1 + Lc2 minus Lp1)

Ep2 xz(Rp1 minus Cp2 minus Lp minus Sc + Sp) + z(1 minus x)(Rp minus Cp2 minus Sc) + x(1 minus z)

middot (Rp minus Cp2 minus Lp + Sp minus Lp1) +(1 minus z)(1 minus x)(Rp minus Cp2 minus Lp)

(6)

+en the average perceived expected value of socialcapitals is as follows

Ep yEp1 +(1 minus y)Ep2 (7)

+erefore the replication dynamic equation of socialcapitals is as follows

F(y) dy

dt y(Ep1 minus Ep) y(1 minus y)(Ep1 minus Ep2)

y(1 minus y)(Cp2 minus Cp1 + Lc2 + xLc1 minus xLc2

minus zLc2 + xLp minus xzLc1 + xzLc2)

(8)

+e first derivative of y is equal to

d(F(y))

dt (1 minus 2y)(Cp2 minus Cp1 + Lc2 + xLc1 minus xLc2

minus zLc2 + xLp minus xzLc1 + xzLc2)

(9)

When F(y) 0 and d(F(y))dylang0 social capitalsrsquostrategy of ldquopositive adoptionrdquo will be in a stable state+erefore there will be F(y) equiv 0 when z Cp2 minus Cp1 + Lc2

+xLc1 minus xLc2 + xLpLc2 + xLc1 minus xLc2 and all the values ofy are in a stable state When zlangCp2 minus Cp1 + Lc2 + xLc1 minus

xLc2+ xLpLc2 + xLc1 minus xLc2 there will be d(F(y))dy|y1lang0 and y 1 is the ESS of the systemWhen zrangCp2 minus Cp1 +

Lc2 + xLc1minus xLc2 + xLpLc2 + xLc1 minus xLc2 there will bed(F(y))dy|y0lang0 and y 0 is the ESS of the system +ephase diagram of social capitalsrsquo behavioral strategy evo-lution is shown in Figure 2 +e surface z Cp2 minus Cp1 +

Lc2 + xLc1 minus xLc2 + xLpLc2 + xLc1 minus xLc2 divides thethree-dimensional space V (x y z| 0lexle 1 0leyle1 0le zle 1) into parts of V1 and V2 In part V1 socialcapitals gradually evolved to y 1 and finally chose thestrategy of ldquopositive adoptionrdquo In part V2 social capitalsgradually evolved to y 0 and finally chose the strategy ofldquopassive adoptionrdquo

323 Replicator Dynamics Equation and Equilibrium PointAnalysis of Contractors According to Table 2 we can get theperceived expected valueEc1 of contractors taking strategy ofldquopositive applicationrdquo and the perceived expected value Ec2of contractors taking strategy of ldquopassive applicationrdquo

Ec1 xy(Rc minus Cc + Sc) + y(1 minus x)(Rc minus Cc + Sc) +(1 minus y)(1 minus x)(Rc minus Cc + Sc)

Rc minus Cc + Sc

Ec2 xy(minusLc minus Lc1) + y(1 minus x)(minusLc2) + x(1 minus y)(minusLc) +(1 minus y)(1 minus x)(0) xyLc2 minus xLc minus yLc minus xyLc

(10)

z

1

1

x

V2

V1

1 y

Figure 1 Phase diagram of governmentsrsquo behavioral strategy evolution

6 Complexity

+en the average perceived expected value of contrac-tors is as follows

Ec zEc1 +(1 minus z)Ec2 (11)

+erefore the replication dynamic equation of con-tractors is as follows

F(z) dz

dt z(Ec1 minus Ec) z(1 minus z)(Ec1 minus Ec2)

z(1 minus z)(Rc minus Cc + Sc + xLc + yLc2 + xyLc1 minus xyLc2)

(12)

+e first derivative of z is equal to

d(F(z))

dz (1 minus 2z)(Rc minus Cc + Sc + xLc + yLc2 + xyLc1 minus xyLc2)

(13)

When F(z) 0 and d(F(z))dzlang0 contractorsrsquo strategyof ldquopositive applicationrdquo will be in a stable state +ereforethere will be F(z) equiv 0 when y Rc minus Cc + Sc + xLcx(Lc2 minus Lc1) minus Lc2 and all the values of z are in a stablestate When ylangRc minus Cc + Sc + xLcx(Lc2 minus Lc1) minus Lc2there will be d(F(z))dz|z1lang0 and z 1 is the ESS of thesystem When yrangRc minus Cc + Sc + xLcx(Lc2 minus Lc1) minus Lc2there will be d(F(z))dz|z0lang0 and z 0 is the ESS of thesystem +e phase diagram of contractorsrsquo behavioralstrategy evolution is shown in Figure 3 +e surface y

Rc minus Cc + Sc + xLcx(Lc2 minus Lc1) minus Lc2 divides the three-dimensional space V (x y z| 0lexle 1 0le yle 1 0lezle 1) into parts of V1 and V2 In part V1 contractorsgradually evolved to z 1 and finally chose the strategy ofldquopositive applicationrdquo In part V2 contractors graduallyevolved to z 0 and finally chose the strategy of ldquopassiveapplicationrdquo

33 Evolutionary Stability Analysis +e Jacobin matrix ofthe system is represented by J

J

J1 J2 J3

J4 J5 J6

J7 J8 J9

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

zf(x)

zx

zf(x)

zy

zf(x)

zz

zf(y)

zx

zf(y)

zy

zf(y)

zz

zf(z)

zx

zf(z)

zy

zf(z)

zz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(14)

Among them

zf(x)

zx (1 minus 2x)(Cg2 minus Cg1 + Lg minus Sp + Lp + Lc minus zLc minus yLp minus zLg)

zf(y)

zy (1 minus 2y)(Cp2 minus Cp1 + Lc2 + xLc1 minus xLc2 minus zLc2 + xLp minus xzLc1 + xzLc2)

zf(z)

zz (1 minus 2z)(Rc minus Cc + Sc + xLc + yLc2 + xyLc1 minus xyLc2)

(15)

+e local stability of equilibrium points is given inTable 3

According to the indirect method of Lyapunov theequilibrium point is asymptotically stable if all its eigen-values of the Jacobian matrix [56] have negative real parts InTable 3 it can be seen that if we want the system evolving tothe ESS point (1 1 1) the following conditions must be metCg + Splt αLg1 Cp1ltCp2 + Lp and CcltRc + Sc

However it should be noted that in the actual PPPprojects the main stakeholders cannot make the best use ofthe existing information in the decision-making because of

their limited rationality +ey usually make decisionsthrough subjective judgment which ultimately leads to de-viation in their behaviors +e deviations will hinder theestablishment of ESS point (1 1 1) and make the systemdifficult to converge to the optimal state +e reasons areanalyzed based on the prospect theory as followstake governments for example the condition that gov-ernments promoting BIM actively is Cg + Splt αLg1 Basedon the prospect theory individuals always overestimate thelow probability loss and underestimate the high probabilitygain namely individuals tend to choose risk aversion in the

z

1

1

x

V2

V11 y

Figure 2 Phase diagram of social capitalsrsquo behavioral strategyevolution

Complexity 7

Table 3 +e eigenvalues of each equilibrium point

Equilibrium pointEigenvalues of Jacobi matrix

λ1 λ2 λ3E(0 0 0) minusCg minus Sp + Lg + Lp + αLg1 Cp2 minus Cp1 + Lc2 Rc minus Cc + ScE(1 0 0) Cg + Sp minus Lg minus Lp minus Lc minus αLg1 Cp2 minus Cp1 + Lc1 + Lp Rc minus Cc + Sc + LcE(0 1 0) minusCg minus Sp + Lc + αLg1 Cp2 minus Cp1 minus Lc2 Rc minus Cc + Sc + Lc2E(0 0 1) minusCg minus Sp + Lc + αLg1 Cp2 minus Cp1 minusRc + Cc + ScE(1 1 0) Cg + Sp minus Lc minus αLg1 minusCp2 minus Cp1 minus Lc1 minus Lp Rc minus Cc + Sc + Lc + Lc1E(1 0 1) Cg + Sp minus Lp minus Lg minus αLg1 Cp2 minus Cp1 + Lp minusRc + Cc minus Sc minus LcE(0 1 1) minusCg minus Sp + αLg1 minusCp2 + Cp1 minusRc + Cc minus Sc minus Lc2E(1 1 1) Cg + Sp minus αLg1 minusCp2 minus Cp1 + Lp minusRc + Cc minus Sc minus Lc minus Lc1

z

1

1

x

V2

V1

1 y

Figure 3 Phase diagram of contractorsrsquo behavioral strategy evolution

111

0908070605

z

108

06y04 05 06 07 08

x

09 1

Cg=20Cg=15Cg=10

(a)

11

1

09

08

07

06

05

z

Cg=20Cg=15Cg=10

1 095 09 085 08 075 07 065 06 055 05x

(b)

Figure 4 Influence of Cg change on the evolutionary game process

8 Complexity

face of gains but tend to prefer risk preference when facinglosses [57] In the practice of PPP projects governmentsoften think that their behavior will not be easily detected bythe superior supervision authority due to the commonexistence of fluke mentality +erefore governments tend tounderestimate the probability α and the value of punishmentLg1 by the superior supervision authority which means thatthe right part of the conditional formula Cg + Splt αLg1 willbe underestimated in practice On the other hand whengovernments choose the active strategy it will definitely paythe promotion cost while choosing the passive strategy willonly have a certain probability of punitive losses Accordingto the prospect theory governments will be more inclined torisk preference that is they prefer to take risks and choosethe passive strategy rather than to bear the deterministic

expenditure +erefore the above reasons will cause certaindifficulties for the establishment of the formulaCg + Splt αLg1

4 Numerical Simulation and Analysis

In order to better analyze the behavior evolution mechanismamong governments social capitals and contractors in BIMadoption this study uses MATLAB R2018a to simulate thetrend and influencing factors of the evolutionary game +evalues of parameters are randomly assigned under the as-sumption which do not represent the real benefits ofstakeholders in PPP projects

According to the above analysis if the system convergesto the ESS point (1 1 1) the following conditions need to be

111

0908070605

z

108

06y04 04

0608

x

1

a=05a=08a=1

(a)

a=05a=08a=1

11

1

09

08

07

06

05

z

11 1 09 08 07 06 05x

(b)

Figure 5 Influence of α change on the evolutionary game process

Lg1=80Lg1=100Lg1=120

111

0908070605

z

108

06y04 04

0608

x

1

(a)

Lg1=80Lg1=100Lg1=120

11

1

09

08

07

06

05

z

11 1 09 08 07 06 05x

(b)

Figure 6 Influence of Lg1 change on the evolutionary game process

Complexity 9

satisfied Cg + Splt αLg1 Cp1ltCp2 + Lp and CcltRc + Sc+e initial values of each parameter are set to x 05 y

05 α 05Cg 20 Sp 10 Lgl 80Cp1 30Cp2

20 Sp 10 Lg1 80Cp1 30Cp2 20 Lp 20Cc

30Rc 40 Sc 10 Lg 10 Lc 10 Lc1 10 LC2 5

41 Influence of Cg1 and Cg2 Changes on the EvolutionaryGame Process Keep the other parameters unchanged andthe influence on the evolution trend of the system is ob-served by changing the values of Cg α and Lg1 as shown inFigures 4(a) 5(a) and 6(a) In Figure 4(b) we can see thatthe convergence rate of x to the ESS point (1 1 1) isaccelerated as the value of Cg goes down +e simulationresults show that reducing the cost of governmentsrsquo strategyof ldquopositive promotionrdquo is helpful to enhance the willingness

of governments to promote the adoption of BIM in PPPprojects In Figures 5(b) and 6(b) we can see that theconvergence rate of x to the ESS point (1 1 1) is acceleratedas the value of α and Lg1 go bigger +e simulation resultsshow that increasing the supervision probability and theseverity of punishment will help increase the willingness ofgovernments to promote BIM application in PPP projects

42 Influence of Cp1 and Lp Changes on the EvolutionaryGame Process Keep the other parameters unchanged andthe influence on the evolution trend of the system is ob-served by changing the values of Cp1 and Lp as shown inFigures 7(a) and 8(a) In Figure 7(b) we can see that theconvergence rate of y to the ESS point (1 1 1) is acceleratedas the value of Cp1 goes down +e simulation results show

111

0908070605

z

108

06y04 05 06 07 08

x

091

Cp1=30Cp1=20Cp1=10

(a)

Cp1=30Cp1=20Cp1=10

11

1

09

08

07

06

05

z

11 1 09 08 07 06 05y

(b)

Figure 7 Influence of Cp1 change on the evolutionary game process

111

0908070605

z

108

06y04 05

06 07 x08 09 1

Lp=20Lp=30Lp=40

(a)

Lp=20Lp=30Lp=40

11

1

09

08

07

06

05

z

11 1 09 08 07 06 05y

(b)

Figure 8 Influence of Lp change on the evolutionary game process

10 Complexity

that reducing the cost of social capitalsrsquo strategy of ldquopositiveadoptionrdquo is helpful to enhance the willingness of socialcapitals to adopt BIM in PPP projects In Figure 8(b) we cansee that the convergence rate of y to the ESS point (1 1 1) isaccelerated as the value of Lp goes bigger +e simulationresults show that increasing the punishment of social cap-italsrsquo strategy of ldquopassive adoptionrdquo is helpful to enhance thewillingness of social capitals to adopt BIM in PPP projects

43 Influence of Cc and Sc Changes on the Evolutionary GameProcess Keep the other parameters unchanged and theinfluence on the evolution trend of the system is observed bychanging the values of Cc and Sc as shown in Figures 9(a)and 10(a) In Figure 9(b) we can see that the convergence

rate of z to the ESS point (1 1 1) is accelerated as the value ofCc goes down+e simulation results show that reducing thecost of contractorsrsquo strategy of ldquopositive applicationrdquo ishelpful to enhance the willingness of contractors to applyBIM in PPP projects In Figure 10(b) we can see that theconvergence rate of z to the ESS point (1 1 1) is acceleratedas the value of Sc goes bigger +e simulation results showthat increasing the punishment of contractorsrsquo strategy ofldquopassive applicationrdquo is helpful to enhance the willingness ofcontractors to apply BIM in PPP projects

5 Conclusions and Management Implications

Under the assumption of bounded rationality this studyconstructs a tripartite evolutionary game model including

111

0908070605

z

108

06y04 05 06 07 08

x

091

Cc=30Cc=20Cc=10

(a)

Cc=30Cc=20Cc=10

11

1

09

08

07

06

05

z

1 09095 08085 07075 06065 05055y

(b)

Figure 9 Influence of Cc change on the evolutionary game process

111

0908070605

z

108

06y04 05 06 07 x

08 091

Sc=20Sc=30Sc=40

(a)

Sc=20Sc=30Sc=40

11

1

09

08

07

06

05

z

1 09095 08085 07075 06065 05055y

(b)

Figure 10 Influence of Sc change on the evolutionary game process

Complexity 11

governments social capitals and contractors +e satisfac-tion conditions that the system converging to the ESS point(1 1 1) are analyzed based on prospect theory and the mainfactors influencing stakeholdersrsquo behavior decision-makingare simulated by MATLAB +e research results will help toclarify the behavior evolution mechanism and the influ-encing factors of the ESS for adopting BIM among stake-holders in PPP projects so as to improve the cross-organization information collaboration efficiency of PPPprojects and promote the informatization of the construc-tion industry +e primary conclusions and managementimplications are as follows

First the cost of BIM promotion by governments and thepunishment from the superior supervision authority whengovernments choose the passive strategy are the key factorsaffecting governmentsrsquo behavior choice When governmentsactively promote the cost reduction of BIM or punishedmore by the superior supervision authority governmentswill choose the active promotion strategy and the behaviorgradually converges to the ESS point (1 1 1)

Second the cost of active adoption of BIM and thepunishment of passive adoption of BIM are the key factorsaffecting the choice of social capitalsrsquo behavior When thecost of active adoption of BIM is reduced or the penalty ofpassive adoption of BIM is increased the intention of socialcapitals to choose active adoption of BIM strategy is in-creased and the behavior of social capitals gradually con-verges to the ESS point (1 1 1)

Finally the cost of active application of BIM and thereward of active application of BIM are the key factorsinfluencing the choice of contractorsrsquo behavior When thecost of the active application of BIM technology is reducedor the penalty of the negative application of BIM technologyis increased the intention of the contractor to choose thestrategy of active application of BIM technology is increasedand the contractorrsquos behavior gradually converges to the ESSpoint (1 1 1)

+e adoption of BIM in PPP projects is a major reform ofthe traditional building mode whichrequires all stakeholders to change the traditionalthinking and actively cooperate In order to promoteBIM adoption among stakeholders of PPP projects thefollowing management implications are proposedImprove the supervision efficiency of superior supervisionauthority on governmentsrsquo promotion of BIM and furtherenrich the supervision means with the help of the media andthe public Increase the punishment for the governmentsrsquopassive promotion of BIM so as to improve the perceivedvalue of governments for regulatory punishment

Strengthen the punishment for the passive adoption ofBIM by social capitals and the reward for the active ap-plication of BIM by contractors Improve the cognition ofsocial capitals and contractors on BIM and other advancedinformation technology through publicity and training andgradually transform the system promotion into habitualadoption

Under the PPP mode the adoption of BIM can bepromoted by the governmentrsquos supervision but the gov-ernmentrsquos behavior also needs supervision +e public is the

users and beneficiaries of PPP projectsrsquo goods and the publicshould also have the right to know and to supervise theimplementation of the projects +erefore in the next stagethe authors will further research the role of public super-vision in the adoption of BIM in PPP projects in order toexplore the evolutionary mechanism and influencing factorsunder more stakeholders

Data Availability

Some or all data models or code generated or used tosupport the findings of this study are available from thecorresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interest

Acknowledgments

+is work was supported by the National Natural ScienceFoundation of China (NSFC) (71801119) in part by theSocial Science Planning Foundation of Liaoning(L18BGL020) and in part by the Basic Scientific ResearchProjects in Institutions of Higher Learning of LiaoningEducation Department (LJ2017QW009)

References

[1] N Carbonara and R Pellegrino ldquoRevenue guarantee inpublic-private partnerships a win-win modelrdquo ConstructionManagement amp Economics vol 36 no 10 pp 584ndash598 2018

[2] L Cheng ldquoDiscussion on the application of PPPmodel in newinfrastructure constructionrdquo Open Journal of Social Sciencesvol 7 no 9 pp 283ndash288 2019

[3] D Wu and S Q Wang ldquoResearch development and trend ofpublic private partnership in Chinardquo Journal of EngineeringManagement vol 28 no 6 pp 75ndash80 2014

[4] K Leidel and H W Alfen Lifecycle-Oriented Risk Manage-ment for PPP-Projects in Public Real Estate pp 587ndash604 EresLos Angeles CA USA 2009

[5] J Yuan X Li Y Ke W Xu Z Xu and M SkibnewskildquoDeveloping a building information modelingndashbased per-formance management system for publicndashprivate partner-shipsrdquo Engineering Construction and ArchitecturalManagement vol 27 no 8 pp 1727ndash1762 2020

[6] S Jayasuriya G Zhang and R J Yang ldquoExploring the impactof stakeholder management strategies on managing issues inPPP projectsrdquo International Journal of Construction Man-agement vol 20 no 6 pp 666ndash678 2020

[7] C X Jiang ldquoCauses and implications of the failure intransportation infrastructure public privatepartnershipsmdashmdasha study of 25 typical PPP casesrdquo Journal ofBeijing Jiaotong University vol 15 no 3 pp 50ndash58 2016

[8] R Osei-Kyei and A P C Chan ldquoReview of studies on thecritical success factors for public-private partnership(PPP) projects from 1990 to 2013rdquo International Journal ofProject Management vol 33 no 6 pp 1335ndash1346 2015

[9] B Zhang Y Wang X Liu et al ldquo+e configuration of PPPfailuremdashmdasha crisp-set qualitative comparative analysisrdquoJournal of Public Administration vol 12 no 04pp 65ndash81+191 2019

12 Complexity

[10] N Costantino and R Pellegrino ldquoPublicndashprivate partnershipimplementation cross-case analysisrdquo Proceedings of the In-stitution of Civil Engineers-management Procurement andLaw vol 168 no 4 pp 164ndash176 2015

[11] P Zhang K Zhan and Y Zhou ldquoGame model of three-partyshared risk allocation for PPP projectsmdashmdashbased on theperspective of incomplete informationrdquo IOP Conference Se-ries Earth and Environmental Science vol 295 no 4pp 42ndash49 2019

[12] N W Young S A Jones and H M Bernstein BuildingInformation Modeling Transforming Design and Constructionto Achieve Greater Industry Productivity McGraw HillConstruction New York NY USA 2008

[13] N D Aziz A H Nawawi and N R M Ariff ldquoBuildinginformation modelling (BIM) in facilities management op-portunities to be considered by facility managersrdquo Procedia-Social and Behavioral Sciences vol 234 pp 353ndash362 2016

[14] W Zou M Kumaraswamy and J Chung ldquoIdentifying thecritical success factors for relationship management in PPPprojectsrdquo International Journal of Project Managementvol 32 no 2 pp 265ndash274 2014

[15] QWang and TWang ldquoResearch on promotion of BIM basedon game analysis of stakeholdersrdquo Sichuan BuildingMaterialsvol 41 no 6 p 272 + 278 2015

[16] Q L Yin Evolutionary Game Analysis on BIM TechnologyDiffusion of Prefabricated Construction Qingdao Universityof Technology Qingdao China 2019

[17] H X Tang J M Cao H S Xu and C F Han ldquoAnalysis onBIM application cooperation behaviors in integrated facilitymanagement organization based on evolutionary game the-oryrdquo Industrial Engineering amp Management vol 25 no 4pp 1ndash8 2020

[18] J R Song ldquoResearch on incentive and countermeasures ofBIM application based on game theoryrdquo Value Engineeringvol 36 no 33 pp 43-44 2017

[19] C Jia Z H Wang and L Zhao ldquoEvolutionary game analysisand simulation for cooperative application of BIM technologyin PPPrdquo Journal of Liaoning Technical University vol 22no 4 pp 270ndash277 2020

[20] K A Tversky ldquoProspect theory an analysis of decision underriskrdquo Econometrica vol 47 no 2 pp 263ndash291 1979

[21] R L Gao and Q Bao ldquoSelection of government supervisionmode during the operational period in PPP projects based onevolutionary game theoryrdquo Operations Research and Man-agement Science vol 28 no 4 pp 155ndash162 2019

[22] S K He G W Liang and J B Meng ldquoMulti-subjectsbenefit game and behavior evolution mechanism of majorengineering based on prospect theoryrdquo Science andTechnology Management Research vol 40 no 5 pp 207ndash214 2020

[23] A Porwal and K N Hewage ldquoBuilding information modeling(BIM) partnering framework for public construction proj-ectsrdquo Automation in Construction vol 31 no 3 pp 204ndash2142013

[24] P E D Love J Liu J Matthews C-P Sing and J SmithldquoFuture proofing PPPs life-cycle performance measurementand building information modellingrdquo Automation in Con-struction vol 56 pp 26ndash35 2015

