the evolution of nonlinear dynamics in political science

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The Evolution of Nonlinear Dynamics in PoliticalScience and Public Administration: Methods,

Modeling and MomentumL. DOUGLAS KIEL*

Professor of Public Administration and Political Economy, School of Social Sciences,University of Texas at Dallas, P.O. Box 830688, Richardson, TX 75083

(Received 2 February 2000)

Evolution is never total adaptation. It always requires destabilization, the reaching out, theself-presentation which offers new symbiotic relations, the risk accompanying innovation,

This paper examines the evolution of the application of nonlinear dynamics and relatedmethods to the study of political science and public administration throughout the 20thcentury. Some analysts understood the importance of nonlinearity to political andadministrative studies in the early part of the century. More recently, a growing numberof scholars understand that the political and administrative worlds are ripe withnonlinearity and thus amenable to nonlinear dynamical techniques and models. Thecurrent state of the application of both discrete and continuous time models in politicalscience and public administration are presented. There is growing momentum inpolitical and public administration studies that may serve to enhance the realism andapplicability of these sciences to a nonlinear world.

Keywords: Political science; Public administration; Public policy; Research methods; Nonlinearity;Complexity; Discrete time models; Continuous time models

It is often difficult to determine if innovation in theacademic enterprise is the result of courageousintent, tempered sagacity, or simple dumb luck bythose with too much time on their hands. Clearlythough, one variable related to innovation is thatsuch novelty is generally produced by a smallgroup that manages to extricate itself from the

dominance of the existing intellectual paradigm. Inshort, evolution occurs along the periphery.

This paper explores and presents the results ofsuch an evolutionary process. Specifically, thispaper examines the evolution and current state ofthe application of nonlinear dynamics to thephenomena of political science and public

*Tel.: (972)-883-2019, Fax: (972) 883-2019, e-mail: [email protected] Jantsch, The Self-Organizing Universe, 1980.

265

266 L.D. KIEL

administration during the 20th century. Thisevolution is the result of a small group ofinnovators in the fields of political science andpublic administration who are just beginning tosee their work disperse among a larger number ofinterested scholars.When examining the evolution of an academic

enterprise it is always of value to remember theKuhnian warning concerning intellectual revolu-tions (Kuhn, 1962). In brief, the incursions of theemergent paradigm will always be slowed by theweight of the existing convention. Given the factthat political science and public administration are

still relatively immature as "mathematical enter-prises" any new method can be seen as either athreat to or an opportunity for the existing body ofmethodological tools. Clearly though the dominantassumptions concerning dynamics in these fieldshave been those of linear relationships and stabledynamics. This reality is more a result of therelative state of statistical and mathematicalsophistication among scholars in these fields thanit is a failure to appreciate the potential fornonlinearity, instability and uncertainty in politicaland administrative phenomena.

This paper is organized into four sections. Thefirst section explores the evolution of consideringand examining nonlinearity in political and publicadministrative phenomena. The second sectiondetails the array of nonlinear methods currentlyemployed in political and public administrativeresearch. These methods cross the categories oftemporal, spatial and functional nonlinearity. Thenext section presents previously published non-

linear mathematical models of political and publicadministrative phenomena. The mathematics ofthe dynamic models presented focus on the more

complex models and methods developed in thefields. This third section also introduces the currentdebate among political scientists as to the relativevalue of discrete versus continuous time models ofpolitical phenomena. Finally, an effort is made todiscern the prospects for the further evolution ofnonlinear methods in understanding political andpublic administrative phenomena.

THE EVOLUTIONOF THE RECOGNITIONOF NONLINEARITY IN POLITICALSCIENCE AND PUBLICADMINISTRATION

Early Recognition of Nonlinearity

Some early 20th century social scientists were

cognizant of the possibility of nonlinearity inpolitical and public administrative phenomena. Asearly as 1928 the quantitative literature in politicalscience evidenced the recognition of the possibilityof nonlinearity in data relevant to the study ofpolitics. Stuart Rice’s, Quantitative Methods in

Politics (1928, p. 106) revealed his appreciation forthe likelihood of the nonlinear nature of thediffusion of political attitudes or what he labeled"the velocity of diffusion". Rice’s language is

hauntingly similar to more recent studies investi-

gating contagion effects in attitudinal formation.Rice accepted the limitations of the then state ofthe art in social statistics to contend with non-

linearity by noting that he was "merely.. suggest-ing one direction in which statistical methods ofdetermining variability may yet throw light uponan important social and political phenomenon"(Rice, 1928, p. 107).

Efforts to discover early evidence of therecognition of nonlinearity in the literature ofpublic administration resulted in Nobel LaureateHerbert Simon’s (1957) appreciation for the non-

linear nature of human behavior. Simon’s book,Models of Man: Mathematical Essays on RationalBehavior in a Social Setting (1957), is comprised ofsixteen of his essays that appeared in previouslypublished journal articles. Simon’s interest inadministration early in his career, in fact hisPh.D. work and early career were devoted to

public administration, is evidenced in the booksorganization into sections including topics such as

"motivation: inducements and contributions" and"rationality and administrative decision-making".Simon was concerned with the interactions

of "social groups" and the effects of outside

DYNAMICS IN POLITICAL SCIENCE 267

influences on the maintenance of these groups. Heassumed nonlinear relationships between the"friendliness" of groups and the "amount ofactivity" of the group as a saturation phenomenain which increased friendliness resulted in dimin-ishing rates of activity. Simon’s mathematicalmodel was defined via the continuous dynamicsof a system of differential equations. It is worthy ofnote that, while rigorous assumptions concerningnonlinearity are seen as a recent event, Simonrecognized the importance of nonlinearity as a

major factor in altering system dynamics as earlyas the 1950s.

