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Volume 1, Issue 1 – July – December – 2017
Volume 4, Issue 7 - July - December - 2020
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
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 1-7
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
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 1-7
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.
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
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 1-7
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.
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
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 1-7
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.
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
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 1-7
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.
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.
References
Capitanescu F., Marvuglia A., Benetto E.,
Ahmadi A., Tiruta L. (2017). Linear
programming-based directed local search for
expensive multi-objective optimization
problems: Application to drinking water
production plants. European Journal of
Operational Research, 262(1), Issue 1, 322-334.
https://doi.org/10.1016/j.ejor.2017.03.057
Hnaien F., Yalaoui F., Mhadhbi A., and
Nourelfath M. (2016) A mixed-integer
programming model for integrated production
and maintenance, IFAC-PapersOnLine. 49(12),
556-561.
https://doi.org/10.1016/j.ifacol.2016.07.694
Krajewski, Lee J; Ritzman, Larry P. &
Malhotra, Manoj K. (2008). Administración de
operaciones. México: Pearson Educación.
Motta C., Da Silva M., Bressan M., Almada B.,
(2016). Mathematical programming-based
approaches for multi-facility glass container
production planning. Computers & Operations
Research. 74 (1), 92-107.
https://doi.org/10.1016/j.cor.2016.02.019
Novas, Juan M. (2016). Modelo MILP para la
programación de la producción en ambientes
job-shop flexibles con división de lotes.
Iberoamerican Journal of Industrial
Engineering, 8(16), 56-72.
Ortiz Triana, Viviana K., Caicedo Rolón,
Álvaro Jr. (2015). Procedimiento para la
programación y control de la producción de una
pequeña empresa. Revista Ingeniería Industrial.
14 (1), 89-104.
Pérez Rodríguez, M. E. (2017). Impacto de la
toma de decisiones basada en los modelos de
investigación de operaciones en la dirección de
proyectos de ingeniería para el sector privado.
barranquilla. Secretaria de Economia. (s.f.).
Secretaria de minería.
Romero Rojas, J. D., Ortiz Triana, V. K.,
Caicedo Rolón, A. J., (2019). La Teoría de
Restricciones y la Optimización como
Herramientas Gerenciales para la Programación
de la Producción. Una Aplicación en la
Industria de Muebles. Revista de métodos
cuantitativos para la economía y la empresa. 27
(1), 74-90.
7
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 1-7
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.
Silva Rodríguez, Julián, Díaz Cárdenas,
Camilo, Galindo Carabalí, Julián. (2017).
Herramientas cuantitativas para la planeación y
programación
de la producción: estado del arte. Ingeniería
Industrial. Actualidad y Nuevas Tendencias,
5(18), 99-114.
Taebok Kim and Christoph H.Glock,
Production planning for a two-stage production
system with multiple parallel machines and
variable production rates. International Journal
of Production Economics. 196(1), 284-292.
https://doi.org/10.1016/j.ijpe.2017.11.018.
8
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 8-15
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
DOI: 10.35429/JMQM.2020.7.4.8.15 Received July 15, 2020; Accepted December 30, 2020
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]).
† Researcher contributing as first author.
© RINOE – Spain www.rinoe.org/spain
9
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 8-15
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
ISSN-On line: 2531-2979
RINOE® All rights reserved.
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
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 8-15
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
ISSN-On line: 2531-2979
RINOE® All rights reserved.
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
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 8-15
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
ISSN-On line: 2531-2979
RINOE® All rights reserved.
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
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 8-15
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
ISSN-On line: 2531-2979
RINOE® All rights reserved.
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
Source: Own computations from the simulations
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%
Source: Own computations from the simulations
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***
R-Square 0.426673 Adj. R-
Square
0.426615
Table 3 OLS regression of final blue population with
respect to initial blue population with a threshold level of
37.5%
Source: Own computations from the simulations
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
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 8-15
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
ISSN-On line: 2531-2979
RINOE® All rights reserved.
Figure 4 Scatter plot of the initial blue population (ai)
and the final blue population (af), for a threshold of 50%
Source: Own computations from the simulations
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***
R-Square 0.600131 Adj. R-
Square
0.600091
Table 4 OLS regression of final blue population with
respect to initial blue population with a threshold level of
50%
Source: Own computations from the simulations
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).
Figure 5 Scatter plot of the initial blue population (ai)
and the final blue population (af), for a threshold of
62.5%
Source: Own computations from the simulations
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***
R-Square 0.875661 Adj. R-
Square
0.875648
Table 5 OLS regression of final blue population with
respect to initial blue population with a threshold level of
62.5%
Source: Own computations from the simulations
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
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 8-15
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
ISSN-On line: 2531-2979
RINOE® All rights reserved.
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
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 8-15
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
ISSN-On line: 2531-2979
RINOE® All rights reserved.
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
United Nations Development Program
(UNDP). (2016). Human development report
2016: Human development for everyone. New
York. Recovered from
http://hdr.undp.org/sites/default/files/2016_hum
an_development_report.pdf
United Nations Development Program
(UNDP). (2018). Human development indices
and indicators 2018 statistical update. New
York. Recovered from
http://hdr.undp.org/sites/default/files/2018_hum
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
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 16-25
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
DOI: 10.35429/JMQM.2020.7.4.16.25 Received July 20, 2020; Accepted December 30, 2020
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.
† Researcher contributing as first author.
© RINOE – Spain www.rinoe.org/spain
17
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 16-25
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
ISSN-On line: 2531-2979
RINOE® All rights reserved.
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
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 16-25
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
ISSN-On line: 2531-2979
RINOE® All rights reserved.
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
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 16-25
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
ISSN-On line: 2531-2979
RINOE® All rights reserved.
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
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 16-25
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
ISSN-On line: 2531-2979
RINOE® All rights reserved.
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
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 16-25
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
ISSN-On line: 2531-2979
RINOE® All rights reserved.
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
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 16-25
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
ISSN-On line: 2531-2979
RINOE® All rights reserved.
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
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 16-25
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
ISSN-On line: 2531-2979
RINOE® All rights reserved.
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.
References
Aldana, E., & López, J. (2018). Estudio
histórico-epistemológico y didáctico de la
parábola. Praxis & Saber, 9(19), 63-88.
https://doi.org/10.19053/22160159.v9.n19.201
8.7922
Artola, E., Mayoral, L., & Benarroch, A.
(2016). Dificultades de aprendizaje de las
representaciones gráficas cartesianas asociadas
a biología de poblaciones en estudiantes de
educación secundaria. Un estudio semiótico.
Revista Eureka Sobre Enseñanza y Divulgación
de Las Ciencias, 13(1), 36-52. Recuperado de
https://revistas.uca.es/index.php/eureka/article/v
iew/2951
Aznar, M., Distéfano, M., Moler, E., & Pesa,
M. (2018). A didactic sequence to improve the
conversion of semiotic representations of
curves and regions of the complex plane.
Uniciencia, 32(1), 46-67.
https://doi.org/10.15359/ru.32-1.4
Bernal, C. (2020). Propuesta para la
innovación del curso de precálculo: funciones,
sus gráficas, dominios y codominios.
Recuperado de
http://funes.uniandes.edu.co/17269/1/Bernal_C
arlos_Eduardo_(2020)_Funciones%2C_sus_gr
%C3%A1ficas%2C_dominios_y_codominios.p
df
Buendía, L., Colás, P., & Hernández, F. (1998).
Métodos de investigación en Psicopedagogía.
España: McGraw Hill.
Campos, C. (2003). La argumentación gráfica
en la transformación de funciones cuadráticas.
Una aproximación socioepistemológica. Tesis
de Maestría no publicada. Centro de
Investigación y Estudios Avanzados del
Instituto Politécnico Nacional. México, D.F.
Campos, M., & Rodríguez, M. (2020). Un
estudio sobre la aprehensión conceptual de las
inecuaciones. Revista Paradigma, 41, 540-570.