[25] Y L Yin Q J Liu and X Wang ldquoResearch on the whole lifecycle supervision system of public projectsmdashmdashbased on thecollaboration of VfM evaluation and BIM technologyrdquo ProjectManagement Technology vol 14 no 07 pp 17ndash20 2016

[26] S Q Wang R Tiong S Ting et al ldquoEvaluation and man-agement of political risks in Chinarsquos BOTprojectsrdquo Journal of

Construction Engineering and Management vol 126 no 3pp 242ndash250 2000

[27] Y M Chen and H F Chen ldquo+e risk management of PPPproject life cycle based on BIM technologyrdquo Project Man-agement Technology vol 15 no 5 pp 51ndash55 2017

[28] G Luo and X X Zhan ldquoResearch on the application of BIM inPPP project whole-life cyclerdquo Technology and EconomicGuide vol 26 no 5 p 168 2018

[29] B L Zhou and C L Lin ldquo+e application of BIM in PPPproject whole-life cyclerdquo Engineering Economy vol 29 no 3pp 51ndash54 2019

[30] J M Smith Evolution and the )eory of Games CambridgeUniversity Press Cambridge UK 1982

[31] J W Weibull Evolutionary Game )eory MIT PressCambridge MA USA 1997

[32] K Binmore ldquoModeling rational players part Irdquo Economicsand Philosophy vol 3 no 2 pp 179ndash214 1987

[33] K Binmore ldquoModeling rational players part IIrdquo Economicsand Philosophy vol 4 1988

[34] P D Taylor and L B Jonker ldquoEvolutionarily stable strategiesand game dynamicsrdquoMathematical Biosciences vol 40 no 1-2 pp 145ndash156 1978

[35] P Young ldquoConventional contractsrdquo )e Review of EconomicStudies vol 65 no 4 pp 773ndash792 1998

[36] D Fudenberg and C Harris ldquoEvolutionary dynamics withaggregate shocksrdquo Journal of Economic )eory vol 57 no 2pp 420ndash441 1992

[37] H Bester W Guth and W Guth ldquoIs altruism evolutionarilystablerdquo Journal of Economic Behavior amp Organization vol 34no 2 pp 193ndash209 1998

[38] K E Boulding R R Nelson and S G Winter ldquoAn evolu-tionary theory of economic changrdquo American Journal ofAgricultural Economics vol 66 no 4 p 535 1984

[39] S Y Xie Economic Game )eory Fudan University PressShanghai China 2002

[40] D Friedman and K C Fung ldquoInternational trade and theinternal organization of firms an evolutionary approachrdquoJournal of International Economics vol 41 no 1-2 pp 113ndash137 1996

[41] Y P Wang and W D Meng ldquoEvolutionary game analysis onco-competition mechanism of supply chainrdquo Journal of In-dustrial Engineering and Engineering Management vol 18no 2 pp 96ndash98 2004

[42] J Gao and Z H Shen ldquoElementary groping for evolutionarygame theory and its application in electricity marketrdquo Au-tomation of Electric Power Systems vol 27 no 18 pp 18ndash212003

[43] Y Zhang and M Guizani Game )eory for Wireless Com-munications and Networking CRC Press Boca Raton FLUSA 2011

[44] F Y Lu ldquoEvolutionary game analysis on environmentalpollution problemrdquo Systems Engineering-)eory amp Practicevol 27 no 9 pp 148ndash152 2007

[45] S L Dreisbach ldquoImpact of psychological research on moraltheory prospect theory and the doctrine of doing andallowingrdquo Dissertations amp )eses-Gradworks University ofCalifornia Santa Cruz CA USA 2012

[46] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Econometrica vol 47 no 2 1979

[47] N C Barberis ldquo+irty years of prospect theory in economicsa review and assessmentrdquo )e Journal of Economic Perspec-tives vol 27 no 1 pp 173ndash196 2013

Complexity 13

[48] S Benartzi and R H +aler ldquoMyopic loss aversion and theequity premium puzzlerdquo Quarterly Journal of Economicsvol 110 no 1 pp 73ndash92 1995

[49] N Barberis and M Huang ldquoStocks as lotteries the impli-cations of probability weighting for security pricesrdquo )eAmerican Economic Review vol 98 no 5 pp 2066ndash21002008

[50] A Frazzini ldquo+e disposition effect and underreaction tonewsrdquo )e Journal of Finance vol 61 no 4 pp 2017ndash20462006

[51] W Y Hu and J S Scott ldquoBehavioral obstacles in the annuitymarketrdquo Financial Analysts Journal vol 63 no 6 pp 71ndash822007

[52] B Koszegi andM Rabin ldquoReference-dependent consumptionplansrdquo )e American Economic Review vol 99 no 3pp 909ndash936 2009

[53] P Vamvakas E E Tsiropoulou and S Papavassiliou ldquoDy-namic spectrum management in 5G wireless networks a real-life modeling approachrdquo in Proceedings of the IEEE INFO-COM 2019-IEEE Conference on Computer CommunicationsIEEE Paris France April 2019

[54] Y N Zhou and J C Liu ldquoStudy on supervision strategy of pppproject considering the participation of higher-level gov-ernmentrdquo Chinese Journal of Management Science vol 1ndash162021 httpsdoiorg1016381jcnkiissn1003-207x20200801

[55] H Q Zhang J C Wang and H L Tao ldquoResearch the de-cision-making behavior of investors based on prospect theoryin the ppp projectrdquo Soft Science vol 32 no 8 pp 129ndash1332018

[56] D Friedman ldquoEvolutionary games in economicsrdquo Econo-metrica vol 59 no 3 pp 637ndash666 1991

[57] Z B Zhao and Q P Man ldquoEvolutionary game analysis of riskmanagement behavior of major infrastructure projects basedon projects based on prospect theoryrdquo Journal of SystemsManagement vol 27 no 1 pp 112ndash120 2018

14 Complexity

Ep1 xz(Rp minus Cp minus Sc + Sp) + z(1 minus x)(Rp minus Cp1 minus Sc) + x(1 minus z)(Rp minus Cp1 + Sp

+ Lc1 minus Lp1) +(1 minus z)(1 minus x)(Rp1 minus Cp1 + Lc2 minus Lp1)

Ep2 xz(Rp1 minus Cp2 minus Lp minus Sc + Sp) + z(1 minus x)(Rp minus Cp2 minus Sc) + x(1 minus z)

middot (Rp minus Cp2 minus Lp + Sp minus Lp1) +(1 minus z)(1 minus x)(Rp minus Cp2 minus Lp)

(6)

+en the average perceived expected value of socialcapitals is as follows

Ep yEp1 +(1 minus y)Ep2 (7)

+erefore the replication dynamic equation of socialcapitals is as follows

F(y) dy

dt y(Ep1 minus Ep) y(1 minus y)(Ep1 minus Ep2)

y(1 minus y)(Cp2 minus Cp1 + Lc2 + xLc1 minus xLc2

minus zLc2 + xLp minus xzLc1 + xzLc2)

(8)

+e first derivative of y is equal to

d(F(y))

dt (1 minus 2y)(Cp2 minus Cp1 + Lc2 + xLc1 minus xLc2

minus zLc2 + xLp minus xzLc1 + xzLc2)

(9)

When F(y) 0 and d(F(y))dylang0 social capitalsrsquostrategy of ldquopositive adoptionrdquo will be in a stable state+erefore there will be F(y) equiv 0 when z Cp2 minus Cp1 + Lc2

+xLc1 minus xLc2 + xLpLc2 + xLc1 minus xLc2 and all the values ofy are in a stable state When zlangCp2 minus Cp1 + Lc2 + xLc1 minus

xLc2+ xLpLc2 + xLc1 minus xLc2 there will be d(F(y))dy|y1lang0 and y 1 is the ESS of the systemWhen zrangCp2 minus Cp1 +

Lc2 + xLc1minus xLc2 + xLpLc2 + xLc1 minus xLc2 there will bed(F(y))dy|y0lang0 and y 0 is the ESS of the system +ephase diagram of social capitalsrsquo behavioral strategy evo-lution is shown in Figure 2 +e surface z Cp2 minus Cp1 +

Lc2 + xLc1 minus xLc2 + xLpLc2 + xLc1 minus xLc2 divides thethree-dimensional space V (x y z| 0lexle 1 0leyle1 0le zle 1) into parts of V1 and V2 In part V1 socialcapitals gradually evolved to y 1 and finally chose thestrategy of ldquopositive adoptionrdquo In part V2 social capitalsgradually evolved to y 0 and finally chose the strategy ofldquopassive adoptionrdquo

323 Replicator Dynamics Equation and Equilibrium PointAnalysis of Contractors According to Table 2 we can get theperceived expected valueEc1 of contractors taking strategy ofldquopositive applicationrdquo and the perceived expected value Ec2of contractors taking strategy of ldquopassive applicationrdquo

Ec1 xy(Rc minus Cc + Sc) + y(1 minus x)(Rc minus Cc + Sc) +(1 minus y)(1 minus x)(Rc minus Cc + Sc)

Rc minus Cc + Sc

Ec2 xy(minusLc minus Lc1) + y(1 minus x)(minusLc2) + x(1 minus y)(minusLc) +(1 minus y)(1 minus x)(0) xyLc2 minus xLc minus yLc minus xyLc

(10)

z

1

1

x

V2

V1

1 y

Figure 1 Phase diagram of governmentsrsquo behavioral strategy evolution

6 Complexity

+en the average perceived expected value of contrac-tors is as follows

Ec zEc1 +(1 minus z)Ec2 (11)

+erefore the replication dynamic equation of con-tractors is as follows

F(z) dz

dt z(Ec1 minus Ec) z(1 minus z)(Ec1 minus Ec2)

z(1 minus z)(Rc minus Cc + Sc + xLc + yLc2 + xyLc1 minus xyLc2)

(12)

+e first derivative of z is equal to

d(F(z))

dz (1 minus 2z)(Rc minus Cc + Sc + xLc + yLc2 + xyLc1 minus xyLc2)

(13)

When F(z) 0 and d(F(z))dzlang0 contractorsrsquo strategyof ldquopositive applicationrdquo will be in a stable state +ereforethere will be F(z) equiv 0 when y Rc minus Cc + Sc + xLcx(Lc2 minus Lc1) minus Lc2 and all the values of z are in a stablestate When ylangRc minus Cc + Sc + xLcx(Lc2 minus Lc1) minus Lc2there will be d(F(z))dz|z1lang0 and z 1 is the ESS of thesystem When yrangRc minus Cc + Sc + xLcx(Lc2 minus Lc1) minus Lc2there will be d(F(z))dz|z0lang0 and z 0 is the ESS of thesystem +e phase diagram of contractorsrsquo behavioralstrategy evolution is shown in Figure 3 +e surface y

Rc minus Cc + Sc + xLcx(Lc2 minus Lc1) minus Lc2 divides the three-dimensional space V (x y z| 0lexle 1 0le yle 1 0lezle 1) into parts of V1 and V2 In part V1 contractorsgradually evolved to z 1 and finally chose the strategy ofldquopositive applicationrdquo In part V2 contractors graduallyevolved to z 0 and finally chose the strategy of ldquopassiveapplicationrdquo

33 Evolutionary Stability Analysis +e Jacobin matrix ofthe system is represented by J

J

J1 J2 J3

J4 J5 J6

J7 J8 J9

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

zf(x)

zx

zf(x)

zy

zf(x)

zz

zf(y)

zx

zf(y)

zy

zf(y)

zz

zf(z)

zx

zf(z)

zy

zf(z)

zz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(14)

Among them

zf(x)

zx (1 minus 2x)(Cg2 minus Cg1 + Lg minus Sp + Lp + Lc minus zLc minus yLp minus zLg)

zf(y)

zy (1 minus 2y)(Cp2 minus Cp1 + Lc2 + xLc1 minus xLc2 minus zLc2 + xLp minus xzLc1 + xzLc2)

zf(z)

zz (1 minus 2z)(Rc minus Cc + Sc + xLc + yLc2 + xyLc1 minus xyLc2)

(15)

+e local stability of equilibrium points is given inTable 3

According to the indirect method of Lyapunov theequilibrium point is asymptotically stable if all its eigen-values of the Jacobian matrix [56] have negative real parts InTable 3 it can be seen that if we want the system evolving tothe ESS point (1 1 1) the following conditions must be metCg + Splt αLg1 Cp1ltCp2 + Lp and CcltRc + Sc

However it should be noted that in the actual PPPprojects the main stakeholders cannot make the best use ofthe existing information in the decision-making because of

their limited rationality +ey usually make decisionsthrough subjective judgment which ultimately leads to de-viation in their behaviors +e deviations will hinder theestablishment of ESS point (1 1 1) and make the systemdifficult to converge to the optimal state +e reasons areanalyzed based on the prospect theory as followstake governments for example the condition that gov-ernments promoting BIM actively is Cg + Splt αLg1 Basedon the prospect theory individuals always overestimate thelow probability loss and underestimate the high probabilitygain namely individuals tend to choose risk aversion in the

z

1

1

x

V2

V11 y

Figure 2 Phase diagram of social capitalsrsquo behavioral strategyevolution

Complexity 7

Table 3 +e eigenvalues of each equilibrium point

Equilibrium pointEigenvalues of Jacobi matrix

λ1 λ2 λ3E(0 0 0) minusCg minus Sp + Lg + Lp + αLg1 Cp2 minus Cp1 + Lc2 Rc minus Cc + ScE(1 0 0) Cg + Sp minus Lg minus Lp minus Lc minus αLg1 Cp2 minus Cp1 + Lc1 + Lp Rc minus Cc + Sc + LcE(0 1 0) minusCg minus Sp + Lc + αLg1 Cp2 minus Cp1 minus Lc2 Rc minus Cc + Sc + Lc2E(0 0 1) minusCg minus Sp + Lc + αLg1 Cp2 minus Cp1 minusRc + Cc + ScE(1 1 0) Cg + Sp minus Lc minus αLg1 minusCp2 minus Cp1 minus Lc1 minus Lp Rc minus Cc + Sc + Lc + Lc1E(1 0 1) Cg + Sp minus Lp minus Lg minus αLg1 Cp2 minus Cp1 + Lp minusRc + Cc minus Sc minus LcE(0 1 1) minusCg minus Sp + αLg1 minusCp2 + Cp1 minusRc + Cc minus Sc minus Lc2E(1 1 1) Cg + Sp minus αLg1 minusCp2 minus Cp1 + Lp minusRc + Cc minus Sc minus Lc minus Lc1

z

1

1

x

V2

V1

1 y

Figure 3 Phase diagram of contractorsrsquo behavioral strategy evolution

111

0908070605

z

108

06y04 05 06 07 08

x

09 1

Cg=20Cg=15Cg=10

(a)

11

1

09

08

07

06

05

z

Cg=20Cg=15Cg=10

1 095 09 085 08 075 07 065 06 055 05x

(b)

Figure 4 Influence of Cg change on the evolutionary game process

8 Complexity

face of gains but tend to prefer risk preference when facinglosses [57] In the practice of PPP projects governmentsoften think that their behavior will not be easily detected bythe superior supervision authority due to the commonexistence of fluke mentality +erefore governments tend tounderestimate the probability α and the value of punishmentLg1 by the superior supervision authority which means thatthe right part of the conditional formula Cg + Splt αLg1 willbe underestimated in practice On the other hand whengovernments choose the active strategy it will definitely paythe promotion cost while choosing the passive strategy willonly have a certain probability of punitive losses Accordingto the prospect theory governments will be more inclined torisk preference that is they prefer to take risks and choosethe passive strategy rather than to bear the deterministic

expenditure +erefore the above reasons will cause certaindifficulties for the establishment of the formulaCg + Splt αLg1

4 Numerical Simulation and Analysis

In order to better analyze the behavior evolution mechanismamong governments social capitals and contractors in BIMadoption this study uses MATLAB R2018a to simulate thetrend and influencing factors of the evolutionary game +evalues of parameters are randomly assigned under the as-sumption which do not represent the real benefits ofstakeholders in PPP projects

According to the above analysis if the system convergesto the ESS point (1 1 1) the following conditions need to be

111

0908070605

z

108

06y04 04

0608

x

1

a=05a=08a=1

(a)

a=05a=08a=1

11

1

09

08

07

06

05

z

11 1 09 08 07 06 05x

(b)

Figure 5 Influence of α change on the evolutionary game process

Lg1=80Lg1=100Lg1=120

111

0908070605

z

108

06y04 04

0608

x

1

(a)

Lg1=80Lg1=100Lg1=120

11

1

09

08

07

06

05

z

11 1 09 08 07 06 05x

(b)

Figure 6 Influence of Lg1 change on the evolutionary game process

Complexity 9

satisfied Cg + Splt αLg1 Cp1ltCp2 + Lp and CcltRc + Sc+e initial values of each parameter are set to x 05 y

05 α 05Cg 20 Sp 10 Lgl 80Cp1 30Cp2

20 Sp 10 Lg1 80Cp1 30Cp2 20 Lp 20Cc

30Rc 40 Sc 10 Lg 10 Lc 10 Lc1 10 LC2 5

41 Influence of Cg1 and Cg2 Changes on the EvolutionaryGame Process Keep the other parameters unchanged andthe influence on the evolution trend of the system is ob-served by changing the values of Cg α and Lg1 as shown inFigures 4(a) 5(a) and 6(a) In Figure 4(b) we can see thatthe convergence rate of x to the ESS point (1 1 1) isaccelerated as the value of Cg goes down +e simulationresults show that reducing the cost of governmentsrsquo strategyof ldquopositive promotionrdquo is helpful to enhance the willingness

of governments to promote the adoption of BIM in PPPprojects In Figures 5(b) and 6(b) we can see that theconvergence rate of x to the ESS point (1 1 1) is acceleratedas the value of α and Lg1 go bigger +e simulation resultsshow that increasing the supervision probability and theseverity of punishment will help increase the willingness ofgovernments to promote BIM application in PPP projects

42 Influence of Cp1 and Lp Changes on the EvolutionaryGame Process Keep the other parameters unchanged andthe influence on the evolution trend of the system is ob-served by changing the values of Cp1 and Lp as shown inFigures 7(a) and 8(a) In Figure 7(b) we can see that theconvergence rate of y to the ESS point (1 1 1) is acceleratedas the value of Cp1 goes down +e simulation results show

111

0908070605

z

108

06y04 05 06 07 08

x

091

Cp1=30Cp1=20Cp1=10

(a)

Cp1=30Cp1=20Cp1=10

11

1

09

08

07

06

05

z

11 1 09 08 07 06 05y

(b)

Figure 7 Influence of Cp1 change on the evolutionary game process

111

0908070605

z

108

06y04 05

06 07 x08 09 1

Lp=20Lp=30Lp=40

(a)

Lp=20Lp=30Lp=40

11

1

09

08

07

06

05

z

11 1 09 08 07 06 05y

(b)

Figure 8 Influence of Lp change on the evolutionary game process

10 Complexity

that reducing the cost of social capitalsrsquo strategy of ldquopositiveadoptionrdquo is helpful to enhance the willingness of socialcapitals to adopt BIM in PPP projects In Figure 8(b) we cansee that the convergence rate of y to the ESS point (1 1 1) isaccelerated as the value of Lp goes bigger +e simulationresults show that increasing the punishment of social cap-italsrsquo strategy of ldquopassive adoptionrdquo is helpful to enhance thewillingness of social capitals to adopt BIM in PPP projects

43 Influence of Cc and Sc Changes on the Evolutionary GameProcess Keep the other parameters unchanged and theinfluence on the evolution trend of the system is observed bychanging the values of Cc and Sc as shown in Figures 9(a)and 10(a) In Figure 9(b) we can see that the convergence

rate of z to the ESS point (1 1 1) is accelerated as the value ofCc goes down+e simulation results show that reducing thecost of contractorsrsquo strategy of ldquopositive applicationrdquo ishelpful to enhance the willingness of contractors to applyBIM in PPP projects In Figure 10(b) we can see that theconvergence rate of z to the ESS point (1 1 1) is acceleratedas the value of Sc goes bigger +e simulation results showthat increasing the punishment of contractorsrsquo strategy ofldquopassive applicationrdquo is helpful to enhance the willingness ofcontractors to apply BIM in PPP projects

5 Conclusions and Management Implications

Under the assumption of bounded rationality this studyconstructs a tripartite evolutionary game model including

111

0908070605

z

108

06y04 05 06 07 08

x

091

Cc=30Cc=20Cc=10

(a)

Cc=30Cc=20Cc=10

11

1

09

08

07

06

05

z

1 09095 08085 07075 06065 05055y

(b)

Figure 9 Influence of Cc change on the evolutionary game process

111

0908070605

z

108

06y04 05 06 07 x

08 091

Sc=20Sc=30Sc=40

(a)

Sc=20Sc=30Sc=40

11

1

09

08

07

06

05

z

1 09095 08085 07075 06065 05055y

(b)

Figure 10 Influence of Sc change on the evolutionary game process

Complexity 11

governments social capitals and contractors +e satisfac-tion conditions that the system converging to the ESS point(1 1 1) are analyzed based on prospect theory and the mainfactors influencing stakeholdersrsquo behavior decision-makingare simulated by MATLAB +e research results will help toclarify the behavior evolution mechanism and the influ-encing factors of the ESS for adopting BIM among stake-holders in PPP projects so as to improve the cross-organization information collaboration efficiency of PPPprojects and promote the informatization of the construc-tion industry +e primary conclusions and managementimplications are as follows

First the cost of BIM promotion by governments and thepunishment from the superior supervision authority whengovernments choose the passive strategy are the key factorsaffecting governmentsrsquo behavior choice When governmentsactively promote the cost reduction of BIM or punishedmore by the superior supervision authority governmentswill choose the active promotion strategy and the behaviorgradually converges to the ESS point (1 1 1)

Second the cost of active adoption of BIM and thepunishment of passive adoption of BIM are the key factorsaffecting the choice of social capitalsrsquo behavior When thecost of active adoption of BIM is reduced or the penalty ofpassive adoption of BIM is increased the intention of socialcapitals to choose active adoption of BIM strategy is in-creased and the behavior of social capitals gradually con-verges to the ESS point (1 1 1)

Finally the cost of active application of BIM and thereward of active application of BIM are the key factorsinfluencing the choice of contractorsrsquo behavior When thecost of the active application of BIM technology is reducedor the penalty of the negative application of BIM technologyis increased the intention of the contractor to choose thestrategy of active application of BIM technology is increasedand the contractorrsquos behavior gradually converges to the ESSpoint (1 1 1)

+e adoption of BIM in PPP projects is a major reform ofthe traditional building mode whichrequires all stakeholders to change the traditionalthinking and actively cooperate In order to promoteBIM adoption among stakeholders of PPP projects thefollowing management implications are proposedImprove the supervision efficiency of superior supervisionauthority on governmentsrsquo promotion of BIM and furtherenrich the supervision means with the help of the media andthe public Increase the punishment for the governmentsrsquopassive promotion of BIM so as to improve the perceivedvalue of governments for regulatory punishment

