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Journal-Mathematical and Quantitave Methods

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Presentation of the Content

In the first chapter we present, Production optimization by using integer linear programming

under a practical approach, by VELARDE-CANTÚ, José Manuel, LÓPEZ-ACOSTA, Mauricio,

CHACARA-MONTES, Allán and RAMÍREZ-CÁRDENAS, Ernesto, with adscription in the

Instituto Tecnológico de Sonora, as a second article we present, Imitation as a social norm that

influences the choice of career and labor participation of women, by QUINTERO-ROJAS, Coralia A.

& VIIANTO, Lari A., with adscription in the Universidad de Guanajuato, as the following article we

present Learning the parabola through semiotic records in university students, by ENCINAS-

PABLOS, Francisco Javier, PERALTA-GARCÍA, Julia Xochilt, CUEVAS-SALAZAR, Omar and

OSORIO-SÁNCHEZ, Mucio, with affiliation at the Instituto Tecnológico de Sonora, as next article

we present Knowledge friction in universities, approach from game theory, by TALAVERA-RUZ,

Marianela, LARA-GÓMEZ, Graciela and VALDEZ-RESÉNDIZ, Macario, with affiliation at the

Universidad Tecnológica de Querétaro.

Content

Article Page

Production optimization by using integer linear programming under a practical approach

VELARDE-CANTÚ, José Manuel, LÓPEZ-ACOSTA, Mauricio, CHACARA-MONTES,

Allán and RAMÍREZ-CÁRDENAS, Ernesto

Instituto Tecnológico de Sonora

1-7

Imitation as a social norm that influences the choice of career and labor participation of

women

QUINTERO ROJAS, Coralia A. & VIIANTO, Lari A.

Universidad de Guanajuato

8-15

Learning the parabola through semiotic records in university students

ENCINAS-PABLOS, Francisco Javier, PERALTA-GARCÍA, Julia Xochilt, CUEVAS-

SALAZAR, Omar and OSORIO-SÁNCHEZ, Mucio

Instituto Tecnológico de Sonora

16-25

Knowledge friction in universities, approach from game theory

TALAVERA-RUZ, Marianela, LARA-GÓMEZ, Graciela and VALDEZ-RESÉNDIZ,

Macario

Universidad Tecnológica de Querétaro

26-50

1

Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 1-7

Production optimization by using integer linear programming under a practical

approach

Optimización de la producción mediante la utilización de la programación lineal

entera bajo un enfoque práctico

VELARDE-CANTÚ, José Manuel†, LÓPEZ-ACOSTA, Mauricio, CHACARA-MONTES, Allán and

RAMÍREZ-CÁRDENAS, Ernesto

Instituto Tecnológico de Sonora, Mexico.

ID 1st Author: José Manuel, Velarde-Cantú / ORC ID: 0000-0002-1697-8551

ID 1st Coauthor: Mauricio, López-Acosta / ORC ID: 0000-0003-3728-9576, Researcher ID Thomson: X-4274-2019

ID 2nd Coauthor: Allán, Chacara-Montes / ORC ID: 0000-0002-0567-0017

ID 3rd Coauthor: Ernesto, Ramírez-Cárdenas / ORC ID: 0000-0002-5248-724X

DOI: 10.35429/JMQM.2020.7.4.1.7 Received July 10, 2020; Accepted December 30, 2020

Abstract

This paper addresses the problem of production

scheduling under a practical approach, which seeks to

find out what would be the product mix to ensure the

company to obtain the most useful, also requires that

these combinations of products obtained from quickly

and efficiently contributing thus to achieve lower costs

associated with production. A specific mathematical

model based on integer linear programming applied

specifically to the product mix is presented, as well as the

results obtained from the practical problem from the use

of the model in integer linear programming, the use of

the software and considering the own conditions of the

problem addressed here.

Production scheduling, PL, Optimization, Product

mix

Resumen

El presente trabajo aborda el problema de la

programación de la producción bajo un enfoque práctico,

el cual busca encontrar cual sería la mezcla de productos

que garantice a la empresa obtener la mayor utilidad, así

mismo se requiere que estas combinaciones de productos

se obtengan de manera rápida y eficiente coadyuvando

así a lograr disminuir los costos asociados a la

producción. Se presenta un modelo matemático

especifico basado en programación lineal entera aplicado

específicamente al de mezcla de productos, así mismo se

muestra los resultados obtenidos del problema practico a

partir de la utilización modelo en programación lineal

entera, la utilización del software y considerando las

condiciones propias del problema aquí abordado.

Programación de la producción, PL, Optimización,

Mezcla de productos

Citation: VELARDE-CANTÚ, José Manuel, LÓPEZ-ACOSTA, Mauricio, CHACARA-MONTES, Allán and RAMÍREZ-

CÁRDENAS, Ernesto. Production optimization by using integer linear programming under a practical approach. Journal-

Mathematical and Quantitative Methods. 2020. 4-7:1-7.

† Researcher contributing as first author.

© RINOE – Spain www.rinoe.org/spain

2


VELARDE-CANTÚ, José Manuel, LÓPEZ-ACOSTA, Mauricio, CHACARA-

MONTES, Allán and RAMÍREZ-CÁRDENAS, Ernesto. Production optimization

by using integer linear programming under a practical approach. Journal-

Mathematical and Quantitative Methods. 2020

ISSN-On line: 2531-2979

RINOE® All rights reserved.

Introduction

Currently, the implementation of tools based on

hard methodologies that seek to make each area

or process more efficient in an organization has

been increasing, these tools are extremely

important in the search for optimization in the

different activities found in each process such

as: warehouse, production, distribution, etc. as

well as in the rest of the areas of the

organization.

With the implementation of the

operations research methodology, it is sought to

verify the hypothesis based on information and

numerical measurements as well as with the

statistical analysis with which different patterns

of behavior can be established considering the

correct use of the organization's available

resources. The decrease in inventory levels,

satisfying demand in a timely manner, in

production, manufacturing only what is

necessary, making these adjustments will

improve the levels of profit provided by the

sales of the different products handled by the

company (Pérez et al, 2017)

When making a decision it is always

very important to have sufficient and quality

information to help us decide on each problem

that may arise in the company, that is why we

need tools, methodologies that provide

information in a systematic way and relevant to

the system under study. Within these tools and

methodologies you can find the entire linear

programming, which is used in multiple real-

life optimization problems, among which the

problem addressed here is identified, which

seeks to find the best combination of

manufacturing products that satisfy each and

every one of the conditions of the problem,

those of available resources such as raw

materials, labor, available time, with the main

objective of maximizing the profits obtained

from the sale of these products.

Considering that the problem faced here

has been widely studied by a large number of

authors and that the product mix optimization

problem is known in the literature, this problem

contemplates small variations compared to its

original form, one of them would be production

of whole units and would be given by the

integer linear programming (PLE), another

variation is to consider the production as

continuous or the products to be produced are

in bulk, etc.

The problem studied here is considered

typical in the literature and its large study does

not delimit the different and varied applications

that it may have, a clear example can be seen in

Krajewski et al., (2008) where they use this

problem to model the programming of

production with the objective of finding the

product mix in a given period, in this modeling

consider market demand restrictions and

available resources. This type of application to

this problem gives us the confidence to

continue developing new mathematical models

to search for optimal solutions.

The main objective of this research is to

build a solution to the problem addressed in a

systematic way that contributes to the increase

in profits. What is expected is to develop an

optimization model according to the available

manufacturing resources, considering the

conditions of its production system, thus

achieving a valuable contribution in the

optimization area by designing and / or

establishing a mathematical model based on

integer linear programming. according to the

special characteristics of the analyzed system.

Methodology

In the literature there is a great variety of

different techniques focused on the search for

the solution to the problem analyzed here, one

of them is the mixed integer linear

programming proposed in the work of Knaien

et al., (2016) which uses it with the main

objective of minimizing the sum of total

production costs considering in them those

related to the maintenance of the machines as

well as the effect of these in the total profit of

the company

In Taebok et al., (2018) propose

deterministic models to define the production

scheduling where the producer wants to know

the quantity to be manufactured of each product

in each of the different machines, which have a

greater capacity than the demand. and where it

is necessary to define the combination of

machines to be used that optimizes operating

costs

In Motta et al., (2016) presents a model

based on mixed integer linear programming to

find the solution to the production scheduling

problem using different facilities.

3








Among the objectives that this work

seeks to achieve are those of meeting demand,

reducing inventory costs and transportation

costs, as well as increasing the use of different

production facilities.

In the literature there are multiple

applications of mixed integer linear

programming, one of them addresses the

problem of production scheduling in flexible

job-shop environments where it considers the

divisions of manufacturing jobs into smaller

sublots, the mathematical model implemented

allows to define if the division of the batches is

necessary, considering an established measure

of development and satisfying the set of

restrictions given by the problem addressed.

The proposal defines the start and end time for

each manufacturing operation that is carried out

on the sublots, in the same way it allocates

these sublots to the appropriate production

equipment (Novas, 2016).

In Capitanescu et al., (2017) the use of

heuristic techniques can be found, which are

widely used for the search for the solution for

this type of problems, in this work the author

proposes a special algorithm using a hybrid

evolutionary method with the main objective of

efficiently solving the problem of production

scheduling under a bi-objective approach.

In Ortiz et al. (2015), an application of

the theory of constraints based on linear

programming can be observed to solve the

problem of production programming, where

through the identification of the different

restrictions and the appropriate approach of the

mathematical model it helps to define the mix

of products to be manufactured that maximizes

profit. Another application of the theory of

constraints to the production scheduling

problem can be found in Romero et al. (2019),

where he uses this technique to achieve an

improvement in the performance of the

production system by determining the

quantities to be manufactured of each product

and the production sequence in the company.

In the work presented by Silva et al.,

(2017), concentrates a great variety of

optimization techniques applied to the problem

of production scheduling, performs a

classification in two groups for these models,

group one in deterministic and group two in

stochastics, in the first they used exact solution

techniques such as linear programming, mixed

integer linear programming, and branching and

boundary algorithms, for those of the second

group they used approximation methods and

stochastic programming, they also mention in

their review that the 82% of the authors used

deterministic models, which provides evidence

of a notable growth in the use of exact solution

techniques.

Based on an extensive analysis of

methods and tools used in the literature, the

decision was made to apply integer linear

programming (PLE) as the best option to meet

the objectives sought by the company, in the

same way the software solver the which is

found in the add-ins option in Excel, that is, due

to its great ease of understanding and applying

it. While it is true that the methodology to be

used is an exact solution search technique

which on some occasions may present some

disadvantages due to the size of the problems,

although for this particular case it does not

present this disadvantage, that is, because the

the problem is considered small, that is why it

will help us find the solution quickly and

efficiently, that is, it will provide us with the

optimal solution. In contrast to heuristic

programming techniques, which help us to find

a solution when the problem is very large or

when the search for this solution takes too long,

for our practical case the use of evolutionary

algorithms was not necessary since the model

the mathematical used and the size of the

problem favored the use of techniques based on

exact algorithms, thus helping the company to

have a quick, efficient solution that guarantees

the maximum return on its profits.

The mathematical model based on PLE

was implemented specifically applied to the

product mix problem, this is because it

currently does not have a methodology that

systematically provides the different quantities

to be produced of each item that allow the

company to maximize its profits.

4








The different parameters, indices and

variables used in the model are defined to later

define the objective function as well as the set

of restrictions that the problem contemplates.

The following section presents the

mathematical model based on PLE applied to

the practical problem addressed in this work.

Mathematical model

Indices:

i: Index used to identify the process of production 𝑖 ∈ {1, … , 𝑚}

𝑗: 𝐼𝑛𝑑𝑒𝑥 𝑢𝑠𝑒𝑑 𝑡𝑜 𝑖𝑑𝑒𝑛𝑡𝑖𝑓𝑦 𝑒𝑎𝑐ℎ 𝑡𝑦𝑝𝑒

𝑜𝑓 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑠 𝑡𝑜 𝑏𝑒 𝑚𝑎𝑛𝑢𝑓𝑎𝑐𝑡𝑢𝑟𝑒𝑑 𝑗∈ {1, … , 𝑛}

Parameters:

𝑚: 𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑟𝑜𝑐𝑒𝑠𝑠𝑒𝑠 𝑢𝑠𝑒𝑑 𝑜𝑛 𝑡ℎ𝑒 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑖𝑜𝑛 𝑙𝑖𝑛𝑒 𝑛: 𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑝𝑟𝑜𝑑𝑢𝑐𝑡𝑠 𝑡𝑜 𝑚𝑎𝑛𝑢𝑓𝑎𝑐𝑡𝑢𝑟𝑒 𝑃𝑖𝑗: 𝑇𝑖𝑚𝑒 𝑖𝑛 𝑚𝑖𝑛𝑢𝑡𝑒𝑠 𝑢𝑠𝑒𝑑 𝑖𝑛 𝑡ℎ𝑒 𝑝𝑟𝑜𝑐𝑒𝑠𝑠 𝑖 𝑡𝑜 𝑚𝑎𝑛𝑢𝑓𝑎𝑐𝑡𝑢𝑟𝑒 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑗 𝑇𝑖: 𝑇𝑜𝑡𝑎𝑙 𝑡𝑖𝑚𝑒 𝑖𝑛 𝑚𝑖𝑛𝑢𝑡𝑒𝑠 𝑎𝑣𝑎𝑖𝑙𝑎𝑏𝑙𝑒 𝑡𝑜 𝑑𝑎𝑦 𝑖𝑛 𝑒𝑎𝑐ℎ 𝑝𝑟𝑜𝑐𝑒𝑠𝑠 𝑖 𝐷𝑗: 𝑀𝑖𝑛𝑖𝑚𝑢𝑚 𝑑𝑎𝑖𝑙𝑦 𝑑𝑒𝑚𝑎𝑛𝑑 𝑡𝑜 𝑝𝑟𝑜𝑑𝑢𝑐𝑒 𝑜𝑓 𝑒𝑎𝑐ℎ 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑗 𝐺𝑗: 𝑃𝑟𝑜𝑓𝑖𝑡 𝑜𝑓 𝑒𝑎𝑐ℎ 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑗

Decision variable:

𝑋𝑗: 𝐴𝑚𝑜𝑢𝑛𝑡 𝑜𝑓 𝑝𝑟𝑜𝑑𝑢𝑐𝑡 𝑗 𝑡𝑜 𝑏𝑒 𝑝𝑟𝑜𝑑𝑢𝑐𝑒𝑑 𝑝𝑒𝑟 𝑑𝑎𝑦

Objective Function:

𝐹. 𝑂 𝑀𝑎𝑥 𝑍 = ∑ 𝐺𝑗𝑋𝑗𝑛 𝑗=1 (1)

Equation 1. Objective function which

calculates the maximum profit with the product

mix

Subject to:

∑ 𝑃𝑖𝑗𝑋𝑗𝑛 𝑗=1 ≤ 𝑇𝑖 ∀ i ∈ {1, … , m} (2)

Equation 2. Calculate not to exceed the

times available in each process for the

manufacture of the products.

∑ 𝑋𝑗

𝑛𝑗=1 ≥ 𝐷𝑗 (3)

Equation 3. It guarantees us to meet the

minimum demand for each type of product

manufactured.

6𝑋𝑗−1 − 𝑋𝑗 = 0 ∀ 𝑗 ∈ {3} (4)

𝑋𝑗−1 − 𝑋𝑗 = 0 ∀ 𝑗 ∈ {9} (5)

2𝑋𝑗−2 − 𝑋𝑗 = 0 ∀ 𝑗 ∈ {10} (6)

Equations 4, 5 and 6 are production

restrictions according to the types of products

to be manufactured

𝑋𝑗 ≥ 0, ∀ 𝑗 ∈ {1, … , 𝑛}, 𝑒𝑛𝑡𝑒𝑟𝑎 (7)

Equation 7, establishes the non-negative

and integer variables for this model.

Results

In this research work, the problem of

production scheduling is considered,

specifically that of product mix, for which a

mathematical model based on PLE was

developed in which it is sought to solve the

central research hypothesis.

The results obtained with this

investigation define the values of the decision

variables considered in the model, as well as

compliance with the series of restrictions

imposed by the problem addressed that together

form the optimal solution to the practical

problem.

The results obtained from the problem

under study are presented, which should show

the optimal values of the variables that

maximize the profits of the company. For this,

relevant data of the problem were considered,

which were obtained considering the needs and

demands of the company where the study was

carried out.

5








The data used to find the solution to the

real problem are: 10 different products to be

manufactured were taken into account, a total

of 15 manufacturing processes were defined

through which each product must pass, the

company works two 8-hour shifts daily which

gives us a total of 960 minutes in both shifts,

tables 1 and 2 show the times required to use

each manufacturing process for each product,

these times were obtained directly from the

engineering department where the standard

times are considered of each manufacturing

process of each of the products considered in

the study, in table 3 you can see the minimum

demand that each of the products has, which

must be satisfied daily, as well as the expected

profit of each one of the products which is

obtained from the difference between the sale

price and the production cost associated with

each of them, in table 4 you can see e l total

time available in minutes for each production

process.

{𝑷𝒊𝒋} {j}

{i} 1 2 3 4 5

1 0.4 0.18 0.12 0.3 0.24

2 1 0.45 0.3 0.75 0.6

3 1.4 0.63 0.42 1.05 0.84

4 1.4 0.63 0.42 1.05 0.84

5 1.8 0.81 0.54 1.35 1.08

6 1.6 0.72 0.48 1.2 0.96

7 1.2 0.54 0.36 0.9 0.72

8 0.6 0.27 0.18 0.45 0.36

9 1.2 0.54 0.36 0.9 0.72

10 0.8 0.36 0.24 0.6 0.48

11 1 0.45 0.3 0.75 0.6

12 1.8 0.81 0.54 1.35 1.08

13 1.6 0.72 0.48 1.2 0.96

14 2.4 1.08 0.72 1.8 1.44

15 1.8 0.81 0.54 1.35 1.08

Table 1 Work times in each process {i} per product {j}

in minutes {Pij}

{𝑃𝑖𝑗} {j}

{i} 6 7 8 9 10

1 0.32 0.26 0.18 0.2 0.24

2 0.8 0.65 0.45 0.5 0.6

3 1.12 0.91 0.63 0.7 0.84

4 1.12 0.91 0.63 0.7 0.84

5 1.44 1.17 0.81 0.9 1.08

6 1.28 1.04 0.72 0.8 0.96

7 0.96 0.78 0.54 0.6 0.72

8 0.48 0.39 0.27 0.3 0.36

9 0.96 0.78 0.54 0.6 0.72

10 0.64 0.52 0.36 0.4 0.48

11 0.8 0.65 0.45 0.5 0.6

12 1.44 1.17 0.81 0.9 1.08

13 1.28 1.04 0.72 0.8 0.96

14 1.92 1.56 1.08 1.2 1.44

15 1.44 1.17 0.81 0.9 1.08

Table 2 Work times in each process {i} per product {j}

in minutes {Pij}

{𝑿𝒋} {Gj} {Dj}

1 20099 4

2 9300 6

3 2505 6

4 14099 4

5 8700 6

6 9000 5

7 13200 6

8 5100 6

9 3299 6

10 1579.8 6

Table 3 Gain {Gj} and minimum demand {Dj} for each

type of product {Xj}

Process {i} {Ti}

1 19.2

2 48

3 67.2

4 67.2

5 86.4

6 76.8

7 57.6

8 28.8

9 57.6

10 38.4

11 48

12 86.4

13 76.8

14 115.2

15 86.4

Table 4 Available time {Ti} to manufacture in each

process {i}

The results obtained with the EXCEL

SOLVER software are presented below, which

suggested that it is necessary to consider that

the decision variables take the following values,

see Table 5 in order to maximize profit.