The Behavioral Revolution and RecentAdvances in Modeling Nonlinear

Dynamics Systems

The behavioral revolution in the social sciencesafter World War II had a profound impact onboth political science and public administration.This behavioral or quantitative revolution broughtstatistical rigor to fields that had relied previouslyon normative debate. Much of the immediate post-War effort was aimed at simply quantifying or

statistically verifying questions of "What exists?".Yet, both of these fields were greatly strengthenedby efforts to move them away from the mereespousal of theory to at least offering the potentialfor the actual testing of theory.Many of the methodological advances in poli-

tical science and public administration during thedecades of the 1960’s and 1970’s involved the pro-liferation of traditional statistical techniques andmethods of ordinary least squares linear modeling.While political science and public administrationdid not attempt to emulate the natural sciences tothe same extent as their economics brethren, mostpolitical scientists and public administrationists didpursue a mechanical and linear statistical paradigmsimilar to the "physics envy" attributed to econo-mists (Mirowski, 1989).One widely recognized effort to produce math-

ematical models of political behavior was LewisFry Richardson’s examination of arms races

(Richardson, 1960a, 1960b). Richardson’s researchwas driven by the historical recognition that WW Ioccurred although the participating nationsseemed not to want war. Richardson thus sawthe conflict as the result of dynamics in theinternational "system" that dominated even theintentions of political leadership groups. This earlymodeling effort thus focused on the "systemeffects" of arms build-ups and the action-reactionsequence taken by adversaries.

Richardson’s work on arms races is also ofparticular interest because it signaled the gapbetween the theoretical recognition of nonlinearityand the computational limitations of much of 20thcentury social science. Richardson’s theoreticalmodels suggested nonlinear specifications due to

assumptions of perceived threat levels of anadversary’s military spending level varying con-

tingent upon these spending levels as a proportionof the target State’s spending level (See Saperstein,1997; Richards, 2000, Introduction). Yet, Richard-son produced a set of linear differential equationsto model these relationships. Clearly, the extantstate of the art inhibited the introduction of non-

linearity in system variable relationships.The decade of the 1980’s evidenced an increase

in the number of efforts to contend with andrecognize nonlinearity in political (Huckfeldt,Kohfeld and Likens, 1982; Axelrod, 1984; Saper-stein, 1986; C. Brown, 1988; Saperstein andMayer-Kress, 1988) and administrative systems(Kiel, 1989). In particular, Robert Axelrod’s, TheEvolution of Cooperation (1984), included agent-based and evolutionary representations of actorsin simulated political environments. While thenonlinearity in Axelrod’s models was not alwaysexplicit, he was attuned to the "nonlinear effects"(See Axelrod, 1984) that were likely to result fromhis simulations of actors altering their survivalstrategies over time. Axelrod’s evolutionary ap-proach coupled with the elegant simplicity of hisuse of the classic prisoner’s dilemma model servedas a foundation for the importance of nonlineareffects in dynamical models of politics. Axelrod’s,work (1984) continues to be cited in a variety of

268 L.D. KIEL

academic disciplines as an example of the value ofiterated models that generate uncertain outcomes.

It was the decade of the 1990’s though thatwitnessed a genuine proliferation in the applica-tion of nonlinear methods to political science(Saperstein, 1990-1992, 1996, 1997; C. Brown,1991, 1993-1995a, 1995b; Richards, 1990, 1992,2000a, 2000b; Mayer-Kress, 1990; T. Brown, 1996;McBurnett, 1996) and public administration (Kiel,1993, 1994; Kiel and Elliott, 1992; Kiel andSeldon, 1998). However, the use of the termproliferation must be viewed in light of thecontinuing dominance of linear and static meth-odologies in political and administrative studies.Political science, however, did produce a widespectrum of scholarship applying nonlinear dy-namics to political phenomenon. The seminalwork in the 1990’s of nonlinear dynamical model-ing in political science though must be seen asCourtney Brown’s 1995 book, Serpents in theSand: Essays on the Nonlinear Nature of Politicsand Human Destiny.

Relative to political science the literature ofpublic administration reveals far fewer efforts touse nonlinear methods of analysis and modelbuilding. This reality, is in part, a reflection of thefact that public administration has a practical sidethat must appeal to practicing public managers.Clearly, most public managers are not preparedfor an academic discussion of nonlinear meth-odologies in the social sciences. L. Douglas Kiel’s1994, Managing Chaos and Complexity in Govern-ment remains as the singular book devoted toinforming both scholars and practitioners of thepotential value of understanding the nonlineardynamics of public organizations and publicmanagement. Kiel has contributed other work onthe dynamics of work (1993) and with hiscolleagues has explored the nonlinearity in budget-ary outlays (Kiel and Elliott, 1992) and the limitson managerial rationality produced by the non-

linearity in service demands on a public organiza-tions (Kiel and Seldon, 1998). While an increasingnumber of public administration scholars arecognizant of the importance of incorporating

nonlinearity into models of public administration(See Morcol, 1997) most of these scholars continueto apply nonlinear dynamics as metaphor (Morcoland Dennard, 2000) rather than as a means formathematical modelling.