Recuperado de
http://funes.uniandes.edu.co/22158/1/Campos2
020Un.pdf
Consejo de Acreditación de la Enseñanza de la
Ingeniería (CACEI). (2018). Marco de
referencia 2018. Recuperado de
http://cacei.org.mx/nvfs/nvfs02/nvfs0210.php
Cruz, F., Báez, J., & Corona-Galindo, M.
(2018). Estrategia de enseñanza y aprendizaje
para el estudio de los elementos característicos
de la parábola. El Cálculo y su Enseñanza,
Enseñanza de las Ciencias y la Matemática.
11(2). 62-82. Recuperado de
https://recacym.org/index.php/recacym/article/v
iew/28
Denardi, V., & Bisognin, E. (2020). Resolução
de Problemas e Representações Semióticas na
Formação Inicial de Professores de Matemática.
Revista De Educação Matemática, 17,
e020022.
https://doi.org/10.37001/remat25269062v17id2
72
Díaz-Barriga, F., & Hernández, G. (2010).
Estrategias docentes para un aprendizaje
significativo. Una interpretación
constructivista. México: Editorial McGraw-
Hill.
24
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 16-25
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
ISSN-On line: 2531-2979
RINOE® All rights reserved.
Díaz, P., & Leyva, E. (2013). Metodología para
determinar la calidad de los instrumentos de
evaluación. Revista Educación Médica
Superior, 27(2), 269-286. Recuperado de
http://www.medigraphic.com/pdfs/educacion/ce
m-2013/cem132n.pdf
Duval, R. (1988). Graphiques et Equations:
l'articulation de deux registres, in Annales de
Didactique et de Sciences Cognitives, n°1, 235-
253. (Versión en español de Blanca M. Parra).
Duval, R. (1998). Registros de representación
semiótica y funcionamiento cognitivo del
pensamiento. En F. Hitt (Ed.), Investigaciones
en Matemática Educativa II (pp. 173-201).
México: Grupo Editorial Iberoamérica.
Duval, R. (2010). Sémiosis, pensée humaine et
activité mathématique. Revista de Educação em
Ciências e Matemáticas, 6(1), 126-143.
Recuperado de
https://dialnet.unirioja.es/servlet/articulo?codig
o=5870410
Duval, R. (2012). Lo esencial de los procesos
cognitivos de comprensión en matemáticas: los
registros de representación semiótica. En U.
Malaspina (Ed.). Resúmenes del VI Coloquio
Internacional de Didáctica de las Matemáticas:
avances y desafíos actuales (pp.14-17). Lima,
Peru: Pontificia Universidad Católica del Perú.
Farabello, S., & Trigueros, M. (2020). La
Transformación de Funciones en el aula de
Física. UNIÓN Revista Iberoamericana de
Educación Matemática, 16(58), 25-47.
Recuperado de
https://union.fespm.es/index.php/UNION/articl
e/view/82/23
García, J., Segovia, I., & Lupiáñez, J. (2011).
Errores y dificultades de estudiantes mexicanos
de primer curso universitario en la resolución
de tareas algebraicas. En J. L. Lupiáñez, M. C.
Cañadas, M. Molina, M. Palarea, y A. Maz
(Eds.), Investigaciones en Pensamiento
Numérico y Algebraico e Historia de la
Matemática y Educación Matemática (pp. 145-
155). Granada: Dpto. Didáctica de la
Matemática, Universidad de Granada.
Recuperado de
http://funes.uniandes.edu.co/2018/1/GarciaSego
viaLupianez2011.pdf
Godino, J., Wihelmi, M., Blanco, T., Contreras,
A., & Giacomone, B. (2016). Análisis de la
actividad matemática mediante dos
herramientas teóricas: Registros de
representación semiótica y configuración
ontosemiótica. Avances de Investigación en
Educación Matemática, 10, 91-110.
Recuperado de
https://www.researchgate.net/publication/31010
0320_Analisis_de_la_actividad_matematica_m
ediante_dos_herramientas_teoricas_Registros_
de_representacion_semiotica_y_configuracion_
ontosemiotica
Guerrero, C., Camacho, M., & Mejía, H.
(2010). Dificultades de los estudiantes en la
interpretación de las soluciones de ecuaciones
diferenciales ordinarias que modelan un
problema. Enseñanza de las ciencias, 28(3),
341-352. Recuperado de
https://www.raco.cat/index.php/Ensenanza/artic
le/view/210804/353412
Hernández, R., Fernández, C., & Baptista, P.
(2014). Metodología de la investigación.
México: McGraw Hill. Sexta edición.
Hitt, F. (2008). Investigaciones en Ambientes
Tecnológicos, Marcos Teóricos y
Metodológicos: Un Punto de Vista Pragmático.
En Pantoja, R. Añorve, E., Cortés, J., y
Osornio, L. (Comp.). Investigaciones y
Propuestas sobre el uso de Tecnología en
Educación Matemática. (pp. 1-20). México:
ITCG.
Instituto Nacional para la Evaluación de la
Educación (INEE). (2019). Informe de
resultados Planea EMS 2017. Recuperado de
https://www.inee.edu.mx/wp-
content/uploads/2019/05/P1D320.pdf
Larrazolo, N., Backhoff, E., & Tirado, F.
(2013). Habilidades de razonamiento
matemático de estudiantes de educación media
superior en México. Revista Mexicana de
Investigación Educativa, 18(59), 1137-1163.
Recuperado de
https://www.comie.org.mx/revista/v2018/rmie/i
ndex.php/nrmie/article/view/283/283
25
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 16-25
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
ISSN-On line: 2531-2979
RINOE® All rights reserved.
Mercedes, A., Pérez, O., & Triana, B. (2017).
Propuesta didáctica basada en múltiples formas
de representación semiótica de los objetos
matemáticos para desarrollar el proceso de
enseñanza aprendizaje del cálculo diferencial.
Revista Academia y Virtualidad, 10(2), 20-30.
https://doi.org/10.18359/ravi.2743
Moreira, M. (2012). ¿Al final, qué es
aprendizaje significativo? Revista Qurriculum,
25, 29-56. Recuperado de:
http://publica.webs.ull.es/upload/REV%20QUR
RICULUM/25%20-%202012/02.pdf
Organización para la Cooperación y el
Desarrollo Económico (OCDE). (2019).
Programa para la evaluación internacional de
alumnos PISA 2018 resultados. Recuperado de
https://www.oecd.org/pisa/publications/PISA20
18_CN_MEX_Spanish.pdf
Peralta, J., Encinas, F., Rojas, J., Cuevas, O.,
Ansaldo, J., & Osorio, M. (2013).
Implementación de la estrategia resolución de
problemas en el aprendizaje del tema
ecuaciones lineales en alumnos de ingeniería.
En Pizá, R., González, M. y Vizcarra, L.
(Comp.). Valoración de Indicadores del
Desempeño Académico. (pp. 124-135).
México: ITSON. Recuperado de
https://www.itson.mx/publicaciones/Documents
/rada/valoraciondeindicadores.pdf
Pizarro, R. (2009). Las TICs en la enseñanza de
las Matemáticas. Buenos Aires Argentina:
Universidad Nacional de la Plata.
Prada, R., Hernández, C., & Jaimes, L. (2017).
Representación semiótica de la noción de
función: concepciones de los estudiantes que
transitan del Colegio a la Universidad.
Panorama, 11(20), 34-44. Recuperado de
https://journal.poligran.edu.co/index.php/panor
ama/article/view/1008/749
Rodríguez, I., & Torrealba, A. (2016).
Dificultades que conducen a errores en el
aprendizaje del lenguaje algebraico en
estudiantes de tercer año de educación media
general. Revista Arjé, 11(20), 416-438.
Recuperado de
http://arje.bc.uc.edu.ve/arj20/art38.pdf
Sánchez, E. (2016). Algunas dificultades de
aprendizaje presentes en el estudio de la
parábola como sección cónica. Repositorio
Digital IDEP/ B. Currículo y prácticas de
enseñanza (pp. 213-230). Bogotá, Colombia.
Recuperado de
https://repositorio.idep.edu.co/handle/001/2342
Santibañez, J. (2011). Manual para la
evaluación del aprendizaje estudiantil.
(Primera edición). México: Editorial Trillas.