{𝑿𝒋} Value obtained by the Optimizer in units

to produce

1 7

2 6

3 36

4 4

5 6

6 5

7 6

8 6

9 6

10 12

Table 5 Solution obtained through the Excel Solver

6








It can be observed (Table 5) the

different values for the decision variables, these

values show that for product 1 we must produce

7 units, for product 2 we must produce 6, for

product 3 we must produce 36 and so on until

reaching to product 10 which must have a

production of 12 units per day in order to obtain

a maximum profit of 588820.6 pesos per day

for the production and sale of the products

manufactured by the company.

Acknowledgments

The work team thanks all the participants for

their interest and collaboration during the data

collection. Thanks also to the university

(Instituto Tecnológico de Sonora) and the

company for their support and the facilities for

the development of the project; This publication

has been financed with resources from

PROFEXCE 2020.

Conclusions

It can be seen that thanks to the correct

implementation of the mathematical model

based on PLE applied to the production

scheduling problem specifically to the product

mix problem in this company, it helped to find

a solution that met all the restrictions, thereby

maximizing profits. Likewise, through this, it

was possible to improve the use of available

resources to produce.

It was also possible to establish a

methodology that provides the company with a

solution to the problem addressed in a

systematic way by defining the amount of

products that must be produced per day,

considering that currently the company does not

use any methodology that helps define the

production results. Presented here are even

more relevant since it provides this product mix

through a methodology based on a quantitative

method, which ensures that we have sufficient

and quality information to make the best

decision that helps the company find the

maximum profit considering the use of

available resources such as: capital, availability

of labor and raw materials as well as the

conditions of the problem addressed.

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Silva Rodríguez, Julián, Díaz Cárdenas,

Camilo, Galindo Carabalí, Julián. (2017).

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Industrial. Actualidad y Nuevas Tendencias,

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https://doi.org/10.1016/j.ijpe.2017.11.018.

8


Imitation as a social norm that influences the choice of career and labor

participation of women

La imitación como norma social que influencia la elección de carrera y de

participación laboral de la mujer

QUINTERO ROJAS, Coralia A.†* & VIIANTO, Lari A.

Universidad de Guanajuato, Economics & Finance, Mexico.

ID 1st Author: Coralia A., Quintero-Rojas / ORC ID: 0000-0003-3994-1775, CVU CONACYT ID: 36503

ID 1st Coauthor: Lari A., Viianto / ORC ID: 0000-0002-8681-3744, CVU CONACYT ID: 343523


Abstract

In Latin America, labor market indicators still show large

gender gaps in access to opportunities and rights. These

inequalities persist despite various policy efforts because

they emanate from a social system that reproduces

stereotypes and gender roles. This contradicts the

development vision of the United Nations and the ILO:

“Without gender equality, sustainable development is

neither development nor sustainable”. This contribution

focuses on the impact of social norms on women's

decisions, regarding career choice and labor participation.

To this end, we have built and simulated an agent-based

model. The model assumes that gender roles are acquired

and reproduced through the environment that surrounds

women. The results of this social simulation suggest that

the environment greatly influences the perception of

women about the roles they must assume and the areas in

which they must play, so it is pertinent to design policies

aiming to change the conceptions of identity and

traditional gender roles, since the construction and

adoption of new and more equitable values is in part

acquired through the imitation of the behavior observed

in a woman's environment.

Social norms, gender inequality, agent-based

modeling

Resumen

En América Latina, los indicadores del mercado laboral

aún muestran grandes brechas de género en el acceso a

oportunidades y derechos. Estas desigualdades persisten

pese a los diversos esfuerzos de política, porque emanan

de un sistema social que reproduce estereotipos y roles de

género. Esto contradice la visión de desarrollo de las

Naciones Unidas y la OIT: “Sin igualdad de género, el

desarrollo sostenible no es desarrollo ni es sostenible”.

Esta contribución se centra en el impacto de las normas

sociales en las decisiones de las mujeres, en cuanto a la

elección de carrera y la participación laboral. Con este

fin, hemos construido y simulado un modelo basado en

agentes. El modelo asume que los roles de género se

adquieren y reproducen a través del entorno que rodea a

las mujeres. Los resultados de esta simulación social

sugieren que el entorno influye en gran medida en la

percepción de las mujeres sobre los roles que deben

asumir y los ámbitos en los que se deben desempeñar,

por lo que es pertinente diseñar políticas que busquen

cambiar las concepciones de identidad y roles de género

tradicionales, ya que la construcción y adopción de

nuevos valores más equitativos, se logra en parte a través

de la imitación del comportamiento observado en el

entorno de la mujer.

Normas sociales, desigualdad de género, modelación

basada en agentes

Citation: QUINTERO ROJAS, Coralia A. & VIIANTO, Lari A. Imitation as a social norm that influences the choice of

career and labor participation of women. Journal-Mathematical and Quantitative Methods. 2020. 4-7:8-15.

* Correspondence to Author (Email: [email protected]).



9



Imitation as a social norm that influences the choice of

career and labor participation of women. Journal-




Introduction

In Latin America, labor market indicators

continue to show large gender gaps in access to

opportunities and rights between men and

women, and this gap has closed only slightly in

the last decade. Gender inequalities persists

despite the various policy efforts of the

governments of the region because they arise

from a social system that

reproduces gender stereotypes and roles. These

structural factors limit women's labor insertion

and represent an obstacle to overcoming

poverty and inequality , as well as to achieving

economic autonomy for women; This problem

has been exacerbated by the global contraction

in growth of last years and has been aggravated

by the economic slowdown derived from the

current COVID-19 pandemic.

Despite the heterogeneity in the region,

most countries exhibit gender gaps in the

employment and occupation rates and, most

notably, in the participation rate. In Mexico, the

total participation rate in 2018 has been around

three percentage points below the Latin

American average, standing respectively at

61.1% and 64.1%. This indicates a high labor

force participation in both cases, but the rates

by gender show large contrasts, both between

regions and within the composition of the labor

force. For example, in the decade from 1998 to

2007, the gender gap in Mexico was

42.1%; that is, only 39.98% of women

participated in the labor force; that is, less than

half of the participation of men (82.08%). In

contrast, the gender gap in Latin America and

the Caribbean was smaller (30.04%), because

the male participation rate was like in Mexico,

but the female participation rate was 9% higher

than in Mexico. In both cases, the gender gap

narrowed over time, basically due to the

increasing participation of women. However, in

2018 the gap is still considerable: 35.1% in

Mexico and 26.5% in Latin America and the

Caribbean.1 This suggest that neither the

market nor public policies have been able to

recruit and retain in the labor market a group of

women who still facing obstacles to improving

their economic conditions and those of their

families; This dynamic is incompatible with the

United Nations and ILO’s vision of

development:

1 Rates obtained from data from the International Labor

Office, ILO Econometric Trend Models

(ilo.org/wesodata).

“Without gender equality, sustainable

development is neither development nor

sustainable” (OIT, Notas para la Igualdad

No. 22, 2017)

The underlying problem is that women

access the public sphere in inferior conditions,

whether economic, social, or cultural

(Astelarra, 2005). They also participate in

segregated spaces; that is, electing jobs,

occupations or professions traditionally

considered “feminine occupations”, which are

characterized by lower social and monetary

value than the “male ones”. In addition, their

participation is also shaped by the place they

occupy in the private sphere and by their roles

as caretakers and unpaid domestic workers.

Since the burden of these home-activities are

not modified, women support double working

hours, with all the difficulties and costs that this

implies. (CEPAL, 2019).

In a wider scope, the Gender Inequality

Index (GII), which captures inequalities that

women face in reproductive health, political

representation and the labor market (1 indicates

complete inequality and 0 indicates perfect

equality) had a total value of 0.441 in

2017 (UNDP, 2018). Likewise, in 2018 the

Global Gender Gap Index (GGGI) ,

which examines the gap between men and

women in four fundamental categories (1

means parity): participation and economic

opportunity, educational attainment, health and

survival, and political empowerment was of

68%, which means that globally there was still

a 32% gap to close. (World Economic Forum,

2018). Finally, in 2017 the average global value

of the Human Development Index (HDI) for

women (0.705) was 4.4% lower than that of

men (UNDP, 2018). In short, large gender

inequalities persist, especially in the political

and economic spheres. These inequalities are

due in part to gender discrimination, which

derives both from the law and from practice. In

the first case, legal and political institutions are

used to perpetuate gender divisions, denying

women access to the same legal rights as

men. In the second case, culture and

tradition are translated into social norms that

significantly influence the configuration of

gender roles. Thereby, the norms and traditions

that distribute most of the unpaid work in the

home to women, limit their participation in the

labor market and even the

girls' access to education (UNDP, 2015).

10








Given that the problem has several

dimensions, this contribution focuses on the

impact of social norms on women 's decisions,

regarding career choice and participation in the

labor market. As social norms encompass a

variety of behaviors and attitudes, in this work

we will simulate a social experiment to study

how a woman's social environment affects her

decision, whether to choose a career, or to

participate or not in the labor market. Our

working hypothesis is the following:

Hypothesis: When a woman is

surrounded by a certain percentage of women

who participate in the labor market, she will be

more likely to participate as well. Similarly,

being surrounded by women with professions

other than those traditionally considered

female, encourages women to choose areas

usually delegated to men. On the contrary,

when the prevailing social norm is that women

work in the private sphere, women tend not to

participate in the labor market or to train

professionally in non-traditional fields, thus

reproducing gender inequalities.

Because of the social nature of this

phenomenon, in our analysis we will use the

Agent Based Modelling. This approach is a

form of computational simulation that allows

creating, analyzing and experimenting with

artificial worlds made up of heterogeneous

agents; the basic principle is to provide the

various types of agents with simple rules of

behavior, in order to investigate how the

interactions among themselves, and

between the agents and their environment, add

up to form the patterns seen in the real world

(Hamill and Gilbert, 2016). The use of this

approach has become popular in the social

sciences, as it allows the analysis of a complex

phenomenon in a simplified representation of

social reality (Wilensky and Rand, 2015),

avoiding the difficulties and ethical

problems that would arise from carrying out

social experiments in the real world (Gilbert,

2008). Since the social norm regarding women’

choices implies the adoption of the cultural

costumes that are observed in their social

environment, we will model it as an imitation

rule. In the case of the labor market

participation, the imitation rule will depends on

the percentage of women in the environment

who carry out activities in the public sphere;

and, in the case of career choice, by the

percentage of women with no-traditionally

female professions.

Our contribution adds to the social

research that seeks to identify the underlying

causes of a phenomenon, which is very

important for the correct design of adequate or

pertinent policies that promote gender equality

and human development. Moreover,

achieving gender equality and the

empowerment of women is the Objective 5 of

the United Nations Agenda 2030 for

Sustainable Development (UNDP, 2016). Our

results suggest that the environment greatly

influences the perception of women about the

roles they must assume and the areas in which

they must play, so it is pertinent to design

policies aiming to change the conceptions of

identity and traditional gender roles, since the

construction and adoption of new and more

equitable values is in part acquired through the

imitation of the behavior observed in a woman's

environment.

The remainder of this document is

organized as follows. In the following section

we develop and simulate the agent-based

model. Next, we run OLS regressions with the

simulated data and discuss the results. Finally,

we present our conclusions.

The Model

The model is constructed in the NetLogo

platform, a free disposable software, provided

by Uri Willensky, that is specifically

constructed for agent-based simulation. The

world is populated by women and is

represented by a 33x33 square lattice where the

1089 patches are agents; each agent is related to

their 8 neighbours. They decide between

assuming a traditional role (such as developing

activities in the private sphere or choosing

"female" professions) or adopting a non-

traditional role (either developing activities in

the public sphere or choosing “male”

professions). We assume a mechanism of

choice strictly social that derives from the

neighbour’s behaviour.

A proportion of the population has

already made their choice. Those who chose a

traditional role are shown in pink, while those

who chose a non-traditional role are shown in

blue; finally, those who have not made their

choice yet are shown in white.

11








The location of the agents who have

already decided is random.

White (undecided) agents observe the

behaviour (that is, the colour) of their eight

neighbours; and each agent will make her

choice according to the most prominent

behaviour (given some threshold) in his

neighbourhood at that given moment. Once the

choice is made it is permanent. This process

continues until there are no more white agents

willing to make a decision. In other words,

undecided agents, which are chosen randomly,

may take a decision in an iterative process,

mimicking the behaviour observed in their

neighbourhood.

Simulations

We run several simulations were the initial

portion of agents who already has made their

choice, ranks from 5% to 50% of total

population (5, 10, 15, 20, 25, 30, 35, 40, 45 and

50%). Threshold ranks between 37.5% (3

neighbours) to 62.5% (5 neighbours) at steps of

12.5% (1 neighbour). The model is simulated

for the 300 different parameter configurations

resulting from these values.

As a matter of illustration, figure 1

shows in panel (a) the initial disposition of the

world, assuming that population is distributed

as: Pink agents: 10%; Blue agents: 10%; and

white agents: 80%. The threshold for white

agents to choose is settled at 37.5%, meaning

that a white agent imitates the behaviour

(colour) that is showed for more than three of

their neighbours.

(a)

(b)

Figure 1 (a) Initial disposition of the world. (b) Final

disposition of the world

Source: Own computations

The simulation starts from these values

and the model proceeds iteratively, randomly

choosing white agents, who check their

neighbourhood and imitate the most

predominant colour in it, according to the given

threshold. At each iteration, the disposition of

the world changes, reflecting the new choices

made, which in turn influences the decisions of

the white agents that will decide in the

subsequent iteration. The iteration stops when

all the white agents have chosen their role. The

panel (b) in Figure 1 shows the distribution of

the world at the end of our simulation, given

these initial parameter values, with the next

proportions: White agents: 0%; Pink agents:

47.4%; and Blue agents: 52.6%. The variations

of these percentage at each iteration is showed

in Figure 2.

The model is simulated for the 300

different parameter configurations considered.

Results are shown in next section.

Figure 2 Distribution of agents at the end of simulation

Source: Own computations

Results

Table 1 shows the correlation between the

initial and final share of each population (pink

or blue).

Initial

blue

Initial

Pink

Final

Blue

Final

Pink

Initial

blue

1.0000 0.0000 0.7486 -0.4506

Initial

Pink

0.0000 1.0000 -0.4495 0.7481

Final

Blue

0.7486 -0.4495 1.0000 -0.6450

Final

Pink

-0.4506 0.7481 -0.6450 1.0000

Table 1 Correlations between initial and final population

values

Source: Own computations from the simulations

0

100

200

300

400

500

600

700

800

900

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

White Blue Pink

12








As expected, for each population, initial

and final shares are strongly correlated; and

each initial population is negatively correlated

with the final share of the other type

population. Due to the symmetry of

simulations, correlations are nearly identical for

each type to himself and to the other type, and

any difference is statistically insignificant given

the randomness of the simulation process. Thus,

we made OLS regressions, using the whole

sample of 30,000 simulations, only for the blue

population, being the results also applicable to

the pink population.

Table 2 shows the results from

regressing the final blue population with respect

to the initial blue population and the threshold

value (as percentages).2

Dep. Var. Final

blue

Coefficient St. Deviation t-

statistic

p-value

Constant 0.368810 0.00510165 72.29 <0.0001

Initial blue 1.39492 0.00663239 210.3 <0.0001

Threshold −0.00640105 9.33858e-05 −68.54 <0.0001

R-Square 0.619953 Adj. R-

Square

0.619928

Table 2 OLS regression of the final blue population with

respect to the initial blue population and the threshold

level


This simple linear model adjusts the

data quite accurately, obtaining an adjusted R-

Square close to 0.62 and all estimated

coefficients are significant at the 1% confidence

level. The coefficient of the initial blue

population is nearly 1.4, implying a spread of

40% of their behaviour among white agents. As

expected, the threshold has a negative impact

on the imitation of behaviour, since it increases

the number of same-type neighbours required to

be imitated by white agents.

Separating for threshold levels

For a threshold of 37.5% (3 neighbours) the

adoption of the blue values for white agents is

easier (remember that the same will occur if we

consider the pink population); this generates a

higher variation in the end population, which is

inversely proportional to the population size, as

is shown in figure 3.

2 The initial pink population is omitted due to the by

construction collinearity with the initial blue

population.

Figure 3 Scatter plot of the initial blue population (ai)

and the final blue population (af), for a threshold of

37.5%


This yields a less accurate regression, as

is shown in table 3.

Dep. Var.

Final blue

Coefficient St. Deviation t-statistic p-value

Constant 0.165275 0.00437999 37.73 <0.0001***

Initial blue 1.21915 0.0141337 86.26 <0.0001***


Square

0.426615

Table 3 OLS regression of final blue population with

respect to initial blue population with a threshold level of

37.5%


As we can observe, the effect of the

initial population on the final population is

lower than before, representing an imitation

effect of roughly a 22% of the initial

population, but the adjusted R-Squared is only

0.42.

A higher threshold (50%, 4 neighbours)

reduces the variance of results, especially for

low levels of initial population, suggesting

more difficulties to influence the white

neighbours. According to the scatter plot in

figure 4, for initial shares of blue population

lower than 10 – 15 % it is harder to have the

concentration of blue neighbours required to be

imitated by white agents.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6

af

ai

13









and the final blue population (af), for a threshold of 50%


However, due to the less variance in the

results, the regression for this threshold level

yields to a slightly better fit of the model, as is

shown in table 4.

Dep. Var.

Final blue

Coefficient St.

Deviation

t-statistic p-value

Constant 0.0325045 0.00389538 8.344 <0.0001***

Initial blue 1.53975 0.0125699 122.5 <0.0001***


Square

0.600091



50%


In this case, the significance levels

remain very high, and the initial share of blue

population has a higher coefficient, yielding to

a final blue population 54% larger than the

initial one. The adjusted R-Square improves to

around 0.6.

Finally, the results for the largest

threshold used in the simulations, 62.5% (5

neighbours), indicate that it is much more

difficult the adoption of the behaviour of blue

agents, since white agents need to have more

than 5 blue neighbours to imitate their

behaviour. In this case, the variance in the final

population is further reduced, and it is needed

an initial blue population higher than 20% to

start noticing the imitation process (see figure

5).


and the final blue population (af), for a threshold of

62.5%


Observe that for an initial population of

5% there is not imitation, and any possible

effect remains quite insignificant up to an initial

population of 25%; for an initial blue

population of 35% and more, there is a notable

expansion of blue population (over 50% of the

final population); this is explained by the

density of the blue population at the beginning

of the process. The lower variance produces a

better fit, but the effect of the initial population

on the final population is lower. Results are

reported in table 5.

Dep.

Var.

Final

blue

Coefficient St.

Deviation

t-

statistic

p-value

Constant −0.0515078 0.00166524 −30.93 <0.0001***

Initial blue

1.42587 0.00537351 265.4 <0.0001***


Square

0.875648



62.5%


The constant has changed sign but still

be highly significant; however, its relevance

diminishes considerably. The model adjustment

is also higher, over 0.87. Finally, the effect of

the initial population is slightly lower but

implies an increase of the blue population

42.6% with respect to the initial population; so,

even if the higher threshold impede the

imitation process, the total effect still

remarkable.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6

af

ai

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6

af

ai

14








Conclusions

In this work we have constructed and simulated

an Agent Based Model to shed light on the

social causes of gender inequality observed in

various areas, such as participation in the labour

market or career choice. The model assumes

that gender roles are acquired and reproduced

through the environment that surrounds women.

The results show that as the population density

increases, the values of each type of population,

whether traditional or less traditional, are more

easily adopted by the population of women who

are about to decide their future (white agents).