Strengthen the punishment for the passive adoption ofBIM by social capitals and the reward for the active ap-plication of BIM by contractors Improve the cognition ofsocial capitals and contractors on BIM and other advancedinformation technology through publicity and training andgradually transform the system promotion into habitualadoption

Under the PPP mode the adoption of BIM can bepromoted by the governmentrsquos supervision but the gov-ernmentrsquos behavior also needs supervision +e public is the

users and beneficiaries of PPP projectsrsquo goods and the publicshould also have the right to know and to supervise theimplementation of the projects +erefore in the next stagethe authors will further research the role of public super-vision in the adoption of BIM in PPP projects in order toexplore the evolutionary mechanism and influencing factorsunder more stakeholders

Data Availability

Some or all data models or code generated or used tosupport the findings of this study are available from thecorresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interest

Acknowledgments

+is work was supported by the National Natural ScienceFoundation of China (NSFC) (71801119) in part by theSocial Science Planning Foundation of Liaoning(L18BGL020) and in part by the Basic Scientific ResearchProjects in Institutions of Higher Learning of LiaoningEducation Department (LJ2017QW009)

References

[1] N Carbonara and R Pellegrino ldquoRevenue guarantee inpublic-private partnerships a win-win modelrdquo ConstructionManagement amp Economics vol 36 no 10 pp 584ndash598 2018

[2] L Cheng ldquoDiscussion on the application of PPPmodel in newinfrastructure constructionrdquo Open Journal of Social Sciencesvol 7 no 9 pp 283ndash288 2019

[3] D Wu and S Q Wang ldquoResearch development and trend ofpublic private partnership in Chinardquo Journal of EngineeringManagement vol 28 no 6 pp 75ndash80 2014

[4] K Leidel and H W Alfen Lifecycle-Oriented Risk Manage-ment for PPP-Projects in Public Real Estate pp 587ndash604 EresLos Angeles CA USA 2009

[5] J Yuan X Li Y Ke W Xu Z Xu and M SkibnewskildquoDeveloping a building information modelingndashbased per-formance management system for publicndashprivate partner-shipsrdquo Engineering Construction and ArchitecturalManagement vol 27 no 8 pp 1727ndash1762 2020

[6] S Jayasuriya G Zhang and R J Yang ldquoExploring the impactof stakeholder management strategies on managing issues inPPP projectsrdquo International Journal of Construction Man-agement vol 20 no 6 pp 666ndash678 2020

[7] C X Jiang ldquoCauses and implications of the failure intransportation infrastructure public privatepartnershipsmdashmdasha study of 25 typical PPP casesrdquo Journal ofBeijing Jiaotong University vol 15 no 3 pp 50ndash58 2016

[8] R Osei-Kyei and A P C Chan ldquoReview of studies on thecritical success factors for public-private partnership(PPP) projects from 1990 to 2013rdquo International Journal ofProject Management vol 33 no 6 pp 1335ndash1346 2015

[9] B Zhang Y Wang X Liu et al ldquo+e configuration of PPPfailuremdashmdasha crisp-set qualitative comparative analysisrdquoJournal of Public Administration vol 12 no 04pp 65ndash81+191 2019

12 Complexity

[10] N Costantino and R Pellegrino ldquoPublicndashprivate partnershipimplementation cross-case analysisrdquo Proceedings of the In-stitution of Civil Engineers-management Procurement andLaw vol 168 no 4 pp 164ndash176 2015

[11] P Zhang K Zhan and Y Zhou ldquoGame model of three-partyshared risk allocation for PPP projectsmdashmdashbased on theperspective of incomplete informationrdquo IOP Conference Se-ries Earth and Environmental Science vol 295 no 4pp 42ndash49 2019

[12] N W Young S A Jones and H M Bernstein BuildingInformation Modeling Transforming Design and Constructionto Achieve Greater Industry Productivity McGraw HillConstruction New York NY USA 2008

[13] N D Aziz A H Nawawi and N R M Ariff ldquoBuildinginformation modelling (BIM) in facilities management op-portunities to be considered by facility managersrdquo Procedia-Social and Behavioral Sciences vol 234 pp 353ndash362 2016

[14] W Zou M Kumaraswamy and J Chung ldquoIdentifying thecritical success factors for relationship management in PPPprojectsrdquo International Journal of Project Managementvol 32 no 2 pp 265ndash274 2014

[15] QWang and TWang ldquoResearch on promotion of BIM basedon game analysis of stakeholdersrdquo Sichuan BuildingMaterialsvol 41 no 6 p 272 + 278 2015

[16] Q L Yin Evolutionary Game Analysis on BIM TechnologyDiffusion of Prefabricated Construction Qingdao Universityof Technology Qingdao China 2019

[17] H X Tang J M Cao H S Xu and C F Han ldquoAnalysis onBIM application cooperation behaviors in integrated facilitymanagement organization based on evolutionary game the-oryrdquo Industrial Engineering amp Management vol 25 no 4pp 1ndash8 2020

[18] J R Song ldquoResearch on incentive and countermeasures ofBIM application based on game theoryrdquo Value Engineeringvol 36 no 33 pp 43-44 2017

[19] C Jia Z H Wang and L Zhao ldquoEvolutionary game analysisand simulation for cooperative application of BIM technologyin PPPrdquo Journal of Liaoning Technical University vol 22no 4 pp 270ndash277 2020

[20] K A Tversky ldquoProspect theory an analysis of decision underriskrdquo Econometrica vol 47 no 2 pp 263ndash291 1979

[21] R L Gao and Q Bao ldquoSelection of government supervisionmode during the operational period in PPP projects based onevolutionary game theoryrdquo Operations Research and Man-agement Science vol 28 no 4 pp 155ndash162 2019

[22] S K He G W Liang and J B Meng ldquoMulti-subjectsbenefit game and behavior evolution mechanism of majorengineering based on prospect theoryrdquo Science andTechnology Management Research vol 40 no 5 pp 207ndash214 2020

[23] A Porwal and K N Hewage ldquoBuilding information modeling(BIM) partnering framework for public construction proj-ectsrdquo Automation in Construction vol 31 no 3 pp 204ndash2142013

[24] P E D Love J Liu J Matthews C-P Sing and J SmithldquoFuture proofing PPPs life-cycle performance measurementand building information modellingrdquo Automation in Con-struction vol 56 pp 26ndash35 2015

[25] Y L Yin Q J Liu and X Wang ldquoResearch on the whole lifecycle supervision system of public projectsmdashmdashbased on thecollaboration of VfM evaluation and BIM technologyrdquo ProjectManagement Technology vol 14 no 07 pp 17ndash20 2016

[26] S Q Wang R Tiong S Ting et al ldquoEvaluation and man-agement of political risks in Chinarsquos BOTprojectsrdquo Journal of

Construction Engineering and Management vol 126 no 3pp 242ndash250 2000

[27] Y M Chen and H F Chen ldquo+e risk management of PPPproject life cycle based on BIM technologyrdquo Project Man-agement Technology vol 15 no 5 pp 51ndash55 2017

[28] G Luo and X X Zhan ldquoResearch on the application of BIM inPPP project whole-life cyclerdquo Technology and EconomicGuide vol 26 no 5 p 168 2018

[29] B L Zhou and C L Lin ldquo+e application of BIM in PPPproject whole-life cyclerdquo Engineering Economy vol 29 no 3pp 51ndash54 2019

[30] J M Smith Evolution and the )eory of Games CambridgeUniversity Press Cambridge UK 1982

[31] J W Weibull Evolutionary Game )eory MIT PressCambridge MA USA 1997

[32] K Binmore ldquoModeling rational players part Irdquo Economicsand Philosophy vol 3 no 2 pp 179ndash214 1987

[33] K Binmore ldquoModeling rational players part IIrdquo Economicsand Philosophy vol 4 1988

[34] P D Taylor and L B Jonker ldquoEvolutionarily stable strategiesand game dynamicsrdquoMathematical Biosciences vol 40 no 1-2 pp 145ndash156 1978

[35] P Young ldquoConventional contractsrdquo )e Review of EconomicStudies vol 65 no 4 pp 773ndash792 1998

[36] D Fudenberg and C Harris ldquoEvolutionary dynamics withaggregate shocksrdquo Journal of Economic )eory vol 57 no 2pp 420ndash441 1992

[37] H Bester W Guth and W Guth ldquoIs altruism evolutionarilystablerdquo Journal of Economic Behavior amp Organization vol 34no 2 pp 193ndash209 1998

[38] K E Boulding R R Nelson and S G Winter ldquoAn evolu-tionary theory of economic changrdquo American Journal ofAgricultural Economics vol 66 no 4 p 535 1984

[39] S Y Xie Economic Game )eory Fudan University PressShanghai China 2002

[40] D Friedman and K C Fung ldquoInternational trade and theinternal organization of firms an evolutionary approachrdquoJournal of International Economics vol 41 no 1-2 pp 113ndash137 1996

[41] Y P Wang and W D Meng ldquoEvolutionary game analysis onco-competition mechanism of supply chainrdquo Journal of In-dustrial Engineering and Engineering Management vol 18no 2 pp 96ndash98 2004

[42] J Gao and Z H Shen ldquoElementary groping for evolutionarygame theory and its application in electricity marketrdquo Au-tomation of Electric Power Systems vol 27 no 18 pp 18ndash212003

[43] Y Zhang and M Guizani Game )eory for Wireless Com-munications and Networking CRC Press Boca Raton FLUSA 2011

[44] F Y Lu ldquoEvolutionary game analysis on environmentalpollution problemrdquo Systems Engineering-)eory amp Practicevol 27 no 9 pp 148ndash152 2007

[45] S L Dreisbach ldquoImpact of psychological research on moraltheory prospect theory and the doctrine of doing andallowingrdquo Dissertations amp )eses-Gradworks University ofCalifornia Santa Cruz CA USA 2012

[46] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Econometrica vol 47 no 2 1979

[47] N C Barberis ldquo+irty years of prospect theory in economicsa review and assessmentrdquo )e Journal of Economic Perspec-tives vol 27 no 1 pp 173ndash196 2013

Complexity 13

[48] S Benartzi and R H +aler ldquoMyopic loss aversion and theequity premium puzzlerdquo Quarterly Journal of Economicsvol 110 no 1 pp 73ndash92 1995

[49] N Barberis and M Huang ldquoStocks as lotteries the impli-cations of probability weighting for security pricesrdquo )eAmerican Economic Review vol 98 no 5 pp 2066ndash21002008

[50] A Frazzini ldquo+e disposition effect and underreaction tonewsrdquo )e Journal of Finance vol 61 no 4 pp 2017ndash20462006

[51] W Y Hu and J S Scott ldquoBehavioral obstacles in the annuitymarketrdquo Financial Analysts Journal vol 63 no 6 pp 71ndash822007

[52] B Koszegi andM Rabin ldquoReference-dependent consumptionplansrdquo )e American Economic Review vol 99 no 3pp 909ndash936 2009

[53] P Vamvakas E E Tsiropoulou and S Papavassiliou ldquoDy-namic spectrum management in 5G wireless networks a real-life modeling approachrdquo in Proceedings of the IEEE INFO-COM 2019-IEEE Conference on Computer CommunicationsIEEE Paris France April 2019

[54] Y N Zhou and J C Liu ldquoStudy on supervision strategy of pppproject considering the participation of higher-level gov-ernmentrdquo Chinese Journal of Management Science vol 1ndash162021 httpsdoiorg1016381jcnkiissn1003-207x20200801

[55] H Q Zhang J C Wang and H L Tao ldquoResearch the de-cision-making behavior of investors based on prospect theoryin the ppp projectrdquo Soft Science vol 32 no 8 pp 129ndash1332018

[56] D Friedman ldquoEvolutionary games in economicsrdquo Econo-metrica vol 59 no 3 pp 637ndash666 1991

[57] Z B Zhao and Q P Man ldquoEvolutionary game analysis of riskmanagement behavior of major infrastructure projects basedon projects based on prospect theoryrdquo Journal of SystemsManagement vol 27 no 1 pp 112ndash120 2018

14 Complexity

+en the average perceived expected value of contrac-tors is as follows

Ec zEc1 +(1 minus z)Ec2 (11)

+erefore the replication dynamic equation of con-tractors is as follows

F(z) dz

dt z(Ec1 minus Ec) z(1 minus z)(Ec1 minus Ec2)

z(1 minus z)(Rc minus Cc + Sc + xLc + yLc2 + xyLc1 minus xyLc2)

(12)

+e first derivative of z is equal to

d(F(z))

dz (1 minus 2z)(Rc minus Cc + Sc + xLc + yLc2 + xyLc1 minus xyLc2)

(13)

When F(z) 0 and d(F(z))dzlang0 contractorsrsquo strategyof ldquopositive applicationrdquo will be in a stable state +ereforethere will be F(z) equiv 0 when y Rc minus Cc + Sc + xLcx(Lc2 minus Lc1) minus Lc2 and all the values of z are in a stablestate When ylangRc minus Cc + Sc + xLcx(Lc2 minus Lc1) minus Lc2there will be d(F(z))dz|z1lang0 and z 1 is the ESS of thesystem When yrangRc minus Cc + Sc + xLcx(Lc2 minus Lc1) minus Lc2there will be d(F(z))dz|z0lang0 and z 0 is the ESS of thesystem +e phase diagram of contractorsrsquo behavioralstrategy evolution is shown in Figure 3 +e surface y

Rc minus Cc + Sc + xLcx(Lc2 minus Lc1) minus Lc2 divides the three-dimensional space V (x y z| 0lexle 1 0le yle 1 0lezle 1) into parts of V1 and V2 In part V1 contractorsgradually evolved to z 1 and finally chose the strategy ofldquopositive applicationrdquo In part V2 contractors graduallyevolved to z 0 and finally chose the strategy of ldquopassiveapplicationrdquo

33 Evolutionary Stability Analysis +e Jacobin matrix ofthe system is represented by J

J

J1 J2 J3

J4 J5 J6

J7 J8 J9

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

zf(x)

zx

zf(x)

zy

zf(x)

zz

zf(y)

zx

zf(y)

zy

zf(y)

zz

zf(z)

zx

zf(z)

zy

zf(z)

zz

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

(14)

Among them

zf(x)

zx (1 minus 2x)(Cg2 minus Cg1 + Lg minus Sp + Lp + Lc minus zLc minus yLp minus zLg)

zf(y)

zy (1 minus 2y)(Cp2 minus Cp1 + Lc2 + xLc1 minus xLc2 minus zLc2 + xLp minus xzLc1 + xzLc2)

zf(z)

zz (1 minus 2z)(Rc minus Cc + Sc + xLc + yLc2 + xyLc1 minus xyLc2)

(15)

+e local stability of equilibrium points is given inTable 3

According to the indirect method of Lyapunov theequilibrium point is asymptotically stable if all its eigen-values of the Jacobian matrix [56] have negative real parts InTable 3 it can be seen that if we want the system evolving tothe ESS point (1 1 1) the following conditions must be metCg + Splt αLg1 Cp1ltCp2 + Lp and CcltRc + Sc

However it should be noted that in the actual PPPprojects the main stakeholders cannot make the best use ofthe existing information in the decision-making because of

their limited rationality +ey usually make decisionsthrough subjective judgment which ultimately leads to de-viation in their behaviors +e deviations will hinder theestablishment of ESS point (1 1 1) and make the systemdifficult to converge to the optimal state +e reasons areanalyzed based on the prospect theory as followstake governments for example the condition that gov-ernments promoting BIM actively is Cg + Splt αLg1 Basedon the prospect theory individuals always overestimate thelow probability loss and underestimate the high probabilitygain namely individuals tend to choose risk aversion in the

z

1

1

x

V2

V11 y

Figure 2 Phase diagram of social capitalsrsquo behavioral strategyevolution

Complexity 7

Table 3 +e eigenvalues of each equilibrium point

Equilibrium pointEigenvalues of Jacobi matrix

λ1 λ2 λ3E(0 0 0) minusCg minus Sp + Lg + Lp + αLg1 Cp2 minus Cp1 + Lc2 Rc minus Cc + ScE(1 0 0) Cg + Sp minus Lg minus Lp minus Lc minus αLg1 Cp2 minus Cp1 + Lc1 + Lp Rc minus Cc + Sc + LcE(0 1 0) minusCg minus Sp + Lc + αLg1 Cp2 minus Cp1 minus Lc2 Rc minus Cc + Sc + Lc2E(0 0 1) minusCg minus Sp + Lc + αLg1 Cp2 minus Cp1 minusRc + Cc + ScE(1 1 0) Cg + Sp minus Lc minus αLg1 minusCp2 minus Cp1 minus Lc1 minus Lp Rc minus Cc + Sc + Lc + Lc1E(1 0 1) Cg + Sp minus Lp minus Lg minus αLg1 Cp2 minus Cp1 + Lp minusRc + Cc minus Sc minus LcE(0 1 1) minusCg minus Sp + αLg1 minusCp2 + Cp1 minusRc + Cc minus Sc minus Lc2E(1 1 1) Cg + Sp minus αLg1 minusCp2 minus Cp1 + Lp minusRc + Cc minus Sc minus Lc minus Lc1

z

1

1

x

V2

V1

1 y

Figure 3 Phase diagram of contractorsrsquo behavioral strategy evolution

111

0908070605

z

108

06y04 05 06 07 08

x

09 1

Cg=20Cg=15Cg=10

(a)

11

1

09

08

07

06

05

z

Cg=20Cg=15Cg=10

1 095 09 085 08 075 07 065 06 055 05x

(b)

Figure 4 Influence of Cg change on the evolutionary game process

8 Complexity

face of gains but tend to prefer risk preference when facinglosses [57] In the practice of PPP projects governmentsoften think that their behavior will not be easily detected bythe superior supervision authority due to the commonexistence of fluke mentality +erefore governments tend tounderestimate the probability α and the value of punishmentLg1 by the superior supervision authority which means thatthe right part of the conditional formula Cg + Splt αLg1 willbe underestimated in practice On the other hand whengovernments choose the active strategy it will definitely paythe promotion cost while choosing the passive strategy willonly have a certain probability of punitive losses Accordingto the prospect theory governments will be more inclined torisk preference that is they prefer to take risks and choosethe passive strategy rather than to bear the deterministic

expenditure +erefore the above reasons will cause certaindifficulties for the establishment of the formulaCg + Splt αLg1

4 Numerical Simulation and Analysis

In order to better analyze the behavior evolution mechanismamong governments social capitals and contractors in BIMadoption this study uses MATLAB R2018a to simulate thetrend and influencing factors of the evolutionary game +evalues of parameters are randomly assigned under the as-sumption which do not represent the real benefits ofstakeholders in PPP projects

According to the above analysis if the system convergesto the ESS point (1 1 1) the following conditions need to be

111

0908070605

z

108

06y04 04

0608

x

1

a=05a=08a=1

(a)

a=05a=08a=1

11

1

09

08

07

06

05

z

11 1 09 08 07 06 05x

(b)

Figure 5 Influence of α change on the evolutionary game process

Lg1=80Lg1=100Lg1=120

111

0908070605

z

108

06y04 04

0608

x

1

(a)

Lg1=80Lg1=100Lg1=120

11

1

09

08

07

06

05

z

11 1 09 08 07 06 05x

(b)

Figure 6 Influence of Lg1 change on the evolutionary game process

Complexity 9

satisfied Cg + Splt αLg1 Cp1ltCp2 + Lp and CcltRc + Sc+e initial values of each parameter are set to x 05 y

05 α 05Cg 20 Sp 10 Lgl 80Cp1 30Cp2

20 Sp 10 Lg1 80Cp1 30Cp2 20 Lp 20Cc

30Rc 40 Sc 10 Lg 10 Lc 10 Lc1 10 LC2 5

41 Influence of Cg1 and Cg2 Changes on the EvolutionaryGame Process Keep the other parameters unchanged andthe influence on the evolution trend of the system is ob-served by changing the values of Cg α and Lg1 as shown inFigures 4(a) 5(a) and 6(a) In Figure 4(b) we can see thatthe convergence rate of x to the ESS point (1 1 1) isaccelerated as the value of Cg goes down +e simulationresults show that reducing the cost of governmentsrsquo strategyof ldquopositive promotionrdquo is helpful to enhance the willingness

of governments to promote the adoption of BIM in PPPprojects In Figures 5(b) and 6(b) we can see that theconvergence rate of x to the ESS point (1 1 1) is acceleratedas the value of α and Lg1 go bigger +e simulation resultsshow that increasing the supervision probability and theseverity of punishment will help increase the willingness ofgovernments to promote BIM application in PPP projects

42 Influence of Cp1 and Lp Changes on the EvolutionaryGame Process Keep the other parameters unchanged andthe influence on the evolution trend of the system is ob-served by changing the values of Cp1 and Lp as shown inFigures 7(a) and 8(a) In Figure 7(b) we can see that theconvergence rate of y to the ESS point (1 1 1) is acceleratedas the value of Cp1 goes down +e simulation results show

111

0908070605

z

108

06y04 05 06 07 08

x

091

Cp1=30Cp1=20Cp1=10

(a)

Cp1=30Cp1=20Cp1=10

11

1

09

08

07

06

05

z

11 1 09 08 07 06 05y

(b)

Figure 7 Influence of Cp1 change on the evolutionary game process

111

0908070605

z

108

06y04 05

06 07 x08 09 1

Lp=20Lp=30Lp=40

(a)

Lp=20Lp=30Lp=40

11

1

09

08

07

06

05

z

11 1 09 08 07 06 05y

(b)

Figure 8 Influence of Lp change on the evolutionary game process

10 Complexity

that reducing the cost of social capitalsrsquo strategy of ldquopositiveadoptionrdquo is helpful to enhance the willingness of socialcapitals to adopt BIM in PPP projects In Figure 8(b) we cansee that the convergence rate of y to the ESS point (1 1 1) isaccelerated as the value of Lp goes bigger +e simulationresults show that increasing the punishment of social cap-italsrsquo strategy of ldquopassive adoptionrdquo is helpful to enhance thewillingness of social capitals to adopt BIM in PPP projects

43 Influence of Cc and Sc Changes on the Evolutionary GameProcess Keep the other parameters unchanged and theinfluence on the evolution trend of the system is observed bychanging the values of Cc and Sc as shown in Figures 9(a)and 10(a) In Figure 9(b) we can see that the convergence

rate of z to the ESS point (1 1 1) is accelerated as the value ofCc goes down+e simulation results show that reducing thecost of contractorsrsquo strategy of ldquopositive applicationrdquo ishelpful to enhance the willingness of contractors to applyBIM in PPP projects In Figure 10(b) we can see that theconvergence rate of z to the ESS point (1 1 1) is acceleratedas the value of Sc goes bigger +e simulation results showthat increasing the punishment of contractorsrsquo strategy ofldquopassive applicationrdquo is helpful to enhance the willingness ofcontractors to apply BIM in PPP projects

5 Conclusions and Management Implications

Under the assumption of bounded rationality this studyconstructs a tripartite evolutionary game model including

111

0908070605

z

108

06y04 05 06 07 08

x

091

Cc=30Cc=20Cc=10

(a)

Cc=30Cc=20Cc=10

11

1

09

08

07

06

05

z

1 09095 08085 07075 06065 05055y

(b)