It is worthy of note to appreciate that a widebody of interdisciplinary scholars are now apply-ing nonlinear techniques to the problems andpotential solutions of public policy (Elliott andKiel, 1997; Elliott and Kiel, 2000). Examples ofpolicy problems investigated from the nonlineardynamical perspective include teen births (Dooley,Hamilton, Cherri, West and Fisher, 1997; Hamil-ton and West, 2000) to earthquake disasterresponse (Comfort, 1999, 2000) to environmentaldamage and cleanup (C. Brown, 1994, 1995a).

Three major texts now exist that include state ofthe art nonlinear methods and mathematics inpolitical science and public policy. These texts areKiel and Elliott’s Chaos Theory in the SocialSciences: Foundation and Applications (1996),Richards’ Political Complexity: Nonlinear Models

of Politics (2000) and Elliott and Kiel’s NonlinearDynamics, Complexity and Public Policy (2000).The titles of the latter two texts just noted alsoreveals the expanding tendency of scholars inpolitical science and public policy to eithersubsume nonlinear dynamics within the morerecognizable field of "complexity" or to considernonlinearity an essential element of the increasinginterest in "complex systems".A search for nonlinear methods as they apply to

politics and public administration still representsa small minority of the literature in these fields.While not all of these emerging methods are effortsto build dynamical models of political phenomenait is clear that an increasing number of politicalscientists are beginning to appreciate the benefitsof interjecting the "realism" of nonlinearity intomodel development. Political scientists and publicadministrationists have also not been as quick astheir fellow scholars in economics to tackle theanalytical challenges of nonlinearity. This is likelydue to two reasons. First, economists in generalare far more mathematically sophisticated than


political scientists. And second, economists havethe benefit of a wider variety of detailed andfrequently produced time series data, for examplestock market ticks or daily exchange rates, than dostudents of politics who may have only time seriesdata based on bi-annual elections or infrequentpublic opinion polls.

to nonlinearity noted by Richards (2000a). How-ever, since the simplification of the complex doesadd-value, at least as a means of adding clarity,Richards’ typology will be used to present apicture of the variety of applications of nonlinearmodels and methods in political science and publicadministration.

THE ARRAY OF METHODSFOR EXPLORING NONLINEARITYIN POLITICAL SCIENCE AND PUBLICADMINISTRATION

Making sense of the array of methods thatincorporate nonlinear relationships applied to

political science and public administration can beconfusing. Political scientists have defined twoframeworks for classifying this body of nonlinearmethods. Brown (1995a) has suggested a dichot-omous approach to classifying nonlinear modelsbased on either functional or longitudinalnonlinearity. More recently, Richards (2000a)has developed a tripartite typology of nonlinearapproaches that enhances Brown’s (1995a)scheme. Richards’ system (2000a) views nonlinearapproaches as: (1) Temporal Nonlinearity, whichsimply focuses on dynamics over time (the same asBrown’s (1995) longitudinal nonlinearity; (2)Spatial Nonlinearity, based on search and optimi-zation methods as "agents" or some other variablemoves over some defined region, or two-dimen-sional computer screen; and (3) Functional Non-linearity, which considers relationships betweenvariables as nonlinear functions.

Naturally, as with most of our efforts to simplifythe complex, even Richards’ scheme fails toprovide a means for making clear distinctionsbetween the varying modeling methods employed.This of course is less the fault of the classificationscheme than it is a reflection of the reality that themodels used often employ more than one of theapproaches to nonlinearity mentioned above. Forexample, a model of adaptive agents moving on alandscape may exhibit all three of the approaches

Temporal Nonlinearity: The Lessons of Time

It is most fitting to start with applications oftemporal nonlinearity in political science andpublic administration since this literature consti-tutes the majority of nonlinear studies in thesefields. Within political science the largest body ofliterature has focused on the subfield of inter-national relations (Saperstein, 1984, 1986, 1998;Wolfson, Puri and Martelli, 1992). The few studiesof temporal nonlinearity in public administrationhave focused on workplace dynamics (Kiel, 1993,1994). Temporal nonlinear models in politicalscience generally are efforts to develop dynamicalsystems models of political phenomena.The literature of nonlinear dynamics in interna-

tional relations has been dominated by the workof physicist, (Alvin Saperstein, 1984, 1986a, 1986b,1988, 1990-1992, 1996, 1997; Saperstein andMayer-Kress, 1988). In fact, his two earliestnonlinear models of international relations ap-peared in the journals Nature (1984) and theAmerican Journal of Physics (1986). The principleconcern of Saperstein’s work is the identificationof the dynamics that lead to war and internationalconflict. In short, Saperstein has examined howvarying nation-state behaviors interact with theresponses of competing states to engender eitherincreased instability or increased stability innation-state relations. Saperstein’s 1984 publica-tion also appears as an early, if not the first, recog-nition of the potential for mathematical chaos inthe temporal dynamics of political phenomena.