Every time a white agent adopts the

values that he observes in his environment, his

decision not only affects him, but also, by

increasing the number of agents of that type of

population, the probability of change of the

remaining white agents is influenced. This is

due to the fact that this social phenomenon

constitutes a complex system in which the

agents are interdependent and the strategies of

one agent affect the strategies of the others;

moreover, the macroeconomic pattern that

emerges from these interactions between

agents, and between agents and their

environment, is more than the sum of individual

actions. Specifically, a complex system is an

organized set of interrelated elements and

processes whose dynamic interaction over time

produces behaviours and macroscopic

regularities (or emergent properties) that cannot

be deduced linearly from the analytical

knowledge of its parts. Thereby, the results of

the regressions generally point to a diffusion-

imitation effect of the values of a group, even

when its initial population is low.

The results of our social simulation

suggest that the environment largely influences

the perception of women about the roles that

they have to assume and the spheres in which

they have to play, so that any gender equality

policies must take into account the social

organization that forms the basis of

discrimination against women, as well as their

role in that social organization. Thus, by

intervening to change the conceptions of

identity and traditional gender roles, the

construction and adoption of new and more

equitable values will be achieved through the

imitation of the behaviour observed in a

woman's environment.

Annexe. The NetLogo program code

Globals [blancosiniciales azulesiniciales rosasiniciales bi%

ai% ri% all blancos1 azules1 rosas1 bf% af% rf%]

patches-own [change? go? vecinosrosas vecinosazules

vecinosblancos]

to set-up

ca

ask patches [set pcolor white]

let p count patches

let r round p * rosas / 100

let a round p * azules / 100

ask n-of r patches [set pcolor pink]

ask n-of a patches with [pcolor = white][set pcolor blue]

iniciales

reset-ticks

end

to iniciales

set blancosiniciales count patches with [pcolor = white]

set azulesiniciales count patches with [pcolor = blue]

set rosasiniciales count patches with [pcolor = pink]

set all count patches

set bi% blancosiniciales / all

set ai% azulesiniciales / all

set ri% azulesiniciales / all

end

to valores

set blancos1 count patches with [pcolor = white]

set azules1 count patches with [pcolor = blue]

set rosas1 count patches with [pcolor = pink]

set bf% blancos1 / all

set af% azules1 / all

set rf% rosas1 / all

end

to contar

ask patches [set vecinosrosas count neighbors with [pcolor =

pink] set vecinosazules count neighbors with [pcolor = blue]

set vecinosblancos count neighbors with [pcolor = white]]

end

to cambiar

ask patches [set change? false set go? false]

ask patches with [pcolor = white][set change? true]

while [any? patches with [change? = true]] [

ask one-of patches with [change? = true] [

set change? false

if (vecinosrosas / 8) * 100 >= porcentaje or

(vecinosazules / 8) * 100 >= porcentaje [

if vecinosrosas > vecinosazules [set pcolor pink set go?

true]

if vecinosrosas < vecinosazules [set pcolor blue set go?

true]

if vecinosrosas = vecinosazules [set pcolor one-of [pink

blue] set go? true]

]]]

end

to continuo

contar

cambiar

valores

tick

if all? patches [go? = false][stop]

end

15








References

Astelarra, J. (2005). Veinte años de políticas de

igualdad, Madrid, Ediciones Cátedra.

Comisión Económica para América Latina y el

Caribe (CEPAL) (2019), “Planes de igualdad

de género en América Latina y el Caribe:

mapas de ruta para el desarrollo”, Observatorio

de Igualdad de Género en América Latina y el

Caribe. Estudios, Nº 1 (LC/PUB.2017/1-

P/Rev.1), Santiago.

Gilbert, N. (2008). Agent-based models

(quantitative applications in the social

sciences). Thousand Oaks, CA: SAGE

Publications, Inc.

Hamill, L., and Gilbert, N. (2016). Agent-based

modelling in economics. Chichester: Wiley.

Oficina Internacional del Trabajo (OIT),

Modelos econométricos de tendencias de la

OIT. Recuperado el 25 de septiembre de 2019

de http://ilo.org/wesodata

Oficina Internacional del Trabajo (OIT) (2017),

Notas para la Igualdad No. 22.

United Nations Development Program

(UNDP). (2015). Human development report

2015: Work for human development. New

York. Recovered from

http://hdr.undp.org/sites/default/files/2015_hum

an_development_report_1.pdf


(UNDP). (2016). Human development report

2016: Human development for everyone. New



an_development_report.pdf


(UNDP). (2018). Human development indices

and indicators 2018 statistical update. New



an_development_statistical_update.pdf

Wilensky, U., and Rand, W. (2014).

Introduction to agent-based modeling.

Cambridge, MA: MIT Press.

World Economic Forum. (2018). The global

gender gap report 2018. Recovered from

http://www3.weforum.org/docs/WEF_GGGR_

2018.pdf.

16


Learning the parabola through semiotic records in university students

Aprendizaje de la parábola mediante registros semióticos en estudiantes

universitarios

ENCINAS-PABLOS, Francisco Javier†, PERALTA-GARCÍA, Julia Xochilt, CUEVAS-SALAZAR,

Omar and OSORIO-SÁNCHEZ, Mucio

Instituto Tecnológico de Sonora. Cd. Obregón Sonora México, 5 de febrero 818 Sur, CP 85000, Mexico

ID 1st Author: Francisco Javier, Encinas-Pablos / ORC ID: 0000-0003-3859-680X, Researcher ID Thomson: S-6944-

2018, CVU CONACYT ID: 947291

ID 1st Co-author: Julia Xochilt, Peralta-García / ORC ID: 0000-0002-2598-5825, arXiv ID Author: 2764706, CVU

CONACYT ID: 89800

ID 2nd Co-author: Omar, Cuevas-Salazar / ORC ID: 0000-0003-0113-0475, Researcher ID Thomson: O-9522-2014,

CVU CONACYT ID: 257896

ID 3rd Co-author: Mucio, Osorio-Sánchez / ORC ID: 0000-0002-5448-8393, Researcher ID Thomson: ABB-7912-2020,

CVU CONACYT ID: 80565


Abstract

Learning achieved by students in their first course of

mathematics at the university reflects a low achievement,

which is especially observed in the topic of the parabola.

Due to this problem, the objective of improving the

academic achievement of students in that topic, through a

didactic strategy based on semiotic representations, was

proposed. To this end, a quantitative inquiry was carried

out, with a pre-test post-test design, on both a control

group and an experimental group. There, a total of 44

students, of an average age of 19 years who were taking

the subject of Mathematics Foundations, participated. It

was found that the gain between the post-test and pre-test

measurement was significantly higher (p<0.01) in the

experimental group with respect to the control group,

where the conventional strategy for the course was being

used. It is concluded that it is possible to improve the

learning of the parabola in students through the strategy

based on semiotic representations, and that it is highly

recommended to apply it for the learning of other

mathematical objects in the basic sciences courses of the

engineering division.

Learning, Parabola, Semiotic Representation

Resumen

Los aprendizajes que logran los estudiantes en el primer

curso de matemáticas de la universidad reflejan un bajo

aprovechamiento y esto se observa sobre todo en el tema

de la parábola. Por ello se planteó el objetivo de mejorar

el aprovechamiento académico de los estudiantes en este

tema a través de una estrategia didáctica basada en las

representaciones semióticas. Con este fin se llevó a cabo

una indagación de tipo cuantitativa, con un diseño

pretest-postest, con un grupo control y otro experimental,

donde participaron un total de 44 estudiantes con una

edad promedio de 19 años que cursaban la asignatura de

Fundamentos de Matemáticas. Se encontró que la

ganancia entre la medida postest y pretest fue

significativamente mayor (p<0.01) en el grupo

experimental con respecto al grupo control donde se

empleó la estrategia convencional del curso. Se concluye

que es posible mejorar el aprendizaje de la parábola en

los estudiantes a través de esta estrategia basada en

representaciones semióticas y que es altamente

recomendable utilizarla en el aprendizaje de otros objetos

matemáticos en los cursos de ciencias básicas de la

división de ingeniería.

Aprendizaje, Parábola, Representación Semiótica

Citation: ENCINAS-PABLOS, Francisco Javier, PERALTA-GARCÍA, Julia Xochilt, CUEVAS-SALAZAR, Omar and

OSORIO-SÁNCHEZ, Mucio. Learning the parabola through semiotic records in university students. Journal-Mathematical

and Quantitative Methods. 2020. 4-7:16-25.



17


ENCINAS-PABLOS, Francisco Javier, PERALTA-GARCÍA, Julia Xochilt, CUEVAS-SALAZAR, Omar and OSORIO-SÁNCHEZ,

Mucio. Learning the parabola through semiotic records in university students. Journal-Mathematical and Quantitative Methods. 2020



Introduction

Most of the students have problems

understanding and using mathematical

knowledge, which is reflected in a low index of

academic performance, this is shown by the

Unesco Institute of Statistics (UIS)… “More

than 617 million children and adolescents do

not they are reaching the minimum levels of

proficiency in reading and mathematics” (UIS,

2017).

Among the standardized tests to

measure the basic learning of students is the

PISA test (Program for International Student

Assessment), developed by the OECD

(Organization for Economic Cooperation and

Development), which shows the skills acquired

by the student in the third year high school in

science, reading and math. In 2015, the

mathematics competencies shown by Mexican

students reached a score of 408, below the

OECD average of 493 points. For 2018 the

performance was 409 and the OECD average

489. Furthermore, in 2015 less than 1% of

Mexican students achieved higher levels of

proficiency (levels 5 and 6), for 2018 1%

reached those levels. So the last PISA test of

2018 compared on average with the previous

ones from 2006 do not show significant

differences (OECD, 2019).

In Mexico there are some tests such as

the EXHCOBA exam (Examination of Basic

Skills and Knowledge), which is applied to

students who graduate from high school and

aspire to enter university. The results obtained

in mathematics confirm that students do not

understand basic concepts, do not have

problem-solving skills, and the knowledge

acquired is related to memorization (Larrazolo,

Backhoff & Tirado, 2013).

Another of the standardized tests

applied at the high school level is the PLANEA

test, in the last evaluation in 2017 it is reported

that the score achieved by high school students

was 500 points which corresponds to Level I.

The students were able to solve problems

related to whole numbers and decimals but did

not show algebraic skills at that level of

development (National Institute for the

Evaluation of Education (INEE), 2019).

It is expected that students at the high

school level achieve a mastery of the algebraic

rules, understand some mathematical functions

establishing its formula and from it its graph

and vice versa, as well as construct and

interpret mathematical models using arithmetic,

algebraic, geometric and variation records, to

understand and analyze various situations

(INEE, 2019). But according to the results

shown by these tests, students fail to achieve

these domains in mathematics.

This is observed in the first mathematics

courses of higher education, the previous

knowledge acquired by the students is very

deficient, which hinders the learning of new

knowledge to achieve a meaningful learning of

mathematics. In the case of this research, an

important topic in mathematics that was

addressed is that of second degree equations.

At the Technological Institute of Sonora

(ITSON), several studies have been prepared to

find out the status of the learning problems of

their students, for example Sotelo, Echeverría

and Ramos (2009) mention that “over 65% of

students drop out of one or more subjects in a

school year”. Following this same line of

research, in another report issued by ITSON, of

all the careers from the 2002 to 2006 generation

it was found that 53% of the students dropped

out of one or more subjects.

In a study with students new to ITSON

engineering careers, it is mentioned that since

2009 a preparatory course has been offered to

students who do not pass a diagnostic math

exam, seeking to improve their skills in topics

related to Algebra, Trigonometry and

Analytical Geometry (Peralta, Encinas, Rojas,

Cuevas, Ansaldo and Osorio, 2013).

The Academy of the Fundamentals of

Mathematics course has reported that the failure

rate has increased from 31% in 2010 to 44.8%

in 2017. While the group average, in the same

period, has gone from 5.3 to 5.7. Results that

have had a negative impact on the indicators of

Educational Programs such as lag and terminal

efficiency, so it is important to make

improvements in the teaching and learning

process in mathematics courses by designing

and implementing new strategies didactics and

in this way have a positive impact on this

problem.

18






The contributions to the field of

educational mathematics by researchers and

teachers with experience in mathematics

learning problems, point out the importance of

the use of various representation systems, as

well as the use of didactic and technological

tools to structure and direct the teaching and

learning of mathematics in general, and in

particular of quadratic equations, Campos

(2003), Hitt (2008), Pizarro (2009), Bernal

(2020), Campos and Rodríguez (2020),

Farabello and Trigueros (2020) ), among others.

It is important to reconsider the role that

teachers have in the teaching of mathematics,

not to be only transmitters of knowledge, but to

consider making an analysis of the difficulties

that students show when they want to acquire

mathematical knowledge, such is the case of the

study of the parable. Therefore, it will be

necessary from the difficulties shown by the

student to promote different teaching strategies,

involving the use of technology in the

classroom and encouraging the student to want

to learn (Sánchez, 2016).

Various investigations in relation to the

parable show how the teacher can consider

other ways of approaching its study in the

classroom. In an investigation carried out for

secondary school teachers, a workshop was

developed and the difficulty of solving

problems related to the parabola was detected in

them, because it was identified that the learning

of this object focuses more attention on the

algebraic treatment than on the geometric

aspect. Therefore, the proposal by Torres,

Advíncula, León and Flores (2019) promotes

the use of concrete materials and the use of

Geogebra, to develop the knowledge of the

parabola as a geometric place and thus study its

properties. In another research carried out by

Aldama and López (2018) they show that in

order to achieve a better understanding of this

mathematical object, it is necessary to

investigate the historical part and origin of the

concept, as well as its didactics and develop

sequences supported with Geogebra. On the

other hand, Cruz, Báez and Corona-Galindo

(2018), in a study related to the teaching and

learning of the parable, propose a didactic

strategy based on meaningful and constructivist

learning, called the Potentially Significant

Teaching Unit. Therefore, he focuses his

attention on this as a geometric place, until the

concepts learned from the parable get the

students to solve a real problem.

It is important to consider the role of the

teacher in the educational process, according to

Díaz-Barriga and Hernández (2010), every

teacher must acquire and deepen a theoretical

framework that helps them understand the

teaching process to achieve meaningful learning

in the student. Being able to reflect on their

own work and promote innovative didactic

strategies.

Objectives

General purpose

Improve the academic achievement of students

in the topic of the parable through a didactic

strategy based on semiotic representations with

the purpose of improving indicators of the

ITSON Engineering Educational Programs.

Specific objectives

− Design a questionnaire based on the

framework of Duval's semiotic

representations, using a table of

specifications, to measure academic

achievement on the subject of the parable.

− Implement an unconventional didactic

strategy in the learning of the parable,

based on Duval's semiotic representations

in an experimental group.

− Apply the designed instrument to the

experimental group and to a control group

exposed to a conventional strategy, at two

different times, before and after the

teaching-learning process.

Justification

Researchers and teachers have been concerned

to understand the problems that students have

in learning mathematics. The process of

teaching and learning requires reformulating the

work of the teacher and the student, involving

new tools for their work in the classroom. The

implementation of new didactic proposals is

intended to improve the understanding of

mathematics (Sánchez, 2016).

19






On the other hand, according to Pizarro

(2009); Campos (2003) and Hitt (2008) the

contributions in the field of educational

mathematics, point out the importance of the

use of various semiotic representation systems,

as well as the use of didactic and technological

tools to restructure and direct teaching and

learning of mathematics, in particular that of

quadratic equations.

The proposal designed in this research

for the teaching and learning of the parable,

intends to improve the understanding of this

topic. Thus, helping to increase the approval

rate, decrease the dropout and lag rates of

students. For what it will positively impact and

to a certain extent, with the terminal efficiency

of the various engineering careers, these

indicators are important for accrediting bodies

such as the CACEI Council for the

Accreditation of Engineering Teaching

(CACEI, 2018).

Theoretical framework

For a learning to be remembered at the required

moment it is necessary that it has been

apprehended in a meaningful way, that is, that

when the learner comes into contact with the

new content to be learned, they do so by

connecting their previous knowledge with that

content. new knowing. It is through this process

that the cognitive structure of the learning

subject deepens and branches, generating new

structures and connections each time it interacts

with more and more learning content. In such a

way that if a part of this structure is activated,

the subject manages to remember other contents

thanks to those connections (Díaz-Barriga &

Hernández, 2010).

For subjects to learn meaningfully, two

things need to happen. The first is that learners

are willing to relate the new content to their

previous knowledge in order to give them

meaning. The second has to do with the content

to be learned, it must have a certain

organization or structure that is potentially

related to the prior knowledge of the learners.

Hence the idea of finding out in advance what

the apprentices already know and teaching

accordingly (Moreira, 2012).

Therefore, it is important to know what

prior knowledge students have when they are

going to be taught new content. Find out what

they already know and what they don't know.

With this knowledge, it is possible to establish

a starting point in the teaching of new content,

in order to promote that students can develop

their learning in a meaningful way.

An important element to consider in the

foundation of a didactic proposal on

mathematical objects is Raymond Duval's

theory of semiotic representation registers,

which are currently considered of utmost

importance in learning mathematics (Duval,

1998, 2010).

Theory of records of semiotic representation

The use of semiotic representations of the

quadratic function offers a variety of very

particular features in each of them, which

facilitate the development of cognitive

activities, when in the teaching and learning

process conversion activities between

representations are promoted: algebraic,

graphical, tabular and verbal.

Duval (1998) mentions that “a writing, a

notation, the strokes, the figures represent

mathematical objects such as: a number, a

function, a vector, a segment, a point, a circle”

and affirms that mathematical objects they

should not be confused with their

representations.

The use of semiotic representations of

Raymond Duval are considered of utmost

importance in learning mathematics (Duval,

1998, 2010). The Parabola can be represented

by various graphic, algebraic, verbal or tabular

registers. Which are necessary in mathematical

activity, in order to interact with the object

(Duval, 1998).

Duval structures in the theory of

semiotic representations three cognitive

activities related to semiotic representation

registers: Formation, Treatment and

Conversion. Substantial and necessary for the

understanding and communication of

mathematics (Godino, Wihelmi, Blanco,

Contreras and Giacomone, 2016; Duval, 1988).

20






Training is an activity associated with

the apprehension of semiotic representations

and has to do with the signs that characterize

each record and their identification rules in the

same record. In the Treatment activity, the

transformation of the representation of the

mathematical object occurs but within the same

record. Finally, Conversion, which is the

transformation of a representation into another

register, whose activity is considered very

essential because it plays a fundamental role in

the conceptualization of represented content

(Duval, 1998, 2012).

The Training activity is exemplified

when there is a representation of the form y =

ax ^ 2 + bx + c, in the algebraic register. And if

this representation is transformed to the form y

= (x + h) ^ 2 + k then it is the Treatment

activity. And the Conversion activity when this

quadratic equation is represented in the

graphical register with a trace, identifying each

element of the equation with the corresponding

element in the graph.

Research methodology

The type of research that was carried out, which

subjects participated in the study, what

instrument was used to gather information and

the procedure or steps carried out in the

investigation is detailed below.

Kind of investigation

The inquiry was quantitative, quasi-

experimental with a pretest-posttest design with

a control group (Buendía, Colás and

Hernández, 1998).

Subjects

The study included 44 students who

were taking the subject Fundamentals of

Mathematics, with an average age of 19 years.

Two groups of already conformed students

were randomly selected, one experimental and

one control. The first was made up of 24

students, 12 of whom were women and 12 men;

the second group consisted of 20 students, of

whom eight were women and 12 men.