Figure 9 Influence of Cc change on the evolutionary game process

111

0908070605

z

108

06y04 05 06 07 x

08 091

Sc=20Sc=30Sc=40

(a)

Sc=20Sc=30Sc=40

11

1

09

08

07

06

05

z

1 09095 08085 07075 06065 05055y

(b)

Figure 10 Influence of Sc change on the evolutionary game process

Complexity 11

governments social capitals and contractors +e satisfac-tion conditions that the system converging to the ESS point(1 1 1) are analyzed based on prospect theory and the mainfactors influencing stakeholdersrsquo behavior decision-makingare simulated by MATLAB +e research results will help toclarify the behavior evolution mechanism and the influ-encing factors of the ESS for adopting BIM among stake-holders in PPP projects so as to improve the cross-organization information collaboration efficiency of PPPprojects and promote the informatization of the construc-tion industry +e primary conclusions and managementimplications are as follows

First the cost of BIM promotion by governments and thepunishment from the superior supervision authority whengovernments choose the passive strategy are the key factorsaffecting governmentsrsquo behavior choice When governmentsactively promote the cost reduction of BIM or punishedmore by the superior supervision authority governmentswill choose the active promotion strategy and the behaviorgradually converges to the ESS point (1 1 1)

Second the cost of active adoption of BIM and thepunishment of passive adoption of BIM are the key factorsaffecting the choice of social capitalsrsquo behavior When thecost of active adoption of BIM is reduced or the penalty ofpassive adoption of BIM is increased the intention of socialcapitals to choose active adoption of BIM strategy is in-creased and the behavior of social capitals gradually con-verges to the ESS point (1 1 1)

Finally the cost of active application of BIM and thereward of active application of BIM are the key factorsinfluencing the choice of contractorsrsquo behavior When thecost of the active application of BIM technology is reducedor the penalty of the negative application of BIM technologyis increased the intention of the contractor to choose thestrategy of active application of BIM technology is increasedand the contractorrsquos behavior gradually converges to the ESSpoint (1 1 1)

+e adoption of BIM in PPP projects is a major reform ofthe traditional building mode whichrequires all stakeholders to change the traditionalthinking and actively cooperate In order to promoteBIM adoption among stakeholders of PPP projects thefollowing management implications are proposedImprove the supervision efficiency of superior supervisionauthority on governmentsrsquo promotion of BIM and furtherenrich the supervision means with the help of the media andthe public Increase the punishment for the governmentsrsquopassive promotion of BIM so as to improve the perceivedvalue of governments for regulatory punishment

Strengthen the punishment for the passive adoption ofBIM by social capitals and the reward for the active ap-plication of BIM by contractors Improve the cognition ofsocial capitals and contractors on BIM and other advancedinformation technology through publicity and training andgradually transform the system promotion into habitualadoption

Under the PPP mode the adoption of BIM can bepromoted by the governmentrsquos supervision but the gov-ernmentrsquos behavior also needs supervision +e public is the

users and beneficiaries of PPP projectsrsquo goods and the publicshould also have the right to know and to supervise theimplementation of the projects +erefore in the next stagethe authors will further research the role of public super-vision in the adoption of BIM in PPP projects in order toexplore the evolutionary mechanism and influencing factorsunder more stakeholders

Data Availability

Some or all data models or code generated or used tosupport the findings of this study are available from thecorresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interest

Acknowledgments

+is work was supported by the National Natural ScienceFoundation of China (NSFC) (71801119) in part by theSocial Science Planning Foundation of Liaoning(L18BGL020) and in part by the Basic Scientific ResearchProjects in Institutions of Higher Learning of LiaoningEducation Department (LJ2017QW009)

References

[1] N Carbonara and R Pellegrino ldquoRevenue guarantee inpublic-private partnerships a win-win modelrdquo ConstructionManagement amp Economics vol 36 no 10 pp 584ndash598 2018

[2] L Cheng ldquoDiscussion on the application of PPPmodel in newinfrastructure constructionrdquo Open Journal of Social Sciencesvol 7 no 9 pp 283ndash288 2019

[3] D Wu and S Q Wang ldquoResearch development and trend ofpublic private partnership in Chinardquo Journal of EngineeringManagement vol 28 no 6 pp 75ndash80 2014

[4] K Leidel and H W Alfen Lifecycle-Oriented Risk Manage-ment for PPP-Projects in Public Real Estate pp 587ndash604 EresLos Angeles CA USA 2009

[5] J Yuan X Li Y Ke W Xu Z Xu and M SkibnewskildquoDeveloping a building information modelingndashbased per-formance management system for publicndashprivate partner-shipsrdquo Engineering Construction and ArchitecturalManagement vol 27 no 8 pp 1727ndash1762 2020

[6] S Jayasuriya G Zhang and R J Yang ldquoExploring the impactof stakeholder management strategies on managing issues inPPP projectsrdquo International Journal of Construction Man-agement vol 20 no 6 pp 666ndash678 2020

[7] C X Jiang ldquoCauses and implications of the failure intransportation infrastructure public privatepartnershipsmdashmdasha study of 25 typical PPP casesrdquo Journal ofBeijing Jiaotong University vol 15 no 3 pp 50ndash58 2016

[8] R Osei-Kyei and A P C Chan ldquoReview of studies on thecritical success factors for public-private partnership(PPP) projects from 1990 to 2013rdquo International Journal ofProject Management vol 33 no 6 pp 1335ndash1346 2015

[9] B Zhang Y Wang X Liu et al ldquo+e configuration of PPPfailuremdashmdasha crisp-set qualitative comparative analysisrdquoJournal of Public Administration vol 12 no 04pp 65ndash81+191 2019

12 Complexity

[10] N Costantino and R Pellegrino ldquoPublicndashprivate partnershipimplementation cross-case analysisrdquo Proceedings of the In-stitution of Civil Engineers-management Procurement andLaw vol 168 no 4 pp 164ndash176 2015

[11] P Zhang K Zhan and Y Zhou ldquoGame model of three-partyshared risk allocation for PPP projectsmdashmdashbased on theperspective of incomplete informationrdquo IOP Conference Se-ries Earth and Environmental Science vol 295 no 4pp 42ndash49 2019

[12] N W Young S A Jones and H M Bernstein BuildingInformation Modeling Transforming Design and Constructionto Achieve Greater Industry Productivity McGraw HillConstruction New York NY USA 2008

[13] N D Aziz A H Nawawi and N R M Ariff ldquoBuildinginformation modelling (BIM) in facilities management op-portunities to be considered by facility managersrdquo Procedia-Social and Behavioral Sciences vol 234 pp 353ndash362 2016

[14] W Zou M Kumaraswamy and J Chung ldquoIdentifying thecritical success factors for relationship management in PPPprojectsrdquo International Journal of Project Managementvol 32 no 2 pp 265ndash274 2014

[15] QWang and TWang ldquoResearch on promotion of BIM basedon game analysis of stakeholdersrdquo Sichuan BuildingMaterialsvol 41 no 6 p 272 + 278 2015

[16] Q L Yin Evolutionary Game Analysis on BIM TechnologyDiffusion of Prefabricated Construction Qingdao Universityof Technology Qingdao China 2019

[17] H X Tang J M Cao H S Xu and C F Han ldquoAnalysis onBIM application cooperation behaviors in integrated facilitymanagement organization based on evolutionary game the-oryrdquo Industrial Engineering amp Management vol 25 no 4pp 1ndash8 2020

[18] J R Song ldquoResearch on incentive and countermeasures ofBIM application based on game theoryrdquo Value Engineeringvol 36 no 33 pp 43-44 2017

[19] C Jia Z H Wang and L Zhao ldquoEvolutionary game analysisand simulation for cooperative application of BIM technologyin PPPrdquo Journal of Liaoning Technical University vol 22no 4 pp 270ndash277 2020

[20] K A Tversky ldquoProspect theory an analysis of decision underriskrdquo Econometrica vol 47 no 2 pp 263ndash291 1979

[21] R L Gao and Q Bao ldquoSelection of government supervisionmode during the operational period in PPP projects based onevolutionary game theoryrdquo Operations Research and Man-agement Science vol 28 no 4 pp 155ndash162 2019

[22] S K He G W Liang and J B Meng ldquoMulti-subjectsbenefit game and behavior evolution mechanism of majorengineering based on prospect theoryrdquo Science andTechnology Management Research vol 40 no 5 pp 207ndash214 2020

[23] A Porwal and K N Hewage ldquoBuilding information modeling(BIM) partnering framework for public construction proj-ectsrdquo Automation in Construction vol 31 no 3 pp 204ndash2142013

[24] P E D Love J Liu J Matthews C-P Sing and J SmithldquoFuture proofing PPPs life-cycle performance measurementand building information modellingrdquo Automation in Con-struction vol 56 pp 26ndash35 2015

[25] Y L Yin Q J Liu and X Wang ldquoResearch on the whole lifecycle supervision system of public projectsmdashmdashbased on thecollaboration of VfM evaluation and BIM technologyrdquo ProjectManagement Technology vol 14 no 07 pp 17ndash20 2016

[26] S Q Wang R Tiong S Ting et al ldquoEvaluation and man-agement of political risks in Chinarsquos BOTprojectsrdquo Journal of

Construction Engineering and Management vol 126 no 3pp 242ndash250 2000

[27] Y M Chen and H F Chen ldquo+e risk management of PPPproject life cycle based on BIM technologyrdquo Project Man-agement Technology vol 15 no 5 pp 51ndash55 2017

[28] G Luo and X X Zhan ldquoResearch on the application of BIM inPPP project whole-life cyclerdquo Technology and EconomicGuide vol 26 no 5 p 168 2018

[29] B L Zhou and C L Lin ldquo+e application of BIM in PPPproject whole-life cyclerdquo Engineering Economy vol 29 no 3pp 51ndash54 2019

[30] J M Smith Evolution and the )eory of Games CambridgeUniversity Press Cambridge UK 1982

[31] J W Weibull Evolutionary Game )eory MIT PressCambridge MA USA 1997

[32] K Binmore ldquoModeling rational players part Irdquo Economicsand Philosophy vol 3 no 2 pp 179ndash214 1987

[33] K Binmore ldquoModeling rational players part IIrdquo Economicsand Philosophy vol 4 1988

[34] P D Taylor and L B Jonker ldquoEvolutionarily stable strategiesand game dynamicsrdquoMathematical Biosciences vol 40 no 1-2 pp 145ndash156 1978

[35] P Young ldquoConventional contractsrdquo )e Review of EconomicStudies vol 65 no 4 pp 773ndash792 1998

[36] D Fudenberg and C Harris ldquoEvolutionary dynamics withaggregate shocksrdquo Journal of Economic )eory vol 57 no 2pp 420ndash441 1992

[37] H Bester W Guth and W Guth ldquoIs altruism evolutionarilystablerdquo Journal of Economic Behavior amp Organization vol 34no 2 pp 193ndash209 1998

[38] K E Boulding R R Nelson and S G Winter ldquoAn evolu-tionary theory of economic changrdquo American Journal ofAgricultural Economics vol 66 no 4 p 535 1984

[39] S Y Xie Economic Game )eory Fudan University PressShanghai China 2002

[40] D Friedman and K C Fung ldquoInternational trade and theinternal organization of firms an evolutionary approachrdquoJournal of International Economics vol 41 no 1-2 pp 113ndash137 1996

[41] Y P Wang and W D Meng ldquoEvolutionary game analysis onco-competition mechanism of supply chainrdquo Journal of In-dustrial Engineering and Engineering Management vol 18no 2 pp 96ndash98 2004

[42] J Gao and Z H Shen ldquoElementary groping for evolutionarygame theory and its application in electricity marketrdquo Au-tomation of Electric Power Systems vol 27 no 18 pp 18ndash212003

[43] Y Zhang and M Guizani Game )eory for Wireless Com-munications and Networking CRC Press Boca Raton FLUSA 2011

[44] F Y Lu ldquoEvolutionary game analysis on environmentalpollution problemrdquo Systems Engineering-)eory amp Practicevol 27 no 9 pp 148ndash152 2007

[45] S L Dreisbach ldquoImpact of psychological research on moraltheory prospect theory and the doctrine of doing andallowingrdquo Dissertations amp )eses-Gradworks University ofCalifornia Santa Cruz CA USA 2012

[46] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Econometrica vol 47 no 2 1979

[47] N C Barberis ldquo+irty years of prospect theory in economicsa review and assessmentrdquo )e Journal of Economic Perspec-tives vol 27 no 1 pp 173ndash196 2013

Complexity 13

[48] S Benartzi and R H +aler ldquoMyopic loss aversion and theequity premium puzzlerdquo Quarterly Journal of Economicsvol 110 no 1 pp 73ndash92 1995

[49] N Barberis and M Huang ldquoStocks as lotteries the impli-cations of probability weighting for security pricesrdquo )eAmerican Economic Review vol 98 no 5 pp 2066ndash21002008

[50] A Frazzini ldquo+e disposition effect and underreaction tonewsrdquo )e Journal of Finance vol 61 no 4 pp 2017ndash20462006

[51] W Y Hu and J S Scott ldquoBehavioral obstacles in the annuitymarketrdquo Financial Analysts Journal vol 63 no 6 pp 71ndash822007

[52] B Koszegi andM Rabin ldquoReference-dependent consumptionplansrdquo )e American Economic Review vol 99 no 3pp 909ndash936 2009

[53] P Vamvakas E E Tsiropoulou and S Papavassiliou ldquoDy-namic spectrum management in 5G wireless networks a real-life modeling approachrdquo in Proceedings of the IEEE INFO-COM 2019-IEEE Conference on Computer CommunicationsIEEE Paris France April 2019

[54] Y N Zhou and J C Liu ldquoStudy on supervision strategy of pppproject considering the participation of higher-level gov-ernmentrdquo Chinese Journal of Management Science vol 1ndash162021 httpsdoiorg1016381jcnkiissn1003-207x20200801

[55] H Q Zhang J C Wang and H L Tao ldquoResearch the de-cision-making behavior of investors based on prospect theoryin the ppp projectrdquo Soft Science vol 32 no 8 pp 129ndash1332018

[56] D Friedman ldquoEvolutionary games in economicsrdquo Econo-metrica vol 59 no 3 pp 637ndash666 1991

[57] Z B Zhao and Q P Man ldquoEvolutionary game analysis of riskmanagement behavior of major infrastructure projects basedon projects based on prospect theoryrdquo Journal of SystemsManagement vol 27 no 1 pp 112ndash120 2018

14 Complexity

Table 3 +e eigenvalues of each equilibrium point

Equilibrium pointEigenvalues of Jacobi matrix

λ1 λ2 λ3E(0 0 0) minusCg minus Sp + Lg + Lp + αLg1 Cp2 minus Cp1 + Lc2 Rc minus Cc + ScE(1 0 0) Cg + Sp minus Lg minus Lp minus Lc minus αLg1 Cp2 minus Cp1 + Lc1 + Lp Rc minus Cc + Sc + LcE(0 1 0) minusCg minus Sp + Lc + αLg1 Cp2 minus Cp1 minus Lc2 Rc minus Cc + Sc + Lc2E(0 0 1) minusCg minus Sp + Lc + αLg1 Cp2 minus Cp1 minusRc + Cc + ScE(1 1 0) Cg + Sp minus Lc minus αLg1 minusCp2 minus Cp1 minus Lc1 minus Lp Rc minus Cc + Sc + Lc + Lc1E(1 0 1) Cg + Sp minus Lp minus Lg minus αLg1 Cp2 minus Cp1 + Lp minusRc + Cc minus Sc minus LcE(0 1 1) minusCg minus Sp + αLg1 minusCp2 + Cp1 minusRc + Cc minus Sc minus Lc2E(1 1 1) Cg + Sp minus αLg1 minusCp2 minus Cp1 + Lp minusRc + Cc minus Sc minus Lc minus Lc1

z

1

1

x

V2

V1

1 y

Figure 3 Phase diagram of contractorsrsquo behavioral strategy evolution

111

0908070605

z

108

06y04 05 06 07 08

x

09 1

Cg=20Cg=15Cg=10

(a)

11

1

09

08

07

06

05

z

Cg=20Cg=15Cg=10

1 095 09 085 08 075 07 065 06 055 05x

(b)

Figure 4 Influence of Cg change on the evolutionary game process

8 Complexity

face of gains but tend to prefer risk preference when facinglosses [57] In the practice of PPP projects governmentsoften think that their behavior will not be easily detected bythe superior supervision authority due to the commonexistence of fluke mentality +erefore governments tend tounderestimate the probability α and the value of punishmentLg1 by the superior supervision authority which means thatthe right part of the conditional formula Cg + Splt αLg1 willbe underestimated in practice On the other hand whengovernments choose the active strategy it will definitely paythe promotion cost while choosing the passive strategy willonly have a certain probability of punitive losses Accordingto the prospect theory governments will be more inclined torisk preference that is they prefer to take risks and choosethe passive strategy rather than to bear the deterministic

expenditure +erefore the above reasons will cause certaindifficulties for the establishment of the formulaCg + Splt αLg1

4 Numerical Simulation and Analysis

In order to better analyze the behavior evolution mechanismamong governments social capitals and contractors in BIMadoption this study uses MATLAB R2018a to simulate thetrend and influencing factors of the evolutionary game +evalues of parameters are randomly assigned under the as-sumption which do not represent the real benefits ofstakeholders in PPP projects

According to the above analysis if the system convergesto the ESS point (1 1 1) the following conditions need to be

111

0908070605

z

108

06y04 04

0608

x

1

a=05a=08a=1

(a)

a=05a=08a=1

11

1

09

08

07

06

05

z

11 1 09 08 07 06 05x

(b)

Figure 5 Influence of α change on the evolutionary game process

Lg1=80Lg1=100Lg1=120

111

0908070605

z

108

06y04 04

0608

x

1

(a)

Lg1=80Lg1=100Lg1=120

11

1

09

08

07

06

05

z

11 1 09 08 07 06 05x

(b)

Figure 6 Influence of Lg1 change on the evolutionary game process

Complexity 9

satisfied Cg + Splt αLg1 Cp1ltCp2 + Lp and CcltRc + Sc+e initial values of each parameter are set to x 05 y

05 α 05Cg 20 Sp 10 Lgl 80Cp1 30Cp2

20 Sp 10 Lg1 80Cp1 30Cp2 20 Lp 20Cc

30Rc 40 Sc 10 Lg 10 Lc 10 Lc1 10 LC2 5

41 Influence of Cg1 and Cg2 Changes on the EvolutionaryGame Process Keep the other parameters unchanged andthe influence on the evolution trend of the system is ob-served by changing the values of Cg α and Lg1 as shown inFigures 4(a) 5(a) and 6(a) In Figure 4(b) we can see thatthe convergence rate of x to the ESS point (1 1 1) isaccelerated as the value of Cg goes down +e simulationresults show that reducing the cost of governmentsrsquo strategyof ldquopositive promotionrdquo is helpful to enhance the willingness

of governments to promote the adoption of BIM in PPPprojects In Figures 5(b) and 6(b) we can see that theconvergence rate of x to the ESS point (1 1 1) is acceleratedas the value of α and Lg1 go bigger +e simulation resultsshow that increasing the supervision probability and theseverity of punishment will help increase the willingness ofgovernments to promote BIM application in PPP projects

42 Influence of Cp1 and Lp Changes on the EvolutionaryGame Process Keep the other parameters unchanged andthe influence on the evolution trend of the system is ob-served by changing the values of Cp1 and Lp as shown inFigures 7(a) and 8(a) In Figure 7(b) we can see that theconvergence rate of y to the ESS point (1 1 1) is acceleratedas the value of Cp1 goes down +e simulation results show

111

0908070605

z

108

06y04 05 06 07 08

x

091

Cp1=30Cp1=20Cp1=10

(a)

Cp1=30Cp1=20Cp1=10

11

1

09

08

07

06

05

z

11 1 09 08 07 06 05y

(b)

Figure 7 Influence of Cp1 change on the evolutionary game process

111

0908070605

z

108

06y04 05

06 07 x08 09 1

Lp=20Lp=30Lp=40

(a)

Lp=20Lp=30Lp=40

11

1

09

08

07

06

05

z

11 1 09 08 07 06 05y

(b)

Figure 8 Influence of Lp change on the evolutionary game process

10 Complexity

that reducing the cost of social capitalsrsquo strategy of ldquopositiveadoptionrdquo is helpful to enhance the willingness of socialcapitals to adopt BIM in PPP projects In Figure 8(b) we cansee that the convergence rate of y to the ESS point (1 1 1) isaccelerated as the value of Lp goes bigger +e simulationresults show that increasing the punishment of social cap-italsrsquo strategy of ldquopassive adoptionrdquo is helpful to enhance thewillingness of social capitals to adopt BIM in PPP projects

43 Influence of Cc and Sc Changes on the Evolutionary GameProcess Keep the other parameters unchanged and theinfluence on the evolution trend of the system is observed bychanging the values of Cc and Sc as shown in Figures 9(a)and 10(a) In Figure 9(b) we can see that the convergence

rate of z to the ESS point (1 1 1) is accelerated as the value ofCc goes down+e simulation results show that reducing thecost of contractorsrsquo strategy of ldquopositive applicationrdquo ishelpful to enhance the willingness of contractors to applyBIM in PPP projects In Figure 10(b) we can see that theconvergence rate of z to the ESS point (1 1 1) is acceleratedas the value of Sc goes bigger +e simulation results showthat increasing the punishment of contractorsrsquo strategy ofldquopassive applicationrdquo is helpful to enhance the willingness ofcontractors to apply BIM in PPP projects

5 Conclusions and Management Implications

Under the assumption of bounded rationality this studyconstructs a tripartite evolutionary game model including

111

0908070605

z

108

06y04 05 06 07 08

x

091

Cc=30Cc=20Cc=10

(a)

Cc=30Cc=20Cc=10

11

1

09

08

07

06

05

z

1 09095 08085 07075 06065 05055y

(b)

Figure 9 Influence of Cc change on the evolutionary game process

111

0908070605

z

108

06y04 05 06 07 x

08 091

Sc=20Sc=30Sc=40

(a)

Sc=20Sc=30Sc=40

11

1

09

08

07

06

05

z

1 09095 08085 07075 06065 05055y

(b)

Figure 10 Influence of Sc change on the evolutionary game process

Complexity 11

governments social capitals and contractors +e satisfac-tion conditions that the system converging to the ESS point(1 1 1) are analyzed based on prospect theory and the mainfactors influencing stakeholdersrsquo behavior decision-makingare simulated by MATLAB +e research results will help toclarify the behavior evolution mechanism and the influ-encing factors of the ESS for adopting BIM among stake-holders in PPP projects so as to improve the cross-organization information collaboration efficiency of PPPprojects and promote the informatization of the construc-tion industry +e primary conclusions and managementimplications are as follows

First the cost of BIM promotion by governments and thepunishment from the superior supervision authority whengovernments choose the passive strategy are the key factorsaffecting governmentsrsquo behavior choice When governmentsactively promote the cost reduction of BIM or punishedmore by the superior supervision authority governmentswill choose the active promotion strategy and the behaviorgradually converges to the ESS point (1 1 1)