Secretaría de Educación Pública. (s.f.). Planea,
resultados nacionales 2017. Recuperado de
http://planea.sep.gob.mx/content/general/docs/2
017/ResultadosNacionalesPlaneaMS2017.PDF
Sotelo, C., Echeverría, C., & Ramos, E. (2009).
Relaciones entre variables motivacionales y
rendimiento académico en estudiantes
universitarios. Memoria Electrónica del X
Congreso Nacional de Investigación Educativa.
Veracruz, México.
Torres, I., Advícula, E., León, J., & Flores, H.
(2019). Estudio de la parábola como lugar
geométrico: una forma de ampliar el
conocimiento especializado del profesor. XV
Conferencia Interamericana de Educación
Matemática. Conferencia llevada a cabo en
Medellín, Colombia. Recuperado de
https://conferencia.ciaem-
redumate.org/index.php/xvciaem/xv/paper/view
File/879/117
UNESCO Institute for Statistics (UIS). (2017).
More Than One-Half of Children and
Adolescents Are Not Learning Worldwide ".
Recuperado de
http://uis.unesco.org/sites/default/files/documen
ts/fs46-more-than-half-children-not-learning-
en-2017.pdf.
26
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 26-50
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
DOI: 10.35429/JMQM.2020.7.4.26.50 Received July 25, 2020; Accepted December 30, 2020
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])
† Researcher contributing as first author.
© RINOE – Spain www.rinoe.org/spain
27
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 26-50
TALAVERA-RUZ, Marianela, LARA-GÓMEZ, Graciela and
VALDEZ-RESÉNDIZ, Macario. Knowledge friction in
universities, approach from game theory. Journal-Mathematical
and Quantitative Methods. 2020
ISSN-On line: 2531-2979
RINOE® All rights reserved.
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
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 26-50
TALAVERA-RUZ, Marianela, LARA-GÓMEZ, Graciela and
VALDEZ-RESÉNDIZ, Macario. Knowledge friction in
universities, approach from game theory. Journal-Mathematical
and Quantitative Methods. 2020
ISSN-On line: 2531-2979
RINOE® All rights reserved.
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
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 26-50
TALAVERA-RUZ, Marianela, LARA-GÓMEZ, Graciela and
VALDEZ-RESÉNDIZ, Macario. Knowledge friction in
universities, approach from game theory. Journal-Mathematical
and Quantitative Methods. 2020
ISSN-On line: 2531-2979
RINOE® All rights reserved.
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?
In an organizational
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.
In an organizational
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
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 26-50
TALAVERA-RUZ, Marianela, LARA-GÓMEZ, Graciela and
VALDEZ-RESÉNDIZ, Macario. Knowledge friction in
universities, approach from game theory. Journal-Mathematical
and Quantitative Methods. 2020
ISSN-On line: 2531-2979
RINOE® All rights reserved.
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
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 26-50
TALAVERA-RUZ, Marianela, LARA-GÓMEZ, Graciela and
VALDEZ-RESÉNDIZ, Macario. Knowledge friction in
universities, approach from game theory. Journal-Mathematical
and Quantitative Methods. 2020
ISSN-On line: 2531-2979
RINOE® All rights reserved.
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
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 26-50
TALAVERA-RUZ, Marianela, LARA-GÓMEZ, Graciela and
VALDEZ-RESÉNDIZ, Macario. Knowledge friction in
universities, approach from game theory. Journal-Mathematical
and Quantitative Methods. 2020
ISSN-On line: 2531-2979
RINOE® All rights reserved.
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
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 26-50
TALAVERA-RUZ, Marianela, LARA-GÓMEZ, Graciela and
VALDEZ-RESÉNDIZ, Macario. Knowledge friction in
universities, approach from game theory. Journal-Mathematical
and Quantitative Methods. 2020
ISSN-On line: 2531-2979
RINOE® All rights reserved.
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
From the perspective of earnings from
payments, these can be expressed as a function
of incentives:
U (R) for a good gain or bad gain scenario.
However, payments should be tied to
perceived dissatisfaction (or disutility) for the
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
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 26-50
TALAVERA-RUZ, Marianela, LARA-GÓMEZ, Graciela and
VALDEZ-RESÉNDIZ, Macario. Knowledge friction in
universities, approach from game theory. Journal-Mathematical
and Quantitative Methods. 2020
ISSN-On line: 2531-2979
RINOE® All rights reserved.
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.
The payment equations:
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
subtracted in payment 3 so that payment 4
obtains a higher utility.
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
investing a greater effort WH or a less effort
WL.
The payment equations:
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
subtracted in payment 5 so that payment 6
obtains a higher utility.
Pay 7 (PN, ANN, WH) is compared
with pay 8 (PN, ANN, WL), corresponding to
the decision of an employee without prestige
PN who does not make an effort to be accepted
ANN, between investing a greater effort WH or
less effort WL.
The payment equations:
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
subtracted in payment 7 so that payment 8
obtains a higher utility.
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
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 26-50
TALAVERA-RUZ, Marianela, LARA-GÓMEZ, Graciela and
VALDEZ-RESÉNDIZ, Macario. Knowledge friction in
universities, approach from game theory. Journal-Mathematical
and Quantitative Methods. 2020
ISSN-On line: 2531-2979
RINOE® All rights reserved.
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
impacts the effort and therefore the disutility
perceived by the employee:
When the acceptance factor is high:
𝐴𝑁𝑌 > 𝐴𝑁𝑁 + (𝑑𝐿
𝐴𝑁𝑌−
𝑑𝐿
𝐴𝑁𝑁)
The impact on disutility is less so it will
subtract less from utility and will be chosen (6).
When the acceptance factor is low:
𝐴𝑁𝑌 < 𝐴𝑁𝑁 + (𝑑𝐿
𝐴𝑁𝑌−
𝑑𝐿
𝐴𝑁𝑁)
The impact on disutility is greater so it
will subtract more from the utility and it will be
chosen (8).
When 𝐴𝑁𝑌 = 𝐴𝑁𝑁 + (𝑑𝐿
𝐴𝑁𝑌−
𝑑𝐿
𝐴𝑁𝑁) you
can choose either of the two payments because
they will be worth the same.
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
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 26-50
TALAVERA-RUZ, Marianela, LARA-GÓMEZ, Graciela and
VALDEZ-RESÉNDIZ, Macario. Knowledge friction in
universities, approach from game theory. Journal-Mathematical
and Quantitative Methods. 2020
ISSN-On line: 2531-2979
RINOE® All rights reserved.
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
𝐴𝑌𝑌 > 𝐴𝑌𝑁 + (𝑑𝐿
𝐴𝑌𝑁−
𝑑𝐿
𝐴𝑌𝑌)
The dominant Strategy would be PY,
AYN, WL When the acceptance factor is low
𝐴𝑌𝑌 < 𝐴𝑌𝑁 + (𝑑𝐿
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
𝐴𝑁𝑌 > 𝐴𝑁𝑁 + (𝑑𝐿
𝐴𝑁𝑁−
𝑑𝐿
𝐴𝑁𝑌)
The dominant Strategy would be PN,
ANN, WL When the acceptance factor is low
𝐴𝑁𝑌 < 𝐴𝑁𝑁 + (𝑑𝐿
𝐴𝑁𝑁−
𝑑𝐿
𝐴𝑁𝑌)
When 𝐴𝑁𝑌 = 𝐴𝑁𝑁 + (𝑑𝐿
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
From the perspective of earnings from
payments, these can be expressed as a function
of incentives:
U (RG) for a good gain scenario
U (RB) for a bad gain scenario
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
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
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 26-50
TALAVERA-RUZ, Marianela, LARA-GÓMEZ, Graciela and
VALDEZ-RESÉNDIZ, Macario. Knowledge friction in
universities, approach from game theory. Journal-Mathematical
and Quantitative Methods. 2020
ISSN-On line: 2531-2979
RINOE® All rights reserved.
Figure 2 Game tree for Scenario 2 Payment plus
incentive (R ') based on the profitability of the project
Source: Self-made
Scenario Solution 2
To solve this dynamic non-cooperative game
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
who strives to be accepted AYY, between
investing a greater effort WH or a less effort
WL.