Instrument

The questionnaire used in the inquiry

consisted of nine items. Three of them were

used for students to reflect the cognitive activity

of Treatment: verbal, algebraic and graphic

respectively. The other six items were designed

for students to show the cognitive activity of

Conversion of the following records: from

verbal to algebraic, from verbal to graphic,

from algebraic to verbal, from algebraic to

graphic, from graphic to verbal and from

graphic to algebraic.

The tasks requested by the reagents

were the following:

R Registry Task

Start-

End

1 V-V The verbal equivalent of a quadratic

equation with a parabola.

2 A-A It asks to pass the expression

𝑦=(𝑥−𝑏)2+𝑐 to 𝑦=𝑎𝑥2+𝑝𝑥+𝑞.

3 G-G Identify over what interval a function

grows.

4 V-A From a sentence identify its equation.

5 A-V Convert the expression

(𝑎−𝑏)(𝑎+𝑏)=𝑎2− 𝑏2 a verbal

statement.

6 V-G From a sentence identify its graph.

7 G-V A graph is shown and its statement is

requested.

8 A-G An equation is offered and asked to

identify its graph.

9 G-A Given a graph identify its equation.

Note: R = Reactive; Ini = initial; End = end

Table 1 Task requested by each instrument reagent

Process

The first step was to develop the

measurement instrument, based on Duval's

(1998) conceptual framework of semiotic

representations. The records were considered:

verbal, algebraic and graphic of the parabola as

they are the most used in the mathematical

literature. A table of specifications was used in

order to achieve the content validity of the

instrument, according to the recommendations

of Santibañez (2011). Subsequently, the

instrument was presented to three experts in the

area to validate its content. A pilot test was then

carried out with a group of students to review

the writing of the items, until it was in its final

version.

21






Subsequently, and in accordance with

the recommendations of Díaz and Leyva

(2013), the difficulty indices of the items were

classified as follows: more than 0.9 would be

considered the easy item, between 0.8 and 0.89

the item would be moderately easy, between 0.7

and 0.79 would be of medium difficulty,

between 0.5 and 0.69 would be moderately

difficult and less than 0.5 would be considered

difficult items. According to the research

design, the instrument was applied to both the

control and the experimental group before

starting the learning of the parable topic

(pretest) and after the teaching process

(posttest). In the control group a conventional

sequence of the course was applied while in the

experimental group the new didactic sequence

was implemented.

With the data collected through the

instrument, it was possible to identify those

Treatments and Conversions with which the

students show difficulties and strengths through

descriptive statistics. The Shapiro Wilk test was

performed to verify that the data met the

normality assumption. It was found that the

pretest and posttest data of the control group

fulfilled this assumption, but not the data of the

experimental group. Therefore, the Wilcoxon

paired comparison test was used to determine

the existence of a significant difference

between the groups, both in the pretest and in

the posttest, as well as in their gain, as

recommended by Hernández, Fernández and

Baptista (2014). With the results obtained, it

was possible to achieve the objective of the

investigation to finally write the corresponding

report.

Results

When applying the measurement instrument at

the pretest moment, it was found that the

groups were not equal in terms of prior

knowledge of the parabola (Table 2). There was

a significant difference between the control and

experimental groups (p <0.05). The control

group obtained an average score higher than the

experimental group from 0.18 to 2.1 points.

This situation may be due to the fact that the

university receives high school students with

very diverse training programs, which is

reflected in a different level of prior knowledge.

Thus, from the outset, the control group showed

a higher starting point than the experimental

group in terms of prior knowledge of the

parable.

Group Pretest Postest Gain

Control 6.10 6.60 0.50

Experimental 4.96 6.88 1.92

Table 2 Average number of correct items in each group

out of a total of nine items

After the pretest and after having

exposed the control and experimental group to

two different didactic sequences, the instrument

was applied again at a posttest moment. Here

both groups showed similar academic

achievement since it was not found that there

was a significant difference between them at

that time (p> 0.05). However, when comparing

each group with itself between the pretest and

posttest, it was found that in the control group

there were no significant advances, since there

was no increase in their score (p> 0.05). Not so

in the experimental group that did show a

highly significant advance in their grade (p

<0.01), which increased on average from 1.26

to 2.57 points. When making a comparison of

the gain between both groups, a highly

significant difference was also found between

them (p <0.01). The experimental group had a

gain greater than that of the control from 0.53

to 2.31 points. This result is consistent with

other studies that have also implemented

learning tasks aimed at students working on

different semiotic representations, such as the

works reported by Prada, Hernández and Jaimes

(2017), Mercedes, Pérez and Triana (2017),

Aznar, Distéfano, Moler and Pesa (2018),

Denardi and Bisognin (2020), among others.

The results obtained by the groups in their

notes, therefore, show that the didactic

sequence based on semiotic representations

offers better learning results than the

conventional didactic sequence of the course.

On the other hand, the result observed in

the experimental group with respect to the gain

in the average of their grade, was also recorded

in the gain obtained by the group in the two

cognitive activities evaluated (Table 3).

Although the pre-test starting point showed that

the control group started with a better position

in the Treatment and Conversion activities than

the experimental group, at the post-test moment

this difference was canceled. This caused that

again, with regard to the gain in the two

cognitive activities that were evaluated, a

significant difference was observed in favor of

the experimental group over the control, with

respect to Treatment (p <0.01), and Conversion

(p <0.05).

22






As the ability to perform record

conversions is evidence of the deep

understanding of the mathematical object Duval

(1988), then the didactic sequence based on

semiotic registers offers in terms of

apprehension greater gain and a steeper

learning curve than the conventional didactic

sequence. of the course of foundations on the

theme of the parable. Due to all of the above,

these results bring with them some practical

implications both for the fundamentals course

and for other subjects in the area, since a

didactic sequence similar to that of this research

could be implemented in other mathematical

objects, with good expectations of success.

Cognitive

activity

Pretest Postest Gain

GC GE GC GE GC GE

Treatment 0.62 0.49 0.63 0.65 0.01 0.16

Conversion 0.71 0.58 0.78 0.82 0.07 0.24

Note. CG = control group; EG = experimental group.

Table 3 Average difficulty index for cognitive activity

In relation to each of the reagents of the

instrument that was used to evaluate the ability

of the subjects to perform Treatments and

Conversions in the graphic, algebraic and

verbal records (Table 4). The results indicate

that at the beginning, at the pretest moment, the

control group had a better starting point than

the experimental group in terms of these

cognitive abilities. Condition that changed at

the post-test moment, where the groups

performed equivalently in most of the tasks.

However, again the gain observed in the

experimental group was higher compared to the

control group in the three Treatment tasks and

in five of the six Conversion tasks. In the

control group (Table 5) the difficulty index did

not change in the Treatment tasks (AA, GG,

VV) and in three Conversion tasks (GA, VG,

GV) between the pretest - posttest

measurements, while in the The experimental

group did show improvements in the difficulty

index in two Treatments (VV, GG) and in five

Conversions (AG, GA, GV, AV, VG). This

offers evidence that the didactic sequence

applied in the experimental group offers better

learning results than the conventional sequence

applied in the control group.

AC Registry Pretest Postest Gain

Start-End GC GE GC GE GC GE

T G-G 0.50 0.29 0.55 0.58 0.05 0.29

T A-A 0.40 0.29 0.35 0.42 -0.05 0.13

T V-V 0.95 0.88 1.00 0.96 0.05 0.08

C G-A 0.60 0.00 0.65 0.79 0.05 0.79

C A-V 0.45 0.24 0.70 0.58 0.25 0.34

C A-G 0.95 0.79 0.85 1.00 -0.10 0.21

C V-G 0.60 0.46 0.60 0.67 0.00 0.21

C G-V 0.95 0.83 1.00 1.00 0.05 0.17

C V-A 0.70 0.88 0.90 0.88 0.20 0.00

Note. Ini = initial; End = end; V = verbal register; A =

algebraic register; G = graphic record; AC = cognitive

activity; T = treatment; C = conversion.

Table 4 Difficulty index of the reagents in the control

(CG) and experimental (EG) group, at the pretest and

posttest time, and the gain between these two

measurements

AC Registry Pretest Postest

Start-End GC GE GC GE

T G-G MD D MD MD

T A-A D D D D

T V-V F MF F F

C G-A MD D MD DM

C A-V D D DM MD

C A-G F DM MF F

C V-G MD D MD MD

C G-V F MF F F

C V-A DM MF F MF

Note. F = easy; MF = moderately easy; DM = medium

difficulty; MD = moderately difficult; D = difficult

Table 5 Difficulty index (according to their

classification) of each item in the control (CG) and

experimental (EG) group, at the pre-test and post-test

time

On the other hand, it was observed that

the Algebraic Treatment (AA) is the most

difficult in both study groups even after the

teaching process, since it did not change its

difficulty index between the pretest and posttest

(Table 5), these problems with algebra coincide

with studies carried out by García, Segovia and

Lupiáñez (2011), Rodríguez and Torrealba

(2016). What makes it clear that algebra is a

content little understood by students, who have

deficiencies in this area since high school, as

shown by the results of the Planea exam

(Ministry of Public Education, n.d.). In addition, three tasks with moderate

difficulty were also found in the experimental

group at the post-test moment. The A-V and V-

G Conversions and the G-G Treatment. The

latter usually represents a problem for students,

who struggle with the reading and interpretation

of graphic records, as also reported by

Guerrero, Camacho and Mejía (2010), Artola,

Mayoral and Benarroch (2016).

23






Therefore, the A-A and G-G

Treatments, as well as the A-V and V-G

Conversions for the next school year are

identified as areas of opportunity in the

teaching of the parable.

Conclusions

From the results obtained, it is concluded that it

was possible to improve the academic

achievement of the students in the theme of the

parable through a didactic strategy based on

semiotic representations. It was achieved that

the students had a steeper learning curve, and in

turn, some areas of opportunity were identified

for the teaching of this mathematical object in

which special attention must be paid in the

future, such as: Treatments in the algebraic

register (AA) and in the graphic register (GG),

as well as in the Conversions from the algebraic

register to the verbal (AV) and from the verbal

register to the graph (VG).

It is recommended to use this didactic

strategy in other subjects of the mathematics

fundamentals course where there are similar

learning problems and also to use it in the

teaching and learning of mathematical objects

in other subjects in the basic sciences area of

the curricula plans of engineering educational

programs.

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26


Knowledge friction in universities, approach from game theory

Fricción del conocimiento en instituciones de educación superior, planteamiento

desde la teoría de juegos

TALAVERA-RUZ, Marianela†*, LARA-GÓMEZ, Graciela and VALDEZ-RESÉNDIZ, Macario

Universidad Tecnológica de Querétaro, Mexico.

ID 1st Author: Marianela, Talavera-Ruz / ORC ID: 0000-0002-9185-4743, Researcher ID Thomson: V-7347-2018, CVU

CONACYT ID: 580789

ID 1st Co-author: Graciela, Lara- Gómez / ORC ID: 0000-0001-9984-7372, CVU CONACYT ID: 99837

ID 2nd Co-author: Macario, Valdez-Reséndiz / ORC ID: 0000-0003-1723-1545, CVU CONACYT ID: 474336


Abstract

In today's market economies, organizations see knowledge

as one of their most valuable and strategic resources and

seek to properly manage it so that it becomes a competitive

advantage (Teece, 1988; Hamel and Prahalad, 1990,

Drucker, 1994; Nonaka and Takeuchi, 1995; Boisot, 1998;

Spender, 1996; Senge, 1990). Although many organizations

make significant investments in technology and tools to

promote knowledge sharing, cultural, behavioral, and

structural aspects are the main determinants of success

(Sharma and Bhattacharya, 2013). Organizational

knowledge processes are, by their nature, generally social

and complex. The behaviors related to sharing knowledge of

organizational agents are full of situations of conflict of

interest or dilemmas in which they receive different

payments based on their strategic decisions. Such situations

can be modeled as games. This article presents the approach

to a particular dilemma, that of the knowledge friction in an

Institution of Higher Education through Game Theory,

describing a non-cooperative game model that allows

showing the scope of said situation according to the

decisions considered to be done by employees and employer

and their related payments, exploring different decision-

making scenarios.

Knowledge management, game theory, knowledge

dilemmas

Resumen

En las economías de mercado actuales, las organizaciones

ven al conocimiento como uno de sus más valiosos y

estratégicos recursos y buscan manejarlo adecuadamente

para que se convierta en ventaja competitiva (Teece, 1988;

Hamel y Prahalad, 1990, Drucker, 1994; Nonaka y

Takeuchi, 1995; Boisot, 1998; Spender, 1996; Senge, 1990).

Aunque muchas organizaciones hacen inversiones

significativas en tecnología y herramientas para promover la

compartición de conocimiento, son los aspectos culturales,

de comportamiento y estructurales los principales

determinantes del éxito (Sharma y Bhattacharya, 2013). Los

procesos de conocimiento organizacional son, por su

naturaleza, generalmente sociales y complejos. Los

comportamientos relacionados con compartir conocimiento

de agentes organizacionales están llenos de situaciones de

conflicto de intereses, o dilemas, en los que ellos perciben

diferentes pagos basados en sus decisiones estratégicas.

Dichas situaciones pueden ser modeladas como juegos. Este

artículo presenta el planteamiento de un dilema en

particular, el de la fricción del conocimiento en una

Institución de Educación Superior a través de Teoría de

Juegos describiendo un modelo de juego no cooperativo que

permite mostrar los alcances de dicha situación acorde a las

decisiones de empleados y empleador y los pagos

relacionados, explorando diferentes escenarios de tomas de

decisión.

Gestión del conocimiento, Teoría de juegos, dilemas del

conocimiento

Citation: TALAVERA-RUZ, Marianela, LARA-GÓMEZ, Graciela and VALDEZ-RESÉNDIZ, Macario. Knowledge

friction in universities, approach from game theory. Journal-Mathematical and Quantitative Methods. 2020. 4-7:26-50.

* Correspondence of the Author (Email: [email protected])



27


TALAVERA-RUZ, Marianela, LARA-GÓMEZ, Graciela and

VALDEZ-RESÉNDIZ, Macario. Knowledge friction in

universities, approach from game theory. Journal-Mathematical

and Quantitative Methods. 2020



Introduction

In today's complex and globalized market

economies, organizations see knowledge as one

of their most valuable and strategic resources

and seek to manage it properly so that it

becomes a competitive advantage (Teece, 1988;

Hamel and Prahalad, 1990, Drucker, 1994;

Nonaka and Takeuchi, 1995; Boisot, 1998;

Spender, 1996; Senge, 1990). Although many

organizations make significant investments in

technology and tools to promote knowledge

sharing, cultural, behavioral, and structural

aspects are the main determinants of success

(Sharma and Bhattacharya, 2013).

Organizational knowledge processes

are, by their nature, generally social and

complex. One of the biggest challenges in

Knowledge Management is behavior change

related to the creation and consumption of

knowledge.

The flow of knowledge refers to the

links between creation and consumption and

occurs at two levels: the intra-organizational

level, which occurs within the limits of the

organization; and the interorganizational level,

which extends outside the organization and

includes external entities such as suppliers,

alliances, business partners, competitors, and

industry regulators (Sharma and Bhattacharya,

2013).

Behaviors related to sharing knowledge

of organizational agents (knowledge of

employees) are fraught with situations of

conflict of interest in which they receive

different payments based on their strategic

decisions. Such situations can be modeled as

games. This article presents the approach to a

particular dilemma, that of the friction of

knowledge in an Institution of Higher

Education through Game Theory, describing a

non-cooperative game model that allows

showing the scope of said situation according to

the decisions made. are considered and related

payouts in the game described.

Knowledge Dilemmas

The Organizational Knowledge Ecosystem

includes situations where agents, as creators

and consumers of knowledge, make strategic

decisions to derive the best possible results for

the organization, however, the search for

personal or group interests may conflict with

the interests of the organization. organization

(Sharma and Bhattacharya, 2013). This is

particularly true in organizations such as

Institutions of Higher Education. Mutual

dependence on such alternatives can lead to

undesirable payoffs for some or all

stakeholders. To capture and articulate the

essence of such complexity in decision making,

the notion of dilemma is introduced into the

knowledge ecosystem.

A Dilemma is understood as a situation

in which it is necessary to choose between

apparently undesirable alternatives. Knowledge

Dilemmas are situations of conflict of interest,

mainly, that can be specified through multiple

perspectives and levels, both strategic,

implementation, cultural, economic and

political. Some key knowledge dilemmas that

characterize situations of conflict of interest

between agents of the organization are the Silos

of knowledge, The Tragedy of the public good,

Frictions of knowledge and the toxicity of

knowledge.

Regarding these dilemmas, the main

factors involved include aspects such as the

perception of power that knowledge grants

(Sharma and Bhattacharya, 2013), the

acceptance of other employees, the prestige of

those involved and decisions related to the

effort to carry out knowledge sharing tasks.

It is difficult to measure the recognition

of an employee, and even more so his social

prestige. In general, an employee is said to have

a certain social value when he offers something

that society appreciates and considers

important. In addition, public opinion considers

that this recognition should be rewarded with a

salary level in accordance with the work

performed (Vaillant, 2007).

28








An important factor to consider in the

analysis of the teacher's situation as an

employee of an Educational Institution is the

respect or prestige they enjoy, because it will

depend on whether they find more or less

difficulties in the development of their tasks

(Vaillant, 2007).

Game Theory for Knowledge Management

Games are taxonomies of strategic situations.

Game Theory is a mathematical derivation that

analyzes the cognitive abilities of the player's

strategies (Camerer, 2003). The main objective

of theoretical reasoning for games is not to

predict the outcome of the game, but to

discover how the game is played and how

rational players who pursue their own interests

are likely to make strategic decisions in

response to the strategies of other participating

players (Polak, 2007).

The rationality assumption suggests that

knowledge workers pursue their individual

interests while playing in the organizational

knowledge ecosystem. Often, according to

Adam Smith's "invisible hand" theory,

individual interests gained through their "best

responses" are expected to lead to superior

results for the organization (Sharma and

Bhattacharya, 2013). Therefore, game models

that describe knowledge dilemmas could be

structured as non-cooperative in nature. The

knowledge dilemma explored in this study

implies a tension between the best for the

organization and the conflict of interest

(Sharma and Bhattacharya, 2013).

Knowledge is a resource that does not

diminish and knowledge transactions offer

potential winning opportunities for all

participants (Sharma and Bhattacharya, 2013).

From a game modeling perspective, such

situations are conceptualized as variable sum.

In terms of research approach, this paper takes

advantage of simple models offered by Game

Theory, such as the exclusive “Principal-

Agent” game model. For simplicity, only two-

player games are considered.

Considerations for applying Game Theory

Game Theory of behavior suggests that factors

such as history and culture influence the actual

behavior of human agents in social situations

(Camerer, 1997; Camerer, 2003; Dufwenberg,

2004). Game theory researchers also recognize

the role of a critical mass of human agents in

initiating change in the social behavior of a

group (Dixit and Nalebuff, 2008). The themes

of communality and conflict of interest

proposed by game theorists are congruent with

theories of social exchange and collective

action, in relation to the organizational

knowledge ecosystem (Ghobadi and D’Ambra,

2011).

The dominant individual rationality

implies retaining the monopoly of knowledge

through hoarding (Chua, 2003) considering

knowledge associated with the perception of

power. In multi-unit organizations, the

asymmetry of knowledge repositories,

authorities, structural and cultural arrangements

between units can result in a poor flow of

knowledge (Tsai, 2002; Gupta and

Govindarajan, 2000). Additionally, the

asymmetry of information in knowledge

exchanges impairs the flow of knowledge in

organizations (Davenport and Prusak, 1998).