Second the cost of active adoption of BIM and thepunishment of passive adoption of BIM are the key factorsaffecting the choice of social capitalsrsquo behavior When thecost of active adoption of BIM is reduced or the penalty ofpassive adoption of BIM is increased the intention of socialcapitals to choose active adoption of BIM strategy is in-creased and the behavior of social capitals gradually con-verges to the ESS point (1 1 1)

Finally the cost of active application of BIM and thereward of active application of BIM are the key factorsinfluencing the choice of contractorsrsquo behavior When thecost of the active application of BIM technology is reducedor the penalty of the negative application of BIM technologyis increased the intention of the contractor to choose thestrategy of active application of BIM technology is increasedand the contractorrsquos behavior gradually converges to the ESSpoint (1 1 1)

+e adoption of BIM in PPP projects is a major reform ofthe traditional building mode whichrequires all stakeholders to change the traditionalthinking and actively cooperate In order to promoteBIM adoption among stakeholders of PPP projects thefollowing management implications are proposedImprove the supervision efficiency of superior supervisionauthority on governmentsrsquo promotion of BIM and furtherenrich the supervision means with the help of the media andthe public Increase the punishment for the governmentsrsquopassive promotion of BIM so as to improve the perceivedvalue of governments for regulatory punishment

Strengthen the punishment for the passive adoption ofBIM by social capitals and the reward for the active ap-plication of BIM by contractors Improve the cognition ofsocial capitals and contractors on BIM and other advancedinformation technology through publicity and training andgradually transform the system promotion into habitualadoption

Under the PPP mode the adoption of BIM can bepromoted by the governmentrsquos supervision but the gov-ernmentrsquos behavior also needs supervision +e public is the

users and beneficiaries of PPP projectsrsquo goods and the publicshould also have the right to know and to supervise theimplementation of the projects +erefore in the next stagethe authors will further research the role of public super-vision in the adoption of BIM in PPP projects in order toexplore the evolutionary mechanism and influencing factorsunder more stakeholders

Data Availability

Some or all data models or code generated or used tosupport the findings of this study are available from thecorresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interest

Acknowledgments

+is work was supported by the National Natural ScienceFoundation of China (NSFC) (71801119) in part by theSocial Science Planning Foundation of Liaoning(L18BGL020) and in part by the Basic Scientific ResearchProjects in Institutions of Higher Learning of LiaoningEducation Department (LJ2017QW009)

References

[1] N Carbonara and R Pellegrino ldquoRevenue guarantee inpublic-private partnerships a win-win modelrdquo ConstructionManagement amp Economics vol 36 no 10 pp 584ndash598 2018

[2] L Cheng ldquoDiscussion on the application of PPPmodel in newinfrastructure constructionrdquo Open Journal of Social Sciencesvol 7 no 9 pp 283ndash288 2019

[3] D Wu and S Q Wang ldquoResearch development and trend ofpublic private partnership in Chinardquo Journal of EngineeringManagement vol 28 no 6 pp 75ndash80 2014

[4] K Leidel and H W Alfen Lifecycle-Oriented Risk Manage-ment for PPP-Projects in Public Real Estate pp 587ndash604 EresLos Angeles CA USA 2009

[5] J Yuan X Li Y Ke W Xu Z Xu and M SkibnewskildquoDeveloping a building information modelingndashbased per-formance management system for publicndashprivate partner-shipsrdquo Engineering Construction and ArchitecturalManagement vol 27 no 8 pp 1727ndash1762 2020

[6] S Jayasuriya G Zhang and R J Yang ldquoExploring the impactof stakeholder management strategies on managing issues inPPP projectsrdquo International Journal of Construction Man-agement vol 20 no 6 pp 666ndash678 2020

[7] C X Jiang ldquoCauses and implications of the failure intransportation infrastructure public privatepartnershipsmdashmdasha study of 25 typical PPP casesrdquo Journal ofBeijing Jiaotong University vol 15 no 3 pp 50ndash58 2016

[8] R Osei-Kyei and A P C Chan ldquoReview of studies on thecritical success factors for public-private partnership(PPP) projects from 1990 to 2013rdquo International Journal ofProject Management vol 33 no 6 pp 1335ndash1346 2015

[9] B Zhang Y Wang X Liu et al ldquo+e configuration of PPPfailuremdashmdasha crisp-set qualitative comparative analysisrdquoJournal of Public Administration vol 12 no 04pp 65ndash81+191 2019

12 Complexity

[10] N Costantino and R Pellegrino ldquoPublicndashprivate partnershipimplementation cross-case analysisrdquo Proceedings of the In-stitution of Civil Engineers-management Procurement andLaw vol 168 no 4 pp 164ndash176 2015

[11] P Zhang K Zhan and Y Zhou ldquoGame model of three-partyshared risk allocation for PPP projectsmdashmdashbased on theperspective of incomplete informationrdquo IOP Conference Se-ries Earth and Environmental Science vol 295 no 4pp 42ndash49 2019

[12] N W Young S A Jones and H M Bernstein BuildingInformation Modeling Transforming Design and Constructionto Achieve Greater Industry Productivity McGraw HillConstruction New York NY USA 2008

[13] N D Aziz A H Nawawi and N R M Ariff ldquoBuildinginformation modelling (BIM) in facilities management op-portunities to be considered by facility managersrdquo Procedia-Social and Behavioral Sciences vol 234 pp 353ndash362 2016

[14] W Zou M Kumaraswamy and J Chung ldquoIdentifying thecritical success factors for relationship management in PPPprojectsrdquo International Journal of Project Managementvol 32 no 2 pp 265ndash274 2014

[15] QWang and TWang ldquoResearch on promotion of BIM basedon game analysis of stakeholdersrdquo Sichuan BuildingMaterialsvol 41 no 6 p 272 + 278 2015

[16] Q L Yin Evolutionary Game Analysis on BIM TechnologyDiffusion of Prefabricated Construction Qingdao Universityof Technology Qingdao China 2019

[17] H X Tang J M Cao H S Xu and C F Han ldquoAnalysis onBIM application cooperation behaviors in integrated facilitymanagement organization based on evolutionary game the-oryrdquo Industrial Engineering amp Management vol 25 no 4pp 1ndash8 2020

[18] J R Song ldquoResearch on incentive and countermeasures ofBIM application based on game theoryrdquo Value Engineeringvol 36 no 33 pp 43-44 2017

[19] C Jia Z H Wang and L Zhao ldquoEvolutionary game analysisand simulation for cooperative application of BIM technologyin PPPrdquo Journal of Liaoning Technical University vol 22no 4 pp 270ndash277 2020

[20] K A Tversky ldquoProspect theory an analysis of decision underriskrdquo Econometrica vol 47 no 2 pp 263ndash291 1979

[21] R L Gao and Q Bao ldquoSelection of government supervisionmode during the operational period in PPP projects based onevolutionary game theoryrdquo Operations Research and Man-agement Science vol 28 no 4 pp 155ndash162 2019

[22] S K He G W Liang and J B Meng ldquoMulti-subjectsbenefit game and behavior evolution mechanism of majorengineering based on prospect theoryrdquo Science andTechnology Management Research vol 40 no 5 pp 207ndash214 2020

[23] A Porwal and K N Hewage ldquoBuilding information modeling(BIM) partnering framework for public construction proj-ectsrdquo Automation in Construction vol 31 no 3 pp 204ndash2142013

[24] P E D Love J Liu J Matthews C-P Sing and J SmithldquoFuture proofing PPPs life-cycle performance measurementand building information modellingrdquo Automation in Con-struction vol 56 pp 26ndash35 2015

[25] Y L Yin Q J Liu and X Wang ldquoResearch on the whole lifecycle supervision system of public projectsmdashmdashbased on thecollaboration of VfM evaluation and BIM technologyrdquo ProjectManagement Technology vol 14 no 07 pp 17ndash20 2016

[26] S Q Wang R Tiong S Ting et al ldquoEvaluation and man-agement of political risks in Chinarsquos BOTprojectsrdquo Journal of

Construction Engineering and Management vol 126 no 3pp 242ndash250 2000

[27] Y M Chen and H F Chen ldquo+e risk management of PPPproject life cycle based on BIM technologyrdquo Project Man-agement Technology vol 15 no 5 pp 51ndash55 2017

[28] G Luo and X X Zhan ldquoResearch on the application of BIM inPPP project whole-life cyclerdquo Technology and EconomicGuide vol 26 no 5 p 168 2018

[29] B L Zhou and C L Lin ldquo+e application of BIM in PPPproject whole-life cyclerdquo Engineering Economy vol 29 no 3pp 51ndash54 2019

[30] J M Smith Evolution and the )eory of Games CambridgeUniversity Press Cambridge UK 1982

[31] J W Weibull Evolutionary Game )eory MIT PressCambridge MA USA 1997

[32] K Binmore ldquoModeling rational players part Irdquo Economicsand Philosophy vol 3 no 2 pp 179ndash214 1987

[33] K Binmore ldquoModeling rational players part IIrdquo Economicsand Philosophy vol 4 1988

[34] P D Taylor and L B Jonker ldquoEvolutionarily stable strategiesand game dynamicsrdquoMathematical Biosciences vol 40 no 1-2 pp 145ndash156 1978

[35] P Young ldquoConventional contractsrdquo )e Review of EconomicStudies vol 65 no 4 pp 773ndash792 1998

[36] D Fudenberg and C Harris ldquoEvolutionary dynamics withaggregate shocksrdquo Journal of Economic )eory vol 57 no 2pp 420ndash441 1992

[37] H Bester W Guth and W Guth ldquoIs altruism evolutionarilystablerdquo Journal of Economic Behavior amp Organization vol 34no 2 pp 193ndash209 1998

[38] K E Boulding R R Nelson and S G Winter ldquoAn evolu-tionary theory of economic changrdquo American Journal ofAgricultural Economics vol 66 no 4 p 535 1984

[39] S Y Xie Economic Game )eory Fudan University PressShanghai China 2002

[40] D Friedman and K C Fung ldquoInternational trade and theinternal organization of firms an evolutionary approachrdquoJournal of International Economics vol 41 no 1-2 pp 113ndash137 1996

[41] Y P Wang and W D Meng ldquoEvolutionary game analysis onco-competition mechanism of supply chainrdquo Journal of In-dustrial Engineering and Engineering Management vol 18no 2 pp 96ndash98 2004

[42] J Gao and Z H Shen ldquoElementary groping for evolutionarygame theory and its application in electricity marketrdquo Au-tomation of Electric Power Systems vol 27 no 18 pp 18ndash212003

[43] Y Zhang and M Guizani Game )eory for Wireless Com-munications and Networking CRC Press Boca Raton FLUSA 2011

[44] F Y Lu ldquoEvolutionary game analysis on environmentalpollution problemrdquo Systems Engineering-)eory amp Practicevol 27 no 9 pp 148ndash152 2007

[45] S L Dreisbach ldquoImpact of psychological research on moraltheory prospect theory and the doctrine of doing andallowingrdquo Dissertations amp )eses-Gradworks University ofCalifornia Santa Cruz CA USA 2012

[46] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Econometrica vol 47 no 2 1979

[47] N C Barberis ldquo+irty years of prospect theory in economicsa review and assessmentrdquo )e Journal of Economic Perspec-tives vol 27 no 1 pp 173ndash196 2013

Complexity 13

[48] S Benartzi and R H +aler ldquoMyopic loss aversion and theequity premium puzzlerdquo Quarterly Journal of Economicsvol 110 no 1 pp 73ndash92 1995

[49] N Barberis and M Huang ldquoStocks as lotteries the impli-cations of probability weighting for security pricesrdquo )eAmerican Economic Review vol 98 no 5 pp 2066ndash21002008

[50] A Frazzini ldquo+e disposition effect and underreaction tonewsrdquo )e Journal of Finance vol 61 no 4 pp 2017ndash20462006

[51] W Y Hu and J S Scott ldquoBehavioral obstacles in the annuitymarketrdquo Financial Analysts Journal vol 63 no 6 pp 71ndash822007

[52] B Koszegi andM Rabin ldquoReference-dependent consumptionplansrdquo )e American Economic Review vol 99 no 3pp 909ndash936 2009

[53] P Vamvakas E E Tsiropoulou and S Papavassiliou ldquoDy-namic spectrum management in 5G wireless networks a real-life modeling approachrdquo in Proceedings of the IEEE INFO-COM 2019-IEEE Conference on Computer CommunicationsIEEE Paris France April 2019

[54] Y N Zhou and J C Liu ldquoStudy on supervision strategy of pppproject considering the participation of higher-level gov-ernmentrdquo Chinese Journal of Management Science vol 1ndash162021 httpsdoiorg1016381jcnkiissn1003-207x20200801

[55] H Q Zhang J C Wang and H L Tao ldquoResearch the de-cision-making behavior of investors based on prospect theoryin the ppp projectrdquo Soft Science vol 32 no 8 pp 129ndash1332018

[56] D Friedman ldquoEvolutionary games in economicsrdquo Econo-metrica vol 59 no 3 pp 637ndash666 1991

[57] Z B Zhao and Q P Man ldquoEvolutionary game analysis of riskmanagement behavior of major infrastructure projects basedon projects based on prospect theoryrdquo Journal of SystemsManagement vol 27 no 1 pp 112ndash120 2018

14 Complexity

face of gains but tend to prefer risk preference when facinglosses [57] In the practice of PPP projects governmentsoften think that their behavior will not be easily detected bythe superior supervision authority due to the commonexistence of fluke mentality +erefore governments tend tounderestimate the probability α and the value of punishmentLg1 by the superior supervision authority which means thatthe right part of the conditional formula Cg + Splt αLg1 willbe underestimated in practice On the other hand whengovernments choose the active strategy it will definitely paythe promotion cost while choosing the passive strategy willonly have a certain probability of punitive losses Accordingto the prospect theory governments will be more inclined torisk preference that is they prefer to take risks and choosethe passive strategy rather than to bear the deterministic

expenditure +erefore the above reasons will cause certaindifficulties for the establishment of the formulaCg + Splt αLg1

4 Numerical Simulation and Analysis

In order to better analyze the behavior evolution mechanismamong governments social capitals and contractors in BIMadoption this study uses MATLAB R2018a to simulate thetrend and influencing factors of the evolutionary game +evalues of parameters are randomly assigned under the as-sumption which do not represent the real benefits ofstakeholders in PPP projects

According to the above analysis if the system convergesto the ESS point (1 1 1) the following conditions need to be

111

0908070605

z

108

06y04 04

0608

x

1

a=05a=08a=1

(a)

a=05a=08a=1

11

1

09

08

07

06

05

z

11 1 09 08 07 06 05x

(b)

Figure 5 Influence of α change on the evolutionary game process

Lg1=80Lg1=100Lg1=120

111

0908070605

z

108

06y04 04

0608

x

1

(a)

Lg1=80Lg1=100Lg1=120

11

1

09

08

07

06

05

z

11 1 09 08 07 06 05x

(b)

Figure 6 Influence of Lg1 change on the evolutionary game process

Complexity 9

satisfied Cg + Splt αLg1 Cp1ltCp2 + Lp and CcltRc + Sc+e initial values of each parameter are set to x 05 y

05 α 05Cg 20 Sp 10 Lgl 80Cp1 30Cp2

20 Sp 10 Lg1 80Cp1 30Cp2 20 Lp 20Cc

30Rc 40 Sc 10 Lg 10 Lc 10 Lc1 10 LC2 5

41 Influence of Cg1 and Cg2 Changes on the EvolutionaryGame Process Keep the other parameters unchanged andthe influence on the evolution trend of the system is ob-served by changing the values of Cg α and Lg1 as shown inFigures 4(a) 5(a) and 6(a) In Figure 4(b) we can see thatthe convergence rate of x to the ESS point (1 1 1) isaccelerated as the value of Cg goes down +e simulationresults show that reducing the cost of governmentsrsquo strategyof ldquopositive promotionrdquo is helpful to enhance the willingness

of governments to promote the adoption of BIM in PPPprojects In Figures 5(b) and 6(b) we can see that theconvergence rate of x to the ESS point (1 1 1) is acceleratedas the value of α and Lg1 go bigger +e simulation resultsshow that increasing the supervision probability and theseverity of punishment will help increase the willingness ofgovernments to promote BIM application in PPP projects

42 Influence of Cp1 and Lp Changes on the EvolutionaryGame Process Keep the other parameters unchanged andthe influence on the evolution trend of the system is ob-served by changing the values of Cp1 and Lp as shown inFigures 7(a) and 8(a) In Figure 7(b) we can see that theconvergence rate of y to the ESS point (1 1 1) is acceleratedas the value of Cp1 goes down +e simulation results show

111

0908070605

z

108

06y04 05 06 07 08

x

091

Cp1=30Cp1=20Cp1=10

(a)

Cp1=30Cp1=20Cp1=10

11

1

09

08

07

06

05

z

11 1 09 08 07 06 05y

(b)

Figure 7 Influence of Cp1 change on the evolutionary game process

111

0908070605

z

108

06y04 05

06 07 x08 09 1

Lp=20Lp=30Lp=40

(a)

Lp=20Lp=30Lp=40

11

1

09

08

07

06

05

z

11 1 09 08 07 06 05y

(b)

Figure 8 Influence of Lp change on the evolutionary game process

10 Complexity

that reducing the cost of social capitalsrsquo strategy of ldquopositiveadoptionrdquo is helpful to enhance the willingness of socialcapitals to adopt BIM in PPP projects In Figure 8(b) we cansee that the convergence rate of y to the ESS point (1 1 1) isaccelerated as the value of Lp goes bigger +e simulationresults show that increasing the punishment of social cap-italsrsquo strategy of ldquopassive adoptionrdquo is helpful to enhance thewillingness of social capitals to adopt BIM in PPP projects

43 Influence of Cc and Sc Changes on the Evolutionary GameProcess Keep the other parameters unchanged and theinfluence on the evolution trend of the system is observed bychanging the values of Cc and Sc as shown in Figures 9(a)and 10(a) In Figure 9(b) we can see that the convergence

rate of z to the ESS point (1 1 1) is accelerated as the value ofCc goes down+e simulation results show that reducing thecost of contractorsrsquo strategy of ldquopositive applicationrdquo ishelpful to enhance the willingness of contractors to applyBIM in PPP projects In Figure 10(b) we can see that theconvergence rate of z to the ESS point (1 1 1) is acceleratedas the value of Sc goes bigger +e simulation results showthat increasing the punishment of contractorsrsquo strategy ofldquopassive applicationrdquo is helpful to enhance the willingness ofcontractors to apply BIM in PPP projects

5 Conclusions and Management Implications

Under the assumption of bounded rationality this studyconstructs a tripartite evolutionary game model including

111

0908070605

z

108

06y04 05 06 07 08

x

091

Cc=30Cc=20Cc=10

(a)

Cc=30Cc=20Cc=10

11

1

09

08

07

06

05

z

1 09095 08085 07075 06065 05055y

(b)

Figure 9 Influence of Cc change on the evolutionary game process

111

0908070605

z

108

06y04 05 06 07 x

08 091

Sc=20Sc=30Sc=40

(a)

Sc=20Sc=30Sc=40

11

1

09

08

07

06

05

z

1 09095 08085 07075 06065 05055y

(b)

Figure 10 Influence of Sc change on the evolutionary game process

Complexity 11

governments social capitals and contractors +e satisfac-tion conditions that the system converging to the ESS point(1 1 1) are analyzed based on prospect theory and the mainfactors influencing stakeholdersrsquo behavior decision-makingare simulated by MATLAB +e research results will help toclarify the behavior evolution mechanism and the influ-encing factors of the ESS for adopting BIM among stake-holders in PPP projects so as to improve the cross-organization information collaboration efficiency of PPPprojects and promote the informatization of the construc-tion industry +e primary conclusions and managementimplications are as follows

First the cost of BIM promotion by governments and thepunishment from the superior supervision authority whengovernments choose the passive strategy are the key factorsaffecting governmentsrsquo behavior choice When governmentsactively promote the cost reduction of BIM or punishedmore by the superior supervision authority governmentswill choose the active promotion strategy and the behaviorgradually converges to the ESS point (1 1 1)

Second the cost of active adoption of BIM and thepunishment of passive adoption of BIM are the key factorsaffecting the choice of social capitalsrsquo behavior When thecost of active adoption of BIM is reduced or the penalty ofpassive adoption of BIM is increased the intention of socialcapitals to choose active adoption of BIM strategy is in-creased and the behavior of social capitals gradually con-verges to the ESS point (1 1 1)

Finally the cost of active application of BIM and thereward of active application of BIM are the key factorsinfluencing the choice of contractorsrsquo behavior When thecost of the active application of BIM technology is reducedor the penalty of the negative application of BIM technologyis increased the intention of the contractor to choose thestrategy of active application of BIM technology is increasedand the contractorrsquos behavior gradually converges to the ESSpoint (1 1 1)

+e adoption of BIM in PPP projects is a major reform ofthe traditional building mode whichrequires all stakeholders to change the traditionalthinking and actively cooperate In order to promoteBIM adoption among stakeholders of PPP projects thefollowing management implications are proposedImprove the supervision efficiency of superior supervisionauthority on governmentsrsquo promotion of BIM and furtherenrich the supervision means with the help of the media andthe public Increase the punishment for the governmentsrsquopassive promotion of BIM so as to improve the perceivedvalue of governments for regulatory punishment

Strengthen the punishment for the passive adoption ofBIM by social capitals and the reward for the active ap-plication of BIM by contractors Improve the cognition ofsocial capitals and contractors on BIM and other advancedinformation technology through publicity and training andgradually transform the system promotion into habitualadoption

Under the PPP mode the adoption of BIM can bepromoted by the governmentrsquos supervision but the gov-ernmentrsquos behavior also needs supervision +e public is the

users and beneficiaries of PPP projectsrsquo goods and the publicshould also have the right to know and to supervise theimplementation of the projects +erefore in the next stagethe authors will further research the role of public super-vision in the adoption of BIM in PPP projects in order toexplore the evolutionary mechanism and influencing factorsunder more stakeholders

Data Availability

Some or all data models or code generated or used tosupport the findings of this study are available from thecorresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interest

Acknowledgments

+is work was supported by the National Natural ScienceFoundation of China (NSFC) (71801119) in part by theSocial Science Planning Foundation of Liaoning(L18BGL020) and in part by the Basic Scientific ResearchProjects in Institutions of Higher Learning of LiaoningEducation Department (LJ2017QW009)

References

[1] N Carbonara and R Pellegrino ldquoRevenue guarantee inpublic-private partnerships a win-win modelrdquo ConstructionManagement amp Economics vol 36 no 10 pp 584ndash598 2018

[2] L Cheng ldquoDiscussion on the application of PPPmodel in newinfrastructure constructionrdquo Open Journal of Social Sciencesvol 7 no 9 pp 283ndash288 2019

[3] D Wu and S Q Wang ldquoResearch development and trend ofpublic private partnership in Chinardquo Journal of EngineeringManagement vol 28 no 6 pp 75ndash80 2014

[4] K Leidel and H W Alfen Lifecycle-Oriented Risk Manage-ment for PPP-Projects in Public Real Estate pp 587ndash604 EresLos Angeles CA USA 2009