The payment equations:
𝑝 [𝑈𝑅𝐺 − 𝐴𝑌𝑌 −𝑑𝐻
𝐴𝑌𝑌
] + (1 − 𝑝) [𝑈𝑅𝐵 − 𝐴𝑌𝑌 −𝑑𝐻
𝐴𝑌𝑌
]
Strategy (𝑃𝑌, 𝐴𝑌𝑌 , 𝑊𝐻) (1)
(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),
corresponding to the decision of an employee
with prestige PY who does not make an effort
to be accepted AYN, between investing a
greater effort WH or less effort WL.
The payment equations:
𝑝 [𝑈𝑅𝐺 − 𝐴𝑌𝑁 −𝑑𝐻
𝐴𝑌𝑁] + (1 − 𝑝) [𝑈𝑅𝐵 − 𝐴𝑌𝑁 −
𝑑𝐻
𝐴𝑌𝑁]
Strategy (𝑃𝑌, 𝐴𝑌𝑁 , 𝑊𝐻) (3)
(1 − 𝑝) [𝑈𝑅𝐺 − 𝐴𝑌𝑁 −𝑑𝐿
𝐴𝑌𝑁] + 𝑝 [𝑈𝑅𝐵 − 𝐴𝑌𝑁 −
𝑑𝐿
𝐴𝑌𝑁]
Strategy (𝑃𝑌, 𝐴𝑌𝑁 , 𝑊𝐿) (4)
Simplifying (3) 𝑝𝑈𝑅𝐺 + (1 − 𝑝)𝑈𝑅𝐵 −
𝐴𝑌𝑁 −𝑑𝐻
𝐴𝑌𝑁
Simplifying (4) (1 − 𝑝)𝑈𝑅𝐺 + 𝑝𝑈𝑅𝐵 −
𝐴𝑌𝑁 −𝑑𝐿
𝐴𝑌𝑁
As 𝑝𝑈𝑅𝐺 > (1 − 𝑝)𝑈𝑅𝐺 but As
(1 − 𝑝)𝑈𝑅𝐵 < 𝑝𝑈𝑅𝐵 y −𝑑𝐻
𝐴𝑌𝑁< −
𝑑𝐿
𝐴𝑌𝑁
subscenarios arise:
𝑅𝐺 >[(1 − 𝑝)𝑈𝑅𝐺 + 𝑝𝑈𝑅𝐵 − (
𝑑𝐿𝐴𝑌𝑁
) − (1 − 𝑝)𝑈𝑅𝐵 + (𝑑𝐻
𝐴𝑌𝑁)]
𝑝𝑈
38
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 26-50
TALAVERA-RUZ, Marianela, LARA-GÓMEZ, Graciela and
VALDEZ-RESÉNDIZ, Macario. Knowledge friction in
universities, approach from game theory. Journal-Mathematical
and Quantitative Methods. 2020
ISSN-On line: 2531-2979
RINOE® All rights reserved.
Being
𝜋2 =[(1 − 𝑝)𝑈𝑅𝐺 + 𝑝𝑈𝑅𝐵 − (
𝑑𝐿𝐴𝑌𝑁
) − (1 − 𝑝)𝑈𝑅𝐵 + (𝑑𝐻
𝐴𝑌𝑁)]
𝑝𝑈
If 𝑅𝐺 > 𝜋2 will be chosen (3) with the
Strategy 𝑃𝑌, 𝐴𝑌𝑁 , 𝑊𝐻
If 𝑅𝐺 < 𝜋2 will be chosen (4) with the
Strategy 𝑃𝑌, 𝐴𝑌𝑁 , 𝑊𝐿
If 𝑅𝐺 = 𝜋2 you can choose either
because both payments are equal.
Now we analyze the case of an
employee without prestige. Pay 5 (PN, ANY,
WH) is compared with pay 6 (PN, AYY, WL),
corresponding to the decision of an employee
without prestige PN who strives to be accepted
AYY, between investing a greater effort WH or
a less effort WL.
The payment equations:
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 − (
𝑑𝐿𝐴𝑁𝑌
) − (1 − 𝑞)𝑈𝑅𝐵 + (𝑑𝐻
𝐴𝑁𝑌)]
𝑞𝑈
If 𝑅𝐺 > 𝜋3 will be chosen (5) with the
Strategy 𝑃𝑁 , 𝐴𝑁𝑌, 𝑊𝐻
If 𝑅𝐺 < 𝜋3 will be chosen (6) with the
Strategy 𝑃𝑁 , 𝐴𝑁𝑌, 𝑊𝐿
If 𝑅𝐺 = 𝜋3 you can choose either
because both payments are equal.
Pay 7 (PN, ANN, WH) is compared
with pay 8 (PN, ANN, WL), corresponding to
the decision of an employee without prestige
PN who does not make an effort to be accepted
ANN, between investing a greater effort WH or
less effort WL.
The payment equations:
𝑞 [𝑈𝑅𝐺 − 𝐴𝑁𝑁 −𝑑𝐻
𝐴𝑁𝑁] + (1 − 𝑞) [𝑈𝑅𝐵 − 𝐴𝑁𝑁 −
𝑑𝐻
𝐴𝑁𝑁]
Strategy (𝑃𝑁 , 𝐴𝑁𝑁 , 𝑊𝐻) (7)
(1 − 𝑞) [𝑈𝑅𝐺 − 𝐴𝑁𝑁 −𝑑𝐿
𝐴𝑁𝑁] + 𝑞 [𝑈𝑅𝐵 − 𝐴𝑁𝑁 −
𝑑𝐿
𝐴𝑁𝑁]
Strategy (𝑃𝑁 , 𝐴𝑁𝑁 , 𝑊𝐿) (8)
Simplifying (7) 𝑞𝑈𝑅𝐺 + (1 − 𝑞)𝑈𝑅𝐵 −
𝐴𝑁𝑁 −𝑑𝐻
𝐴𝑁𝑁
Simplifying (8) (1 − 𝑞)𝑈𝑅𝐺 + 𝑞𝑈𝑅𝐵 −
𝐴𝑁𝑁 −𝑑𝐿
𝐴𝑁𝑁
As 𝑞𝑈𝑅𝐺 > (1 − 𝑞)𝑈𝑅𝐺 but As
(1 − 𝑞)𝑈𝑅𝐵 < 𝑞𝑈𝑅𝐵 y −𝑑𝐻
𝐴𝑁𝑁< −
𝑑𝐿
𝐴𝑁𝑁 sub
scenarios arise:
𝑅𝐺 >[(1 − 𝑞)𝑈𝑅𝐺 + 𝑞𝑈𝑅𝐵 − (
𝑑𝐿𝐴𝑁𝑁
) − (1 − 𝑞)𝑈𝑅𝐵 + (𝑑𝐻
𝐴𝑁𝑁)]
𝑞𝑈
Being
𝜋4 =[(1 − 𝑝)𝑈𝑅𝐺 + 𝑝𝑈𝑅𝐵 − (
𝑑𝐿𝐴𝑁𝑁
) − (1 − 𝑝)𝑈𝑅𝐵 + (𝑑𝐻
𝐴𝑁𝑁)]
𝑝𝑈
If 𝑅𝐺 > 𝜋4 will be chosen (7) with the
Strategy 𝑃𝑁 , 𝐴𝑁𝑁 , 𝑊𝐻
If 𝑅𝐺 < 𝜋4 will be chosen (8) with the
Strategy 𝑃𝑁 , 𝐴𝑁𝑁 , 𝑊𝐿
If 𝑅𝐺 = 𝜋4 you can choose either
because both payments are equal.
Now it is analyzed When 𝑅𝐺 >𝜋1, 𝜋2, 𝜋3, 𝜋4
Payment (1) is compared with payment
(3):
39
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 26-50
TALAVERA-RUZ, Marianela, LARA-GÓMEZ, Graciela and
VALDEZ-RESÉNDIZ, Macario. Knowledge friction in
universities, approach from game theory. Journal-Mathematical
and Quantitative Methods. 2020
ISSN-On line: 2531-2979
RINOE® All rights reserved.