The flow of knowledge in organizations

depends heavily on the creation of new

knowledge through voluntary contribution and

the transfer of such knowledge by sharing it to

be used as needed (Sharma and Bhattacharya,

2013). However, when the possession of

knowledge is associated with a sense of holding

power, knowledge agents may not be willing to

share it. Instead, knowledge agents make

decisions about the investments of limited time

and effort required to exchange knowledge

based on the interests they perceive as their

own (Davenport and Prusak, 1998). In this way,

knowledge sharing situations reflect conflicts of

interest, where individuals make strategic

decisions between contributing or accumulating

their knowledge depending on the benefits

perceived in the exchanges (Sharma and

Bhattacharya, 2013).

29








The asymmetry of information in

knowledge exchanges can spread inefficiency

throughout the knowledge ecosystem

(Davenport and Prusak, 1998; von Hippel,

1994; Hansen and Nohria, 2004). This

information asymmetry affects the perceived

value of knowledge by the receiver and

consequently, the effectiveness of the

knowledge flow. Knowledge exchanges

generally occur with incomplete information

between two players: the knowledge seeker and

the knowledge provider; Furthermore, there are

operational inefficiencies in knowledge transfer

because the nature of knowledge and its value

is uncertain in the sharing stage (Sharma and

Bhattacharya, 2013). Together, the three

problem areas contribute to a dilemma that

affects the process of knowledge creation and

sharing in the organization, producing

disaggregated knowledge packages or

knowledge silos.

Knowledge friction

As organizations seek to compare their

knowledge and replicate best practices within

their limits, such knowledge transfer may be

inhibited by contingent factors such as

similarity of context, motivational disposition,

strength of relationships, and absorptive

capacities (Szulanski, 1996; O 'Dell and

Grayson, 1998; Argote and Ingram, 2000;

Perrin, Rolland and Stanley, 2007). In this

sense, a game is proposed contemplating the

lack of adoption of organizational knowledge as

a representation of the Friction Dilemma of

knowledge.

Methodology

The information used to propose the scenarios

and games that are developed below are based

on the professional experience of ten years of

members of an Educational Institution and the

participation in different projects related to the

sharing of knowledge in professional services,

participatory observation and non-participatory,

and relationships with stakeholders. The

modeling of the game is proposed based on

different scenarios that are described from

elements of Game Theory.

Dimension Key questions for

game analysis

Implications for the

knowledge ecosystem

Players Who are the players

in the game?

In an organizational

ecosystem, the players

are all the members

involved in knowledge

processes such as

knowledge creation,

transfer and

application. In the case

of analysis, the players

are: the Director of a

Division and the

“Agent” (research

professor) required for

the task of sharing

knowledge in a project.

Value

added

What is the

additional value that

each player brings

to the game over the

other players?


context, the reputation

or prestige of an

employee who is a

source of knowledge

can add value for him

(Sharma and

Bhattacharya, 2013)

and influence the

success of the project.

Rules Are the rules fixed

or are they

manipulable? In the

case of the proposed

game, the Director

has the power to set

the rules. In a

strategic sense, the

player who makes

the first move can

create advantage for

himself, as in the

case of the proposed

game.


knowledge ecosystem,

there is no set of

universal rules. They

can be organizational

policies that require

knowledge

contribution as an

evaluation criterion.

But in practice, it is the

organizational culture

that determines

whether some of the

dilemmas permeate the

entire organization.

Tactics What is the

perception of the

different players in

the game? Players'

perception of the

game is one of pure

competition (win-

lose) and influences

the tactics that

players will adopt in

the game.

Under what

circumstances do

players (co) create,

transfer and apply

knowledge to

maximize their

payouts. These

constitute the set of

tactics to formulate.

Scope What is the scope of

the game? The

scope is set by the

Director in terms of

the delimitation of

the factors to be

considered for

payments and the

choice of agent

based on prestige.

The scope of the game

is limited to the

barriers of the

organization. The

scope of the game can

be changed by

allowing cross-

organizational

knowledge flows such

as partnerships,

alliances, and

outsourcing.

Table 1 Dimensions and implications for game analysis

Source: Own elaboration based on Sharma and

Bhattacharya (2013)

30








As described in Table 1, the additional

value that each player brings to the game with

respect to the other players can include aspects

such as the prestige, acceptance and perception

of value of the knowledge to be shared. This

suggests how players can increase their own

profits and limit the payouts of other players,

for example, by distinguishing between those of

higher and lower prestige.

If one or more players have the power to

manipulate the rules using strategic moves, it

impacts the added value and tactics of the

players (Sharma and Bhattacharya, 2013). In a

strategic sense, the player who makes the first

move can create advantage for himself. In an

organizational knowledge ecosystem, there is

no set of universal rules, (Sharma and

Bhattacharya, 2013), they can be organizational

policies that require knowledge contribution as

an evaluation criterion. But in practice, it is the

organizational culture that determines whether

some of the dilemmas permeate the entire

organization (Sharma and Bhattacharya, 2013).

Regarding the scope of the game,

players can change it by expanding or reducing

the boundaries of the game.

Dimension Key questions for game analysis

Players Organizational leadership that intends

to disseminate knowledge-based

practices or assets within the

organization and to the employees or

groups involved in sharing such

knowledge.

Value added Leadership can add value by

formulating appropriate incentive

schemes according to the level of effort

of employees and their prestige and by

limiting the added value of employees

through incentives subject to project

results.

Rules Fixed rules like work agreements.

However, reinforcing the adoption of

knowledge-based practices through

contractual arrangements is not entirely

feasible in the context of an

organization.

Tactics Players related to the adoption of

certain knowledge perceive the

activities linked to additional effort

levels and evaluate the payments of

additional efforts against the incentives

offered to them for such activities.

Scope Game boundaries can range from small

groups to the entire organization.

Table 2 Dimensions of the Friction of Knowledge game

Source: Own elaboration based on Sharma and

Bhattacharya (2013)

Strategic situation (game model)

Strategic parameters for modeling: Two

player game, sequential movement, variable

sum game

As suggested in the knowledge friction

dilemma, management's best interest is

achieved through the transfer and application of

organizational knowledge throughout the

company. However, the actions of groups

involved in the practice of such objectives may

not be aligned with the organizational good.

Therefore, a strategic situation arises due to the

sticky nature of knowledge. It is generally

difficult, if not impossible for the organization

to monitor and control the application of

knowledge (particularly the transfer and reuse

of valuable tacit knowledge and relational

capital) (Sharma and Bhattacharya, 2013). In

Game Theory, this situation with endogenous

uncertainty, where a player is unable to observe

the actions that another player takes, is called

"moral hazards" or "moral risk", that is, the

conflict of collective knowledge where no one

feels responsible. In the scenarios proposed, the

Director cannot monitor each of the agent's

actions, so he decides to act based, not on the

process, but on the final results.

From a Game Theory perspective, it

seeks to analyze the scenario and build effective

incentive mechanisms that can cause teams to

act in line with the best interest of the

ecosystem. The reference model is the

"Principal-Agent" in which a manager

(Principal) hires an employee (Agent) to carry

out a project. In this case, the Principal

represents the Director of a Division of a

Higher Education Institution and the agent

represents a professor in charge of sharing

knowledge from the completion of a project.

Both the manager and the employee can choose

two strategies and there are two consequences

for each employee action:

The Director (“Principal”) decides the

level of compensation to offer the employee:

R (fixed compensation)

R’ (salary plus incentive depending on the

result of the project)

RP (salary plus incentive depending on the

prestige of the agent)

31








R’P (salary plus incentive depending on

the prestige of the agent and the result of the

project)

When the employee is chosen by the

Director, and the project is assigned, he must

make two important decisions: working to be

accepted and the level of effort that will be put

into the project.

The employee responds to the manager's

decision by applying a greater or lesser effort to

be accepted in the group to which he will

transmit the knowledge. If the employee has

prestige:

AYY (greater effort)

AYN (less effort)

If the employee does not have prestige:

ANY (greater effort)

ANN (less effort)

Once the employee makes this decision,

they must decide whether to apply more or less

effort to the assigned project:

WH (greater effort)

WL (less effort)

The employee chooses after the

manager's decision and his decision is not clear

and observable to the manager. The manager

cannot judge the employee's efforts, so the

incentive payment is given with respect to how

observable results or prestige may be, or a

combination of both. Therefore, the Director's

net payments include the two possible levels: G

and B.

From the perspective of earnings from

payments, these can be expressed as a function

of incentives: U(R) or U(R’) or U(RP) or U(R’P)

depending on the type of incentive to pay.

However, payments should be tied to

perceived dissatisfaction (or disutility) for the

level of effort the employee needs to spend (the

difficulty) of achieving the incentive. Assuming

this disutility as dH when the employee makes

a great effort and dL when the employee makes

less effort, the net payments for the employee

are given at the levels given by the different

profits that are generated in each proposed

scenario.

From the concept of "nature" or

serendipity, which determines the additional

profitability to the employee's selected strategy,

it is considered that for each project, there is a

probability that said project will be good or

successful, or bad, due to circumstances that are

outside of the players' decisions, for which

probabilities associated with the success or

failure of the projects are added:

p and (1-p) for the case of the election

of prestigious employees.

q and (1-q) in the case of the election of

employees without prestige.

Type Symbol Variable Meaning

Gain G Good Project

Gain

Profit obtained

from a good

project, which can be

translated into

money or in kind, such as

download hours

for projects, incentives for

publications or

attendance at conferences,

among others.

B Bad Project Gain

Profit obtained from a bad

project.

Payment

scheme chosen

by the

Director

R Fixed payment Fixed payment scheme,

independent of

the level of profit of the

project and the

prestige of the employee.

R ' Payment based

on the profit

obtained from the project

Payment

scheme based

on the profit obtained in

project G or B.

RP Pay based on prestige

Payment scheme based

on the prestige

of the PS or PN employee

where the result

of the project is not considered.

32








R'P Pay based on

prestige and

profit

Payment

scheme based

on the prestige of the employee

and the profit

obtained in project G or B.

Employee

prestige

PY Prestigious

employee

Employee with

a prestigious level.

PN Employee

without prestige

Employee

without prestige

for the project. Acceptance

stress factor

AY Effort factor to

be accepted

Effort factor

that the

employee applies to be

accepted by the

other employees to

whom he will

share the knowledge.

AN Factor of no

effort to be accepted

No effort factor

that the employee does

not apply to be

accepted by the other

employees to

whom they will share the

knowledge.

This factor negatively

impacts the

effort you will have to have in

carrying out the

project.

Utility for the

employee

U Employee profit Profit generated by the payment

granted by the Director.

Employee

disutility

dH High disutility Disutility

perceived by

the employee as high due to the

high effort

applied to obtain the

desired profit. It

is a perception of loss by the

employee.

dL Low disutility Disutility perceived by

the employee as

low due to the low effort

applied to

obtain the desired profit. It

is a perception

of loss on the part of the

employee.

N Prestige level Level of

prestige that the employee

obtains for a good project or

that he loses for

a bad project.

p Probability of success for PY

Probability of success when

there is prestige

q Probability of success for PN

Probability of success when

there is no

prestige

Table 3 Variables and descriptors

Source: Self made

From the theoretical analysis of the

variables and the game conditions observed in

reality, the conditions of Table 4 are proposed.

Condition Interpretation

G>B A good project has a greater profit than

a bad project.

B>R Profit from a bad project should at least

be able to pay for the employee

incentive.

1>AYY

>AYN>ANY

>ANN

Accepting an employee implies less

effort to share knowledge.

RG > RB The incentive for a good project is

greater than the incentive for a bad

project.

dH > dL A greater effort is perceived by the

employee as a higher disutility than a

lesser effort. Employee perceived value

will always be high when they try less.

RPY > RPN The incentive to pay based on prestige

is greater for a more prestigious

employee.

RPG > RG >

RPG>RPB

RPB > RB

The incentive to pay based on prestige

and profit is greater for a more

prestigious employee on a project that

goes well.

p > q The probability of a project going well

is higher for a prestigious employee

than for a non-prestigious employee.

Table 4 Conditions for the game

Source: Self made

Assumptions in the different scenarios

Scenario 1) Fixed payment (R) per project

This scenario raises the Director's proposal to

make a fixed “payment” for the project,

regardless of whether it goes wrong or not, or

the effort applied or the level of prestige that

the employee possesses. Each employee choice

has an impact on the profit of the project. Since

the Director cannot judge the employee's

efforts, he will pay the incentive based on the

profitability achieved (R), so the Director's

profit will be determined by how well the

project succeeds.

33








Figure 1 Game tree for Scenario 1 Fixed pay (R) per

project

Source: Own elaboration

Knowledge is considered a public good

with its “non-exclusion” and “non-rivalry”

characteristics, and, like all public good, it is

susceptible to underinvestment (Sharma and

Bhattacharya, 2013). As the consumption of

knowledge can be enjoyed without any

contribution, the strategy of maximizing

benefits for any agent in the organization is the

“free ride” (Hardin, 1993; Cabrera, 2002). This

leads to a suboptimal for the organizational

knowledge ecosystem and presents a social

dilemma (Sharma and Bhattacharya, 2013).

This dilemma leads to a paradoxical situation

where individual rationality towards the

maximization of self-interest leads to collective

irrationality in the overexploitation of common

resources, producing a suboptimal for the entire

organization. This situation is identical to the

"Tragedy of the public good", for the sharing of

public goods described by Hardin (1993) and

popularized by Lessig (2002), so it can be

considered under this type of game. Knowledge

agents or players must decide how much of the

knowledge they possess to invest in sharing it

so that everyone “enjoys” that knowledge

(common welfare).

Each player has a pool of knowledge ei

that they can contribute (invest) and each player

must decide how much of this knowledge they

will share (invest) gi: 0 ≤ gi ≤ ei. The payment

function of player i is given by: ei - gi, which

are the player's personal interest investments.

We assume the same amount of

knowledge that can be shared, for each player,

which will be reflected in the effort W that the

player includes in carrying out the project.

The net payments for the “Principal” in

the two levels are, respectively:

G - R

B – R



of incentives:

U (R) for a good gain or bad gain scenario.



level of effort W that the employee needs to

spend (the difficulty) of achieving the incentive.

Assuming that disutility as dH when the

employee puts in a great effort and dL when the

employee tries less, the net payments to the

employee at the two levels are, respectively:

U(R) - dH

U(R) - dL

Scenario Solution 1

To solve this dynamic non-cooperative game

with complete information, it is necessary to

compare the payouts based on the last decisions

made, that is, "backwards".

It begins by comparing the

corresponding payments for a prestigious

employee. Pay 1 (PY, AYY, WH) is compared

with pay 2 (PY, AYY, WL), corresponding to

the decision of an employee with prestige PY

who strives to be accepted AYY, between

investing a greater effort WH or a less effort

WL.

The payment equations:

Yij = α + ∑ βhXhij

r

h=1

+ uj + eij

p [UR – 𝐴𝑌𝑌 –𝑑𝐻

𝐴𝑌𝑌] + (1 − 𝑝) [𝑈𝑅 − 𝐴𝑌𝑌 −

𝑑𝐻

𝐴𝑌𝑌]

34








Strategy (𝑃𝑌, 𝐴𝑌𝑌 , 𝑊𝐻) (1)

(1 − 𝑝) [𝑈𝑅 − 𝐴𝑌𝑌 −𝑑𝐿

𝐴𝑌𝑌] + 𝑝 [𝑈𝑅 − 𝐴𝑌𝑌 −

𝑑𝐿

𝐴𝑌𝑌]

Strategy (𝑃𝑌, 𝐴𝑌𝑌 , 𝑊𝐿) (2)

Simplifying (1) 𝑈𝑅 − 𝐴𝑌𝑌 −𝑑𝐻

𝐴𝑌𝑌

Simplifying (2) 𝑈𝑅 − 𝐴𝑌𝑌 −𝑑𝐿

𝐴𝑌𝑌*

As −𝑑𝐻

𝐴𝑌𝑌< −

𝑑𝐿

𝐴𝑌𝑌, a larger number is

subtracted in payment 1 so that payment 2

obtains a higher utility.

Now pay 3 (PY, AYN, WH) is

compared with pay 2 (PY, AYN, WL),

corresponding to the decision of an employee

with prestige PY who does not strive to be

accepted AYN, between investing a greater

effort WH or less effort WL.


𝑝 [𝑈𝑅 − 𝐴𝑌𝑁 −𝑑𝐻

𝐴𝑌𝑁] + (1 − 𝑝) [𝑈𝑅 − 𝐴𝑌𝑁 −

𝑑𝐻

𝐴𝑌𝑁]

Strategy (𝑃𝑌, 𝐴𝑌𝑁 , 𝑊𝐻) (3)

(1 − 𝑝) [𝑈𝑅 − 𝐴𝑌𝑁 −𝑑𝐿

𝐴𝑌𝑁] + 𝑝 [𝑈𝑅 − 𝐴𝑌𝑁 −

𝑑𝐿

𝐴𝑌𝑁]

Strategy (𝑃𝑌, 𝐴𝑌𝑁 , 𝑊𝐿) (4)

Simplifying (3) 𝑈𝑅 − 𝐴𝑌𝑁 −𝑑𝐻

𝐴𝑌𝑁

Simplifying (4) 𝑈𝑅 − 𝐴𝑌𝑁 −𝑑𝐿

𝐴𝑌𝑁*

As −𝑑𝐻

𝐴𝑌𝑁< −

𝑑𝐿

𝐴𝑌𝑁, a larger number is



Now the corresponding payments are

compared for the case of an employee without

prestige. Pay 5 (PN, ANY, WH) is compared

with pay 6 (PN, ANY, WL), corresponding to

the decision of an employee without prestige

PN who strives to be accepted ANY, between


WL.


𝑝 [𝑈𝑅 − 𝐴𝑁𝑌 −𝑑𝐻

𝐴𝑁𝑌] + (1 − 𝑝) [𝑈𝑅 − 𝐴𝑁𝑌 −

𝑑𝐻

𝐴𝑁𝑌]

Strategy (𝑃𝑁 , 𝐴𝑁𝑌, 𝑊𝐻) (5)

(1 − 𝑝) [𝑈𝑅 − 𝐴𝑁𝑌 −𝑑𝐿

𝐴𝑁𝑌] + 𝑝 [𝑈𝑅 − 𝐴𝑁𝑌 −

𝑑𝐿

𝐴𝑁𝑌]

Strategy (𝑃𝑁 , 𝐴𝑁𝑌, 𝑊𝐿) (6)

Simplifying (5) 𝑈𝑅 − 𝐴𝑁𝑌 −𝑑𝐻

𝐴𝑁𝑌

Simplifying (6) 𝑈𝑅 − 𝐴𝑁𝑌 −𝑑𝐿

𝐴𝑁𝑌*

As −𝑑𝐻

𝐴𝑁𝑌< −

𝑑𝐿

𝐴𝑁𝑌, a larger number is



Pay 7 (PN, ANN, WH) is compared

with pay 8 (PN, ANN, WL), corresponding to


PN who does not make an effort to be accepted

ANN, between investing a greater effort WH or

less effort WL.