It is also worthy of note that political scientistssuch as Diana Richards (1990, 1992) have alsoexamined the potential for mathematical chaos toarise out of the process of decision-making during

270 L.D. KIEL

international crises. In short, the iterative andinterconnected process of strategy making can leadto chaotic dynamics as political actors try to

respond to the cues of other actors.Several political scientists have also explored the

dynamics of electoral behavior using nonlinearmethods (Huckfedlt, Kohfeld and Likens, 1982;C. Brown, 1991, 1993, 1995a; McBurnett, 1996).These studies range from Brown’s (1991) study ofelectoral volatility to his later study of an electorallandslide (1993, 1995a). Brown (1993, 1995a) uses

phase diagrams as a source for both hypothesisgeneration and testing. A related field in politicalstudies, public opinion has also been also beenexplored by (Huckfeldt, Kohfeld and Likens,1982) and McBurnett (1996). Mebane (2000) hasshown through the application of differentialequations that small changes in a congressionalchallenger’s quality can destabilize an incumbent’selectoral advantage.More recent efforts to examine nonlinear

temporal dynamics in political phenomena includeefforts such as Brooks, Hinich and Molyneaux’s(2000) investigation of how the relationshipsbetween political and economic events affectinternational exchange rates. This study was aimedat developing a means of "episodic nonlinear eventdetection" (Brooks, Hinich and Molyneaux, 2000)that, in this case, assesses nonlinearities inexchange rates and then looks "back" to identifysalient political and economic events that alteredexchange rates. Lohmann (2000) has examined thenonlinearities that occur over time as humaninformational exchange results in "informationalcascades" that generate instabilities in socialrelationships and outcomes. Richards (2000b)has also examined the nonlinearities that occur,via a game theoretical approach that reveals thatthe choices of international actors may not alwaysconverge to an equilibrium.

In public administration, authors have largelyfocused on nonlinearities that occur in either intraor inter-organizational relationships. For example,Kiel (1993, 1994) has examined the dynamics oforganizational outputs in light of instabilities and

cycles that arise between employee time on taskand results. Kiel (1994) has also examined theuncertainties of organizational improvement ef-forts when nonlinearities exist in the relationshipsbetween employee behavior, managerial expecta-tions and outcomes. Groundbreaking work in thearea of inter-organizational behavior has beenconducted by Comfort (1999, 2000). Comfort hasexamined the interactions of the multiple agenciesthat respond during the peak of environmentaldisasters. This research has revealed both the sensi-tivity of and importance of information networksin effective governmental response to disasters.

Spatial Nonlinearity: Orderand Disorder in Space

Studies of spatial nonlinearities in politics andpublic administration generally involve theexamination of the behavior of either an adaptiveor rational agent that seeks some optimal outcomebased on their spatial relationships to other agents:Optimal outcomes may be defined as economic,behavioral, or policy outcomes that reflect thepreferences of agents.The classic work in political science examining

such spatial dynamics is Schelling’s study, Micro-

motives and Macrobehavior (1978). Schelling’sresearch eventuated in the conclusion that choicesmade by individuals to live in communities withother individuals of a like race, tends to inevitablyresult in segregated communities. Axelrod (1984,1997) has also developed groundbreaking work inthe dynamics of alliance building based on thepeaks and valleys of landscape theory. More recentanalysis relevant to politics is Epstein and Axtell’s(1996) agent-based modeling of how agents con-

sume social resources on a "sugarscape".Another simulation effort indicative of such

spatial analysis is Kollman, Page and Miller’s

(2000) model of how "rational" citizens mayoptimize their own utility by relocating theirhousehold based on their preferences for localpublic policies. A more general approach pre-sented by Bennett (2000) reveals the potential


application of spatial nonlinearity on a variety ofpolitical landscapes. These applications largelyrevolve around coalition-building as either na-

tion-states or individual agents engage in searchregimens to discover policy preferences on a

landscape that results in movement to regimeswith like preferences.

Comfort’s (1999, 2000) study of earthquakedisasters and the concomitant response by govern-ment emergency and disaster relief agencies is a

landmark in applying spatial nonlinear models to

public administration behavior. Her researchexplores how actors on the actual physical land-scape of a city behave under crisis conditions. Theimportance of this research is signaled by its likelyvalue as a means for optimizing the organizationof emergency resources during disasters.Ongoing efforts to expand the array of spatial

nonlinear methods to political and administrativestudies are likely to be enhanced by the introduc-tion of journal outlets such as the Journal ofArtificial Societies and Social Simulation, recentlyestablished at the University of Surrey. The factthat this journal is electronic and computer-basedraises interesting possibilities for social sciencespatial modeling. For example, one can imagine ina not too distant future that such articles will allowreaders to run published simulations in real-timewhile also providing researchers the opportunityto tune model parameters with the aim ofexploring the multiple possibilities such modelsshould incorporate. Kiel (2000) has suggested thatpolitical and public administrative research may begreatly enhanced by expanded efforts to applyspatial models to the political and organizationallandscapes that adaptive agents must explore tofind optimal solutions to the challenges of politicaland organizational life.