𝑝 [𝑈𝑅𝐺 − 𝐴𝑌𝑌 −𝑑𝐻
𝐴𝑌𝑌] + (1 − 𝑝) [𝑈𝑅𝐵 − 𝐴𝑌𝑌 −
𝑑𝐻
𝐴𝑌𝑌]
Strategy (𝑃𝑌, 𝐴𝑌𝑌 , 𝑊𝐻) (1)
𝑝 [𝑈𝑅𝐺 − 𝐴𝑌𝑁 −𝑑𝐻
𝐴𝑌𝑁] + (1 − 𝑝) [𝑈𝑅𝐵 − 𝐴𝑌𝑁 −
𝑑𝐻
𝐴𝑌𝑁]
Strategy (𝑃𝑌, 𝐴𝑌𝑁 , 𝑊𝐻) (3)
Simplifying (1) 𝑝𝑈𝑅𝐺 + (1 − 𝑝)𝑈𝑅𝐵 −
𝐴𝑌𝑌 −𝑑𝐻
𝐴𝑌𝑌
Simplifying (3) 𝑝𝑈𝑅𝐺 + (1 − 𝑝)𝑈𝑅𝐵 −
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
impacts the effort and therefore the disutility
perceived by the employee:
When the acceptance factor is high:
𝐴𝑌𝑌 > 𝐴𝑌𝑁 + (𝑑𝐻
𝐴𝑌𝑁−
𝑑𝐻
𝐴𝑌𝑌)
The impact on disutility is less so it will
subtract less from utility and will be chosen (1).
When the acceptance factor is low:
𝐴𝑌𝑌 < 𝐴𝑌𝑁 + (𝑑𝐻
𝐴𝑌𝑁−
𝑑𝐻
𝐴𝑌𝑌)
The impact on disutility is greater so it
will subtract more from utility and will be
chosen (3).
When 𝐴𝑌𝑌 = 𝐴𝑌𝑁 + (𝑑𝐻
𝐴𝑌𝑁−
𝑑𝐻
𝐴𝑌𝑌) you
can choose either of the two payments because
they will be worth the same.
To analyze the case of the employee
without prestige, the payment (5) is compared
with the payment (7):
𝑞 [𝑈𝑅𝐺 − 𝐴𝑁𝑌 −𝑑𝐻
𝐴𝑁𝑌] + (1 − 𝑞) [𝑈𝑅𝐵 − 𝐴𝑌𝑌 −
𝑑𝐻
𝐴𝑁𝑌]
Strategy (𝑃𝑁 , 𝐴𝑁𝑌, 𝑊𝐻) (5)
𝑞 [𝑈𝑅𝐺 − 𝐴𝑁𝑁 −𝑑𝐻
𝐴𝑁𝑁] + (1 − 𝑞) [𝑈𝑅𝐵 − 𝐴𝑁𝑁 −
𝑑𝐻
𝐴𝑁𝑁]
Strategy (𝑃𝑁 , 𝐴𝑁𝑁 , 𝑊𝐻) (7)
Simplifying (5) 𝑞𝑈𝑅𝐺 + (1 − 𝑞)𝑈𝑅𝐵 −
𝐴𝑁𝑌 −𝑑𝐻
𝐴𝑁𝑌
Simplifying (7) 𝑞𝑈𝑅𝐺 + (1 − 𝑞)𝑈𝑅𝐵 −
𝐴𝑁𝑁 −𝑑𝐻
𝐴𝑁𝑁
−𝐴𝑁𝑌 −𝑑𝐻
𝐴𝑁𝑌 (5)
−𝐴𝑁𝑁 −𝑑𝐻
𝐴𝑁𝑁 (7)
As 𝐴𝑁𝑌 > 𝐴𝑁𝑁 but 𝑑𝐻
𝐴𝑁𝑌<
𝑑𝐻
𝐴𝑁𝑁
Subscenaries arise based on the acceptance
factor. Thus the acceptance factor significantly
impacts the effort and therefore the disutility
perceived by the employee:
When the acceptance factor is high:
𝐴𝑁𝑌 > 𝐴𝑁𝑁 + (𝑑𝐻
𝐴𝑁𝑁−
𝑑𝐻
𝐴𝑁𝑌)
The impact on disutility is less so it will
subtract less from the utility and will be chosen
𝑃𝑁 , 𝐴𝑁𝑌, 𝑊𝐻 (5)
When the acceptance factor is low:
𝐴𝑁𝑌 < 𝐴𝑁𝑁 + (𝑑𝐻
𝐴𝑁𝑁−
𝑑𝐻
𝐴𝑁𝑌)
The impact on disutility is greater so it
will subtract more from the utility and will be
chosen
𝑃𝑁 , 𝐴𝑁𝑁 , 𝑊𝐻 (7)
When 𝐴𝑁𝑌 = 𝐴𝑁𝑁 + (𝑑𝐻
𝐴𝑁𝑁−
𝑑𝐻
𝐴𝑁𝑌)you
can choose either of the two payments because
they will be worth the same.
Now it is analyzed When 𝑅𝐺 <𝜋1, 𝜋2, 𝜋3, 𝜋4
Payment (2) is compared with payment
(4):
40
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 26-50
TALAVERA-RUZ, Marianela, LARA-GÓMEZ, Graciela and
VALDEZ-RESÉNDIZ, Macario. Knowledge friction in
universities, approach from game theory. Journal-Mathematical
and Quantitative Methods. 2020
ISSN-On line: 2531-2979
RINOE® All rights reserved.
(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]
Strategy (𝑃𝑌, 𝐴𝑌𝑁 , 𝑊𝐿) (4)
Simplifying (2) (1 − 𝑝)𝑈𝑅𝐺 + 𝑝𝑈𝑅𝐵 −
𝐴𝑌𝑌 −𝑑𝐿
𝐴𝑌𝑌
Simplifying (4) (1 − 𝑝)𝑈𝑅𝐺 + 𝑝𝑈𝑅𝐵 −
𝐴𝑌𝑁 −𝑑𝐿
𝐴𝑌𝑁
−𝐴𝑌𝑌 −𝑑𝐿
𝐴𝑌𝑌 (2)
−𝐴𝑌𝑁 −𝑑𝐿
𝐴𝑌𝑁 (4)
As 𝐴𝑌𝑌 > 𝐴𝑌𝑁 but 𝑑𝐿
𝐴𝑌𝑌<
𝑑𝐿
𝐴𝑌𝑁
Subscenaries arise based on the acceptance
factor. Thus 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 the utility and will be chosen
𝑃𝑌, 𝐴𝑌𝑌 , 𝑊𝐿 (2)
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 will be
chosen
𝑃𝑌, 𝐴𝑌𝑁 , 𝑊𝐿 (4)
When𝐴𝑌𝑌 = 𝐴𝑌𝑁 + (𝑑𝐿
𝐴𝑌𝑁−
𝑑𝐿
𝐴𝑌𝑌) you
can choose either of the two payments because
they will be worth the same.