[5] J Yuan X Li Y Ke W Xu Z Xu and M SkibnewskildquoDeveloping a building information modelingndashbased per-formance management system for publicndashprivate partner-shipsrdquo Engineering Construction and ArchitecturalManagement vol 27 no 8 pp 1727ndash1762 2020

[6] S Jayasuriya G Zhang and R J Yang ldquoExploring the impactof stakeholder management strategies on managing issues inPPP projectsrdquo International Journal of Construction Man-agement vol 20 no 6 pp 666ndash678 2020

[7] C X Jiang ldquoCauses and implications of the failure intransportation infrastructure public privatepartnershipsmdashmdasha study of 25 typical PPP casesrdquo Journal ofBeijing Jiaotong University vol 15 no 3 pp 50ndash58 2016

[8] R Osei-Kyei and A P C Chan ldquoReview of studies on thecritical success factors for public-private partnership(PPP) projects from 1990 to 2013rdquo International Journal ofProject Management vol 33 no 6 pp 1335ndash1346 2015

[9] B Zhang Y Wang X Liu et al ldquo+e configuration of PPPfailuremdashmdasha crisp-set qualitative comparative analysisrdquoJournal of Public Administration vol 12 no 04pp 65ndash81+191 2019

12 Complexity

[10] N Costantino and R Pellegrino ldquoPublicndashprivate partnershipimplementation cross-case analysisrdquo Proceedings of the In-stitution of Civil Engineers-management Procurement andLaw vol 168 no 4 pp 164ndash176 2015

[11] P Zhang K Zhan and Y Zhou ldquoGame model of three-partyshared risk allocation for PPP projectsmdashmdashbased on theperspective of incomplete informationrdquo IOP Conference Se-ries Earth and Environmental Science vol 295 no 4pp 42ndash49 2019

[12] N W Young S A Jones and H M Bernstein BuildingInformation Modeling Transforming Design and Constructionto Achieve Greater Industry Productivity McGraw HillConstruction New York NY USA 2008

[13] N D Aziz A H Nawawi and N R M Ariff ldquoBuildinginformation modelling (BIM) in facilities management op-portunities to be considered by facility managersrdquo Procedia-Social and Behavioral Sciences vol 234 pp 353ndash362 2016

[14] W Zou M Kumaraswamy and J Chung ldquoIdentifying thecritical success factors for relationship management in PPPprojectsrdquo International Journal of Project Managementvol 32 no 2 pp 265ndash274 2014

[15] QWang and TWang ldquoResearch on promotion of BIM basedon game analysis of stakeholdersrdquo Sichuan BuildingMaterialsvol 41 no 6 p 272 + 278 2015

[16] Q L Yin Evolutionary Game Analysis on BIM TechnologyDiffusion of Prefabricated Construction Qingdao Universityof Technology Qingdao China 2019

[17] H X Tang J M Cao H S Xu and C F Han ldquoAnalysis onBIM application cooperation behaviors in integrated facilitymanagement organization based on evolutionary game the-oryrdquo Industrial Engineering amp Management vol 25 no 4pp 1ndash8 2020

[18] J R Song ldquoResearch on incentive and countermeasures ofBIM application based on game theoryrdquo Value Engineeringvol 36 no 33 pp 43-44 2017

[19] C Jia Z H Wang and L Zhao ldquoEvolutionary game analysisand simulation for cooperative application of BIM technologyin PPPrdquo Journal of Liaoning Technical University vol 22no 4 pp 270ndash277 2020

[20] K A Tversky ldquoProspect theory an analysis of decision underriskrdquo Econometrica vol 47 no 2 pp 263ndash291 1979

[21] R L Gao and Q Bao ldquoSelection of government supervisionmode during the operational period in PPP projects based onevolutionary game theoryrdquo Operations Research and Man-agement Science vol 28 no 4 pp 155ndash162 2019

[22] S K He G W Liang and J B Meng ldquoMulti-subjectsbenefit game and behavior evolution mechanism of majorengineering based on prospect theoryrdquo Science andTechnology Management Research vol 40 no 5 pp 207ndash214 2020

[23] A Porwal and K N Hewage ldquoBuilding information modeling(BIM) partnering framework for public construction proj-ectsrdquo Automation in Construction vol 31 no 3 pp 204ndash2142013

[24] P E D Love J Liu J Matthews C-P Sing and J SmithldquoFuture proofing PPPs life-cycle performance measurementand building information modellingrdquo Automation in Con-struction vol 56 pp 26ndash35 2015

[25] Y L Yin Q J Liu and X Wang ldquoResearch on the whole lifecycle supervision system of public projectsmdashmdashbased on thecollaboration of VfM evaluation and BIM technologyrdquo ProjectManagement Technology vol 14 no 07 pp 17ndash20 2016

[26] S Q Wang R Tiong S Ting et al ldquoEvaluation and man-agement of political risks in Chinarsquos BOTprojectsrdquo Journal of

Construction Engineering and Management vol 126 no 3pp 242ndash250 2000

[27] Y M Chen and H F Chen ldquo+e risk management of PPPproject life cycle based on BIM technologyrdquo Project Man-agement Technology vol 15 no 5 pp 51ndash55 2017

[28] G Luo and X X Zhan ldquoResearch on the application of BIM inPPP project whole-life cyclerdquo Technology and EconomicGuide vol 26 no 5 p 168 2018

[29] B L Zhou and C L Lin ldquo+e application of BIM in PPPproject whole-life cyclerdquo Engineering Economy vol 29 no 3pp 51ndash54 2019

[30] J M Smith Evolution and the )eory of Games CambridgeUniversity Press Cambridge UK 1982

[31] J W Weibull Evolutionary Game )eory MIT PressCambridge MA USA 1997

[32] K Binmore ldquoModeling rational players part Irdquo Economicsand Philosophy vol 3 no 2 pp 179ndash214 1987

[33] K Binmore ldquoModeling rational players part IIrdquo Economicsand Philosophy vol 4 1988

[34] P D Taylor and L B Jonker ldquoEvolutionarily stable strategiesand game dynamicsrdquoMathematical Biosciences vol 40 no 1-2 pp 145ndash156 1978

[35] P Young ldquoConventional contractsrdquo )e Review of EconomicStudies vol 65 no 4 pp 773ndash792 1998

[36] D Fudenberg and C Harris ldquoEvolutionary dynamics withaggregate shocksrdquo Journal of Economic )eory vol 57 no 2pp 420ndash441 1992

[37] H Bester W Guth and W Guth ldquoIs altruism evolutionarilystablerdquo Journal of Economic Behavior amp Organization vol 34no 2 pp 193ndash209 1998

[38] K E Boulding R R Nelson and S G Winter ldquoAn evolu-tionary theory of economic changrdquo American Journal ofAgricultural Economics vol 66 no 4 p 535 1984

[39] S Y Xie Economic Game )eory Fudan University PressShanghai China 2002

[40] D Friedman and K C Fung ldquoInternational trade and theinternal organization of firms an evolutionary approachrdquoJournal of International Economics vol 41 no 1-2 pp 113ndash137 1996

[41] Y P Wang and W D Meng ldquoEvolutionary game analysis onco-competition mechanism of supply chainrdquo Journal of In-dustrial Engineering and Engineering Management vol 18no 2 pp 96ndash98 2004

[42] J Gao and Z H Shen ldquoElementary groping for evolutionarygame theory and its application in electricity marketrdquo Au-tomation of Electric Power Systems vol 27 no 18 pp 18ndash212003

[43] Y Zhang and M Guizani Game )eory for Wireless Com-munications and Networking CRC Press Boca Raton FLUSA 2011

[44] F Y Lu ldquoEvolutionary game analysis on environmentalpollution problemrdquo Systems Engineering-)eory amp Practicevol 27 no 9 pp 148ndash152 2007

[45] S L Dreisbach ldquoImpact of psychological research on moraltheory prospect theory and the doctrine of doing andallowingrdquo Dissertations amp )eses-Gradworks University ofCalifornia Santa Cruz CA USA 2012

[46] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Econometrica vol 47 no 2 1979

[47] N C Barberis ldquo+irty years of prospect theory in economicsa review and assessmentrdquo )e Journal of Economic Perspec-tives vol 27 no 1 pp 173ndash196 2013

Complexity 13

[48] S Benartzi and R H +aler ldquoMyopic loss aversion and theequity premium puzzlerdquo Quarterly Journal of Economicsvol 110 no 1 pp 73ndash92 1995

[49] N Barberis and M Huang ldquoStocks as lotteries the impli-cations of probability weighting for security pricesrdquo )eAmerican Economic Review vol 98 no 5 pp 2066ndash21002008

[50] A Frazzini ldquo+e disposition effect and underreaction tonewsrdquo )e Journal of Finance vol 61 no 4 pp 2017ndash20462006

[51] W Y Hu and J S Scott ldquoBehavioral obstacles in the annuitymarketrdquo Financial Analysts Journal vol 63 no 6 pp 71ndash822007

[52] B Koszegi andM Rabin ldquoReference-dependent consumptionplansrdquo )e American Economic Review vol 99 no 3pp 909ndash936 2009

[53] P Vamvakas E E Tsiropoulou and S Papavassiliou ldquoDy-namic spectrum management in 5G wireless networks a real-life modeling approachrdquo in Proceedings of the IEEE INFO-COM 2019-IEEE Conference on Computer CommunicationsIEEE Paris France April 2019

[54] Y N Zhou and J C Liu ldquoStudy on supervision strategy of pppproject considering the participation of higher-level gov-ernmentrdquo Chinese Journal of Management Science vol 1ndash162021 httpsdoiorg1016381jcnkiissn1003-207x20200801

[55] H Q Zhang J C Wang and H L Tao ldquoResearch the de-cision-making behavior of investors based on prospect theoryin the ppp projectrdquo Soft Science vol 32 no 8 pp 129ndash1332018

[56] D Friedman ldquoEvolutionary games in economicsrdquo Econo-metrica vol 59 no 3 pp 637ndash666 1991

[57] Z B Zhao and Q P Man ldquoEvolutionary game analysis of riskmanagement behavior of major infrastructure projects basedon projects based on prospect theoryrdquo Journal of SystemsManagement vol 27 no 1 pp 112ndash120 2018

14 Complexity

satisfied Cg + Splt αLg1 Cp1ltCp2 + Lp and CcltRc + Sc+e initial values of each parameter are set to x 05 y

05 α 05Cg 20 Sp 10 Lgl 80Cp1 30Cp2

20 Sp 10 Lg1 80Cp1 30Cp2 20 Lp 20Cc

30Rc 40 Sc 10 Lg 10 Lc 10 Lc1 10 LC2 5

41 Influence of Cg1 and Cg2 Changes on the EvolutionaryGame Process Keep the other parameters unchanged andthe influence on the evolution trend of the system is ob-served by changing the values of Cg α and Lg1 as shown inFigures 4(a) 5(a) and 6(a) In Figure 4(b) we can see thatthe convergence rate of x to the ESS point (1 1 1) isaccelerated as the value of Cg goes down +e simulationresults show that reducing the cost of governmentsrsquo strategyof ldquopositive promotionrdquo is helpful to enhance the willingness

of governments to promote the adoption of BIM in PPPprojects In Figures 5(b) and 6(b) we can see that theconvergence rate of x to the ESS point (1 1 1) is acceleratedas the value of α and Lg1 go bigger +e simulation resultsshow that increasing the supervision probability and theseverity of punishment will help increase the willingness ofgovernments to promote BIM application in PPP projects

42 Influence of Cp1 and Lp Changes on the EvolutionaryGame Process Keep the other parameters unchanged andthe influence on the evolution trend of the system is ob-served by changing the values of Cp1 and Lp as shown inFigures 7(a) and 8(a) In Figure 7(b) we can see that theconvergence rate of y to the ESS point (1 1 1) is acceleratedas the value of Cp1 goes down +e simulation results show

111

0908070605

z

108

06y04 05 06 07 08

x

091

Cp1=30Cp1=20Cp1=10

(a)

Cp1=30Cp1=20Cp1=10

11

1

09

08

07

06

05

z

11 1 09 08 07 06 05y

(b)

Figure 7 Influence of Cp1 change on the evolutionary game process

111

0908070605

z

108

06y04 05

06 07 x08 09 1

Lp=20Lp=30Lp=40

(a)

Lp=20Lp=30Lp=40

11

1

09

08

07

06

05

z

11 1 09 08 07 06 05y

(b)

Figure 8 Influence of Lp change on the evolutionary game process

10 Complexity

that reducing the cost of social capitalsrsquo strategy of ldquopositiveadoptionrdquo is helpful to enhance the willingness of socialcapitals to adopt BIM in PPP projects In Figure 8(b) we cansee that the convergence rate of y to the ESS point (1 1 1) isaccelerated as the value of Lp goes bigger +e simulationresults show that increasing the punishment of social cap-italsrsquo strategy of ldquopassive adoptionrdquo is helpful to enhance thewillingness of social capitals to adopt BIM in PPP projects

43 Influence of Cc and Sc Changes on the Evolutionary GameProcess Keep the other parameters unchanged and theinfluence on the evolution trend of the system is observed bychanging the values of Cc and Sc as shown in Figures 9(a)and 10(a) In Figure 9(b) we can see that the convergence

rate of z to the ESS point (1 1 1) is accelerated as the value ofCc goes down+e simulation results show that reducing thecost of contractorsrsquo strategy of ldquopositive applicationrdquo ishelpful to enhance the willingness of contractors to applyBIM in PPP projects In Figure 10(b) we can see that theconvergence rate of z to the ESS point (1 1 1) is acceleratedas the value of Sc goes bigger +e simulation results showthat increasing the punishment of contractorsrsquo strategy ofldquopassive applicationrdquo is helpful to enhance the willingness ofcontractors to apply BIM in PPP projects

5 Conclusions and Management Implications

Under the assumption of bounded rationality this studyconstructs a tripartite evolutionary game model including

111

0908070605

z

108

06y04 05 06 07 08

x

091

Cc=30Cc=20Cc=10

(a)

Cc=30Cc=20Cc=10

11

1

09

08

07

06

05

z

1 09095 08085 07075 06065 05055y

(b)

Figure 9 Influence of Cc change on the evolutionary game process

111

0908070605

z

108

06y04 05 06 07 x

08 091

Sc=20Sc=30Sc=40

(a)

Sc=20Sc=30Sc=40

11

1

09

08

07

06

05

z

1 09095 08085 07075 06065 05055y

(b)

Figure 10 Influence of Sc change on the evolutionary game process

Complexity 11

governments social capitals and contractors +e satisfac-tion conditions that the system converging to the ESS point(1 1 1) are analyzed based on prospect theory and the mainfactors influencing stakeholdersrsquo behavior decision-makingare simulated by MATLAB +e research results will help toclarify the behavior evolution mechanism and the influ-encing factors of the ESS for adopting BIM among stake-holders in PPP projects so as to improve the cross-organization information collaboration efficiency of PPPprojects and promote the informatization of the construc-tion industry +e primary conclusions and managementimplications are as follows

First the cost of BIM promotion by governments and thepunishment from the superior supervision authority whengovernments choose the passive strategy are the key factorsaffecting governmentsrsquo behavior choice When governmentsactively promote the cost reduction of BIM or punishedmore by the superior supervision authority governmentswill choose the active promotion strategy and the behaviorgradually converges to the ESS point (1 1 1)

Second the cost of active adoption of BIM and thepunishment of passive adoption of BIM are the key factorsaffecting the choice of social capitalsrsquo behavior When thecost of active adoption of BIM is reduced or the penalty ofpassive adoption of BIM is increased the intention of socialcapitals to choose active adoption of BIM strategy is in-creased and the behavior of social capitals gradually con-verges to the ESS point (1 1 1)

Finally the cost of active application of BIM and thereward of active application of BIM are the key factorsinfluencing the choice of contractorsrsquo behavior When thecost of the active application of BIM technology is reducedor the penalty of the negative application of BIM technologyis increased the intention of the contractor to choose thestrategy of active application of BIM technology is increasedand the contractorrsquos behavior gradually converges to the ESSpoint (1 1 1)

+e adoption of BIM in PPP projects is a major reform ofthe traditional building mode whichrequires all stakeholders to change the traditionalthinking and actively cooperate In order to promoteBIM adoption among stakeholders of PPP projects thefollowing management implications are proposedImprove the supervision efficiency of superior supervisionauthority on governmentsrsquo promotion of BIM and furtherenrich the supervision means with the help of the media andthe public Increase the punishment for the governmentsrsquopassive promotion of BIM so as to improve the perceivedvalue of governments for regulatory punishment

Strengthen the punishment for the passive adoption ofBIM by social capitals and the reward for the active ap-plication of BIM by contractors Improve the cognition ofsocial capitals and contractors on BIM and other advancedinformation technology through publicity and training andgradually transform the system promotion into habitualadoption

Under the PPP mode the adoption of BIM can bepromoted by the governmentrsquos supervision but the gov-ernmentrsquos behavior also needs supervision +e public is the

users and beneficiaries of PPP projectsrsquo goods and the publicshould also have the right to know and to supervise theimplementation of the projects +erefore in the next stagethe authors will further research the role of public super-vision in the adoption of BIM in PPP projects in order toexplore the evolutionary mechanism and influencing factorsunder more stakeholders

Data Availability

Some or all data models or code generated or used tosupport the findings of this study are available from thecorresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interest

Acknowledgments

+is work was supported by the National Natural ScienceFoundation of China (NSFC) (71801119) in part by theSocial Science Planning Foundation of Liaoning(L18BGL020) and in part by the Basic Scientific ResearchProjects in Institutions of Higher Learning of LiaoningEducation Department (LJ2017QW009)

References

[1] N Carbonara and R Pellegrino ldquoRevenue guarantee inpublic-private partnerships a win-win modelrdquo ConstructionManagement amp Economics vol 36 no 10 pp 584ndash598 2018

[2] L Cheng ldquoDiscussion on the application of PPPmodel in newinfrastructure constructionrdquo Open Journal of Social Sciencesvol 7 no 9 pp 283ndash288 2019

[3] D Wu and S Q Wang ldquoResearch development and trend ofpublic private partnership in Chinardquo Journal of EngineeringManagement vol 28 no 6 pp 75ndash80 2014

[4] K Leidel and H W Alfen Lifecycle-Oriented Risk Manage-ment for PPP-Projects in Public Real Estate pp 587ndash604 EresLos Angeles CA USA 2009

[5] J Yuan X Li Y Ke W Xu Z Xu and M SkibnewskildquoDeveloping a building information modelingndashbased per-formance management system for publicndashprivate partner-shipsrdquo Engineering Construction and ArchitecturalManagement vol 27 no 8 pp 1727ndash1762 2020

[6] S Jayasuriya G Zhang and R J Yang ldquoExploring the impactof stakeholder management strategies on managing issues inPPP projectsrdquo International Journal of Construction Man-agement vol 20 no 6 pp 666ndash678 2020

[7] C X Jiang ldquoCauses and implications of the failure intransportation infrastructure public privatepartnershipsmdashmdasha study of 25 typical PPP casesrdquo Journal ofBeijing Jiaotong University vol 15 no 3 pp 50ndash58 2016

[8] R Osei-Kyei and A P C Chan ldquoReview of studies on thecritical success factors for public-private partnership(PPP) projects from 1990 to 2013rdquo International Journal ofProject Management vol 33 no 6 pp 1335ndash1346 2015

[9] B Zhang Y Wang X Liu et al ldquo+e configuration of PPPfailuremdashmdasha crisp-set qualitative comparative analysisrdquoJournal of Public Administration vol 12 no 04pp 65ndash81+191 2019

12 Complexity

[10] N Costantino and R Pellegrino ldquoPublicndashprivate partnershipimplementation cross-case analysisrdquo Proceedings of the In-stitution of Civil Engineers-management Procurement andLaw vol 168 no 4 pp 164ndash176 2015

[11] P Zhang K Zhan and Y Zhou ldquoGame model of three-partyshared risk allocation for PPP projectsmdashmdashbased on theperspective of incomplete informationrdquo IOP Conference Se-ries Earth and Environmental Science vol 295 no 4pp 42ndash49 2019

[12] N W Young S A Jones and H M Bernstein BuildingInformation Modeling Transforming Design and Constructionto Achieve Greater Industry Productivity McGraw HillConstruction New York NY USA 2008

[13] N D Aziz A H Nawawi and N R M Ariff ldquoBuildinginformation modelling (BIM) in facilities management op-portunities to be considered by facility managersrdquo Procedia-Social and Behavioral Sciences vol 234 pp 353ndash362 2016

[14] W Zou M Kumaraswamy and J Chung ldquoIdentifying thecritical success factors for relationship management in PPPprojectsrdquo International Journal of Project Managementvol 32 no 2 pp 265ndash274 2014

[15] QWang and TWang ldquoResearch on promotion of BIM basedon game analysis of stakeholdersrdquo Sichuan BuildingMaterialsvol 41 no 6 p 272 + 278 2015

[16] Q L Yin Evolutionary Game Analysis on BIM TechnologyDiffusion of Prefabricated Construction Qingdao Universityof Technology Qingdao China 2019

[17] H X Tang J M Cao H S Xu and C F Han ldquoAnalysis onBIM application cooperation behaviors in integrated facilitymanagement organization based on evolutionary game the-oryrdquo Industrial Engineering amp Management vol 25 no 4pp 1ndash8 2020

[18] J R Song ldquoResearch on incentive and countermeasures ofBIM application based on game theoryrdquo Value Engineeringvol 36 no 33 pp 43-44 2017

[19] C Jia Z H Wang and L Zhao ldquoEvolutionary game analysisand simulation for cooperative application of BIM technologyin PPPrdquo Journal of Liaoning Technical University vol 22no 4 pp 270ndash277 2020

[20] K A Tversky ldquoProspect theory an analysis of decision underriskrdquo Econometrica vol 47 no 2 pp 263ndash291 1979

[21] R L Gao and Q Bao ldquoSelection of government supervisionmode during the operational period in PPP projects based onevolutionary game theoryrdquo Operations Research and Man-agement Science vol 28 no 4 pp 155ndash162 2019

[22] S K He G W Liang and J B Meng ldquoMulti-subjectsbenefit game and behavior evolution mechanism of majorengineering based on prospect theoryrdquo Science andTechnology Management Research vol 40 no 5 pp 207ndash214 2020

[23] A Porwal and K N Hewage ldquoBuilding information modeling(BIM) partnering framework for public construction proj-ectsrdquo Automation in Construction vol 31 no 3 pp 204ndash2142013

[24] P E D Love J Liu J Matthews C-P Sing and J SmithldquoFuture proofing PPPs life-cycle performance measurementand building information modellingrdquo Automation in Con-struction vol 56 pp 26ndash35 2015

[25] Y L Yin Q J Liu and X Wang ldquoResearch on the whole lifecycle supervision system of public projectsmdashmdashbased on thecollaboration of VfM evaluation and BIM technologyrdquo ProjectManagement Technology vol 14 no 07 pp 17ndash20 2016

[26] S Q Wang R Tiong S Ting et al ldquoEvaluation and man-agement of political risks in Chinarsquos BOTprojectsrdquo Journal of

