Reflexive Change of the Manager's Objective Function

Mikhail Savelyev

1 a

, Andrey Savchenko

, Vladimir Koretsky

and Anna Koshcheeva

Association for Methodological Support of Business Activity and Social Development Mitra, Pushkinskaya street, 241,

Izhevsk, Russia

Kalashnikov Izhevsk State Technical University, Studencheskaya street, 7, Izhevsk, Russia

Udmurt State University, Universitetskaya, street 1, Izhevsk, Russia

Keywords: Game theory, reflexive logic, economic agents, an objective function.

Abstract: Game-theoretic models of the theory of reflexive games and logical-reflexive models of algebraic methods

for modelling reflexive processes are investigated. To solve the problem of changing the objective functions

of economic agents in different cultural environments, the structural method of searching for the

correspondence of mathematical models to the six levels of reflexivity of a manager known in the theory of

the school's activity of general methodology was applied. It is shown that the change in the agent's objective

function, which is not available in game-theoretic models, is possible when an economic agent gains the level

of problem addressing in models of reflexive logic, as well as at higher levels of reflexivity. Game-theoretic

models are a special case of logical-reflexive models and do not allow phenomenologically reliable modelling

of the reflexive processes of agents. The use of logical-reflexive models gives a broader scope of research and

more accurate tools for modelling reflexive processes.

1 INTRODUCTION

In game-theoretic modelling of the behavior of

institutional agents, researchers traditionally proceed

from the invariability of the objective functions of

agents. It is assumed that the objective functions

specified in the models a priori reflect the interests of

these agents. These interests are considered constant

and immutable. When interests change, a new model

is built with a new objective function. Empirical

experience often gives a different pattern: interests

can change having made a decision. This is due to the

fact that the real agent has an idea of all possible

benefits deriving from the decision. And in the

presence of certain opportunities, an agent can switch

from one goal to another. From the point of view of

modelling, such an objective function of the agent is

multi-criteria and multifactorial. In this case,

individual criteria and factors are elastic only to

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certain external conditions, and in the absence of

these conditions, the agent is not subjected to them.

Modelling innovative development in different

cultural environments this principle is even more

essential. The types of culture are shown in Тable. 1.

Table 1: The structure of social values in the main cultures

of the world. Types of culture.

Principles of

social self-

identification

Equality Inequality

Individualism

Freedom: Western

Culture

Alienating culture

Justice: Middle

Eastern and

Indian Cultures

Isolating culture

Collectivism

Trust, solidarity:

Eastern European

(Eurasian), Latin

American, African,

Southeast Asian

cultures

ualizin

culture

Virtue (Mercy

and Respect):

East Asian

Culture

Consolidating

culture

Source: Savelyev, 2015.

104

Savelyev, M., Savchenko, A., Koretsky, V. and Koshcheeva, A.

Reﬂexive Change of the Manager’s Objective Function.

DOI: 10.5220/0010693900003169

In Proceedings of the International Scientiﬁc-Practical Conference "Ensuring the Stability and Security of Socio-Economic Systems: Overcoming the Threats of the Crisis Space" (SES 2021),

pages 104-110

ISBN: 978-989-758-546-3

In the study of the objective attitudes of managers,

the needs of a manager invariant in the studied

cultural environments were revealed as power and

career. But considering the constituent elements of

this objective attitude, significant differences were

revealed depending on the cultural environment.

Researchers among the interests of managers

identified income, social guarantees, psychic income

(public reputation), patronage, the results of the

institution's activities, the freedom to make changes

and the freedom to manage the institution. When

considering the types of cultural environments, the

interests of managers dominating in these

environments were revealed (see Table. 2).

Table 2: The structure of the manager's motives in different

cultural environments and the leading elements of objective

function.

Principles of

social self-

determination

Equality Inequality

Individualism

Residual

income

The likelihood of

meeting the

interests of allies

(the likelihood of

staying in power)

Collectivism

The likelihood

of growth in

the size of the

organization

employment /

investment /

sales

The likelihood of

meeting the

interests of allies

* the likelihood of

growth

Source: Savelyev, 2020.