Functional Nonlinearity: The Huntfor Nonlinearity

Studies of Functional nonlinearity in politicalscience and public administration generally includeattempts to either discern nonlinearities in existing

data or to identify instances where nonlinear andchaotic behavior has the potential to occur. Whilesuch studies in political science remain in an

embryonic stage instances of such research do exist.One area of study directed at examining

functional forms in political phenomena includesthe comparison on neural network models withtraditional linear statistical methods. Zeng’s (2000)research with neural networks reveals the super-iority of these models in terms of data-fitting totraditional statistical methods such as linear

regression and logit models. Bearce (2000) hasalso shown the superiority of neural networksrelative to traditional statistics while examining theefficacy of economic sanctions in the internationalarena. Another example of the analysis of func-tional nonlinearity is Schrodt’s (2000) use ofhidden Markov models to explore patterns ininternational crises.

If the hunt for nonlinearity in political scienceis still in its embryonic stage then the hunt fornonlinearity in public administration still waitsfor fertilization. While nonlinear approaches havebeen employed in public administration modeling,much of the work examining nonlinear dynamicsin public administration, assume functional non-

linearity, yet epistemologically remain at the meta-phorical stage of exploration (See Morcol andDennard, 2000).One example of the search for functional

nonlinearity in the literature of public administra-tion is evidenced in Kiel and Seldon’s (1998) study.This research examined government service re-

quests in an effort to determine the mathematicalstructure of the evolution of such service requests.Another more sophisticated effort includes Car-penter’s (2000) examination of bureaucratic agencylifetimes. This research used nonlinear estimatorsto test the "hazards" to agency survival.While those research efforts categorized as

explorations of "functional nonlinearity" remaina minority of nonlinear studies in political andadministrative research this area has considerablepotential payoffs. Knowledge of the basic mathe-matical structure of political and administrative

272 L.D. KIEL

networks may eventuate in the discovery ofunderlying patterns that may generate knowledgein areas such as expected levels of uncertainty thatmay lead to improved theory in political andadministrative studies.

DISCRETE AND CONTINUOUS TIMEMODELS IN POLITICAL SCIENCEAND PUBLIC ADMINISTRATION

The work of political scientists and public admin-istrationists over the last twenty years does showa growing body of literature using mathematicalmodels that incorporate nonlinearity. While this isa growing body of literature it remains as a ratherclear and unique niche for research.

It is worthy of note that a divergence of opinionexists among political scientists as to the relativemerits of continuous and discrete time models ofnonlinear political phenomena. It must be remem-bered that at this point in the evolution of politicalscience as an intellectual enterprise a minority ofpolitical scientists possess advanced mathematicalskills. If we define knowledge of calculus as suchan advanced skill, then only a minority of politicalscientists have a strong knowledge of calculus.This group of scholars though are quite familiarwith differential equations and models of con-tinuous time. Perhaps, unfortunately these samescholars are less familiar with the discrete timemodels of difference equations. Thus, continuoustime models relying on differential equationsdominate both the dynamical and nonlineardynamical models developed by political scientists.

Courtney Brown (1995a), is perhaps the mostvocal proponent, among political scientists, of thesuperior value of continuous time models relativeto discrete models. Brown believes that the timedistance between most data points in political data,be it the time span between the taking of publicopinion polls or the distance from one congres-sional election to another, neglects the reality thatnonlinear behavior may occur between data (time)points. Brown sees this problem as particularly

acute when considering the widespread use ofcross-sectional studies in political science. Fromthis perspective, only continuous time models can

capture fluid and potentially nonlinear politicalphenomena such as attitudinal shifts.

Naturally, the reality of modeling is that all ofour known methods include inherent limitations.Clearly, discrete time models are limited byunknown or uncollected data that exist betweenthe data points collected in typical political studies.Yet, discrete time models offer the potential to

produce a wider set of qualitative behaviors overtime relative to continuous time models (Huck-feldt, Kohfeld and Likens, 1982, p. 90). Thusdiscrete models are capable of producing a largerrange of behaviors that are more likely to simulatethe complexity of the social realm.

Perhaps, more importantly discrete modelssatisfy the demands of Occam’s razor. In short,discrete models represent more parsimoniousapproaches that define the "elegance in simplicity"that should motivate the model building process.This recognition is important for two reasons.

First, the expanding recognition that complexbehavior may be generated by "simple rules"emphasizes the potential importance of the use ofdiscrete time models. And second, the slightlymore approachable and understandable mathe-matics of difference equations, relative to differ-ential equations, may energize the mathematicallydisenfranchised to investigate the potential appli-cations of discrete dynamical models. Yet, con-tinuous time models continue to serve as thedominant modality not only due to the strength ofBrown’s (1995a) argument but also because thesemethods are simply more widely recognized andknown among political scientists.