Construction Engineering and Management vol 126 no 3pp 242ndash250 2000

[27] Y M Chen and H F Chen ldquo+e risk management of PPPproject life cycle based on BIM technologyrdquo Project Man-agement Technology vol 15 no 5 pp 51ndash55 2017

[28] G Luo and X X Zhan ldquoResearch on the application of BIM inPPP project whole-life cyclerdquo Technology and EconomicGuide vol 26 no 5 p 168 2018

[29] B L Zhou and C L Lin ldquo+e application of BIM in PPPproject whole-life cyclerdquo Engineering Economy vol 29 no 3pp 51ndash54 2019

[30] J M Smith Evolution and the )eory of Games CambridgeUniversity Press Cambridge UK 1982

[31] J W Weibull Evolutionary Game )eory MIT PressCambridge MA USA 1997

[32] K Binmore ldquoModeling rational players part Irdquo Economicsand Philosophy vol 3 no 2 pp 179ndash214 1987

[33] K Binmore ldquoModeling rational players part IIrdquo Economicsand Philosophy vol 4 1988

[34] P D Taylor and L B Jonker ldquoEvolutionarily stable strategiesand game dynamicsrdquoMathematical Biosciences vol 40 no 1-2 pp 145ndash156 1978

[35] P Young ldquoConventional contractsrdquo )e Review of EconomicStudies vol 65 no 4 pp 773ndash792 1998

[36] D Fudenberg and C Harris ldquoEvolutionary dynamics withaggregate shocksrdquo Journal of Economic )eory vol 57 no 2pp 420ndash441 1992

[37] H Bester W Guth and W Guth ldquoIs altruism evolutionarilystablerdquo Journal of Economic Behavior amp Organization vol 34no 2 pp 193ndash209 1998

[38] K E Boulding R R Nelson and S G Winter ldquoAn evolu-tionary theory of economic changrdquo American Journal ofAgricultural Economics vol 66 no 4 p 535 1984

[39] S Y Xie Economic Game )eory Fudan University PressShanghai China 2002

[40] D Friedman and K C Fung ldquoInternational trade and theinternal organization of firms an evolutionary approachrdquoJournal of International Economics vol 41 no 1-2 pp 113ndash137 1996

[41] Y P Wang and W D Meng ldquoEvolutionary game analysis onco-competition mechanism of supply chainrdquo Journal of In-dustrial Engineering and Engineering Management vol 18no 2 pp 96ndash98 2004

[42] J Gao and Z H Shen ldquoElementary groping for evolutionarygame theory and its application in electricity marketrdquo Au-tomation of Electric Power Systems vol 27 no 18 pp 18ndash212003

[43] Y Zhang and M Guizani Game )eory for Wireless Com-munications and Networking CRC Press Boca Raton FLUSA 2011

[44] F Y Lu ldquoEvolutionary game analysis on environmentalpollution problemrdquo Systems Engineering-)eory amp Practicevol 27 no 9 pp 148ndash152 2007

[45] S L Dreisbach ldquoImpact of psychological research on moraltheory prospect theory and the doctrine of doing andallowingrdquo Dissertations amp )eses-Gradworks University ofCalifornia Santa Cruz CA USA 2012

[46] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Econometrica vol 47 no 2 1979

[47] N C Barberis ldquo+irty years of prospect theory in economicsa review and assessmentrdquo )e Journal of Economic Perspec-tives vol 27 no 1 pp 173ndash196 2013

Complexity 13

[48] S Benartzi and R H +aler ldquoMyopic loss aversion and theequity premium puzzlerdquo Quarterly Journal of Economicsvol 110 no 1 pp 73ndash92 1995

[49] N Barberis and M Huang ldquoStocks as lotteries the impli-cations of probability weighting for security pricesrdquo )eAmerican Economic Review vol 98 no 5 pp 2066ndash21002008

[50] A Frazzini ldquo+e disposition effect and underreaction tonewsrdquo )e Journal of Finance vol 61 no 4 pp 2017ndash20462006

[51] W Y Hu and J S Scott ldquoBehavioral obstacles in the annuitymarketrdquo Financial Analysts Journal vol 63 no 6 pp 71ndash822007

[52] B Koszegi andM Rabin ldquoReference-dependent consumptionplansrdquo )e American Economic Review vol 99 no 3pp 909ndash936 2009

[53] P Vamvakas E E Tsiropoulou and S Papavassiliou ldquoDy-namic spectrum management in 5G wireless networks a real-life modeling approachrdquo in Proceedings of the IEEE INFO-COM 2019-IEEE Conference on Computer CommunicationsIEEE Paris France April 2019

[54] Y N Zhou and J C Liu ldquoStudy on supervision strategy of pppproject considering the participation of higher-level gov-ernmentrdquo Chinese Journal of Management Science vol 1ndash162021 httpsdoiorg1016381jcnkiissn1003-207x20200801

[55] H Q Zhang J C Wang and H L Tao ldquoResearch the de-cision-making behavior of investors based on prospect theoryin the ppp projectrdquo Soft Science vol 32 no 8 pp 129ndash1332018

[56] D Friedman ldquoEvolutionary games in economicsrdquo Econo-metrica vol 59 no 3 pp 637ndash666 1991

[57] Z B Zhao and Q P Man ldquoEvolutionary game analysis of riskmanagement behavior of major infrastructure projects basedon projects based on prospect theoryrdquo Journal of SystemsManagement vol 27 no 1 pp 112ndash120 2018

14 Complexity

that reducing the cost of social capitalsrsquo strategy of ldquopositiveadoptionrdquo is helpful to enhance the willingness of socialcapitals to adopt BIM in PPP projects In Figure 8(b) we cansee that the convergence rate of y to the ESS point (1 1 1) isaccelerated as the value of Lp goes bigger +e simulationresults show that increasing the punishment of social cap-italsrsquo strategy of ldquopassive adoptionrdquo is helpful to enhance thewillingness of social capitals to adopt BIM in PPP projects

43 Influence of Cc and Sc Changes on the Evolutionary GameProcess Keep the other parameters unchanged and theinfluence on the evolution trend of the system is observed bychanging the values of Cc and Sc as shown in Figures 9(a)and 10(a) In Figure 9(b) we can see that the convergence

rate of z to the ESS point (1 1 1) is accelerated as the value ofCc goes down+e simulation results show that reducing thecost of contractorsrsquo strategy of ldquopositive applicationrdquo ishelpful to enhance the willingness of contractors to applyBIM in PPP projects In Figure 10(b) we can see that theconvergence rate of z to the ESS point (1 1 1) is acceleratedas the value of Sc goes bigger +e simulation results showthat increasing the punishment of contractorsrsquo strategy ofldquopassive applicationrdquo is helpful to enhance the willingness ofcontractors to apply BIM in PPP projects

5 Conclusions and Management Implications

Under the assumption of bounded rationality this studyconstructs a tripartite evolutionary game model including

111

0908070605

z

108

06y04 05 06 07 08

x

091

Cc=30Cc=20Cc=10

(a)

Cc=30Cc=20Cc=10

11

1

09

08

07

06

05

z

1 09095 08085 07075 06065 05055y

(b)

Figure 9 Influence of Cc change on the evolutionary game process

111

0908070605

z

108

06y04 05 06 07 x

08 091

Sc=20Sc=30Sc=40

(a)

Sc=20Sc=30Sc=40

11

1

09

08

07

06

05

z

1 09095 08085 07075 06065 05055y

(b)

Figure 10 Influence of Sc change on the evolutionary game process

Complexity 11

governments social capitals and contractors +e satisfac-tion conditions that the system converging to the ESS point(1 1 1) are analyzed based on prospect theory and the mainfactors influencing stakeholdersrsquo behavior decision-makingare simulated by MATLAB +e research results will help toclarify the behavior evolution mechanism and the influ-encing factors of the ESS for adopting BIM among stake-holders in PPP projects so as to improve the cross-organization information collaboration efficiency of PPPprojects and promote the informatization of the construc-tion industry +e primary conclusions and managementimplications are as follows

First the cost of BIM promotion by governments and thepunishment from the superior supervision authority whengovernments choose the passive strategy are the key factorsaffecting governmentsrsquo behavior choice When governmentsactively promote the cost reduction of BIM or punishedmore by the superior supervision authority governmentswill choose the active promotion strategy and the behaviorgradually converges to the ESS point (1 1 1)

Second the cost of active adoption of BIM and thepunishment of passive adoption of BIM are the key factorsaffecting the choice of social capitalsrsquo behavior When thecost of active adoption of BIM is reduced or the penalty ofpassive adoption of BIM is increased the intention of socialcapitals to choose active adoption of BIM strategy is in-creased and the behavior of social capitals gradually con-verges to the ESS point (1 1 1)

Finally the cost of active application of BIM and thereward of active application of BIM are the key factorsinfluencing the choice of contractorsrsquo behavior When thecost of the active application of BIM technology is reducedor the penalty of the negative application of BIM technologyis increased the intention of the contractor to choose thestrategy of active application of BIM technology is increasedand the contractorrsquos behavior gradually converges to the ESSpoint (1 1 1)

+e adoption of BIM in PPP projects is a major reform ofthe traditional building mode whichrequires all stakeholders to change the traditionalthinking and actively cooperate In order to promoteBIM adoption among stakeholders of PPP projects thefollowing management implications are proposedImprove the supervision efficiency of superior supervisionauthority on governmentsrsquo promotion of BIM and furtherenrich the supervision means with the help of the media andthe public Increase the punishment for the governmentsrsquopassive promotion of BIM so as to improve the perceivedvalue of governments for regulatory punishment

Strengthen the punishment for the passive adoption ofBIM by social capitals and the reward for the active ap-plication of BIM by contractors Improve the cognition ofsocial capitals and contractors on BIM and other advancedinformation technology through publicity and training andgradually transform the system promotion into habitualadoption

Under the PPP mode the adoption of BIM can bepromoted by the governmentrsquos supervision but the gov-ernmentrsquos behavior also needs supervision +e public is the

users and beneficiaries of PPP projectsrsquo goods and the publicshould also have the right to know and to supervise theimplementation of the projects +erefore in the next stagethe authors will further research the role of public super-vision in the adoption of BIM in PPP projects in order toexplore the evolutionary mechanism and influencing factorsunder more stakeholders

Data Availability

Some or all data models or code generated or used tosupport the findings of this study are available from thecorresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interest

Acknowledgments

+is work was supported by the National Natural ScienceFoundation of China (NSFC) (71801119) in part by theSocial Science Planning Foundation of Liaoning(L18BGL020) and in part by the Basic Scientific ResearchProjects in Institutions of Higher Learning of LiaoningEducation Department (LJ2017QW009)

References

[1] N Carbonara and R Pellegrino ldquoRevenue guarantee inpublic-private partnerships a win-win modelrdquo ConstructionManagement amp Economics vol 36 no 10 pp 584ndash598 2018

[2] L Cheng ldquoDiscussion on the application of PPPmodel in newinfrastructure constructionrdquo Open Journal of Social Sciencesvol 7 no 9 pp 283ndash288 2019

[3] D Wu and S Q Wang ldquoResearch development and trend ofpublic private partnership in Chinardquo Journal of EngineeringManagement vol 28 no 6 pp 75ndash80 2014

[4] K Leidel and H W Alfen Lifecycle-Oriented Risk Manage-ment for PPP-Projects in Public Real Estate pp 587ndash604 EresLos Angeles CA USA 2009

[5] J Yuan X Li Y Ke W Xu Z Xu and M SkibnewskildquoDeveloping a building information modelingndashbased per-formance management system for publicndashprivate partner-shipsrdquo Engineering Construction and ArchitecturalManagement vol 27 no 8 pp 1727ndash1762 2020

[6] S Jayasuriya G Zhang and R J Yang ldquoExploring the impactof stakeholder management strategies on managing issues inPPP projectsrdquo International Journal of Construction Man-agement vol 20 no 6 pp 666ndash678 2020

[7] C X Jiang ldquoCauses and implications of the failure intransportation infrastructure public privatepartnershipsmdashmdasha study of 25 typical PPP casesrdquo Journal ofBeijing Jiaotong University vol 15 no 3 pp 50ndash58 2016

[8] R Osei-Kyei and A P C Chan ldquoReview of studies on thecritical success factors for public-private partnership(PPP) projects from 1990 to 2013rdquo International Journal ofProject Management vol 33 no 6 pp 1335ndash1346 2015

[9] B Zhang Y Wang X Liu et al ldquo+e configuration of PPPfailuremdashmdasha crisp-set qualitative comparative analysisrdquoJournal of Public Administration vol 12 no 04pp 65ndash81+191 2019

12 Complexity

[10] N Costantino and R Pellegrino ldquoPublicndashprivate partnershipimplementation cross-case analysisrdquo Proceedings of the In-stitution of Civil Engineers-management Procurement andLaw vol 168 no 4 pp 164ndash176 2015

[11] P Zhang K Zhan and Y Zhou ldquoGame model of three-partyshared risk allocation for PPP projectsmdashmdashbased on theperspective of incomplete informationrdquo IOP Conference Se-ries Earth and Environmental Science vol 295 no 4pp 42ndash49 2019

[12] N W Young S A Jones and H M Bernstein BuildingInformation Modeling Transforming Design and Constructionto Achieve Greater Industry Productivity McGraw HillConstruction New York NY USA 2008

[13] N D Aziz A H Nawawi and N R M Ariff ldquoBuildinginformation modelling (BIM) in facilities management op-portunities to be considered by facility managersrdquo Procedia-Social and Behavioral Sciences vol 234 pp 353ndash362 2016

[14] W Zou M Kumaraswamy and J Chung ldquoIdentifying thecritical success factors for relationship management in PPPprojectsrdquo International Journal of Project Managementvol 32 no 2 pp 265ndash274 2014

[15] QWang and TWang ldquoResearch on promotion of BIM basedon game analysis of stakeholdersrdquo Sichuan BuildingMaterialsvol 41 no 6 p 272 + 278 2015

[16] Q L Yin Evolutionary Game Analysis on BIM TechnologyDiffusion of Prefabricated Construction Qingdao Universityof Technology Qingdao China 2019

[17] H X Tang J M Cao H S Xu and C F Han ldquoAnalysis onBIM application cooperation behaviors in integrated facilitymanagement organization based on evolutionary game the-oryrdquo Industrial Engineering amp Management vol 25 no 4pp 1ndash8 2020

[18] J R Song ldquoResearch on incentive and countermeasures ofBIM application based on game theoryrdquo Value Engineeringvol 36 no 33 pp 43-44 2017

[19] C Jia Z H Wang and L Zhao ldquoEvolutionary game analysisand simulation for cooperative application of BIM technologyin PPPrdquo Journal of Liaoning Technical University vol 22no 4 pp 270ndash277 2020

[20] K A Tversky ldquoProspect theory an analysis of decision underriskrdquo Econometrica vol 47 no 2 pp 263ndash291 1979

[21] R L Gao and Q Bao ldquoSelection of government supervisionmode during the operational period in PPP projects based onevolutionary game theoryrdquo Operations Research and Man-agement Science vol 28 no 4 pp 155ndash162 2019

[22] S K He G W Liang and J B Meng ldquoMulti-subjectsbenefit game and behavior evolution mechanism of majorengineering based on prospect theoryrdquo Science andTechnology Management Research vol 40 no 5 pp 207ndash214 2020

[23] A Porwal and K N Hewage ldquoBuilding information modeling(BIM) partnering framework for public construction proj-ectsrdquo Automation in Construction vol 31 no 3 pp 204ndash2142013

[24] P E D Love J Liu J Matthews C-P Sing and J SmithldquoFuture proofing PPPs life-cycle performance measurementand building information modellingrdquo Automation in Con-struction vol 56 pp 26ndash35 2015

[25] Y L Yin Q J Liu and X Wang ldquoResearch on the whole lifecycle supervision system of public projectsmdashmdashbased on thecollaboration of VfM evaluation and BIM technologyrdquo ProjectManagement Technology vol 14 no 07 pp 17ndash20 2016

[26] S Q Wang R Tiong S Ting et al ldquoEvaluation and man-agement of political risks in Chinarsquos BOTprojectsrdquo Journal of

Construction Engineering and Management vol 126 no 3pp 242ndash250 2000

[27] Y M Chen and H F Chen ldquo+e risk management of PPPproject life cycle based on BIM technologyrdquo Project Man-agement Technology vol 15 no 5 pp 51ndash55 2017

[28] G Luo and X X Zhan ldquoResearch on the application of BIM inPPP project whole-life cyclerdquo Technology and EconomicGuide vol 26 no 5 p 168 2018

[29] B L Zhou and C L Lin ldquo+e application of BIM in PPPproject whole-life cyclerdquo Engineering Economy vol 29 no 3pp 51ndash54 2019

[30] J M Smith Evolution and the )eory of Games CambridgeUniversity Press Cambridge UK 1982

[31] J W Weibull Evolutionary Game )eory MIT PressCambridge MA USA 1997

[32] K Binmore ldquoModeling rational players part Irdquo Economicsand Philosophy vol 3 no 2 pp 179ndash214 1987

[33] K Binmore ldquoModeling rational players part IIrdquo Economicsand Philosophy vol 4 1988

[34] P D Taylor and L B Jonker ldquoEvolutionarily stable strategiesand game dynamicsrdquoMathematical Biosciences vol 40 no 1-2 pp 145ndash156 1978

[35] P Young ldquoConventional contractsrdquo )e Review of EconomicStudies vol 65 no 4 pp 773ndash792 1998

[36] D Fudenberg and C Harris ldquoEvolutionary dynamics withaggregate shocksrdquo Journal of Economic )eory vol 57 no 2pp 420ndash441 1992

[37] H Bester W Guth and W Guth ldquoIs altruism evolutionarilystablerdquo Journal of Economic Behavior amp Organization vol 34no 2 pp 193ndash209 1998

[38] K E Boulding R R Nelson and S G Winter ldquoAn evolu-tionary theory of economic changrdquo American Journal ofAgricultural Economics vol 66 no 4 p 535 1984

[39] S Y Xie Economic Game )eory Fudan University PressShanghai China 2002

[40] D Friedman and K C Fung ldquoInternational trade and theinternal organization of firms an evolutionary approachrdquoJournal of International Economics vol 41 no 1-2 pp 113ndash137 1996

[41] Y P Wang and W D Meng ldquoEvolutionary game analysis onco-competition mechanism of supply chainrdquo Journal of In-dustrial Engineering and Engineering Management vol 18no 2 pp 96ndash98 2004

[42] J Gao and Z H Shen ldquoElementary groping for evolutionarygame theory and its application in electricity marketrdquo Au-tomation of Electric Power Systems vol 27 no 18 pp 18ndash212003

[43] Y Zhang and M Guizani Game )eory for Wireless Com-munications and Networking CRC Press Boca Raton FLUSA 2011

[44] F Y Lu ldquoEvolutionary game analysis on environmentalpollution problemrdquo Systems Engineering-)eory amp Practicevol 27 no 9 pp 148ndash152 2007

[45] S L Dreisbach ldquoImpact of psychological research on moraltheory prospect theory and the doctrine of doing andallowingrdquo Dissertations amp )eses-Gradworks University ofCalifornia Santa Cruz CA USA 2012

[46] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Econometrica vol 47 no 2 1979

[47] N C Barberis ldquo+irty years of prospect theory in economicsa review and assessmentrdquo )e Journal of Economic Perspec-tives vol 27 no 1 pp 173ndash196 2013

Complexity 13

[48] S Benartzi and R H +aler ldquoMyopic loss aversion and theequity premium puzzlerdquo Quarterly Journal of Economicsvol 110 no 1 pp 73ndash92 1995

[49] N Barberis and M Huang ldquoStocks as lotteries the impli-cations of probability weighting for security pricesrdquo )eAmerican Economic Review vol 98 no 5 pp 2066ndash21002008

[50] A Frazzini ldquo+e disposition effect and underreaction tonewsrdquo )e Journal of Finance vol 61 no 4 pp 2017ndash20462006

[51] W Y Hu and J S Scott ldquoBehavioral obstacles in the annuitymarketrdquo Financial Analysts Journal vol 63 no 6 pp 71ndash822007

[52] B Koszegi andM Rabin ldquoReference-dependent consumptionplansrdquo )e American Economic Review vol 99 no 3pp 909ndash936 2009

[53] P Vamvakas E E Tsiropoulou and S Papavassiliou ldquoDy-namic spectrum management in 5G wireless networks a real-life modeling approachrdquo in Proceedings of the IEEE INFO-COM 2019-IEEE Conference on Computer CommunicationsIEEE Paris France April 2019

[54] Y N Zhou and J C Liu ldquoStudy on supervision strategy of pppproject considering the participation of higher-level gov-ernmentrdquo Chinese Journal of Management Science vol 1ndash162021 httpsdoiorg1016381jcnkiissn1003-207x20200801

[55] H Q Zhang J C Wang and H L Tao ldquoResearch the de-cision-making behavior of investors based on prospect theoryin the ppp projectrdquo Soft Science vol 32 no 8 pp 129ndash1332018

[56] D Friedman ldquoEvolutionary games in economicsrdquo Econo-metrica vol 59 no 3 pp 637ndash666 1991

[57] Z B Zhao and Q P Man ldquoEvolutionary game analysis of riskmanagement behavior of major infrastructure projects basedon projects based on prospect theoryrdquo Journal of SystemsManagement vol 27 no 1 pp 112ndash120 2018

14 Complexity

governments social capitals and contractors +e satisfac-tion conditions that the system converging to the ESS point(1 1 1) are analyzed based on prospect theory and the mainfactors influencing stakeholdersrsquo behavior decision-makingare simulated by MATLAB +e research results will help toclarify the behavior evolution mechanism and the influ-encing factors of the ESS for adopting BIM among stake-holders in PPP projects so as to improve the cross-organization information collaboration efficiency of PPPprojects and promote the informatization of the construc-tion industry +e primary conclusions and managementimplications are as follows

First the cost of BIM promotion by governments and thepunishment from the superior supervision authority whengovernments choose the passive strategy are the key factorsaffecting governmentsrsquo behavior choice When governmentsactively promote the cost reduction of BIM or punishedmore by the superior supervision authority governmentswill choose the active promotion strategy and the behaviorgradually converges to the ESS point (1 1 1)

Second the cost of active adoption of BIM and thepunishment of passive adoption of BIM are the key factorsaffecting the choice of social capitalsrsquo behavior When thecost of active adoption of BIM is reduced or the penalty ofpassive adoption of BIM is increased the intention of socialcapitals to choose active adoption of BIM strategy is in-creased and the behavior of social capitals gradually con-verges to the ESS point (1 1 1)

Finally the cost of active application of BIM and thereward of active application of BIM are the key factorsinfluencing the choice of contractorsrsquo behavior When thecost of the active application of BIM technology is reducedor the penalty of the negative application of BIM technologyis increased the intention of the contractor to choose thestrategy of active application of BIM technology is increasedand the contractorrsquos behavior gradually converges to the ESSpoint (1 1 1)

+e adoption of BIM in PPP projects is a major reform ofthe traditional building mode whichrequires all stakeholders to change the traditionalthinking and actively cooperate In order to promoteBIM adoption among stakeholders of PPP projects thefollowing management implications are proposedImprove the supervision efficiency of superior supervisionauthority on governmentsrsquo promotion of BIM and furtherenrich the supervision means with the help of the media andthe public Increase the punishment for the governmentsrsquopassive promotion of BIM so as to improve the perceivedvalue of governments for regulatory punishment