The objective function of a manager in different

cultures is more elastic to peculiar to the

corresponding culture variables (Savelyev, 2020):

𝑀𝐼𝐵  𝐹𝑄  𝑃



𝑃



,1

where: MIB is the future income of the manager; FQ

- residual income, defined as the remainder of the

organization's income without income of employees

and investors; P

inc

- the likelihood of growth in the

size of the organization in terms of employment,

investment or sales; P

- the probability of satisfying

the interests of allies (the probability of staying in

power).

It is assumed that the favorable change of manager

will first react the dominant element of the objective

function, and adverse changes - the most distant from

the dominant interests in a particular cultural

environment element of the objective function. For

example, in an isolating culture for favorable events,

the manager will first of all share the additional

income with allies, and in case of an unfavorable

change, will reduce the size of the organization in

parameters that are not in the interests of allies. In an

equalizing culture, if the situation is favorable, the

organization will grow to the maximum possible size,

and if the situation is unfavorable, it will maintain the

size of the organization even if the income of allies is

reduced.

This behavior can be modelled only if ones

formally describe the agent's ideas about the future

before the decision is made and the logical operators

for choosing the objective parameter of the objective

function. And this tool is absent in traditional game-

theoretic models but is being actively developed in

logical-reflexive models.

In general, the combined application of the

methods of game theory and reflexive logic brings

closer the resolution of the epistemological gap

between empirical studies of the influence of culture

on the activities of economic entities and the activities

of territorial authorities and the institutional

description of their activities, if causally

predetermined supra-constitutional rules are included

in the modelled institutional systems.

These systems make it possible to design

institutional provisions for the management of

economic entities and territories in the conditions of

completed industrialization and the transition to an

innovative economy in accordance with the specifics

of the culture of society, and, thereby, lay the

foundation for the fundamental provision of the

ideologeme of the “multipolar world” and a

reasonable refusal from institutional unification

under the implementation of the global project

“unipolar world”. For the mathematical modelling of

institutional systems and the process of designing

institutional transformations, it is necessary to

develop a theory of reflexive processes, involving the

combined application of the methods of game theory

and reflexive logic.

Algebraic methods for modelling reflexive

processes were developed by V. Lefebvre (Lefebvre,

1965, 1966, 1967, 1973, 1991, 1997, 2009; Lefebvre,

1982, 1992, 1998, 2008).

As noted in one of this works (Lefebvre, 2009), a

significant contribution to the theoretical

understanding of reflection was made by T. Taran,

who built a Boolean valued model of the choice of

social norms (Taran, 1998, 2001); V. Krylov, who

studied the problems of axiomatics of reflexive

models (Krylov, 2000); Yu. Schrader, who

considered continuous-valued logics as languages of

Reﬂexive Change of the Manager’s Objective Function

105

reflection and proposed formal models of reflexive

structures and (Schrader, 1999). The contribution of

T. Taran Ramming to the theory of reflexive

processes can, with no exaggeration, be called the

second most important after the results of V.

Lefebvre. Her research and research have

significantly expanded the capabilities of reflexive

models.

The theory of reflexive games as a new class of

game-theoretic models describing the interaction of

subjects (agents) making decisions based on the

hierarchy of ideas about parameters and ideas about

representations was built in the works of D.

Novikova, A. Chkhartishvili, M. Gubko (Novikov,

2003, 2004, 2013; Chkhartishvili, 2004; Gubko,

2002).

Reflexive logic, despite the active calls of V.

Lefebvre, is not used in economic and mathematical

methods. In our opinion, this situation has several

reasons.