Dynamical Models in Discrete Time

The following two sections of this article presentdynamical models in political science and publicadministration. The reader will note the asymme-try in the display of three mathematical modelsfrom the domain of political science and only one


from public administration. This asymmetry isintentional and shows that considerably morework in nonlinear dynamical modeling has beendone in the field of political science relative topublic administration. The practical side of publicadministration may limit such efforts as manypublic administration scholars believe that theirwork must have relevance to practitioners whorarely find "value" in such models.As noted previously, the most prolific publisher

in the applications of nonlinear dynamics topolitical phenomena is the physicist Alvin Saper-stein. Dr. Saperstein’s works have focused on thepotentialities for arms races to produce unstableinternational dynamics and thus the likelihoodof war. Saperstein has ventured to examine a

variety of questions based on the assumption ofnonlinear relationships between competing powers(1996).One of the more macroscopic and consistent

questions in international relations concernswhether democracies or autocracies are relativelymore prone to war. Saperstein handles the ques-tion of political structure and the likelihood of warwith the production of a "procurement recursionrelation" (1996, p. 156) between two nation states.These relations are based on a "fear and loathing"coefficient founded on each nation’s perception ofthe military procurement behavior of its compe-titor nation. In short, each nation’s (nation X)arms procurement at time Xt+ is a proportionalresponse to the amount its arms stocks wereexceeded by its competitor’s (nation Y) armsstocks at time Yt, thus: Y-X. The fear andloathing coefficient a thus serves to constitute theproportionality constant of arms buildup of axxfor nation X and axx for arms buildup in nation Y.The reality of economic constraints is also

included in Saperstein’s nonlinear adaptation ofa Richardson-like arms race model. Since arma-ment procurement cannot expand more than thetotal economy will allow a smooth economic cut-off function is included in the model. Thus Cx isthe maximum arms expenditure for nation X.Thus the smooth economic cut-off function is

(Saperstein, 1996, p. 156):

zxx/c )o( zxx/c ).

The unit step function is 0 thus, (Off/)= for

/> 1, 0 otherwise). Finally, build-down ofarmaments can occur if a nation’s confidencewarrants such action. The confidence coefficientfor build-down, thus results in a proportionalityparameter that is the inverse of the fear andloathing coefficient.Thus this tit-for-tat model of arms procurement

results in the following set of recursion relations(Eq. (1)). Build-ups and build-downs thus changebetween X and Y based on arms stock in theprevious year:

Xn+l Xn -q- axyYn(Yn Xn)O(1 [X / Xn]/Cx)O(Y X,)

----Xn(Xn- rn)O(Xn- rn)axy

Yn+l rn -t-ayxXn(Xn- Yn)O(1 -[Y. + 1- Y.]/Cy)O(X.- Y.)

---Yo(Yo -x.)o(Y. -x.)axy

Saperstein iterated these relations using a

spreadsheet starting with arbitrary initial condi-tions for both X and Y. Iterating the model to a

large n (n= 100) he then calculated Lyapunovcoefficients. Since large values of the fear andloathing coefficient result in more arms build-upand thus a greater region of instability, Sapersteinconcludes that international systems with moreautocratic states are more likely to generatewar than systems with more democratic states.This us due to the likelihood of democratic statessharing information regarding national intent.A relatively sophisticated effort to apply non-

linear dynamic modeling to the subject matter ofpublic administration was presented by Huckfeldt,Kohfeld, and Likens in 1982 (1982, pp. 65-81).These models focus on "budgetary dynamics", anelemental aspect of the subject matter of public

274 L.D. KIEL

administration. These scholars developed a dis-crete model of budgetary dynamics based on thecompetitive interdependence, yet restricted com-

petition, between government agencies as agenciescompete for financial resources. A basic predator/prey model is produced which in the vernacular ofpolitics may be seen as a "zero-sum" game. Thebudgetary gain, or victory, of one agency likelyoccurs at the financial loss to another agency. Amodel consistent with the budgeting literature of"incremental" increases or decreases in budgets isused to constrain explosive dynamics.

Five basic assumptions drive the models pre-sented below (Huckfeldt, Kohfeld and Likens,1982). Initially, the inherent limits of budgetaryresources constrains any agency’s budget share tosome upper limit. Next, some level of budgetaryconstraint is necessary since agencies are rarelyeliminated. Third, some agencies can be assumedto have a lower limit on resources. For example,national defense can always assume some share ofgovernment spending. Fourth, the reality of bud-geting reveals that agencies generally "satisfice"and never obtain either maximal or minimal levelsof financial resources. Finally, future levels ofagency spending are generally contingent uponcurrent funding levels.The coupled cubic equations resulting from

these assumed budgetary dynamics are shown fortwo agencies, X and Y, in (Eq. (2)).

zx r, r,)(2)

Both of these equations include four variables thatenhance model dynamics. The variable L repre-sents the natural agency budget upper limit, whileB represents a lower limit of funding. The terms pand c represent parameter values for an agency’saverage rate of success in securing resources (p)and (c) a discounting factor that represents the acharacteristic rate of the current budgetary successof the competing agencies, X and Y.The logic of the equations thus follows that AXt

represents the change in the rate of budgetary

"success" of agency X, that is proportional to a

typical rate Px to the difference between theagency’s upper limit, Lx, and the agency’s currentlevel of funding, Xt. This result is combined withthe discounting factor cy, that results fromcompetition with agency Y and functions propor-tionally to the budget level of Xt. The parameter cythus functions as a characteristic and proportionalrate of the funding level of each agency at thediscrete time period. The second minus sign in theequation reveals the zero-sum nature of the modelby noting that agency Y’s budget necessarily takesfrom the budget of agency X. The extreme righthand-side of the equation shows that if any agencybudget is decreasing towards its lower bound and[Xt-Bx] is becoming increasingly small then theinteraction between the agency competition andthus budgets decreases.The multiple outcomes generated by this model

are quite intriguing. Nine identifiable equilibriamay exist with this iterative model. These stabiliz-ing results range from those when both agenciesare abolished to more likely instances of stabilitywhen both agencies reach a stable equilibrium offunding between their natural upper and lowerlimits of funding.