To analyze the case of the employee
without prestige, payment (6) is compared with
payment (8):
(1 − 𝑞) [𝑈𝑅𝐺 − 𝐴𝑁𝑌 −𝑑𝐿
𝐴𝑁𝑌] + 𝑞 [𝑈𝑅𝐵 − 𝐴𝑌𝑌 −
𝑑𝐿
𝐴𝑁𝑌]
Strategy (𝑃𝑁 , 𝐴𝑁𝑌, 𝑊𝐿) (6)
(1 − 𝑞) [𝑈𝑅𝐺 − 𝐴𝑁𝑁 −𝑑𝐿
𝐴𝑁𝑁] + 𝑞 [𝑈𝑅𝐵 − 𝐴𝑁𝑁 −
𝑑𝐿
𝐴𝑁𝑁]
Strategy (𝑃𝑁 , 𝐴𝑁𝑁, 𝑊𝐿) (8)
Simplifying (6) (1 − 𝑞)𝑈𝑅𝐺 + 𝑞𝑈𝑅𝐵 −
𝐴𝑁𝑌 −𝑑𝐿
𝐴𝑁𝑌
Simplifying (8) (1 − 𝑞)𝑈𝑅𝐺 + 𝑞𝑈𝑅𝐵 −
𝐴𝑁𝑁 −𝑑𝐿
𝐴𝑁𝑁
−𝐴𝑁𝑌 −𝑑𝐿
𝐴𝑁𝑌 (6)
−𝐴𝑁𝑁 −𝑑𝐿
𝐴𝑁𝑁 (8)
As 𝐴𝑁𝑌 > 𝐴𝑁𝑁 but 𝑑𝐿
𝐴𝑁𝑌<
𝑑𝐿
𝐴𝑁𝑁
Subscenaries arise based on the acceptance
factor. Thus, the acceptance factor significantly
impacts the effort and therefore the disutility
perceived by the employee:
When the acceptance factor is high:
𝐴𝑁𝑌 > 𝐴𝑁𝑁 + (𝑑𝐿
𝐴𝑁𝑁−
𝑑𝐿
𝐴𝑁𝑌)
The impact on disutility is less so it will
subtract less from the utility and will be chosen
𝑃𝑁 , 𝐴𝑁𝑌, 𝑊𝐿 (6)
When the acceptance factor is low:
𝐴𝑁𝑌 < 𝐴𝑁𝑁 + (𝑑𝐿
𝐴𝑁𝑁−
𝑑𝐿
𝐴𝑁𝑌)
The impact on disutility is greater so it
will subtract more from the utility and will be
chosen
𝑃𝑁 , 𝐴𝑁𝑁, 𝑊𝐿 (8)
41
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 26-50
TALAVERA-RUZ, Marianela, LARA-GÓMEZ, Graciela and
VALDEZ-RESÉNDIZ, Macario. Knowledge friction in
universities, approach from game theory. Journal-Mathematical
and Quantitative Methods. 2020
ISSN-On line: 2531-2979
RINOE® All rights reserved.
When 𝐴𝑁𝑌 = 𝐴𝑁𝑁 + (𝑑𝐿
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
From the perspective of earnings from
payments, these can be expressed as a function
of incentives:
U (RPY) for a good profit scenario
U (RPN) for a bad profit scenario
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
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
the two levels are, respectively:
U (RPY) - dH
U (RPN) - dH
U (RPY) – dL
42
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 26-50
TALAVERA-RUZ, Marianela, LARA-GÓMEZ, Graciela and
VALDEZ-RESÉNDIZ, Macario. Knowledge friction in
universities, approach from game theory. Journal-Mathematical
and Quantitative Methods. 2020
ISSN-On line: 2531-2979
RINOE® All rights reserved.
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
Sub-scenarios then arise:
When G > 𝜋6 will be chosen (𝑃𝑌)
When G < 𝜋6 will be chosen (𝑃𝑁)
When G=𝜋6 anyone can be chosen
since both payments are equal.
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
the two levels are, respectively:
G - RPG
B - RPB
From the perspective of earnings from
payments, these can be expressed as a function
of incentives:
43
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 26-50
TALAVERA-RUZ, Marianela, LARA-GÓMEZ, Graciela and
VALDEZ-RESÉNDIZ, Macario. Knowledge friction in
universities, approach from game theory. Journal-Mathematical
and Quantitative Methods. 2020
ISSN-On line: 2531-2979
RINOE® All rights reserved.
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.
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
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
To solve this dynamic non-cooperative game
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),
corresponding to the decision of an employee
with prestige P_Y who strives to be accepted
A_YY, between investing a greater effort W_H
or a least effort W_L.
The payment equations:
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
because both payments are equal.
44
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 26-50
TALAVERA-RUZ, Marianela, LARA-GÓMEZ, Graciela and
VALDEZ-RESÉNDIZ, Macario. Knowledge friction in
universities, approach from game theory. Journal-Mathematical
and Quantitative Methods. 2020
ISSN-On line: 2531-2979
RINOE® All rights reserved.
Now payment 3 (P_Y, A_YN, W_H) is
compared with payment 4 (P_Y, A_YN, W_L),
corresponding to the decision of an employee
with prestige PY who does not make an effort
to be accepted AYN, between investing a
greater effort WH or less effort WL.
The payment equations:
𝑝 [𝑈𝑅𝑃𝐺 − 𝐴𝑌𝑁 −𝑑𝐻
𝐴𝑌𝑁] + (1 − 𝑝) [𝑈𝑅𝑃𝐵 − 𝐴𝑌𝑁 −
𝑑𝐻
𝐴𝑌𝑁]
Strategy (𝑃𝑌, 𝐴𝑌𝑁 , 𝑊𝐻) (3)
(1 − 𝑝) [𝑈𝑅𝑃𝐺 − 𝐴𝑌𝑁 −𝑑𝐿
𝐴𝑌𝑁] + 𝑝 [𝑈𝑅𝑃𝐵 − 𝐴𝑌𝑁 −
𝑑𝐿
𝐴𝑌𝑁]
Strategy (𝑃𝑌, 𝐴𝑌𝑁 , 𝑊𝐿) (4)
Simplifying (3) 𝑝𝑈𝑅𝑃𝐺 + (1 −
𝑝)𝑈𝑅𝑃𝐵 − 𝐴𝑌𝑁 −𝑑𝐻
𝐴𝑌𝑁
Simplifying (4) (1 − 𝑝)𝑈𝑅𝑃𝐺 +
𝑝𝑈𝑅𝑃𝐵 − 𝐴𝑌𝑁 −𝑑𝐿
𝐴𝑌𝑁
As 𝑝𝑈𝑅𝐺 > (1 − 𝑝)𝑈𝑅𝐺 but As
(1 − 𝑝)𝑈𝑅𝐵 < 𝑝𝑈𝑅𝐵 y −𝑑𝐻
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
because both payments are equal.
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
who strives to be accepted AYY, between
investing a greater effort WH or a less effort
WL.
The payment equations:
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, 𝑊𝐻) (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 + 𝑝𝑈𝑅𝐵 − (
𝑑𝐿𝐴𝑁𝑌
) − (1 − 𝑞)𝑈𝑅𝐵 + (𝑑𝐻
𝐴𝑁𝑌)]
𝑝𝑈
Being
𝜋9 =[(1 − 𝑞)𝑈𝑅𝐺 + 𝑝𝑈𝑅𝐵 − (
𝑑𝐿𝐴𝑁𝑌
) − (1 − 𝑞)𝑈𝑅𝐵 + (𝑑𝐻
𝐴𝑁𝑌)]
𝑝𝑈
If 𝑅𝐺 > 𝜋9 will be chosen (5) with the
Strategy 𝑃𝑁 , 𝐴𝑁𝑌, 𝑊𝐻
If 𝑅𝐺 < 𝜋9 will be chosen (6) with the
Strategy 𝑃𝑁 , 𝐴𝑁𝑌, 𝑊𝐿
If 𝑅𝐺 = 𝜋9 you can choose either
because both payments are equal.
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.
The payment equations:
𝑞 [𝑈𝑅𝐺 − 𝐴𝑁𝑁 −𝑑𝐻
𝐴𝑁𝑁] + (1 − 𝑞) [𝑈𝑅𝐵 − 𝐴𝑁𝑁 −
𝑑𝐻
𝐴𝑁𝑁]
Strategy (𝑃𝑁 , 𝐴𝑁𝑁 , 𝑊𝐻) (7)
(1 − 𝑞) [𝑈𝑅𝐺 − 𝐴𝑁𝑁 −𝑑𝐿
𝐴𝑁𝑁] + 𝑞 [𝑈𝑅𝐵 − 𝐴𝑁𝑁 −
𝑑𝐿
𝐴𝑁𝑁]
Strategy (𝑃𝑁 , 𝐴𝑁𝑁 , 𝑊𝐿) (8)
45
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 26-50
TALAVERA-RUZ, Marianela, LARA-GÓMEZ, Graciela and
VALDEZ-RESÉNDIZ, Macario. Knowledge friction in
universities, approach from game theory. Journal-Mathematical
and Quantitative Methods. 2020
ISSN-On line: 2531-2979
RINOE® All rights reserved.