𝑝 [𝑈𝑅 − 𝐴𝑁𝑁 −𝑑𝐻

𝐴𝑁𝑁

] + (1 − 𝑝) [𝑈𝑅 − 𝐴𝑁𝑁 −𝑑𝐻

𝐴𝑁𝑁

]

Strategy (𝑃𝑁 , 𝐴𝑁𝑁 , 𝑊𝐻) (7)

(1 − 𝑝) [𝑈𝑅 − 𝐴𝑁𝑁 −𝑑𝐿

𝐴𝑁𝑁

] + 𝑝 [𝑈𝑅 − 𝐴𝑁𝑁 −𝑑𝐿

𝐴𝑁𝑁

]

Strategy (𝑃𝑁 , 𝐴𝑁𝑁 , 𝑊𝐿) (8)

Simplifying (7) 𝑈𝑅 − 𝐴𝑁𝑁 −𝑑𝐻

𝐴𝑁𝑁

Simplifying (8) 𝑈𝑅 − 𝐴𝑁𝑁 −𝑑𝐿

𝐴𝑁𝑁*

As −𝑑𝐻

𝐴𝑁𝑁< −

𝑑𝐿

𝐴𝑁𝑁, a larger number is



Regarding the decisions of the

prestigious employee, now PS must decide

whether to choose AYY or AYN

The simplified payment equations are:

𝑈𝑅 − 𝐴𝑌𝑌 −𝑑𝐿

𝐴𝑌𝑌 (2)

𝑈𝑅 − 𝐴𝑌𝑁 −𝑑𝐿

𝐴𝑌𝑁 (4)

35








We proceed to compare:

−𝐴𝑌𝑌 −𝑑𝐿

𝐴𝑌𝑌 y −𝐴𝑌𝑁 −

𝑑𝐿

𝐴𝑌𝑁

As 𝐴𝑌𝑌 > 𝐴𝑌𝑁 but 𝑑𝐿

𝐴𝑌𝑌< 𝑑𝐿/𝐴𝑌𝑁

subscenarios arise based on the acceptance

factor.

As the acceptance factor significantly

impacts the effort and therefore the disutility

perceived by the employee:

When the acceptance factor is high:

𝐴𝑌𝑌 > 𝐴𝑌𝑁 + (𝑑𝐿

𝐴𝑌𝑁−

𝑑𝐿

𝐴𝑌𝑌)

The impact on disutility is less so it will

subtract less from utility and (2) will be chosen.

When the acceptance factor is low:

𝐴𝑌𝑌 < 𝐴𝑌𝑁 + (𝑑𝐿

𝐴𝑌𝑁−

𝑑𝐿

𝐴𝑌𝑌)

The impact on disutility is greater so it

will subtract more from the utility and it will be

chosen (4).

When 𝐴𝑌𝑌 = 𝐴𝑌𝑁 + (𝑑𝐿

𝐴𝑌𝑁−

𝑑𝐿

𝐴𝑌𝑌) you

can choose either of the two payments because

they will be worth the same.

As for the decisions of the employee

without prestige, now PN must decide whether

to choose AYY or AYN

The simplified payment equations are:

𝑈𝑅 − 𝐴𝑁𝑌 −𝑑𝐿

𝐴𝑁𝑌* (6)

𝑈𝑅 − 𝐴𝑁𝑁 −𝑑𝐿

𝐴𝑁𝑁* (8)

We proceed to compare:

−𝐴𝑁𝑌 −𝑑𝐿

𝐴𝑁𝑌 y −𝐴𝑁𝑁 −

𝑑𝐿

𝐴𝑁𝑁

As 𝐴𝑁𝑌 > 𝐴𝑁𝑁 but 𝑑𝐿

𝐴𝑁𝑌< 𝑑𝐿/𝐴𝑁𝑁

subgames arise based on the acceptance factor.

As the acceptance factor significantly




𝐴𝑁𝑌 > 𝐴𝑁𝑁 + (𝑑𝐿

𝐴𝑁𝑌−

𝑑𝐿

𝐴𝑁𝑁)


subtract less from utility and will be chosen (6).


𝐴𝑁𝑌 < 𝐴𝑁𝑁 + (𝑑𝐿

𝐴𝑁𝑌−

𝑑𝐿

𝐴𝑁𝑁)


will subtract more from the utility and it will be

chosen (8).

When 𝐴𝑁𝑌 = 𝐴𝑁𝑁 + (𝑑𝐿

𝐴𝑁𝑌−

𝑑𝐿

𝐴𝑁𝑁) you



Now the Director decides whether to

elect PY or PN. For the Director, the incentive

to pay is the same but its utility is not, since p

and q are different (p> q). This is because an

employee with higher prestige is more likely to

obtain a better result from a project than an

employee with less prestige as stated in the

Theory and has been observed empirically.

Payments for the Director are:

(PY) p(G-R) + (1-p) (B-R)

(PN) q(G-R) + (1-q) (B-R)

Simplifying:

(PY) pG + (1-p) B

(PN) qG + (1-q) B

As p>q but (1-p) < (1-q), the payments

will depend on the values that p and q take:

When 𝐺 >[𝑞𝐺+(1−𝑞)𝐺−(1−𝑝)𝐵]

𝑝 will be

chosen PY

When 𝐺 <[𝑞𝐺+(1−𝑞)𝐺−(1−𝑝)𝐵]

𝑝 will be

chosen PN

36








When 𝐺 =[𝑞𝐺+(1−𝑞)𝐺−(1−𝑝)𝐵]

𝑝 you can

choose either of the two payments because they

will be worth the same.

So the solution of the game would be:

When 𝐺 >[𝑞𝐺+(1−𝑞)𝐺−(1−𝑝)𝐵]

𝑝

The dominant Strategy would be PY,

AYY, WL When the acceptance factor is high


𝐴𝑌𝑁−

𝑑𝐿

𝐴𝑌𝑌)

The dominant Strategy would be PY,

AYN, WL When the acceptance factor is low


𝐴𝑌𝑁−

𝑑𝐿

𝐴𝑌𝑌)

When 𝐴𝑌𝑌 > 𝐴𝑌𝑁 + (𝑑𝐿

𝐴𝑌𝑁−

𝑑𝐿

𝐴𝑌𝑌) you

can choose either of the two strategies because

the payouts will be worth the same.

When 𝐺 <[𝑞𝐺+(1−𝑞)𝐺−(1−𝑝)𝐵]

𝑝

The dominant Strategy would be PN,

ANY, WL When the acceptance factor is high


𝐴𝑁𝑁−

𝑑𝐿

𝐴𝑁𝑌)

The dominant Strategy would be PN,

ANN, WL When the acceptance factor is low


𝐴𝑁𝑁−

𝑑𝐿

𝐴𝑁𝑌)


𝐴𝑁𝑌−

𝑑𝐿

𝐴𝑁𝑁) you

can choose either of the two strategies because

the payouts will be worth the same.

When 𝐺 =[𝑞𝐺+(1−𝑞)𝐺−(1−𝑝)𝐵]

𝑝 You can

choose either of the PY or PN strategies with

the winning payments for the employee,

because the Director's payments will be worth

the same.

Scenario 2. Payment plus incentive (R ')

based on the profitability of the project

In the case of this scenario, each employee

option has an impact on the profit of the project

depending on the profitability of the project.

Since the manager cannot judge the

employee's efforts, he will pay the incentive

based on the profitability achieved, so the

incentive will be based on whether the project

had a good or a bad profit. For example, the

manager pays:

− An incentive from RG When the earning

level is G and

− An incentive from RB When profit is B

− Therefore, the net payments to the

manager at the two levels are,

respectively:

G - RG

B - RB



of incentives:

U (RG) for a good gain scenario

U (RB) for a bad gain scenario





that disutility as dH When the employee puts in

a lot of effort and dL When the employee puts

in less effort, the net payments for the employee

at the two levels are, respectively:

U(RG) - dH

U(RB) - dL

37








Figure 2 Game tree for Scenario 2 Payment plus

incentive (R ') based on the profitability of the project

Source: Self-made

Scenario Solution 2


with complete information, you start by

comparing the corresponding payouts for a

prestigious employee.

Pay 1 (PY, AYY, WH) is compared

with pay 2 (PY, AYY, WL), corresponding to

the decision of an employee with prestige PY



WL.


𝑝 [𝑈𝑅𝐺 − 𝐴𝑌𝑌 −𝑑𝐻

𝐴𝑌𝑌

] + (1 − 𝑝) [𝑈𝑅𝐵 − 𝐴𝑌𝑌 −𝑑𝐻

𝐴𝑌𝑌

]


(1 − 𝑝) [𝑈𝑅𝐺 − 𝐴𝑌𝑌 −𝑑𝐿

𝐴𝑌𝑌

] + 𝑝 [𝑈𝑅𝐵 − 𝐴𝑌𝑌 −𝑑𝐿

𝐴𝑌𝑌

]

Strategy (𝑃𝑌, 𝐴𝑌𝑌 , 𝑊𝐿) (2)

Simplifying (1) 𝑝𝑈𝑅𝐺 + (1 − 𝑝)𝑈𝑅𝐵 −

𝐴𝑌𝑌 −𝑑𝐻

𝐴𝑌𝑌

Simplifying (2) (1 − 𝑝)𝑈𝑅𝐺 + 𝑝𝑈𝑅𝐵 −

𝐴𝑌𝑌 −𝑑𝐿

𝐴𝑌𝑌

As 𝑝𝑈𝑅𝐺 > (1 − 𝑝)𝑈𝑅𝐺 but As

(1 − 𝑝)𝑈𝑅𝐵 < 𝑝𝑈𝑅𝐵 y −𝑑𝐻

𝐴𝑌𝑌< −

𝑑𝐿

𝐴𝑌𝑌 sub

scenarios arise:

𝑅𝐺 >[(1 − 𝑝)𝑈𝑅𝐺 + 𝑝𝑈𝑅𝐵 − (

𝑑𝐿𝐴𝑌𝑌

) − (1 − 𝑝)𝑈𝑅𝐵 + (𝑑𝐻𝐴𝑌𝑌

)]

𝑝𝑈

Being

𝜋1 =[(1 − 𝑝)𝑈𝑅𝐺 + 𝑝𝑈𝑅𝐵 − (

𝑑𝐿𝐴𝑌𝑌

) − (1 − 𝑝)𝑈𝑅𝐵 + (𝑑𝐻𝐴𝑌𝑌

)]

𝑝𝑈

If 𝑅𝐺 > 𝜋1 will be chosen (1) with the

Strategy 𝑃𝑌, 𝐴𝑌𝑌, 𝑊𝐻

If 𝑅𝐺 < 𝜋1 will be chosen (2) with the

Strategy 𝑃𝑌, 𝐴𝑌𝑌, 𝑊𝐿

If 𝑅𝐺 = 𝜋1 you can choose either

because both payments are equal.

Now pay 3 (P_Y, A_YN, W_H) is

compared with pay 4 (P_Y, A_YN, W_L),


with prestige PY who does not make an effort

to be accepted AYN, between investing a

greater effort WH or less effort WL.


𝑝 [𝑈𝑅𝐺 − 𝐴𝑌𝑁 −𝑑𝐻

𝐴𝑌𝑁] + (1 − 𝑝) [𝑈𝑅𝐵 − 𝐴𝑌𝑁 −

𝑑𝐻

𝐴𝑌𝑁]


(1 − 𝑝) [𝑈𝑅𝐺 − 𝐴𝑌𝑁 −𝑑𝐿

𝐴𝑌𝑁] + 𝑝 [𝑈𝑅𝐵 − 𝐴𝑌𝑁 −

𝑑𝐿

𝐴𝑌𝑁]



𝐴𝑌𝑁 −𝑑𝐻

𝐴𝑌𝑁


𝐴𝑌𝑁 −𝑑𝐿

𝐴𝑌𝑁



𝐴𝑌𝑁< −

𝑑𝐿

𝐴𝑌𝑁

subscenarios arise:

𝑅𝐺 >[(1 − 𝑝)𝑈𝑅𝐺 + 𝑝𝑈𝑅𝐵 − (

𝑑𝐿𝐴𝑌𝑁

) − (1 − 𝑝)𝑈𝑅𝐵 + (𝑑𝐻

𝐴𝑌𝑁)]

𝑝𝑈

38








Being

𝜋2 =[(1 − 𝑝)𝑈𝑅𝐺 + 𝑝𝑈𝑅𝐵 − (

𝑑𝐿𝐴𝑌𝑁

) − (1 − 𝑝)𝑈𝑅𝐵 + (𝑑𝐻

𝐴𝑌𝑁)]

𝑝𝑈


Strategy 𝑃𝑌, 𝐴𝑌𝑁 , 𝑊𝐻


Strategy 𝑃𝑌, 𝐴𝑌𝑁 , 𝑊𝐿



Now we analyze the case of an

employee without prestige. Pay 5 (PN, ANY,

WH) is compared with pay 6 (PN, AYY, WL),


without prestige PN who strives to be accepted

AYY, between investing a greater effort WH or

a less effort WL.


𝑞 [𝑈𝑅𝐺 − 𝐴𝑁𝑌 −𝑑𝐻

𝐴𝑁𝑌] + (1 − 𝑞) [𝑈𝑅𝐵 − 𝐴𝑁𝑌 −

𝑑𝐻

𝐴𝑁𝑌]

Strategy (𝑃𝑌, 𝐴𝑁𝑌, 𝑊𝐻) (5)

(1 − 𝑞) [𝑈𝑅𝐺 − 𝐴𝑁𝑌 −𝑑𝐿

𝐴𝑁𝑌] + 𝑞[𝑈𝑅𝐵 − 𝐴𝑁𝑌 − 𝑑𝐿/𝐴𝑁𝑌]

Strategy (𝑃𝑌, 𝐴𝑁𝑌, 𝑊𝐿) (6)

Simplifying (5) 𝑞𝑈𝑅𝐺 + (1 − 𝑞)𝑈𝑅𝐵 −

𝐴𝑁𝑌 −𝑑𝐻

𝐴𝑁𝑌

Simplifying (6) (1 − 𝑞)𝑈𝑅𝐺 + 𝑞𝑈𝑅𝐵 −

𝐴𝑁𝑌 −𝑑𝐿

𝐴𝑁𝑌

As 𝑞𝑈𝑅𝐺 > (1 − 𝑞)𝑈𝑅𝐺 but As

(1 − 𝑞)𝑈𝑅𝐵 < 𝑞𝑈𝑅𝐵 y −𝑑𝐻

𝐴𝑁𝑌< −

𝑑𝐿

𝐴𝑁𝑌 sub

scenarios arise:

𝑅𝐺 >[(1 − 𝑞)𝑈𝑅𝐺 + 𝑞𝑈𝑅𝐵 − (

𝑑𝐿𝐴𝑁𝑌

) − (1 − 𝑞)𝑈𝑅𝐵 + (𝑑𝐻

𝐴𝑁𝑌)]

𝑞𝑈

Being

𝜋3 =[(1 − 𝑞)𝑈𝑅𝐺 + 𝑞𝑈𝑅𝐵 − (


) − (1 − 𝑞)𝑈𝑅𝐵 + (𝑑𝐻

𝐴𝑁𝑌)]

𝑞𝑈


Strategy 𝑃𝑁 , 𝐴𝑁𝑌, 𝑊𝐻


Strategy 𝑃𝑁 , 𝐴𝑁𝑌, 𝑊𝐿



Pay 7 (PN, ANN, WH) is compared

with pay 8 (PN, ANN, WL), corresponding to


PN who does not make an effort to be accepted

ANN, between investing a greater effort WH or

less effort WL.


𝑞 [𝑈𝑅𝐺 − 𝐴𝑁𝑁 −𝑑𝐻

𝐴𝑁𝑁] + (1 − 𝑞) [𝑈𝑅𝐵 − 𝐴𝑁𝑁 −

𝑑𝐻

𝐴𝑁𝑁]


(1 − 𝑞) [𝑈𝑅𝐺 − 𝐴𝑁𝑁 −𝑑𝐿

𝐴𝑁𝑁] + 𝑞 [𝑈𝑅𝐵 − 𝐴𝑁𝑁 −

𝑑𝐿

𝐴𝑁𝑁]



𝐴𝑁𝑁 −𝑑𝐻

𝐴𝑁𝑁


𝐴𝑁𝑁 −𝑑𝐿

𝐴𝑁𝑁



𝐴𝑁𝑁< −

𝑑𝐿

𝐴𝑁𝑁 sub

scenarios arise:

𝑅𝐺 >[(1 − 𝑞)𝑈𝑅𝐺 + 𝑞𝑈𝑅𝐵 − (

𝑑𝐿𝐴𝑁𝑁

) − (1 − 𝑞)𝑈𝑅𝐵 + (𝑑𝐻

𝐴𝑁𝑁)]

𝑞𝑈

Being

𝜋4 =[(1 − 𝑝)𝑈𝑅𝐺 + 𝑝𝑈𝑅𝐵 − (


) − (1 − 𝑝)𝑈𝑅𝐵 + (𝑑𝐻

𝐴𝑁𝑁)]

𝑝𝑈


Strategy 𝑃𝑁 , 𝐴𝑁𝑁 , 𝑊𝐻


Strategy 𝑃𝑁 , 𝐴𝑁𝑁 , 𝑊𝐿



Now it is analyzed When 𝑅𝐺 >𝜋1, 𝜋2, 𝜋3, 𝜋4

Payment (1) is compared with payment

(3):

39








𝑝 [𝑈𝑅𝐺 − 𝐴𝑌𝑌 −𝑑𝐻

𝐴𝑌𝑌] + (1 − 𝑝) [𝑈𝑅𝐵 − 𝐴𝑌𝑌 −

𝑑𝐻

𝐴𝑌𝑌]


𝑝 [𝑈𝑅𝐺 − 𝐴𝑌𝑁 −𝑑𝐻

𝐴𝑌𝑁] + (1 − 𝑝) [𝑈𝑅𝐵 − 𝐴𝑌𝑁 −

𝑑𝐻

𝐴𝑌𝑁]



𝐴𝑌𝑌 −𝑑𝐻

𝐴𝑌𝑌


𝐴𝑌𝑁 −𝑑𝐻

𝐴𝑌𝑁

−𝐴𝑌𝑌 − 𝑑𝐻/𝐴𝑌𝑌 (1)

−𝐴𝑌𝑁 − 𝑑𝐻/𝐴𝑌𝑁 (3)

As 𝐴𝑌𝑌 > 𝐴𝑌𝑁 but 𝑑𝐻

𝐴𝑌𝑌<

𝑑𝐻

𝐴𝑌𝑌

Subscenaries arise based on the acceptance

factor. Thus, the acceptance factor significantly




𝐴𝑌𝑌 > 𝐴𝑌𝑁 + (𝑑𝐻

𝐴𝑌𝑁−

𝑑𝐻

𝐴𝑌𝑌)




𝐴𝑌𝑌 < 𝐴𝑌𝑁 + (𝑑𝐻

𝐴𝑌𝑁−

𝑑𝐻

𝐴𝑌𝑌)


will subtract more from utility and will be

chosen (3).