First, it will call into question the paradigm of the

mainstream of Western economics, in particular the

assumption of the rationality of agent behavior and of

methodological individualism. In the case of applying

reflexive logic, rationality will be broken up into

many logics of different cultures, and the rejection of

methodological individualism will cause the “specter

of communism” and the assumption of the existence

of many ideal economic systems corresponding to

different cultures of the world. In the mainstream,

these innovations are ideologically taboo, because the

mainstream is used as an ideological means in hybrid

warfare, so is the means of determining the rules of

international economic policy.

Secondly, the basis for the development of the

theory of reflexive games was the hierarchical

representation of the players' information reflection.

Such a representation set the information structure as

a kind of infinite tree, which makes the algorithmic

problem of optimizing the choice of strategy in a

reflexive game practically impracticable. In this

regard, the creation of a universal algebraic apparatus

that corresponds to the processes of reflection

actually presented in different cultures and allows one

to model these reflexive processes, including in

institutional design, is significant for the development

of scientific knowledge and relevant for addressing

practical management tasks. For example, it is known

that the solution of the problem of modelling the

emergence of organizations in the form of a game to

develop the rules of the game stumbles upon the

problem of “bad infinity” (“reflexive candle”), which

requires its solution by logical-algebraic methods on

the basis of actually existing reflexive processes

studied in cognitive psychology.

Thirdly, on the one hand, there are no empirical

studies of reflexive processes in economic and

political agents that confirm the models of the theory

of reflexive games, and on the other hand, the concept

of self-reflection (self-reflection or reflection of the

first kind (Novikov, 2013)) lies outside the scope of

research. As indicated in this work: “If the only

reflexive subject is an economic agent that seeks to

maximize its objective function by choosing one of

the ethically acceptable actions, then natural reality is

included in the objective function as a parameter, and

the results of reflection (ideas about ideas, etc.)

objective function arguments are not. Then self-

reflection is “unnecessary” because it does not change

the action chosen by the agent."

There is a bulk of references dedicated to the

general methodology (system-thought-activity

methodology) for the level of development of the

subject's reflexive abilities to be classified. The

theory of activity classifies group formation

according to their 6 levels of cohesion (Anisimov,

2001):

1. sympathy (antipathy) or in life, - lack of

reflection,

2. coordination of the results of reflection of

behavior and about the opposition, - situational

reflection with a minimum of problematization and

random regulation,

3. reflection of the process of implementing

norms (addressing problems), - full-fledged "three-

sided" reflection with problematization and

renormalization of unrealizable norms,

4. the grounds for organizing reflection, - the

use of concepts (for example, science) in reflection,

5. the way of developing the foundations for

organizing reflection, - using values as the means of

problematizing a situation and concepts (for example,

a particular methodology of science),

6. search for a way to develop the foundations

of the organization of reflection - problematization

not only of the situation and concepts but also of the

values themselves (for example, a general

methodology).

The penultimate level assumes that the result of

reflection is influenced by the subject's relationship to

the cultural environment, while the latter assumes a

conscious choice of conformity to culture.

The purpose of this study is to determine the level

of development of the agent's reflexive abilities when

one is able to change the objective function and what

logical-reflexive model can describe this process.

Research hypothesis: the logical-reflexive models

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106

developed by previous researchers allow describing

the process of changing the agent's objective function.

 determine within which logical-reflexive

models it is possible to change the objective function

of the agent;

 to determine the algorithm for changing the

investigated objective function of the manager.

2 MATERIALS AND METHODS

The research is mostly based on the structural method

(Gretsky, 1983). It is planned to be applied in relation

to existing logical-reflexive models in order to find the

correspondence of these models to 6 levels of

reflexivity.

In the works (Lefebvre, 2007, 2008; Lefebvre,

2008), a model is consistently introduced into

consideration, which allows one to build hypotheses

about the individual choices of group members that

influence each other, and it is explained how this

model can be used in practice. It is noted (Lefebvre,

2007) that the constructed model is based on the

construction described by V.Lefebvre (Lefebvre,

1982, 2007), as well as on the work performed by

Taran (Taran, 2001), in which the reflexive model was

extended to Boolean algebra with many elements.