Simplifying (7) 𝑞𝑈𝑅𝐺 + (1 − 𝑞)𝑈𝑅𝐵 −
𝐴𝑁𝑁 −𝑑𝐻
𝐴𝑁𝑁
Simplifying (8) (1 − 𝑞)𝑈𝑅𝐺 + 𝑞𝑈𝑅𝐵 −
𝐴𝑁𝑁 −𝑑𝐿
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 − (
𝑑𝐿𝐴𝑁𝑁
) − (1 − 𝑞)𝑈𝑅𝐵 + (𝑑𝐻
𝐴𝑁𝑁)]
𝑞𝑈
Being 𝜋10
=[(1 − 𝑞)𝑈𝑅𝐺 + 𝑝𝑈𝑅𝐵 − (
𝑑𝐿𝐴𝑁𝑁
) − (1 − 𝑞)𝑈𝑅𝐵 + (𝑑𝐻
𝐴𝑁𝑁)]
𝑞𝑈
If 𝑅𝐺 > 𝜋10 will be chosen (7) with the
Strategy (𝑃𝑁 , 𝐴𝑁𝑁 , 𝑊𝐻)
If 𝑅𝐺 < 𝜋10 will be chosen (8) with the
Strategy (𝑃𝑁 , 𝐴𝑁𝑁 , 𝑊𝐿)
If 𝑅𝐺 = 𝜋10 you can choose either
because both payments are equal.
Now it is analyzed when 𝑹𝑷𝑮 >𝝅𝟕, 𝝅𝟖, 𝝅𝟗, 𝝅𝟏𝟎
Payment (1) is compared with payment
(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
Subscenaries arise based on the acceptance
factor. Thus the acceptance factor significantly
impacts the effort and therefore the disutility
perceived by the employee:
When the acceptance factor is high:
𝐴𝑌𝑌 > 𝐴𝑌𝑁 + (𝑑𝐻
𝐴𝑌𝑁−
𝑑𝐻
𝐴𝑌𝑌)
The impact on disutility is less so it will
subtract less from utility and will be chosen (1).
When the acceptance factor is low:
𝐴𝑌𝑌 < 𝐴𝑌𝑁 + (𝑑𝐻
𝐴𝑌𝑁−
𝑑𝐻
𝐴𝑌𝑌)
The impact on disutility is greater so it
will subtract more from utility and will be
chosen (3).
When 𝐴𝑌𝑌 = 𝐴𝑌𝑁 + (𝑑𝐻
𝐴𝑌𝑁−
𝑑𝐻
𝐴𝑌𝑌) you
can choose either of the two payments because
they will be worth the same.
To analyze the case of the employee
without prestige, the payment (5) is compared
with the payment (7):
𝑞 [𝑈𝑅𝐺 − 𝐴𝑁𝑌 −𝑑𝐻
𝐴𝑁𝑌] + (1 − 𝑞) [𝑈𝑅𝐵 − 𝐴𝑌𝑌 −
𝑑𝐻
𝐴𝑁𝑌]
Strategy (𝑃𝑁 , 𝐴𝑁𝑌, 𝑊𝐻) (5)
𝑞 [𝑈𝑅𝐺 − 𝐴𝑁𝑁 −𝑑𝐻
𝐴𝑁𝑁] + (1 − 𝑞) [𝑈𝑅𝐵 − 𝐴𝑁𝑁 −
𝑑𝐻
𝐴𝑁𝑁]
Strategy (𝑃𝑁 , 𝐴𝑁𝑁 , 𝑊𝐻) (7)
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
Subscenaries arise based on the acceptance
factor. Thus the acceptance factor significantly
impacts the effort and therefore the disutility
perceived by the employee:
46
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 26-50
TALAVERA-RUZ, Marianela, LARA-GÓMEZ, Graciela and
VALDEZ-RESÉNDIZ, Macario. Knowledge friction in
universities, approach from game theory. Journal-Mathematical
and Quantitative Methods. 2020
ISSN-On line: 2531-2979
RINOE® All rights reserved.
When the acceptance factor is high:
𝐴𝑁𝑌 > 𝐴𝑁𝑁 + (𝑑𝐻
𝐴𝑁𝑁−
𝑑𝐻
𝐴𝑁𝑌)
The impact on disutility is less so it will
subtract less from the utility and will be chosen
𝑃𝑁 , 𝐴𝑁𝑌, 𝑊𝐻 (5)
When the acceptance factor is low:
𝐴𝑁𝑌 < 𝐴𝑁𝑁 + (𝑑𝐻
𝐴𝑁𝑁−
𝑑𝐻
𝐴𝑁𝑌)
The impact on disutility is greater so it
will subtract more from the utility and will be
chosen
𝑃𝑁 , 𝐴𝑁𝑁 , 𝑊𝐻 (7)
When 𝐴𝑁𝑌 = 𝐴𝑁𝑁 + (𝑑𝐻
𝐴𝑁𝑁−
𝑑𝐻
𝐴𝑁𝑌) you
can choose either of the two payments because
they will be worth the same.
Now it is analyzed When 𝑹𝑮 <𝝅𝟏, 𝝅𝟐, 𝝅𝟑, 𝝅𝟒
Payment (2) is compared with payment
(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]
Strategy (𝑃𝑌, 𝐴𝑌𝑁 , 𝑊𝐿) (4)
Simplifying (2)(1 − 𝑝)𝑈𝑅𝑃𝐺 +
𝑝𝑈𝑅𝐵 − 𝐴𝑌𝑌 −𝑑𝐿
𝐴𝑌𝑌
Simplifying (4) (1 − 𝑝)𝑈𝑅𝑃𝐺 +
𝑝𝑈𝑅𝐵 − 𝐴𝑌𝑁 −𝑑𝐿
𝐴𝑌𝑁
−𝐴𝑌𝑌 −𝑑𝐿
𝐴𝑌𝑌 (2)
−𝐴𝑌𝑁 −𝑑𝐿
𝐴𝑌𝑁 (4)
As 𝐴𝑌𝑌 > 𝐴𝑌𝑁 but 𝑑𝐿
𝐴𝑌𝑌<
𝑑𝐿
𝐴𝑌𝑁
Subscenaries arise based on the acceptance
factor. Thus, 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 the utility and will be chosen
𝑃𝑌, 𝐴𝑌𝑌 , 𝑊𝐿 (2)
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 will be
chosen
𝑃𝑌, 𝐴𝑌𝑁 , 𝑊𝐿 (4)
When 𝐴𝑌𝑌 = 𝐴𝑌𝑁 + (𝑑𝐿
𝐴𝑌𝑁−
𝑑𝐿
𝐴𝑌𝑌) you
can choose either of the two payments because
they will be worth the same.
To analyze the case of the employee
without prestige, payment (6) is compared with
payment (8):
(1 − 𝑞) [𝑈𝑅𝐺 − 𝐴𝑁𝑌 −𝑑𝐿
𝐴𝑁𝑌] + 𝑞 [𝑈𝑅𝐵 − 𝐴𝑌𝑌 −
𝑑𝐿
𝐴𝑁𝑌]
Strategy (𝑃𝑁 , 𝐴𝑁𝑌, 𝑊𝐿) (6)
(1 − 𝑞) [𝑈𝑅𝐺 − 𝐴𝑁𝑁 −𝑑𝐿
𝐴𝑁𝑁] + 𝑞 [𝑈𝑅𝐵 − 𝐴𝑁𝑁 −
𝑑𝐿
𝐴𝑁𝑁]
Strategy (𝑃𝑁 , 𝐴𝑁𝑁 , 𝑊𝐿) (8)
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
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 26-50
TALAVERA-RUZ, Marianela, LARA-GÓMEZ, Graciela and
VALDEZ-RESÉNDIZ, Macario. Knowledge friction in
universities, approach from game theory. Journal-Mathematical
and Quantitative Methods. 2020
ISSN-On line: 2531-2979
RINOE® All rights reserved.