Dynamical Models in Continuous Time

As noted previously, continuous time methods arethe dominant method for modeling nonlineardynamical systems in political and governmentalstudies. Professor Courtney Brown’s leadershipposition in nonlinear dynamical modeling war-rants the exegesis of the following two nonlinearmodels of political phenomena that he developed.The series of equations depicted in (Eq. (3)) is

Brown’s (1994) effort to develop a model ofenvironmental damage based on partisan controland public opinion. In short, this models depictshow partisan political changes in the holding ofthe U.S. presidency coupled with citizen attitudestoward environmental cleanup and expectedpublic policies can serve to inhibit or engenderenvironmental damage. This model assumes that


democratic regimes will accept more social costs tolimit environmental damage while republican heldwhitehouses will focus on economic growth lead-ing to greater environmental damage.

This model includes three basic state variables.These variables are environmental damage (X),public concern for the environment (Y) andspending for environmental cleanup (Z). The first

equation expresses the change in environmentaldamage. The three parameter values represent (r)the pollution growth rate, (p) the effectiveness ofgovernment policies to reduce environmentaldamage, and (k) a decay value that reducesenvironmental damage based on current valuesof that damage.The second equation reveals the change in

public concern for the environment. The variable

Xod defines a lagged value for environmentalconcern since public opinion usually reacts torather than predicts environmental damage. This

equation shows the shifting dynamic betweenpublic concern and costs revealing that as costsrise public concern for the environment is likely todiminish.

Equation three reveals the dynamics of govern-ment environmental spending. Again, a similarlogic as developed in the second equation follows.As public concern and environmental damageboth increase spending increases to some limit.Increases in spending that reach near this limitvalue then serve to diminish public concern result-ing in decreasing spending on the environment.These first three equations interact over time andcontinuously in which cyclical behavior amongall of the state variables is a potential behavior.The final equation reveals how partisan goals

(simplified to mean control of the presidency)function to change environmental policy. Thevariables gdem and grov represent the "ideal policyresponse goal" (Brown, 1994, p. 299) of either a

democratic or republican administration. Remem-bering that p is a parameter for the effectiveness ofgovernment policy, then e is a parameter thatdefines the speed of policy changes toward thepartisan goals of the standing administration.

Given nonlinear assumptions concerning thepotential for environmental damage embedded inBrown’s variables, the graphical results of hismodels results are quite stark. Shifts in policyregimes supported by varying levels of publicsupport and concomitant government spendingcan create dramatic changes in environmentaldamage. If the interaction between public concernand government spending for the environment is

the oft-noted metaphorical "butterfly", then theresulting metaphorical "tornado" is the reality ofdamage to an environmental system in whichnonlinear interactions are just beginning to beunderstood.

dX/dt rX(1 X) pXY kX

dY/dt Xold (1 Y) Z

dz/dt xr( z) zdp/dt ep(gdem + grep --P)

(3)

Another field within political science that is

amenable to dynamical modeling is electoralstudies. Electoral studies is a ripe area for non-linear modeling since elections are viewed long-itudinally and generate numerous data pointsranging from polling data to changes in partisanattachments. Again, Courtney Brown has devel-oped an elegant model of an electoral phenomen-on, that of an electoral landslide. In particular,Brown has attempted to model the U.S. nationalDemocratic landslide of 1964 (1993, 1995a).

In the model shown (Eq. (4)) below Brown(1995a, pp. 55-83) presents mirror-image equa-tions of the dynamics of democratic and repub-lican party support. The result of both equationsis the change in either (D) the proportion of theelectorate supporting the Democratic party or (R)the proportion of the electorate supporting theRepublican party. These equations are best under-stood by first reviewing the variables in the secondset of parentheses on the right hand side ofthe equations. The small letters throughout theequations represent parameter values defining thelimits of and changes in partisan support. Thus

276 L.D. KIEL

the term q(D/R) refers to the proportion of theelectorate of each party affiliation. The term wDRis a multiplicative term defining the probabilityof interaction between members of the diverseparties altering individual party attachments,much like a contagion effect. The term uD definesthe growth in Democratic support based on pre-existing levels of Democratic support.The first parenthesis in this equation in-

corporates the "bunching" effect that occurs asone party tends to dominate an election cycle. Thenumber specifies the model in a normal non-accelerated form, while the parameter j reveals theacceleration in momentum of party support. Theterm yD2 includes the parameter y that serves tomediate the acceleration that results from thesquaring of D that represents the "bunching" or

bandwagon affect.The variable vN is the proportion of nonvoting

eligible voters that is parameterized by theconstant v, since this bloc of nonvoters is generallystable. Finally, the remainder of the equationoutside of the brackets (1 -D)D defines the growthand decay of party support using the logisticmodality.

