Modeling of economic systems using game theory

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Ivan Franko National University of Lviv

Modeling of economic systems using game theory

Mariana Vdovyn Doctor of Philosophy, Associate Professor, Department of Statistics

Larysa Zomchak Doctor of Philosophy, Associate Professor, Department of Economic Cybernetics

Taras Panchyshyn Doctor of Philosophy, Associate Professor, Department of Statistics

Lviv, Ukraine

Abstract

The research paper deals with the problems of evaluation, analysis and modeling of economic systems. The article gives the description of a game theory as one of the modeling methods. Decision-making rules and criteria, including Hurwicz, Wald, Hodges-Lehmann, Bayes-Laplace, Savage, as well as the minimum dispersion criteria, are characterized. The analysis of the economic system was carried out at the micro and macro levels. At the micro level, game theory modeling of business entities of various types of economic activity was used according to economic risk indicators, at the macro level, an analysis of the distribution of the countries of Central and Eastern Europe according to indicators of foreign economic activity was carried out using game theory methods.

Keywords: game theory, modeling, single-player games, economic system, economic indicator, Hurwicz's rule, Wald's rule, Hodges-Lehmann rule, Savage criterion, Bayes-Laplace rule.

Introduction

Formulation of the problem. Modeling of complex economic systems involves the use of economic-mathematical and statistical modeling methods. The variety of methods and approaches is very large, and it is quite difficult to choose one, especially if it is necessary to study phenomena and processes at the micro and macro levels simultaneously. However, there is no doubt about the need to choose a specific method of analysis for the relevant economic system.

Analysis of recent research and publications. A large number of scientific works are devoted to the study and modeling of complex economic systems, in particular, Morton A. and Suarez M. in [1] describe in detail various types of models, and Baranovska L. and Bukovskiy O. [2] offer game-theoretic methods of modeling. In [3] the indicators of the marketing strategies efficiency were considered, in [4] scientific solutions regarding the identification of influence factors in the economic models were proposed, in [5] statistical estimation and analysis of economic processes was carried out, and in [6] the evolutionary model and simulation research of collaborative innovation network, and complex systems modeling in Economics and Finance, were offered. Game theory for modeling various economic systems is used in [8-11].

The overwhelming majority of scholars study and model special objects of research and, this is correct, but it is also important to describe, analyze and carefully research special modeling methods, such as game theory methods, which make it possible to model economic processes at different levels of research and different economic systems .

Highlighting previously unresolved parts of the overall problem. The different process of functioning of economic systems necessitates the use of different modeling methods. Game-theoretic methods of modeling, in particular games with nature, require special attention, since any economic system functions in conditions of certain uncertainty and risk. Therefore, it is important to focus on those methods that involve a random component, and how nature acts randomly and unpredictably.

Formulation of the goals of the article (statement of the task). The purpose of the research is the analysis of various economic systems, in particular at the micro- and macro-levels, using game theory methods.

Research methods. The research paper uses statistical games as one of the game theory methods for modeling complex economic systems. Special attention is paid to such rules and criteria as Hurwicz, Hodges-Lehmann, Wald and Bayes-Laplace.

Presentation of the main research material

Game theory is a theory of mathematical models of decision-making under conditions of conflict or uncertainty. Single-player games, also called statistical games, are matrix games in which one player is a person or a group of people who share a common goal, with player A being the statistician and player B being “nature”. Generally, the other opponent could be "nature" or some other system that does not have a capability to choose responses to the actions of player A. “Nature” is considered as a set of external conditions under which the statistician must make decisions. Nature or the economy is indifferent to winning and does not try to mislead or exploit the bad moves of statistics. Player can use m strategies A_lfA₂,....,A_m, nature can realize n different states B₁,B₂,....,B_n. Statisticians can know the probabilities qj with which nature realizes its states Pj. Acting against nature, the statistician can use both pure strategies A_t and mixed strategies 5^= (p_lfp₂,...., Pm)- If the statistician has the opportunity to quantitatively estimate (by the value a,-_?) the consequences of applying each of his pure strategies Ai at any state of nature Pj, then the game can be defined by a payment matrix (table 1).

Table 1 Schematic representation of the payment matrix

Elements a_tj are the winnings of player A, which are not losses of nature B . When choosing the optimal strategy, different criteria and rules are used. game wald dispersion

A systematized list of decision-making criteria and rules based on single-player games and their brief characteristics are given in Table 2.

Table 2 Characteristics of decision-making rules and criteria

Source: authors' construct

We believe that the use of a combination of various criteria and rules will provide an opportunity to correctly analyze the functioning of economic systems.

The analysis will be carried out at the micro and macro levels. First, we will analyze the distribution of various types of economic activity according to certain economic indicators of enterprise activity, and then we will analyze the distribution of countries according to the indicators of foreign trade. At the micro level, we will consider the methods of game modeling in terms of types of economic activity, since it makes it possible to describe in detail the functioning of the economic system, namely the activity of economic entities of Ukraine, as well as the patterns of its development, the relationships between the elements of this system, etc.

Let's consider the states of nature and alternatives, which are represented by four economic indicators of the activity of enterprises and organizations of Ukraine and types of economic activity.

So, the alternatives are presented - by types of economic activity according to KVED [12]: x₁ -- agriculture, forestry and fishing, x₂ - industry, *₃ - construction, *₄ - wholesale and retail trade; repair of motor vehicles and motorcycles, *₅ - transport, warehousing, postal and courier activities, *₆ - temporary accommodation and catering; x₇ - information and telecommunications, *₈ - financial and insurance activities, x₉ - real estate transactions, x₁₀ - professional, scientific and technical activities;*!! - activities in the field of administrative and auxiliary services, x₁₂ - education, *13 - health care and provision of social assistance, *_i4 - art, sports, entertainment and recreation and x₁₅ - provision of other types of services.

The states of nature are indicators that characterize financial, investment and production activities, such as: y₄ - financial result (balance) before taxation, UAH million, y₂ - net profit (loss), UAH million, y₃ - capital investment for by types of economic activity, UAH million, y₄ - volume of manufactured products, UAH thousand. Obviously, in order to make decisions in conditions of uncertainty the input information is advisable to set in matrix form, the rows of which correspond to possible alternatives (types of economic activity), and the columns correspond to states of “nature” (economic indicators of enterprise activity) (Table 3).

Table 3. Value of states of nature for each of the alternatives in 2020

*Source: authors' construct based on the data of the State Statistics Service [13]

Сertamly, we standardize the data. Standardization and norming are described in [14]. In the article, we standardize: as the ratio, which shows the difference between the actual and average values m numerator and the root-mean-square deviation in denominator. Standardized information on types of economic activity according to the above indicators in 2020 is presented by matrix R:

R=

Fig. 1 Matrix of standardized data on organizations of various types of economic activity according to indicators of economic risk

Each alternative and each state of nature corresponds to a result (consequence), which determines the gain (or loss) when choosing a given alternative and implementing a given state.

To select the best alternatives, we will use various decision-making criteria and rules, such as Bayes-Laplace, Hurwicz, Wald, Hodges-Lehmann.

Based on the calculated four risk assessment criteria, an integral assessment can be made (Table 4). First, we will rank the types of economic activity according to economic indicators in Ukraine for each criteria and rules separately. In general, by the method of the sum of coefficients, let's sum up the standardized values for each type of activity according to four criteria and rules, we will get a total score. Having a total score for each type of activity, we will calculate an integral assessment (final rating).

Table 4 Integrated assessment of the activities of organizations by types of economic activity in Ukraine