As 𝐴𝑁𝑌 > 𝐴𝑁𝑁 but 𝑑𝐿
𝐴𝑁𝑌<
𝑑𝐿
𝐴𝑁𝑁 Sub-
sceneries arise based on the acceptance factor.
Thus, the acceptance factor significantly
impacts the effort and therefore the disutility
perceived by the employee:
When the acceptance factor is high:
𝐴𝑁𝑌 > 𝐴𝑁𝑁 + (𝑑𝐿
𝐴𝑁𝑁−
𝑑𝐿
𝐴𝑁𝑌)
The impact on disutility is less so it will
subtract less from the utility and will be chosen
(𝑃𝑁 , 𝐴𝑁𝑌, 𝑊𝐿) (6)
When the acceptance factor is low:
𝐴𝑁𝑌 < 𝐴𝑁𝑁 + (𝑑𝐿
𝐴𝑁𝑁−
𝑑𝐿
𝐴𝑁𝑌)
The impact on disutility is greater so it
will subtract more from the utility and will be
chosen
(𝑃𝑁 , 𝐴𝑁𝑌, 𝑊𝐿) (8)
When 𝐴𝑁𝑌 = 𝐴𝑁𝑁 + (𝑑𝐿
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 − 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
Sub-scenarios then arise:
When 𝐺 > 𝜋11 will be chosen (PY)
When 𝐺 < 𝜋11 will be chosen (PN)
When 𝐺 = 𝜋11 anyone can be chosen
since both payments are equal.
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
𝐴𝑁𝑌, the dominant Strategy will be
(𝑃𝑁 , 𝐴𝑁𝑁 , 𝑊𝐻)
ó
If 𝑅𝑃𝐺 < 𝜋8 y 𝐴𝑁𝑌 > 𝐴𝑁𝑁 +𝑑𝐻
𝐴𝑁𝑁−
𝑑𝐻
𝐴𝑁𝑌, the dominant Strategy will be
(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
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 26-50
TALAVERA-RUZ, Marianela, LARA-GÓMEZ, Graciela and
VALDEZ-RESÉNDIZ, Macario. Knowledge friction in
universities, approach from game theory. Journal-Mathematical
and Quantitative Methods. 2020
ISSN-On line: 2531-2979
RINOE® All rights reserved.
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
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 26-50
TALAVERA-RUZ, Marianela, LARA-GÓMEZ, Graciela and
VALDEZ-RESÉNDIZ, Macario. Knowledge friction in
universities, approach from game theory. Journal-Mathematical
and Quantitative Methods. 2020
ISSN-On line: 2531-2979
RINOE® All rights reserved.
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.
References
Alter, S. (2006) Goals and tactics on the dark
side of knowledge management, in:
Proceedings of the 39th Hawaii International
Conference on System Sciences (HICSS’06),
Hawaii, USA, January 4–7.
Argote, L. y Ingram, P. (2000) Knowledge
transfer: a basis for competitive advantage in
firms, Organizational Behavior and Human
Decision Processes 82 (1) 150–169.
Bolton, G.E., Ockenfels, A. (2000) ERC: a
theory of equity, reciprocity, and competition,
The American Economic Review 90 (1) 166–
193.
Boisot, M. (1998) Knowledge Assets: Securing
Competitive Advantage in the Information
Economy, Oxford University Press, New York.
Bryant, A. (2006) Knowledge management –
the ethics of the agora or the mechanisms of the
market? in: Proceedings of the 39th Annual
Hawaii International Conference on System
Sciences (HICSS’06), Hawaii, USA, January
4–7.
Cabrera, A. y Cabrera, E.F. (2002) Knowledge-
sharing dilemmas, Organization Studies 23,
687–710.
Chua, A. (2003) Knowledge sharing: a game
people play, Aslib Proceedings: New
Information Perspectives 55 117–129. [21]
W.M.
Cohen, D.A. Levinthal (1990) Absorptive
capacity: a new perspective on learning and
innovation, Administrative Science Quarterly
35 128– 152.
Davenport, T.H., Prusak, L. (1998) Working
Knowledge: How Organizations Manage What
They Know, Harvard Business School Press,
Boston, MA.
Dougherty, D. (1992) Interpretive barriers to
successful product innovation in large firms,
Organization Science 3, 179–202.
Camerer, C.F. (1997) Progress in behavioral
game theory, The Journal of Economic
Perspectives 11, 167–188.
Camerer, C.F. (2003) Behavioral Game
Theory: Experiments in Strategic Interaction,
Princeton University Press, Princeton, NJ.
Dufwenberg, M. y Kirchsteiger, G. (2004) A
theory of sequential reciprocity, Games and
Economic Behavior 47, 268–298.
Drucker, P.F. (1994) The Age of Social
Transformation, The Atlantic Monthly, New
York.
Ghobadi, S. y D’Ambra, J. (2011) Coopetitive
knowledge sharing: an analytical review of
literature, The Electronic Journal of Knowledge
Management 9, 307–317.
50
Article Journal-Mathematical and Quantitative Methods December 2020 Vol.4 No.7 26-50
TALAVERA-RUZ, Marianela, LARA-GÓMEZ, Graciela and
VALDEZ-RESÉNDIZ, Macario. Knowledge friction in
universities, approach from game theory. Journal-Mathematical
and Quantitative Methods. 2020
ISSN-On line: 2531-2979
RINOE® All rights reserved.
Gupta, A.K. y Govindarajan, V. (2000)
Knowledge flows within multinational
corporations, Strategic Management Journal 21,
473–496.
Hamel, G. y Prahalad, C.K. (1990) The core
competence of the corporation, Harvard
Business Review 68, 79–91.
Hardin, G. (1993) The tragedy of the commons,
Science 16, 1243–1248.
Hinds, P.J. y Pfeffer, J. (2003) Why
Organizations Don’t ‘‘Know What They
Know’’: Cognitive and Motivational Factors
Affecting the Transfer of Expertise, MIT Press,
Cambridge, MA.
Loebecke, C., Van Fenema, P.C. y Powell, P.
(1999) Co-opetition and knowledge transfer,
ACM SIGMIS Database 30, 14–25.
Minsky, M. (1994) Negative expertise,
International Journal of Expert Systems 7, 13–
18.
Nonaka, I. y Konno, N. (1998) The concept of
‘‘Ba’’: building a foundation for knowledge
creation, California Management Review 40,
40–54.
O’Dell, C. y Grayson, C.J. (1998) If only we
knew what we know, California Management
Review 40, 154–174.
Perrin, A., Rolland, y N., Stanley, T. (2007)
Achieving best practices transfer across
countries, Journal of Knowledge Management
11, 156–166.
Pfeffer, J., Sutton, R.I. (1999) The Knowing-
Doing Gap: How Smart Companies Turn
Knowledge into Action, Harvard Business
School Press, Cambridge, MA.
Senge, P.M. (1990) The Fifth Discipline: The
Art and Practice of the Learning Organization,
Currency, New York.
Sharma, R.S. y Bhattacharya, S. (2013)
Knowledge dilemmas within organizations:
Resolutions from game theory, In Knowledge-
Based Systems, Volume 45, Pages 100-113,
ISSN 0950-7051,
https://doi.org/10.1016/j.knosys.2013.02.011.(h
ttp://www.sciencedirect.com/science/article/pii/
S0950705113000762)
Spender, J.C. (1996) Competitive advantage
from tacit knowledge? Unpacking the concept
and its strategic implications, in: B. Moingeon,
A. Edmondson (Eds.), Organizational Learning
and Competitive Advantage, Sage Publications,
Thousand Oaks, CA, pp. 56–73.
Szulanski, G. (1996) Exploring internal
stickiness: impediments to the transfer of best
practice within the firm, Strategic Management
Journal 17, 27–43 (winter special issue).
Teece, D.J. (1988) Capturing value from
knowledge assets: the new economy, markets
for know-how, and intangible assets, California
Management Review 40, 55–79.
Yang, H.L. y Wu, T.C.T. (2008) Knowledge
sharing in an organization, Technological
Forecasting and Social Change 75, 1128–1156.
Vaillant, D. (2010). La identidad docente. La
importancia del profesorado. Revista
Novedades educativas, 234, 4-11.
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