The work (Taran, 2001) considers the basic

principles of constructing reflexive models of

behavior in a situation of choice. First, based on the

reflexive model of bipolar choice, V.A. Lefebvre

(Lefebvre, 2003) naturally builds a logical model of

the subject's reflexive behavior, formalized in

Boolean logic

= <{0,1}, &, ˅, ¬ >, gives an

ethical interpretation of the reflexive choice and

indicates that, depending on the domain of definition

variables, various models of reflexive choice can be

built. Further, we propose a generalization of V.

Lefebvre's model in the form of a vector Boolean

model, in which the choice-making activity of the

subject under certain conditions is regulated by

certain norms, models of choice are built on

multivalued Boolean scales, correlating with a set of

actions characterizing the behavior of the subject in

society, regulated by a system of certain norms. In

this study, we rely on logical-reflexive models

proposed by T. Taran (Taran, 1998, 2001, 2004;

Taran, 1998, 2001).

The basic model of reflexive choice developed by

V. Lefebvre, represents the behavior of the agent in

the form of a hierarchical reflexive structure of the

factors (Taran, 1998: 51; Taran, 2004):

Behavior  feeling

з

desire

The decision-making by the agent is considered as

a sequence of binary choices (elementary decision-

making acts). Each elementary choice is made on a

binary scale with two options: positive and negative.

The agent is under pressure from the outside world to

choose one of the options and acts as an operator: at

each moment in time, an agent can choose one of the

options (perform some action: positive (modelled by

a Boolean value 1) or negative (modelled by a

Boolean value 0)). This pressure of the outside world

is not realized by the subject, but subconsciously

makes them do a choice. These perceptions bring to

the mind of the agent opinion about the pressure of

the external world - subjective knowledge, which

does not always coincide with reality (it is included

in the “self-image” that the subject has). This “self-

image” also includes the subject's ideas about

themself: a conscious image of themself, which forms

the subject's intentions to choose one or another

option, which can be considered as personal desires.

Under the influence of the pressure of the outside

world, the subject's idea of this pressure and

intentions the agent goes amenable to making a

choice, accessible to an external observer. The images

available to the subject is visible only for an external

observer. For the subject, they seem to be some kind

of reality. The apparent behavior of the subject is

reality for an external observer. The self-image is a

reality for the subject. The self-image that the self-

image has is the subject's idea of reality.

The reflexive structure is described by a function

of the form:

f (x

)  𝑥











(2),

where 𝑥



describes the pressure of the outside world

towards one of the options; 𝑥



- the subject's idea of

the pressure of the outside world; 𝑥



–– subject's

intentions. The function f (x

) describes the state

of the subject at the moment of choice.

The expression 𝑎



is a logical implication: 𝑏→

𝑎 , so function (2) can also be represented by the

logical formula

𝑋







𝑥



→𝑥





→𝑥



, (3)

where 𝑥



is the perception of the pressure of the

external world (unconscious perception by a person

of impulses from the external environment); 𝑥



assessment of the subject's psychological attitude (the

expected pressure of the outside world, which is

formed on the basis of a person's past knowledge and

psychological attitude); variable 𝑥



is an assessment

of intentions (plans and desires that reflect a person's

Reﬂexive Change of the Manager’s Objective Function

107

perceptions of a complicated situation and about

himself in this situation), the implication 𝑥



→𝑥



the subject's self-assessment (“self-image”).

The scope of these variables and functions is a set

of type 2={0,1} (a Boolean lattice of two elements (0

and 1)) with the operations of disjunction,

conjunction, negation and implication ( 𝑏→𝑎

𝑏 ∨ 𝑎. Therefore 𝑋



is a Boolean function 𝑓: 2



→

2 (Kuznetsov, 2008).

This model can be generalized as a vector Boolean

model, in which (3) is a function 𝑓: 2



→2,, where



n is a Boolean lattice of binary vectors of length n

(this generalization was proposed by T. Taran).