These two differential equations thus representan interdependent system. Parameter estimates ofthe equations were produced using nonlinearregression techniques. These estimates were de-rived from the approximately 3000 counties in theU.S. for presidential election years 1960 and 1964.For non-southern countries Brown’s model ex-

plained approximately 80 percent of the variancefor both Democrats and Republicans (Brown,1995a, p. 66). The fit for the southern U.S.countries was not as convincing, however, thismay have to do with the longer standing institu-tionalization of party line voting in the South.

dD/dt [(1 -+-jD / yD2)(q(D/R)

+ wDR + uD) + vN](1 D)D

dR/dt [(1 + pR + sR2)(Z(R/D)

+ aRD + eR) + gN](1 R)R

(4)

In general, Brown’s work focuses on the kindsof data that are most available to politicalscientists interested in nonlinear modeling. Thesedata are the attitudinal and electoral studies thatare essential to understanding the nonlinearity thatappears inherent in an open and democraticsociety in which multiple interests compete todefine agendas and shape attitudes. Improvedknowledge in these areas is likely to be one of theworthwhile outcomes of an expanded applicationof nonlinear dynamical models of political andpublic administrative phenomena.

CONCLUSION VENTURINGA PREDICTIONIN AN UNPREDICTABLE WORLD

When venturing a guess concerning the evolutionof the academic enterprise one must be cognizantof Thomas Kuhn’s (1962) admonition that scien-tific revolutions are more matters of incrementaladvance rather than rapid change. The evolutionof nonlinear dynamical modeling in politicalscience and public administration seems to typifyKuhn’s view of time and change. Each of thescholars engaged in the "nonlinear enterprise"likely can speak eloquently of their own stories offighting off the slings and arrows that come withtrying to publish papers using nonlinear methods.Thus whether one argues for nonlinear dynamicsas "revolution" or for the "incremental advance-ment" as a definition of nonlinear dynamics,resistance exists either through ignorance or viathe human tendency to stick with that which isknown and comfortable.At present few doctoral programs in political

science of public administration include nonlineardynamical methods of analysis in methods courses.Graduate students receiving training in nonlinearmethods usually receive such training either by a

serendipitous acquaintance with a faculty memberexploring such topics or, in even more rare

circumstances, by the student’s own effort to seekout a graduate program in which faculty are


recognized experts in nonlinear methods of analy-sis. In order to resolve this problem organizedefforts on the part of political scientists and publicadministrationists are necessary to promote great-er knowledge of nonlinear methods and modelsand to promote such methods in both professionalmeetings and in graduate methods curricula.On a more optimistic note, an effort is underway

in public administration to push scholars todevelop more sophisticated and mathematicalmodels (See Lynn, 2000). Morcol and Dennard’s(2000) forthcoming edited volume applying the"new sciences" to public administration, whilegenerally devoid of mathematical models, mayalso serve as a theoretical base for the developmentof nonlinear dynamical models in public adminis-tration. The recent publication of a nonlinearmodel of general public management (O’Toole andMeier, 1999) in a leading journal in the field againshows the growing recognition of the importanceand potential benefit of such modeling effort.One lingering challenge for incorporating more

nonlinear research and modeling into political andpublic administrative research, properly noted byRichards (2000), is the lack of "industry stan-dards" for nonlinear analysis. There simply is nogenerally agreed upon set of specific nonlinearmethods that is generally accepted by socialscientists. This reality makes it difficult for manyjunior scholars to venture into a field that lacks thecomfort of the accepted conventions of traditionalstatistical approaches.

Perhaps the most hopeful note for an increase inthe recognition of the mathematics of nonlinearityin political and public administration phenomenais signaled in the number of established scholarswho contributed to the recent edited book onnonlinear methods in political science by DianaRichards (2000). The fact that three editedvolumes Kiel and Elliott (1996); Richards (2000);Elliot and Kiel (2000) now exist also shows that a

body of literature is developing that can serve as abasis for emerging scholars to take upon thechallenge of nonlinearity in politics and publicadministration.

The increasing recognition of the relevance ofthe sciences of complexity to political science andpublic administration should also serve to advancethe nonlinear agenda. The Princeton Universityseries on Complexity has already contributedSystem Effects (1997); Axelrods’ The Complexity

of Cooperation (1997), and Cederman’s EmergentActors in World Politics: How States and Nations

Develop and Dissolve (1997). John Holland’s(1995) suggestion that agent-based simulationserve as a "flight simulator" for public policyactors may also serve to move the nonlinearagenda forward.

Furthermore, the expanding array of methodsemployed by political scientists and public admin-istration scholars and the increasing level oftechnical sophistication of these methods suggestthat momentum for nonlinear analysis is increas-ing (See Richards, 2000). The increasing num-

ber of papers that examine agent-based behavior

(See Richards, 2000) also holds considerablepromise for meeting Epstein and Axtell’s challengeto build "social science from the ground up"(1996).

Finally, it is increasingly apparent that theevolution of life on the planet is a result ofnonlinear interactions that generate instabilities indissipative systems. These instabilities may resultin complexities in social systems behavior thatcomprise the very challenges of attempting tounderstand the vagaries of political and adminis-trative phenomena (Schieve and Allen, 1982). It isalso clear that understanding these complexitieswill be a major source of the power of nation-states in the future (Pagels, 1988). In a world inwhich information leaks and technology is readilytransported these social resources of informationand technology may expedite nonlinear amplifica-tions that threaten both local and global stability.Given this reality, political scientists and publicadministrationists are likely to have an increasinginterest in studying the nonlinear dynamics of a

world in which nonlinearity poses both an

opportunity for and a threat to institutions,nation-states and the global community.

278 L.D. KIEL

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