Strengthen the punishment for the passive adoption ofBIM by social capitals and the reward for the active ap-plication of BIM by contractors Improve the cognition ofsocial capitals and contractors on BIM and other advancedinformation technology through publicity and training andgradually transform the system promotion into habitualadoption

Under the PPP mode the adoption of BIM can bepromoted by the governmentrsquos supervision but the gov-ernmentrsquos behavior also needs supervision +e public is the

users and beneficiaries of PPP projectsrsquo goods and the publicshould also have the right to know and to supervise theimplementation of the projects +erefore in the next stagethe authors will further research the role of public super-vision in the adoption of BIM in PPP projects in order toexplore the evolutionary mechanism and influencing factorsunder more stakeholders

Data Availability

Some or all data models or code generated or used tosupport the findings of this study are available from thecorresponding author upon request

Conflicts of Interest

+e authors declare that there are no conflicts of interest

Acknowledgments

+is work was supported by the National Natural ScienceFoundation of China (NSFC) (71801119) in part by theSocial Science Planning Foundation of Liaoning(L18BGL020) and in part by the Basic Scientific ResearchProjects in Institutions of Higher Learning of LiaoningEducation Department (LJ2017QW009)

References

[1] N Carbonara and R Pellegrino ldquoRevenue guarantee inpublic-private partnerships a win-win modelrdquo ConstructionManagement amp Economics vol 36 no 10 pp 584ndash598 2018

[2] L Cheng ldquoDiscussion on the application of PPPmodel in newinfrastructure constructionrdquo Open Journal of Social Sciencesvol 7 no 9 pp 283ndash288 2019

[3] D Wu and S Q Wang ldquoResearch development and trend ofpublic private partnership in Chinardquo Journal of EngineeringManagement vol 28 no 6 pp 75ndash80 2014

[4] K Leidel and H W Alfen Lifecycle-Oriented Risk Manage-ment for PPP-Projects in Public Real Estate pp 587ndash604 EresLos Angeles CA USA 2009

[5] J Yuan X Li Y Ke W Xu Z Xu and M SkibnewskildquoDeveloping a building information modelingndashbased per-formance management system for publicndashprivate partner-shipsrdquo Engineering Construction and ArchitecturalManagement vol 27 no 8 pp 1727ndash1762 2020

[6] S Jayasuriya G Zhang and R J Yang ldquoExploring the impactof stakeholder management strategies on managing issues inPPP projectsrdquo International Journal of Construction Man-agement vol 20 no 6 pp 666ndash678 2020

[7] C X Jiang ldquoCauses and implications of the failure intransportation infrastructure public privatepartnershipsmdashmdasha study of 25 typical PPP casesrdquo Journal ofBeijing Jiaotong University vol 15 no 3 pp 50ndash58 2016

[8] R Osei-Kyei and A P C Chan ldquoReview of studies on thecritical success factors for public-private partnership(PPP) projects from 1990 to 2013rdquo International Journal ofProject Management vol 33 no 6 pp 1335ndash1346 2015

[9] B Zhang Y Wang X Liu et al ldquo+e configuration of PPPfailuremdashmdasha crisp-set qualitative comparative analysisrdquoJournal of Public Administration vol 12 no 04pp 65ndash81+191 2019

12 Complexity

[10] N Costantino and R Pellegrino ldquoPublicndashprivate partnershipimplementation cross-case analysisrdquo Proceedings of the In-stitution of Civil Engineers-management Procurement andLaw vol 168 no 4 pp 164ndash176 2015

[11] P Zhang K Zhan and Y Zhou ldquoGame model of three-partyshared risk allocation for PPP projectsmdashmdashbased on theperspective of incomplete informationrdquo IOP Conference Se-ries Earth and Environmental Science vol 295 no 4pp 42ndash49 2019

[12] N W Young S A Jones and H M Bernstein BuildingInformation Modeling Transforming Design and Constructionto Achieve Greater Industry Productivity McGraw HillConstruction New York NY USA 2008

[13] N D Aziz A H Nawawi and N R M Ariff ldquoBuildinginformation modelling (BIM) in facilities management op-portunities to be considered by facility managersrdquo Procedia-Social and Behavioral Sciences vol 234 pp 353ndash362 2016

[14] W Zou M Kumaraswamy and J Chung ldquoIdentifying thecritical success factors for relationship management in PPPprojectsrdquo International Journal of Project Managementvol 32 no 2 pp 265ndash274 2014

[15] QWang and TWang ldquoResearch on promotion of BIM basedon game analysis of stakeholdersrdquo Sichuan BuildingMaterialsvol 41 no 6 p 272 + 278 2015

[16] Q L Yin Evolutionary Game Analysis on BIM TechnologyDiffusion of Prefabricated Construction Qingdao Universityof Technology Qingdao China 2019

[17] H X Tang J M Cao H S Xu and C F Han ldquoAnalysis onBIM application cooperation behaviors in integrated facilitymanagement organization based on evolutionary game the-oryrdquo Industrial Engineering amp Management vol 25 no 4pp 1ndash8 2020

[18] J R Song ldquoResearch on incentive and countermeasures ofBIM application based on game theoryrdquo Value Engineeringvol 36 no 33 pp 43-44 2017

[19] C Jia Z H Wang and L Zhao ldquoEvolutionary game analysisand simulation for cooperative application of BIM technologyin PPPrdquo Journal of Liaoning Technical University vol 22no 4 pp 270ndash277 2020

[20] K A Tversky ldquoProspect theory an analysis of decision underriskrdquo Econometrica vol 47 no 2 pp 263ndash291 1979

[21] R L Gao and Q Bao ldquoSelection of government supervisionmode during the operational period in PPP projects based onevolutionary game theoryrdquo Operations Research and Man-agement Science vol 28 no 4 pp 155ndash162 2019

[22] S K He G W Liang and J B Meng ldquoMulti-subjectsbenefit game and behavior evolution mechanism of majorengineering based on prospect theoryrdquo Science andTechnology Management Research vol 40 no 5 pp 207ndash214 2020

[23] A Porwal and K N Hewage ldquoBuilding information modeling(BIM) partnering framework for public construction proj-ectsrdquo Automation in Construction vol 31 no 3 pp 204ndash2142013

[24] P E D Love J Liu J Matthews C-P Sing and J SmithldquoFuture proofing PPPs life-cycle performance measurementand building information modellingrdquo Automation in Con-struction vol 56 pp 26ndash35 2015

[25] Y L Yin Q J Liu and X Wang ldquoResearch on the whole lifecycle supervision system of public projectsmdashmdashbased on thecollaboration of VfM evaluation and BIM technologyrdquo ProjectManagement Technology vol 14 no 07 pp 17ndash20 2016

[26] S Q Wang R Tiong S Ting et al ldquoEvaluation and man-agement of political risks in Chinarsquos BOTprojectsrdquo Journal of

Construction Engineering and Management vol 126 no 3pp 242ndash250 2000

[27] Y M Chen and H F Chen ldquo+e risk management of PPPproject life cycle based on BIM technologyrdquo Project Man-agement Technology vol 15 no 5 pp 51ndash55 2017

[28] G Luo and X X Zhan ldquoResearch on the application of BIM inPPP project whole-life cyclerdquo Technology and EconomicGuide vol 26 no 5 p 168 2018

[29] B L Zhou and C L Lin ldquo+e application of BIM in PPPproject whole-life cyclerdquo Engineering Economy vol 29 no 3pp 51ndash54 2019

[30] J M Smith Evolution and the )eory of Games CambridgeUniversity Press Cambridge UK 1982

[31] J W Weibull Evolutionary Game )eory MIT PressCambridge MA USA 1997

[32] K Binmore ldquoModeling rational players part Irdquo Economicsand Philosophy vol 3 no 2 pp 179ndash214 1987

[33] K Binmore ldquoModeling rational players part IIrdquo Economicsand Philosophy vol 4 1988

[34] P D Taylor and L B Jonker ldquoEvolutionarily stable strategiesand game dynamicsrdquoMathematical Biosciences vol 40 no 1-2 pp 145ndash156 1978

[35] P Young ldquoConventional contractsrdquo )e Review of EconomicStudies vol 65 no 4 pp 773ndash792 1998

[36] D Fudenberg and C Harris ldquoEvolutionary dynamics withaggregate shocksrdquo Journal of Economic )eory vol 57 no 2pp 420ndash441 1992

[37] H Bester W Guth and W Guth ldquoIs altruism evolutionarilystablerdquo Journal of Economic Behavior amp Organization vol 34no 2 pp 193ndash209 1998

[38] K E Boulding R R Nelson and S G Winter ldquoAn evolu-tionary theory of economic changrdquo American Journal ofAgricultural Economics vol 66 no 4 p 535 1984

[39] S Y Xie Economic Game )eory Fudan University PressShanghai China 2002

[40] D Friedman and K C Fung ldquoInternational trade and theinternal organization of firms an evolutionary approachrdquoJournal of International Economics vol 41 no 1-2 pp 113ndash137 1996

[41] Y P Wang and W D Meng ldquoEvolutionary game analysis onco-competition mechanism of supply chainrdquo Journal of In-dustrial Engineering and Engineering Management vol 18no 2 pp 96ndash98 2004

[42] J Gao and Z H Shen ldquoElementary groping for evolutionarygame theory and its application in electricity marketrdquo Au-tomation of Electric Power Systems vol 27 no 18 pp 18ndash212003

[43] Y Zhang and M Guizani Game )eory for Wireless Com-munications and Networking CRC Press Boca Raton FLUSA 2011

[44] F Y Lu ldquoEvolutionary game analysis on environmentalpollution problemrdquo Systems Engineering-)eory amp Practicevol 27 no 9 pp 148ndash152 2007

[45] S L Dreisbach ldquoImpact of psychological research on moraltheory prospect theory and the doctrine of doing andallowingrdquo Dissertations amp )eses-Gradworks University ofCalifornia Santa Cruz CA USA 2012

[46] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Econometrica vol 47 no 2 1979

[47] N C Barberis ldquo+irty years of prospect theory in economicsa review and assessmentrdquo )e Journal of Economic Perspec-tives vol 27 no 1 pp 173ndash196 2013

Complexity 13

[48] S Benartzi and R H +aler ldquoMyopic loss aversion and theequity premium puzzlerdquo Quarterly Journal of Economicsvol 110 no 1 pp 73ndash92 1995

[49] N Barberis and M Huang ldquoStocks as lotteries the impli-cations of probability weighting for security pricesrdquo )eAmerican Economic Review vol 98 no 5 pp 2066ndash21002008

[50] A Frazzini ldquo+e disposition effect and underreaction tonewsrdquo )e Journal of Finance vol 61 no 4 pp 2017ndash20462006

[51] W Y Hu and J S Scott ldquoBehavioral obstacles in the annuitymarketrdquo Financial Analysts Journal vol 63 no 6 pp 71ndash822007

[52] B Koszegi andM Rabin ldquoReference-dependent consumptionplansrdquo )e American Economic Review vol 99 no 3pp 909ndash936 2009

[53] P Vamvakas E E Tsiropoulou and S Papavassiliou ldquoDy-namic spectrum management in 5G wireless networks a real-life modeling approachrdquo in Proceedings of the IEEE INFO-COM 2019-IEEE Conference on Computer CommunicationsIEEE Paris France April 2019

[54] Y N Zhou and J C Liu ldquoStudy on supervision strategy of pppproject considering the participation of higher-level gov-ernmentrdquo Chinese Journal of Management Science vol 1ndash162021 httpsdoiorg1016381jcnkiissn1003-207x20200801

[55] H Q Zhang J C Wang and H L Tao ldquoResearch the de-cision-making behavior of investors based on prospect theoryin the ppp projectrdquo Soft Science vol 32 no 8 pp 129ndash1332018

[56] D Friedman ldquoEvolutionary games in economicsrdquo Econo-metrica vol 59 no 3 pp 637ndash666 1991

[57] Z B Zhao and Q P Man ldquoEvolutionary game analysis of riskmanagement behavior of major infrastructure projects basedon projects based on prospect theoryrdquo Journal of SystemsManagement vol 27 no 1 pp 112ndash120 2018

14 Complexity

[10] N Costantino and R Pellegrino ldquoPublicndashprivate partnershipimplementation cross-case analysisrdquo Proceedings of the In-stitution of Civil Engineers-management Procurement andLaw vol 168 no 4 pp 164ndash176 2015

[11] P Zhang K Zhan and Y Zhou ldquoGame model of three-partyshared risk allocation for PPP projectsmdashmdashbased on theperspective of incomplete informationrdquo IOP Conference Se-ries Earth and Environmental Science vol 295 no 4pp 42ndash49 2019

[12] N W Young S A Jones and H M Bernstein BuildingInformation Modeling Transforming Design and Constructionto Achieve Greater Industry Productivity McGraw HillConstruction New York NY USA 2008

[13] N D Aziz A H Nawawi and N R M Ariff ldquoBuildinginformation modelling (BIM) in facilities management op-portunities to be considered by facility managersrdquo Procedia-Social and Behavioral Sciences vol 234 pp 353ndash362 2016

[14] W Zou M Kumaraswamy and J Chung ldquoIdentifying thecritical success factors for relationship management in PPPprojectsrdquo International Journal of Project Managementvol 32 no 2 pp 265ndash274 2014

[15] QWang and TWang ldquoResearch on promotion of BIM basedon game analysis of stakeholdersrdquo Sichuan BuildingMaterialsvol 41 no 6 p 272 + 278 2015

[16] Q L Yin Evolutionary Game Analysis on BIM TechnologyDiffusion of Prefabricated Construction Qingdao Universityof Technology Qingdao China 2019

[17] H X Tang J M Cao H S Xu and C F Han ldquoAnalysis onBIM application cooperation behaviors in integrated facilitymanagement organization based on evolutionary game the-oryrdquo Industrial Engineering amp Management vol 25 no 4pp 1ndash8 2020

[18] J R Song ldquoResearch on incentive and countermeasures ofBIM application based on game theoryrdquo Value Engineeringvol 36 no 33 pp 43-44 2017

[19] C Jia Z H Wang and L Zhao ldquoEvolutionary game analysisand simulation for cooperative application of BIM technologyin PPPrdquo Journal of Liaoning Technical University vol 22no 4 pp 270ndash277 2020

[20] K A Tversky ldquoProspect theory an analysis of decision underriskrdquo Econometrica vol 47 no 2 pp 263ndash291 1979

[21] R L Gao and Q Bao ldquoSelection of government supervisionmode during the operational period in PPP projects based onevolutionary game theoryrdquo Operations Research and Man-agement Science vol 28 no 4 pp 155ndash162 2019

[22] S K He G W Liang and J B Meng ldquoMulti-subjectsbenefit game and behavior evolution mechanism of majorengineering based on prospect theoryrdquo Science andTechnology Management Research vol 40 no 5 pp 207ndash214 2020

[23] A Porwal and K N Hewage ldquoBuilding information modeling(BIM) partnering framework for public construction proj-ectsrdquo Automation in Construction vol 31 no 3 pp 204ndash2142013

[24] P E D Love J Liu J Matthews C-P Sing and J SmithldquoFuture proofing PPPs life-cycle performance measurementand building information modellingrdquo Automation in Con-struction vol 56 pp 26ndash35 2015

[25] Y L Yin Q J Liu and X Wang ldquoResearch on the whole lifecycle supervision system of public projectsmdashmdashbased on thecollaboration of VfM evaluation and BIM technologyrdquo ProjectManagement Technology vol 14 no 07 pp 17ndash20 2016

[26] S Q Wang R Tiong S Ting et al ldquoEvaluation and man-agement of political risks in Chinarsquos BOTprojectsrdquo Journal of

Construction Engineering and Management vol 126 no 3pp 242ndash250 2000

[27] Y M Chen and H F Chen ldquo+e risk management of PPPproject life cycle based on BIM technologyrdquo Project Man-agement Technology vol 15 no 5 pp 51ndash55 2017

[28] G Luo and X X Zhan ldquoResearch on the application of BIM inPPP project whole-life cyclerdquo Technology and EconomicGuide vol 26 no 5 p 168 2018

[29] B L Zhou and C L Lin ldquo+e application of BIM in PPPproject whole-life cyclerdquo Engineering Economy vol 29 no 3pp 51ndash54 2019

[30] J M Smith Evolution and the )eory of Games CambridgeUniversity Press Cambridge UK 1982

[31] J W Weibull Evolutionary Game )eory MIT PressCambridge MA USA 1997

[32] K Binmore ldquoModeling rational players part Irdquo Economicsand Philosophy vol 3 no 2 pp 179ndash214 1987

[33] K Binmore ldquoModeling rational players part IIrdquo Economicsand Philosophy vol 4 1988

[34] P D Taylor and L B Jonker ldquoEvolutionarily stable strategiesand game dynamicsrdquoMathematical Biosciences vol 40 no 1-2 pp 145ndash156 1978

[35] P Young ldquoConventional contractsrdquo )e Review of EconomicStudies vol 65 no 4 pp 773ndash792 1998

[36] D Fudenberg and C Harris ldquoEvolutionary dynamics withaggregate shocksrdquo Journal of Economic )eory vol 57 no 2pp 420ndash441 1992

[37] H Bester W Guth and W Guth ldquoIs altruism evolutionarilystablerdquo Journal of Economic Behavior amp Organization vol 34no 2 pp 193ndash209 1998

[38] K E Boulding R R Nelson and S G Winter ldquoAn evolu-tionary theory of economic changrdquo American Journal ofAgricultural Economics vol 66 no 4 p 535 1984

[39] S Y Xie Economic Game )eory Fudan University PressShanghai China 2002

[40] D Friedman and K C Fung ldquoInternational trade and theinternal organization of firms an evolutionary approachrdquoJournal of International Economics vol 41 no 1-2 pp 113ndash137 1996

[41] Y P Wang and W D Meng ldquoEvolutionary game analysis onco-competition mechanism of supply chainrdquo Journal of In-dustrial Engineering and Engineering Management vol 18no 2 pp 96ndash98 2004

[42] J Gao and Z H Shen ldquoElementary groping for evolutionarygame theory and its application in electricity marketrdquo Au-tomation of Electric Power Systems vol 27 no 18 pp 18ndash212003

[43] Y Zhang and M Guizani Game )eory for Wireless Com-munications and Networking CRC Press Boca Raton FLUSA 2011

[44] F Y Lu ldquoEvolutionary game analysis on environmentalpollution problemrdquo Systems Engineering-)eory amp Practicevol 27 no 9 pp 148ndash152 2007

[45] S L Dreisbach ldquoImpact of psychological research on moraltheory prospect theory and the doctrine of doing andallowingrdquo Dissertations amp )eses-Gradworks University ofCalifornia Santa Cruz CA USA 2012

[46] D Kahneman and A Tversky ldquoProspect theory an analysis ofdecision under riskrdquo Econometrica vol 47 no 2 1979

[47] N C Barberis ldquo+irty years of prospect theory in economicsa review and assessmentrdquo )e Journal of Economic Perspec-tives vol 27 no 1 pp 173ndash196 2013

Complexity 13

[48] S Benartzi and R H +aler ldquoMyopic loss aversion and theequity premium puzzlerdquo Quarterly Journal of Economicsvol 110 no 1 pp 73ndash92 1995

[49] N Barberis and M Huang ldquoStocks as lotteries the impli-cations of probability weighting for security pricesrdquo )eAmerican Economic Review vol 98 no 5 pp 2066ndash21002008

[50] A Frazzini ldquo+e disposition effect and underreaction tonewsrdquo )e Journal of Finance vol 61 no 4 pp 2017ndash20462006

[51] W Y Hu and J S Scott ldquoBehavioral obstacles in the annuitymarketrdquo Financial Analysts Journal vol 63 no 6 pp 71ndash822007

[52] B Koszegi andM Rabin ldquoReference-dependent consumptionplansrdquo )e American Economic Review vol 99 no 3pp 909ndash936 2009

[53] P Vamvakas E E Tsiropoulou and S Papavassiliou ldquoDy-namic spectrum management in 5G wireless networks a real-life modeling approachrdquo in Proceedings of the IEEE INFO-COM 2019-IEEE Conference on Computer CommunicationsIEEE Paris France April 2019

[54] Y N Zhou and J C Liu ldquoStudy on supervision strategy of pppproject considering the participation of higher-level gov-ernmentrdquo Chinese Journal of Management Science vol 1ndash162021 httpsdoiorg1016381jcnkiissn1003-207x20200801

[55] H Q Zhang J C Wang and H L Tao ldquoResearch the de-cision-making behavior of investors based on prospect theoryin the ppp projectrdquo Soft Science vol 32 no 8 pp 129ndash1332018

[56] D Friedman ldquoEvolutionary games in economicsrdquo Econo-metrica vol 59 no 3 pp 637ndash666 1991

[57] Z B Zhao and Q P Man ldquoEvolutionary game analysis of riskmanagement behavior of major infrastructure projects basedon projects based on prospect theoryrdquo Journal of SystemsManagement vol 27 no 1 pp 112ndash120 2018

14 Complexity

[48] S Benartzi and R H +aler ldquoMyopic loss aversion and theequity premium puzzlerdquo Quarterly Journal of Economicsvol 110 no 1 pp 73ndash92 1995

[49] N Barberis and M Huang ldquoStocks as lotteries the impli-cations of probability weighting for security pricesrdquo )eAmerican Economic Review vol 98 no 5 pp 2066ndash21002008

[50] A Frazzini ldquo+e disposition effect and underreaction tonewsrdquo )e Journal of Finance vol 61 no 4 pp 2017ndash20462006

[51] W Y Hu and J S Scott ldquoBehavioral obstacles in the annuitymarketrdquo Financial Analysts Journal vol 63 no 6 pp 71ndash822007

[52] B Koszegi andM Rabin ldquoReference-dependent consumptionplansrdquo )e American Economic Review vol 99 no 3pp 909ndash936 2009

[53] P Vamvakas E E Tsiropoulou and S Papavassiliou ldquoDy-namic spectrum management in 5G wireless networks a real-life modeling approachrdquo in Proceedings of the IEEE INFO-COM 2019-IEEE Conference on Computer CommunicationsIEEE Paris France April 2019

[54] Y N Zhou and J C Liu ldquoStudy on supervision strategy of pppproject considering the participation of higher-level gov-ernmentrdquo Chinese Journal of Management Science vol 1ndash162021 httpsdoiorg1016381jcnkiissn1003-207x20200801

[55] H Q Zhang J C Wang and H L Tao ldquoResearch the de-cision-making behavior of investors based on prospect theoryin the ppp projectrdquo Soft Science vol 32 no 8 pp 129ndash1332018

[56] D Friedman ldquoEvolutionary games in economicsrdquo Econo-metrica vol 59 no 3 pp 637ndash666 1991

[57] Z B Zhao and Q P Man ldquoEvolutionary game analysis of riskmanagement behavior of major infrastructure projects basedon projects based on prospect theoryrdquo Journal of SystemsManagement vol 27 no 1 pp 112ndash120 2018

14 Complexity