Suppose the subject has no plans. Then the

simplest model of the subject, when they readiness for

action to choose only under the pressure of the

external world and psychological attitude, has the

form:

𝑋



𝑓



𝑥



,𝑥





𝑥



→𝑥



(4)

This expression is 0 only if 𝑥



= 1, 𝑥



= 0. This

means that having no plans, the subject chooses the

negative option in the only case: if the pressure of the

outside world inclines one to choose the negative

option, and the subject imagines this pressure as

positive so the expectations regarding the situation

are overestimated. A subject with no intention is

called a primal subject, and the choice made at the

second stage of reflection is called a primal choice.

The primitive subject is not aware of their own ideas

about the world, does not have a model of the world

that generates intentions and changes behavior.

3 RESULTS AND DISCUSSION

Within this work, the variable 𝑥



from (2) and (3)

correlates with the needs and interests of the manager

described in formula (1). And the formula (1) itself is

the formula of intentions.

Within the considered model of choice, the

objective function of a manager in different cultural

environments corresponds to the variable 𝑥



formula (2). Note that the formulae of the remaining

agents and limited resources are constraints

(environmental pressure), and our ideas about the

norm correspond to the variable 𝑥



That is, at the first level of reflexivity, the variable

𝑥



corresponds to the assigning stereotypes of the

subject's behavior (at this level, other subjects and

objects do not differ but are perceived as objects of

consumption). At the second level of reflexivity, the

variable 𝑥



corresponds to the expectation from

interaction with other agents (here already other

subjects and objects differ in the sense that the subject

agrees with them or enters into opposition).

At the first and second levels, any agent does not

yet have intentions, since at the first level there is still

no normative activity, there are only consumer

stereotypes, and at the second level we also have

ideas about other subjects and interaction with them.

Thus, the agent’s readiness for action is described by

formula (4).

Starting from the third level, the agent's readiness

for action is described by formula (3), which includes

the variable 𝑥



. Here the objective function of the

manager already arises, which is intention. At this

level of reflexivity, the knowledge variable includes

only the agent's subjective idea of the previous norm,

which agent changes during the reflexive process.

This means that the variable 𝑥



is an argument 𝑥



, in

other words, there is a functional dependence of 𝑥



𝑥



, because the agent uses the relationship between

the new objective function and the norm of activity.

At the fourth level, the agent has an idea not only

of the previous unrealizable norm but also of an

abstract description of the situation, that is, a concept

that includes scientific concepts (theories). In this

case, the agent has the opportunity to choose the

objective function, albeit so far without criteria.

At the fifth level, the subject introduces into the

element of subjective knowledge not only ideas about

an executable norm and an abstract description of the

situation but also value grounds, which, in addition to

an abstract description of the situation, are applicable

to the entire class of being of a given agent, they are

universal, describe wants and certainly motivate

agents to take action. Thus, it is only at this level that

the agent's knowledge includes abstract idea of own

desires. This element becomes a criterion for

choosing possible norms, and not random but

conscious. In other words, thanks to this element, the

choice from possible conceptual norms is carried out

consciously, and not accidentally.

At the sixth level, the subject is able to think of

their own wants (at the abstract level) and develops

technologies for changing the existing reality. Hence

the agents can assess their moral standards, treat them

critically, and for this, they need to generate a new

logic.

4 CONCLUSIONS

The inclusion of game-theoretic models in reflexive

logic made it possible to come to the conclusion that

a change in the objective function of an agent is

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possible starting from the third level of development

of reflexive abilities - the level of the problem. This

process can be described within the framework of

logical-reflexive models considered in the works of

V. Lefebvre and T. Taran.

Thus, models of reflexive logic provide a broader

scope of research and more accurate tools for

modelling reflexive processes. This confirms the

statement of V. Lefebvre, that a reflexive player can

“outplay” a player using the classical maximin, and

game-theoretic models are a special case of logical-

reflexive models in which the objective functions are

unchanged.

ACKNOWLEDGEMENTS

The reported study was funded by RFBR, project

number 20-010-00869.

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