When 𝐴𝑌𝑌 = 𝐴𝑌𝑁 + (𝑑𝐻

𝐴𝑌𝑁−

𝑑𝐻

𝐴𝑌𝑌) you



To analyze the case of the employee

without prestige, the payment (5) is compared

with the payment (7):


𝐴𝑁𝑌] + (1 − 𝑞) [𝑈𝑅𝐵 − 𝐴𝑌𝑌 −

𝑑𝐻

𝐴𝑁𝑌]



𝐴𝑁𝑁] + (1 − 𝑞) [𝑈𝑅𝐵 − 𝐴𝑁𝑁 −

𝑑𝐻

𝐴𝑁𝑁]




𝐴𝑁𝑌



𝐴𝑁𝑁

−𝐴𝑁𝑌 −𝑑𝐻

𝐴𝑁𝑌 (5)

−𝐴𝑁𝑁 −𝑑𝐻

𝐴𝑁𝑁 (7)

As 𝐴𝑁𝑌 > 𝐴𝑁𝑁 but 𝑑𝐻

𝐴𝑁𝑌<

𝑑𝐻

𝐴𝑁𝑁


factor. Thus the acceptance factor significantly




𝐴𝑁𝑌 > 𝐴𝑁𝑁 + (𝑑𝐻

𝐴𝑁𝑁−

𝑑𝐻

𝐴𝑁𝑌)


subtract less from the utility and will be chosen

𝑃𝑁 , 𝐴𝑁𝑌, 𝑊𝐻 (5)


𝐴𝑁𝑌 < 𝐴𝑁𝑁 + (𝑑𝐻

𝐴𝑁𝑁−

𝑑𝐻

𝐴𝑁𝑌)


will subtract more from the utility and will be

chosen

𝑃𝑁 , 𝐴𝑁𝑁 , 𝑊𝐻 (7)

When 𝐴𝑁𝑌 = 𝐴𝑁𝑁 + (𝑑𝐻

𝐴𝑁𝑁−

𝑑𝐻

𝐴𝑁𝑌)you



Now it is analyzed When 𝑅𝐺 <𝜋1, 𝜋2, 𝜋3, 𝜋4


(4):

40








(1 − 𝑝) [𝑈𝑅𝐺 − 𝐴𝑌𝑌 −𝑑𝐿

𝐴𝑌𝑌] + 𝑝 [𝑈𝑅𝐵 − 𝐴𝑌𝑌 −

𝑑𝐿

𝐴𝑌𝑌]

Strategy (𝑃𝑌, 𝐴𝑌𝑌, 𝑊𝐿) (2)

(1 − 𝑝) [𝑈𝑅𝐺 − 𝐴𝑌𝑁 −𝑑𝐿

𝐴𝑌𝑁] + 𝑝 [𝑈𝑅𝐵 − 𝐴𝑌𝑁 −

𝑑𝐿

𝐴𝑌𝑁]



𝐴𝑌𝑌 −𝑑𝐿

𝐴𝑌𝑌


𝐴𝑌𝑁 −𝑑𝐿

𝐴𝑌𝑁


𝐴𝑌𝑌 (2)

−𝐴𝑌𝑁 −𝑑𝐿

𝐴𝑌𝑁 (4)


𝐴𝑌𝑌<

𝑑𝐿

𝐴𝑌𝑁







𝐴𝑌𝑁−

𝑑𝐿

𝐴𝑌𝑌)



𝑃𝑌, 𝐴𝑌𝑌 , 𝑊𝐿 (2)



𝐴𝑌𝑁−

𝑑𝐿

𝐴𝑌𝑌)



chosen

𝑃𝑌, 𝐴𝑌𝑁 , 𝑊𝐿 (4)

When𝐴𝑌𝑌 = 𝐴𝑌𝑁 + (𝑑𝐿

𝐴𝑌𝑁−

𝑑𝐿

𝐴𝑌𝑌) you




without prestige, payment (6) is compared with

payment (8):


𝐴𝑁𝑌] + 𝑞 [𝑈𝑅𝐵 − 𝐴𝑌𝑌 −

𝑑𝐿

𝐴𝑁𝑌]




𝑑𝐿

𝐴𝑁𝑁]

Strategy (𝑃𝑁 , 𝐴𝑁𝑁, 𝑊𝐿) (8)



𝐴𝑁𝑌



𝐴𝑁𝑁

−𝐴𝑁𝑌 −𝑑𝐿

𝐴𝑁𝑌 (6)

−𝐴𝑁𝑁 −𝑑𝐿

𝐴𝑁𝑁 (8)


𝐴𝑁𝑌<

𝑑𝐿

𝐴𝑁𝑁







𝐴𝑁𝑁−

𝑑𝐿

𝐴𝑁𝑌)



𝑃𝑁 , 𝐴𝑁𝑌, 𝑊𝐿 (6)



𝐴𝑁𝑁−

𝑑𝐿

𝐴𝑁𝑌)



chosen

𝑃𝑁 , 𝐴𝑁𝑁, 𝑊𝐿 (8)

41









𝐴𝑁𝑁−

𝑑𝐿

𝐴𝑁𝑌) you can

choose either of the two payments because they

will be worth the same.

Now the Director will choose the best

payment for him.

(PY) 𝑝(𝐺 − 𝑅𝐺) + (1 − 𝑝)(𝐵 − 𝑅𝐵)

(PN) 𝑞(𝐺 − 𝑅𝐺) + (1 − 𝑞)(𝐵 − 𝑅𝐵)

Simplifying:

(PY) 𝑝𝐺 − 𝑝𝑅𝐺 − 𝑝𝐵 + 𝑝𝑅𝐵

(PN) 𝑞𝐺 − 𝑞𝑅𝐺 − 𝑞𝐵 + 𝑞𝑅𝐵

𝐺 >[𝑞𝐺 − 𝑞𝑅𝐺 − 𝑞𝐵 + 𝑞𝑅𝐵 + 𝑝𝑅𝐺 + 𝑝𝐵 − 𝑝𝑅𝐵]

𝑝

𝜋5 =[𝑞𝐺 − 𝑞𝑅𝐺 − 𝑞𝐵 + 𝑞𝑅𝐵 + 𝑝𝑅𝐺 + 𝑝𝐵 − 𝑝𝑅𝐵]

𝑝

Sub-scenarios then arise:

When G > 𝜋5 will be chosen (PY)

When G < 𝜋5 will be chosen (PN)

When G= 𝜋5 anyone can be chosen

since both payments are equal.

In this way,

When G > 𝜋5

If 𝑅𝐺 > 𝜋1 y 𝐴𝑌𝑌 > 𝐴𝑌𝑁 +𝑑𝐻

𝐴𝑌𝑁−

𝑑𝐻

𝐴𝑌𝑌,

the dominant Strategy will be 𝑃𝑌, 𝐴𝑌𝑌 , 𝑊𝐻

ó

If 𝑅𝐺 > 𝜋1 y 𝐴𝑌𝑌 < 𝐴𝑌𝑁 +𝑑𝐻

𝐴𝑌𝑁−

𝑑𝐻

𝐴𝑌𝑌,

the dominant Strategy will be 𝑃𝑌, 𝐴𝑌𝑁 , 𝑊𝐻

ó

If 𝑅𝐺 < 𝜋1 y 𝐴𝑌𝑌 > 𝐴𝑌𝑁 +𝑑𝐻

𝐴𝑌𝑁−

𝑑𝐻

𝐴𝑌𝑌,

the dominant Strategy will be 𝑃𝑌, 𝐴𝑌𝑌 , 𝑊𝐿

ó

If 𝑅𝐺 < 𝜋1 y 𝐴𝑌𝑌 < 𝐴𝑌𝑁 +𝑑𝐻

𝐴𝑌𝑁−

𝑑𝐻

𝐴𝑌𝑌,

the dominant Strategy will be 𝑃𝑌, 𝐴𝑌𝑁 , 𝑊𝐿

When G<𝜋5

If 𝑅𝐺 > 𝜋2 y 𝐴𝑁𝑌 > 𝐴𝑁𝑁 +𝑑𝐻

𝐴𝑁𝑁−

𝑑𝐻

𝐴𝑁𝑌,

the dominant Strategy will be 𝑃𝑁 , 𝐴𝑌𝑌, 𝑊𝐻

ó

If 𝑅𝐺 > 𝜋2 y 𝐴𝑁𝑌 < 𝐴𝑁𝑁 +𝑑𝐻

𝐴𝑁𝑁−

𝑑𝐻

𝐴𝑁𝑌,

the dominant Strategy will be 𝑃𝑁 , 𝐴𝑌𝑁 , 𝑊𝐻

ó

If 𝑅𝐺 < 𝜋2 y 𝐴𝑁𝑌 > 𝐴𝑁𝑁 +𝑑𝐻

𝐴𝑁𝑁−

𝑑𝐻

𝐴𝑁𝑌,

the dominant Strategy will be 𝑃𝑁 , 𝐴𝑌𝑌, 𝑊𝐿

ó

If 𝑅𝐺 < 𝜋2 y 𝐴𝑁𝑌 < 𝐴𝑁𝑁 +𝑑𝐻

𝐴𝑁𝑁−

𝑑𝐻

𝐴𝑁𝑌,

the dominant Strategy will be 𝑃𝑁 , 𝐴𝑌𝑁 , 𝑊𝐿

Scenario 3 Payment plus incentive (R ')

based on the prestige of the employee

In the case of this scenario, the payment is

given solely based on the prestige of the

employee. Since the manager cannot judge the

employee's efforts, he will pay the incentive

based on the employee's prestige. For example,

the manager pays:

− An RPY incentive When the selected

employee has prestige.

− An RPN incentive When the selected

employee has no prestige.

Therefore, the net payments to the

manager at the two levels are, respectively:

G - RPY

G - RPN

B – RPY

B – RPN



of incentives:

U (RPY) for a good profit scenario

U (RPN) for a bad profit scenario





that disutility As dH When the employee puts

in a lot of effort and dL When the employee

tries less, the net payments to the employee at


U (RPY) - dH

U (RPN) - dH

U (RPY) – dL

42








U (RPN) - dL

Figure 3 Scenario 3 Payment plus incentive (R ') based

on the prestige of the employee

Source: Self-made

For this case, the solution of the best

payments for employees is the same as in scenario

1. For the Director, it is compared:

(𝑃𝑌) 𝑝(𝐺 − 𝑅𝑃𝑌) + (1 − 𝑝)(𝐵 − 𝑅𝑃𝑌)

(𝑃𝑁) 𝑞(𝐺 − 𝑅𝑃𝑁) + (1 − 𝑞)(𝐵 − 𝑅𝑃𝑁)

Simplifying:

(𝑃𝑌) 𝑝𝐺 + (1 − 𝑝)𝐵 − 𝑅𝑃𝑌

(𝑃𝑁) 𝑞𝐺 + (1 − 𝑞)𝐵 − 𝑅𝑃𝑁

𝐺 >[𝑞𝐺 + (1 − 𝑞)𝐵 − 𝑅𝑃𝑁 − (1 − 𝑝)𝐵 + 𝑅𝑃𝑌]

𝑝

𝜋6 =[𝑞𝐺 + (1 − 𝑞)𝐵 − 𝑅𝑃𝑁 − (1 − 𝑝)𝐵 + 𝑅𝑃𝑌]

𝑝


When G > 𝜋6 will be chosen (𝑃𝑌)

When G < 𝜋6 will be chosen (𝑃𝑁)

When G=𝜋6 anyone can be chosen


In this way,

When G > 𝜋6

If 𝐴𝑌𝑌 > 𝐴𝑌𝑁 +𝑑𝐿

𝐴𝑌𝑁−

𝑑𝐿

𝐴𝑌𝑌 the dominant

Strategy will be 𝑃𝑌,𝐴𝑌𝑌, 𝑊𝐿

ó

If 𝐴𝑌𝑌 < 𝐴𝑌𝑁 +𝑑𝐿

𝐴𝑌𝑁−

𝑑𝐿

𝐴𝑌𝑌 the dominant

Strategy will be 𝑃𝑌,𝐴𝑌𝑁 , 𝑊𝐿

ó

If 𝐴𝑌𝑌 = 𝐴𝑌𝑁 +𝑑𝐿

𝐴𝑌𝑁−

𝑑𝐿

𝐴𝑌𝑌 you can

choose any payment as both payments are

equal.

When G < 𝜋6

If 𝐴𝑁𝑌 > 𝐴𝑁𝑁 +𝑑𝐿

𝐴𝑁𝑁−

𝑑𝐿

𝐴𝑁𝑌 the

dominant Strategy will be 𝑃𝑁,𝐴𝑌𝑌 , 𝑊𝐿

ó

If 𝐴𝑁𝑌 < 𝐴𝑁𝑁 +𝑑𝐿

𝐴𝑁𝑁−

𝑑𝐿

𝐴𝑁𝑌 the

dominant Strategy will be 𝑃𝑁,𝐴𝑌𝑁 , 𝑊𝐿

ó

If 𝐴𝑁𝑌 = 𝐴𝑁𝑁 +𝑑𝐿

𝐴𝑁𝑁−

𝑑𝐿

𝐴𝑁𝑌 You can

choose any PN, AYY, WL or PN, AYN, WL

payment since both payments are the same.

When G = 𝜋6 you can choose any

payment from WL (2, 4, 6 u 8).

Scenario 4 Payment plus incentive (R’P)

based on the profitability of the project

In the case of this scenario, each employee

option has an impact on the profit of the project

depending on the profitability of the project and

the prestige. Since the manager cannot judge

the employee's efforts, he will pay the incentive

based on the profitability achieved, so the

incentive will be based on whether the project

had a good or a bad profit and based on

prestige. The higher the prestige, the

corresponding prestige pay will be higher. For

example, the manager pays:

An RPG Incentive When Earning Level

is G and

An RPB Incentive When Profit is B

Therefore, the net payments to the manager at


G - RPG

B - RPB



of incentives:

43








U(RPG) for a good profit scenario for an

employee with high prestige.

U(RG) for a good profit scenario for an

employee with low prestige.

U(RPB) for a low profit scenario for a

high prestige employee.

U(RB) for a low profit scenario for a low

prestige employee.





that disutility As dH When the employee puts

in a great effort and dL When the employee

puts in less effort, the net payments for the

employee at the two levels are, respectively:

U(RPG) - dH

U(RPG) - dL

U(RG) - dH

U(RG) - dL

U(RBG) - dH

U(RBG) - dL

U(RB) - dH

U(RB) - dL

Figure 4 Game tree for Scenario 4 Payment plus

incentive based on project profitability and prestige.

Source: Self-made

Scenario Solution 4


with complete information, you start by

comparing the corresponding payouts for a

prestigious employee.

Payment 1 (P_Y, A_YY, W_H) is

compared with payment 2 (P_Y, A_YY, W_L),


with prestige P_Y who strives to be accepted

A_YY, between investing a greater effort W_H

or a least effort W_L.


𝑝 [𝑈𝑅𝑃𝐺 − 𝐴𝑌𝑌 −𝑑𝐻

𝐴𝑌𝑌] + (1 − 𝑝) [𝑈𝑅𝑃𝐵 − 𝐴𝑌𝑌 −

𝑑𝐻

𝐴𝑌𝑌]

Strategy (𝑃𝑌, 𝐴𝑌𝑌,𝑊𝐻) (1)

(1 − 𝑝) [𝑈𝑅𝑃𝐺 − 𝐴𝑌𝑌 −𝑑𝐿

𝐴𝑌𝑌] + 𝑝 [𝑈𝑅𝑃𝐵 − 𝐴𝑌𝑌 −

𝑑𝐿

𝐴𝑌𝑌]

Strategy (𝑃𝑌, 𝐴𝑌𝑌,𝑊𝐿) (2)

Simplifying (1) 𝑝𝑈𝑅𝑃𝐺 + (1 −

𝑝)𝑈𝑅𝑃𝐵 − 𝐴𝑌𝑌 −𝑑𝐻

𝐴𝑌𝑌

Simplifying (2) (1 − 𝑝)𝑈𝑅𝑃𝐺 +

𝑝𝑈𝑅𝑃𝐵 − 𝐴𝑌𝑌 −𝑑𝐿

𝐴𝑌𝑌

As 𝑝𝑈𝑅𝑃𝐺 > (1 − 𝑝)𝑈𝑅𝑃𝐺 but As

(1 − 𝑝)𝑈𝑅𝑃𝐵 < 𝑝𝑈𝑅𝑃𝐵 y −𝑑𝐻

𝐴𝑌𝑌< −

𝑑𝐿

𝐴𝑌𝑌, sub

scenarios arise:

𝑅𝑃𝐺 >[(1 − 𝑝)𝑈𝑅𝑃𝐺 + 𝑝𝑈𝑅𝑃𝐵 − (

𝑑𝐿

𝐴𝑌𝑌) − (1 − 𝑝)𝑈𝑅𝑃𝐵 + (

𝑑𝐻

𝐴𝑌𝑌)]

𝑝𝑈

Being PG

𝜋7 =[(1 − 𝑝)𝑈𝑅𝑃𝐺 + 𝑝𝑈𝑅𝑃𝐵 − (

𝑑𝐿

𝐴𝑌𝑌) − (1 − 𝑝)𝑈𝑅𝑃𝐵 + (

𝑑𝐻

𝐴𝑌𝑌)]

𝑝𝑈

If 𝑅𝑃𝐺 > 𝜋7 will be chosen (1) with

the Strategy 𝑃𝑌 , 𝐴𝑌𝑌, 𝑊𝐻

If 𝑅𝑃𝐺 < 𝜋7 will be chosen (2) with

the Strategy 𝑃𝑌 , 𝐴𝑌𝑌, 𝑊𝐿

If 𝑅𝑃𝐺 = 𝜋7you can choose either


44








Now payment 3 (P_Y, A_YN, W_H) is

compared with payment 4 (P_Y, A_YN, W_L),


with prestige PY who does not make an effort

to be accepted AYN, between investing a

greater effort WH or less effort WL.


𝑝 [𝑈𝑅𝑃𝐺 − 𝐴𝑌𝑁 −𝑑𝐻

𝐴𝑌𝑁] + (1 − 𝑝) [𝑈𝑅𝑃𝐵 − 𝐴𝑌𝑁 −

𝑑𝐻

𝐴𝑌𝑁]


(1 − 𝑝) [𝑈𝑅𝑃𝐺 − 𝐴𝑌𝑁 −𝑑𝐿

𝐴𝑌𝑁] + 𝑝 [𝑈𝑅𝑃𝐵 − 𝐴𝑌𝑁 −

𝑑𝐿

𝐴𝑌𝑁]


Simplifying (3) 𝑝𝑈𝑅𝑃𝐺 + (1 −

𝑝)𝑈𝑅𝑃𝐵 − 𝐴𝑌𝑁 −𝑑𝐻

𝐴𝑌𝑁


𝑝𝑈𝑅𝑃𝐵 − 𝐴𝑌𝑁 −𝑑𝐿

𝐴𝑌𝑁



𝐴𝑌𝑁< −𝑑𝐿/𝐴𝑌𝑁, sub

scenarios arise:

𝑅𝐺 >[(1 − 𝑝)𝑈𝑅𝑃𝐺 + 𝑝𝑈𝑅𝑃𝐵 − (

𝑑𝐿

𝐴𝑌𝑁) − (1 − 𝑝)𝑈𝑅𝑃𝐵 + (

𝑑𝐻

𝐴𝑌𝑁)]

𝑝𝑈

Being

𝜋8 =[(1 − 𝑝)𝑈𝑅𝑃𝐺 + 𝑝𝑈𝑅𝑃𝐵 − (

𝑑𝐿

𝐴𝑌𝑁) − (1 − 𝑝)𝑈𝑅𝑃𝐵 + (

𝑑𝐻

𝐴𝑌𝑁)]

𝑝𝑈

If 𝑅𝑃𝐺 > 𝜋8 will be chosen (3) with the

Strategy 𝑃𝑌, 𝐴𝑌𝑁 , 𝑊𝐻

If 𝑅𝑃𝐺 < 𝜋8 will be chosen (4) with the

Strategy 𝑃𝑌, 𝐴𝑌𝑁 , 𝑊𝐿

If 𝑅𝑃𝐺 = 𝜋8 you can choose either


Now it is analyzed the case of an

employee without prestige. Payment 5 (P_N,

A_NY, W_H) is compared with payment 6

(P_N, A_YY, W_L), corresponding to the

decision of an employee without prestige PN



WL.



𝐴𝑁𝑌] + (1 − 𝑞) [𝑈𝑅𝐵 − 𝐴𝑁𝑌 −

𝑑𝐻

𝐴𝑁𝑌]



𝐴𝑁𝑌] + 𝑞 [𝑈𝑅𝐵 − 𝐴𝑁𝑌 −

𝑑𝐿

𝐴𝑁𝑌]




𝐴𝑁𝑌



𝐴𝑁𝑌



𝐴𝑁𝑌< −

𝑑𝐿

𝐴𝑁𝑌, sub

scenarios arise:

𝑅𝐺 >[(1 − 𝑞)𝑈𝑅𝐺 + 𝑝𝑈𝑅𝐵 − (


) − (1 − 𝑞)𝑈𝑅𝐵 + (𝑑𝐻

𝐴𝑁𝑌)]

𝑝𝑈

Being

𝜋9 =[(1 − 𝑞)𝑈𝑅𝐺 + 𝑝𝑈𝑅𝐵 − (


) − (1 − 𝑞)𝑈𝑅𝐵 + (𝑑𝐻

𝐴𝑁𝑌)]

𝑝𝑈


Strategy 𝑃𝑁 , 𝐴𝑁𝑌, 𝑊𝐻


Strategy 𝑃𝑁 , 𝐴𝑁𝑌, 𝑊𝐿



Payment 7 (P_N, A_ (NN,) W_H) is

compared with payment 8 (P_N, A_ (NN,)

W_L), corresponding to the decision of an

employee without prestige PN who does not

make an effort to be accepted ANN, between

invest more effort WH or less effort WL.



𝐴𝑁𝑁] + (1 − 𝑞) [𝑈𝑅𝐵 − 𝐴𝑁𝑁 −

𝑑𝐻

𝐴𝑁𝑁]




𝑑𝐿

𝐴𝑁𝑁]


45










𝐴𝑁𝑁



𝐴𝑁𝑁

As 𝑝𝑈𝑅𝐺 > (1 − 𝑞)𝑈𝑅𝐺 but As

(1 − 𝑞)𝑈𝑅𝐵 < 𝑝𝑈𝑅𝐵 y −𝑑𝐻

𝐴𝑁𝑁< −

𝑑𝐿

𝐴𝑁𝑁, sub

scenarios arise:

𝑅𝐺 >[(1 − 𝑞)𝑈𝑅𝐺 + 𝑝𝑈𝑅𝐵 − (


) − (1 − 𝑞)𝑈𝑅𝐵 + (𝑑𝐻

𝐴𝑁𝑁)]

𝑞𝑈

Being 𝜋10

=[(1 − 𝑞)𝑈𝑅𝐺 + 𝑝𝑈𝑅𝐵 − (


) − (1 − 𝑞)𝑈𝑅𝐵 + (𝑑𝐻

𝐴𝑁𝑁)]

𝑞𝑈


Strategy (𝑃𝑁 , 𝐴𝑁𝑁 , 𝑊𝐻)


Strategy (𝑃𝑁 , 𝐴𝑁𝑁 , 𝑊𝐿)



Now it is analyzed when 𝑹𝑷𝑮 >𝝅𝟕, 𝝅𝟖, 𝝅𝟗, 𝝅𝟏𝟎


(3):

𝑝 [𝑈𝑅𝑃𝐺 − 𝐴𝑌𝑌 −𝑑𝐻

𝐴𝑌𝑌] + (1 − 𝑝) [𝑈𝑅𝑃𝐵 − 𝐴𝑌𝑌 −

𝑑𝐻

𝐴𝑌𝑌]

Strategy 𝑃𝑌, 𝐴𝑌𝑌, 𝑊𝐻 (1)

𝑝 [𝑈𝑅𝑃𝐺 − 𝐴𝑌𝑁 −𝑑𝐻

𝐴𝑌𝑁] + (1 − 𝑝) [𝑈𝑅𝑃𝐵 − 𝐴𝑌𝑁 −

𝑑𝐻

𝐴𝑌𝑁]

Strategy 𝑃𝑌, 𝐴𝑌𝑁 , 𝑊𝐻 (3)

Simplifying (1)

𝑝𝑈𝑅𝑃𝐺 + (1 − 𝑝)𝑈𝑅𝑃𝐵 − 𝐴𝑌𝑌 −𝑑𝐻

𝐴𝑌𝑌

−𝐴𝑌𝑌 −𝑑𝐻

𝐴𝑌𝑌 (1)

−𝐴𝑌𝑁 −𝑑𝐻

𝐴𝑌𝑁 (3)

As 𝐴𝑌𝑌 > 𝐴𝑌𝑁 but 𝑑𝐻

𝐴𝑌𝑌<

𝑑𝐻

𝐴𝑌𝑁






𝐴𝑌𝑌 > 𝐴𝑌𝑁 + (𝑑𝐻

𝐴𝑌𝑁−

𝑑𝐻

𝐴𝑌𝑌)




𝐴𝑌𝑌 < 𝐴𝑌𝑁 + (𝑑𝐻

𝐴𝑌𝑁−

𝑑𝐻

𝐴𝑌𝑌)


will subtract more from utility and will be

chosen (3).

When 𝐴𝑌𝑌 = 𝐴𝑌𝑁 + (𝑑𝐻

𝐴𝑌𝑁−

𝑑𝐻

𝐴𝑌𝑌) you




without prestige, the payment (5) is compared

with the payment (7):


𝐴𝑁𝑌] + (1 − 𝑞) [𝑈𝑅𝐵 − 𝐴𝑌𝑌 −

𝑑𝐻

𝐴𝑁𝑌]



𝐴𝑁𝑁] + (1 − 𝑞) [𝑈𝑅𝐵 − 𝐴𝑁𝑁 −

𝑑𝐻

𝐴𝑁𝑁]


Simplifying (5)

𝑞𝑈𝑅𝐺 + (1 − 𝑞)𝑈𝑅𝐵 − 𝐴𝑁𝑌 −𝑑𝐻

𝐴𝑁𝑌

Simplifying (7)

𝑞𝑈𝑅𝐺 + (1 − 𝑞)𝑈𝑅𝐵 − 𝐴𝑁𝑁 −𝑑𝐻

𝐴𝑁𝑁

(5) −𝐴𝑁𝑌 −𝑑𝐻

𝐴𝑁𝑌

(7) −𝐴𝑁𝑁 −𝑑𝐻

𝐴𝑁𝑁

As 𝐴𝑁𝑌 > 𝐴𝑁𝑁 but 𝑑𝐻

𝐴𝑁𝑌<

𝑑𝐻

𝐴𝑁𝑁





46









𝐴𝑁𝑌 > 𝐴𝑁𝑁 + (𝑑𝐻

𝐴𝑁𝑁−

𝑑𝐻

𝐴𝑁𝑌)



𝑃𝑁 , 𝐴𝑁𝑌, 𝑊𝐻 (5)


𝐴𝑁𝑌 < 𝐴𝑁𝑁 + (𝑑𝐻

𝐴𝑁𝑁−

𝑑𝐻

𝐴𝑁𝑌)



chosen

𝑃𝑁 , 𝐴𝑁𝑁 , 𝑊𝐻 (7)

When 𝐴𝑁𝑌 = 𝐴𝑁𝑁 + (𝑑𝐻

𝐴𝑁𝑁−

𝑑𝐻

𝐴𝑁𝑌) you



Now it is analyzed When 𝑹𝑮 <𝝅𝟏, 𝝅𝟐, 𝝅𝟑, 𝝅𝟒


(4):

(1 − 𝑝) [𝑈𝑅𝑃𝐺 − 𝐴𝑌𝑌 −𝑑𝐿

𝐴𝑌𝑌] + 𝑝 [𝑈𝑅𝑃𝐵 − 𝐴𝑌𝑌 −

𝑑𝐿

𝐴𝑌𝑌]

Strategy (𝑃𝑌, 𝐴𝑌𝑌,𝑊𝐿) (2)

(1 − 𝑝) [𝑈𝑅𝑃𝐺 − 𝐴𝑌𝑁 −𝑑𝐿

𝐴𝑌𝑁] + 𝑝 [𝑈𝑅𝑃𝐵 − 𝐴𝑌𝑁 −

𝑑𝐿

𝐴𝑌𝑁]


Simplifying (2)(1 − 𝑝)𝑈𝑅𝑃𝐺 +

𝑝𝑈𝑅𝐵 − 𝐴𝑌𝑌 −𝑑𝐿

𝐴𝑌𝑌


𝑝𝑈𝑅𝐵 − 𝐴𝑌𝑁 −𝑑𝐿

𝐴𝑌𝑁


𝐴𝑌𝑌 (2)

−𝐴𝑌𝑁 −𝑑𝐿

𝐴𝑌𝑁 (4)


𝐴𝑌𝑌<

𝑑𝐿

𝐴𝑌𝑁







𝐴𝑌𝑁−

𝑑𝐿

𝐴𝑌𝑌)



𝑃𝑌, 𝐴𝑌𝑌 , 𝑊𝐿 (2)



𝐴𝑌𝑁−

𝑑𝐿

𝐴𝑌𝑌)



chosen

𝑃𝑌, 𝐴𝑌𝑁 , 𝑊𝐿 (4)

When 𝐴𝑌𝑌 = 𝐴𝑌𝑁 + (𝑑𝐿

𝐴𝑌𝑁−

𝑑𝐿

𝐴𝑌𝑌) you




without prestige, payment (6) is compared with

payment (8):


𝐴𝑁𝑌] + 𝑞 [𝑈𝑅𝐵 − 𝐴𝑌𝑌 −

𝑑𝐿

𝐴𝑁𝑌]




𝑑𝐿

𝐴𝑁𝑁]


Simplifying (6) (1 − 𝑞)𝑈𝑅𝐺 +

𝑞𝑈𝑅𝐵 − 𝐴𝑁𝑌 −𝑑𝐿

𝐴𝑁𝑌

Simplifying (8) (1 − 𝑞)𝑈𝑅𝐺 +

𝑞𝑈𝑅𝐵 − 𝐴𝑁𝑁 −𝑑𝐿

𝐴𝑁𝑁

(6) −𝐴𝑁𝑌 −𝑑𝐿

𝐴𝑁𝑌

(8) −𝐴𝑁𝑁 −𝑑𝐿

𝐴𝑁𝑁

47









𝐴𝑁𝑌<

𝑑𝐿

𝐴𝑁𝑁 Sub-

sceneries arise based on the acceptance factor.

Thus, the acceptance factor significantly





𝐴𝑁𝑁−

𝑑𝐿

𝐴𝑁𝑌)



(𝑃𝑁 , 𝐴𝑁𝑌, 𝑊𝐿) (6)



𝐴𝑁𝑁−

𝑑𝐿

𝐴𝑁𝑌)



chosen

(𝑃𝑁 , 𝐴𝑁𝑌, 𝑊𝐿) (8)


𝐴𝑁𝑁−

𝑑𝐿

𝐴𝑁𝑌)you



Now the Director will choose the best

payment for him.

(PY) 𝑝(𝐺 − 𝑅𝑃𝐺) + (1 − 𝑝)(𝐵 − 𝑅𝑃𝐵)

(PN) 𝑞(𝐺 − 𝑅𝐺) + (1 − 𝑞)(𝐵 − 𝑅𝐵)

Simplifying:

(PY) 𝑝𝐺 − 𝑝𝑅𝑃𝐺 − 𝑅𝑃𝐵 − 𝑝𝐵 + 𝑝𝑅𝑃𝐵

(PN) 𝑞𝐺 − 𝑞𝑅𝐺 − 𝑅𝐵 − 𝑞𝐵 + 𝑞𝑅𝐵

𝐺 >[𝑞𝐺 − 𝑞𝑅𝐺 − 𝑅𝐵 − 𝑞𝐵 + 𝑞𝑅𝐵 + 𝑝𝑅𝑃𝐺 + 𝑅𝑃𝐵 + 𝑝𝐵 − 𝑝𝑅𝑃𝐵]

𝑝

𝜋11 =[𝑞𝐺 − 𝑞𝑅𝐺 − 𝑅𝐵 − 𝑞𝐵 + 𝑞𝑅𝐵 + 𝑝𝑅𝑃𝐺 + 𝑅𝑃𝐵 + 𝑝𝐵 − 𝑝𝑅𝑃𝐵]

𝑝


When 𝐺 > 𝜋11 will be chosen (PY)

When 𝐺 < 𝜋11 will be chosen (PN)

When 𝐺 = 𝜋11 anyone can be chosen


In this way,

When 𝐺 > 𝜋11

If 𝑅𝑃𝐺 > 𝜋7 y 𝐴𝑌𝑌 > 𝐴𝑌𝑁 +𝑑𝐻

𝐴𝑌𝑁−

𝑑𝐻

𝐴𝑌𝑌,

the dominant Strategy will be (𝑃𝑌, 𝐴𝑌𝑌 , 𝑊𝐻)

ó

If 𝑅𝑃𝐺 > 𝜋7 y 𝐴𝑌𝑌 < 𝐴𝑌𝑁 +𝑑𝐻

𝐴𝑌𝑁−

𝑑𝐻

𝐴𝑌𝑌,

the dominant Strategy will be (𝑃𝑌, 𝐴𝑌𝑁 , 𝑊𝐻)

ó

If 𝑅𝑃𝐺 < 𝜋7 y 𝐴𝑌𝑌 > 𝐴𝑌𝑁 +𝑑𝐻

𝐴𝑌𝑁−

𝑑𝐻

𝐴𝑌𝑌,

the dominant Strategy will be (𝑃𝑌, 𝐴𝑌𝑌 , 𝑊𝐿)

ó

If 𝑅𝑃𝐺 < 𝜋7 y 𝐴𝑌𝑌 < 𝐴𝑌𝑁 +𝑑𝐻

𝐴𝑌𝑁−

𝑑𝐻

𝐴𝑌𝑌,

the dominant Strategy will be (𝑃𝑌, 𝐴𝑌𝑁 , 𝑊𝐿)

ó

When 𝐺 < 𝜋11

If 𝑅𝑃𝐺 > 𝜋8 y 𝐴𝑁𝑌 > 𝐴𝑁𝑁 +𝑑𝐻

𝐴𝑁𝑁−

𝑑𝐻

𝐴𝑁𝑌, the dominant Strategy will be

(𝑃𝑁 , 𝐴𝑁𝑌, 𝑊𝐻)

ó

If If 𝑅𝑃𝐺 > 𝜋8 y 𝐴𝑁𝑌 < 𝐴𝑁𝑁 +𝑑𝐻

𝐴𝑁𝑁−

𝑑𝐻


(𝑃𝑁 , 𝐴𝑁𝑁 , 𝑊𝐻)

ó

If 𝑅𝑃𝐺 < 𝜋8 y 𝐴𝑁𝑌 > 𝐴𝑁𝑁 +𝑑𝐻

𝐴𝑁𝑁−

𝑑𝐻


(𝑃𝑁 , 𝐴𝑁𝑌, 𝑊𝐿)

ó

If 𝑅𝑃𝐺 < 𝜋8 y 𝐴𝑁𝑌 < 𝐴𝑁𝑁 +𝑑𝐻

𝐴𝑌𝑁−

𝑑𝐻

𝐴𝑁𝑌,

the dominant Strategy will be (𝑃𝑁 , 𝐴𝑁𝑁 , 𝑊𝐿)

ó

When 𝐺 = 𝜋11 Any option may be

chosen based on the employee's conditions

previously presented.

Results

As can be seen in the scenarios presented, the

rational decisions of employees and employers

in situations of knowledge friction can be

represented by non-cooperative dynamic games

with Principal-Agent situations.

48








These games can be played considering

conditions of prestige and not prestige of the

employee, decisions of effort by acceptance or

rejection and decisions to make a greater or

lesser effort. The variables included are

variables supported by the Theory of

Knowledge Management and Organizational

Culture, as well as the observation of cases

through ten years in an Institution. The risk of

scenarios that are affected by external factors,

represented by the probabilities of generating

good or bad projects, makes evident the need to

continue adjusting the modeling to reality.

Managers, based on the expected profit from

the project, must decide if to pay more in order

to increase the chances of success of major

projects. In Institutions where employee

prestige is an important factor for employees,

and there are significant differences in their

perception of who, as the project leader, shares

the information, a scenario with a fixed

incentive does not motivate to make an effort

for acceptance or for the realization of the

project, since the result does not matter,

considering a fixed payment. However, when

the profit difference between a good project and

a bad project is very low and the chances of the

project going well for various reasons is high,

scenario 1 might be the most suitable for those

involved in the game. Scenario 2 is a scenario

that presents a more favorable incentive in

situations where there are no significant

differences in the level of prestige, and there are

significant differences in the profitability of the

projects, clearly distinguishing between a

project that goes well and one that goes bad.

When the probability of success increases

significantly due to the effort made, it is more

advisable to choose an incentive that motivates

you to carry out a good project together with a

higher payment. In scenario 2, the Director

would consider the importance of a higher

profit and the greater probability of success that

a prestigious employee gives him, so he would

choose a prestigious employee over a non-

prestigious employee When the rest of the

conditions are equal. Scenario 3 shows a similar

scenario to scenario 1 and As the payment is

not linked to profitability, as long as the

probabilities of success are similar, the

employee will always choose the minimum

effort. When the chances of success are higher

and exceed the pay difference of a prestigious

employee, the Director will choose an

employee without prestige because the payment

received by the Director will be higher.

Scenario 4 shows a more adjustable

approach to situations where the profit

difference between good projects and bad

projects is significant and there are more

variables that can reinforce the probability of

success through more attractive incentives

depending on the employee's conditions, such

As the prestige that the experience of successful

projects gives, the effort to be accepted and the

effort in carrying out the project.

Acknowledgments

The authors thank the Technological University

of Querétaro and the Autonomous University of

Querétaro for the facilities provided.

Conclusions

When projects do not make a profit difference

for a good or bad project, Directors will choose

to pay fixed incentives. What happens in

scenarios like these, on the employee's side, is

that the tragedy of the public good can occur.

Scenario 1 shows an example of this When

there is an equal probability of obtaining a good

or a bad project, since the employee will always

choose the minimum effort for the same

payment. The same occurs with scenario 3,

when the incentive is not linked to the win, with

the difference that, in repetitive games, the

circumstances could change including the

variable of increasing or decreasing prestige for

the following games that could impact the

expected future profit, but further analysis is

required. Incentives based on results are much

more recommendable When the difference in

profits between a good and a bad project is

significant since the utility of both parties will

exceed the perceived disutilities and then the

effort will be perceived as more rewarding.

It is difficult for organizations to

monitor and control the application of tacit

knowledge by their employees. Analysis of the

game model suggests that moral hazard

compounds the problem and requires additional

initiatives. To combat these types of events, it is

necessary to change employee behaviors so that

they are incentivized to put more effort into

knowledge initiatives.

49








Incentives help reinforce positive

behaviors and culture (Wong, 2005). Extrinsic

reward schemes, as well as economic ones, can

decrease intrinsic motivation (As the

recognition of their peers) (Frey and Jegen,

2001), so it is recommended to favor

associations and contributions that are expected

by employees to favor better behaviors in

knowledge sharing.

Coinciding with Zhang et al. (2012)

favoring and disseminating the performance of

highly visible tasks among employees impacts

on a behavior that contributes to employee

knowledge. For this reason, it is recommended

that organizations design balanced incentive

mechanisms, incorporating both extrinsic and

intrinsic incentives.

It is recommended, according to Yang

and Wu (2008), the design of incentives that

rewards each action of knowledge contribution,

since it is more effective than periodic

performance reviews to motivate knowledge

behaviors.

By embedding informal mentoring and

coaching into employee behavior routines, an

institutional “gift culture” can be fostered,

promoting collaboration (Gratton & Erickson,

2007). Informal networks within the

organization such As communities of practice

can promote the exchange of tacit knowledge

and collaboration in knowledge sharing and

reinforce the prestige status of participating

employees and could decrease the effort related

to convincing other employees that the

knowledge that you want to share is accepted.

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