Game theory is applied to. Mathematical game theory. Examples of recording and solving games from life
This article discusses the application of game theory in economics. Game theory is a branch of mathematical economics. It develops recommendations on the rational action of the participants in the process when their interests do not coincide. Game theory helps businesses make the best decision in a conflict situation.
- Active operations of commercial banks and their accounting
- Improving the formation of a capital repair fund in apartment buildings
- Legal regulation of the issues of assessing the quality of public (municipal) services provided in Russia
Game theory and economics are inextricably linked, as game theory problem solving methods help determine the best strategy for various economic situations. So how is the concept of "game theory" characterized?
Game theory is a mathematical theory of decision making under conflict conditions. Game theory is an important part of operations research theory that studies the issues of decision making in conflict situations.
Game theory is a branch of mathematical economics. The goal of game theory is to develop recommendations for the rational action of the participants in the process when their interests do not coincide, that is, in a conflict situation. The game is a model of a conflict situation. The players in the economy are the partners who take part in the conflict. The result of conflict is win or loss.
In general, the conflict takes place in different areas of human interest: in economics, sociology, political science, biology, cybernetics, military affairs. Most often, game theory and conflict situations are applied in economics. For each player, there is a specific set of strategies that the player can apply. Intersecting, the strategies of several players create a certain situation where each player gets a certain result (win or lose). When choosing a strategy, it is important to consider not only getting the maximum gain for yourself, but also the possible steps of the enemy, and their impact on the situation as a whole.
In order to improve the quality, as well as the efficiency of economic decisions made in the conditions of market relations and uncertainty, game theory methods can be reasonably applied.
In economic situations, games may have complete information or incomplete information. Most often, economists are faced with incomplete information to make decisions. Therefore, it is necessary to make decisions in conditions of uncertainty, as well as in conditions of a certain risk. When solving economic problems (situations), they usually face one-move and multi-move games. The number of strategies can be finite or infinite.
Game theory in economics mainly uses matrix or rectangular games, for which a payoff matrix is compiled (Table 1).
Table 1. The payoff matrix of the game
This concept should be defined. The payoff matrix of the game is a matrix that shows the payment of one player to another, provided that the first player chooses strategy Ai, the second - Bi.
What is the goal of solving economic problems with the help of game theory? To solve an economic problem is to find the optimal strategy for the first and second players and find the price of the game.
Let's solve the economic problem, compiled by me.
In city G, there are two competing companies (Sladkiy Mir and Sladkoezhka) that are engaged in the production of chocolate. Both companies can produce milk chocolate and dark chocolate. Let's designate the strategy of the company "Sweet world" as Аi, the company "Sweet tooth" - Вi. We calculate the efficiency for all possible combinations of the strategies of the companies "Sweet world" and "Sweet tooth" and build a payment matrix (Table 2).
Table 2. The payoff matrix of the game
This payoff matrix does not have a saddle point, so it is solved in mixed strategies.
U1 \u003d (a22-a21) / (a11 + a22-a21-a12) \u003d (6-3) / (5 + 6-3-4) \u003d 0.75.
U2 \u003d (a11-a12) / (a11 + a22-a21-a12) \u003d (5-4) / (5 + 6-3-4) \u003d 0.25.
Z1 \u003d (a22-a12) / (a11 + a22-a21-a12) \u003d (6-4) / (5 + 6-3-4) \u003d 0.4.
Z2 \u003d (a11-a21) / (a11 + a22-a21-a12) \u003d (5-3) / (5 + 6-3-4) \u003d 0.6.
Game price = (a11*a22-a12*a21) / (a11+a22-a21-a12) = (5*6-4*3) / (5+6-3-4) = 4.5.
We can say that the company "Sladkiy Mir" should distribute the production of chocolate as follows: 75% of the total production should be given to the production of milk chocolate, and 25% to the production of dark chocolate. The Sladkoezhka company should produce 40% milk chocolate and 60% bitter chocolate.
Game theory deals with decision-making in conflict situations by two or more reasonable opponents, each of which seeks to optimize their decisions at the expense of others.
Thus, in this article, the application of game theory in economics was considered. In economics, there are often moments when it is necessary to make an optimal decision, and there are several options for making decisions. Game theory helps to make decisions in a conflict situation. Game theory in economics can help determine the optimal output for the enterprise, the optimal payment of insurance premiums, etc.
Bibliography
- Belolipetsky, A. A. Economic and mathematical methods [Text]: textbook for students. Higher Proc. Institutions / A. A. Belolipetsky, V. A. Gorelik. - M.: Publishing Center "Academy", 2010. - 368 p.
- Luginin, O. E. Economic-mathematical methods and models: theory and practice with problem solving [Text]: study guide / O. E. Luginin, V. N. Fomishina. - Rostov n / D: Phoenix, 2009. - 440 p.
- Nevezhin, V.P. Game Theory. Examples and tasks [Text]: textbook / V. P. Nevezhin. – M.: FORUM, 2012. – 128 p.
- Sliva, I. I. Application of the game theory method for solving economic problems [Text] / I. I. Sliva // Proceedings of the Moscow State Technical University MAMI. - 2013. - No. 1. - S. 154-162.
BELARUSIAN STATE UNIVERSITY
FACULTY OF ECONOMICS
CHAIR…
Game theory and its application in economics
course project
2nd year student
departments "Management"
scientific adviser
Minsk, 2010
1. Introduction. page 3
2. Basic concepts of game theory p.4
3. Presentation of games page 7
4. Types of games p.9
5. Application of game theory in economics p.14
6. Problems of practical application in management p.21
7. Conclusion p.23
List of references page 24
1. INTRODUCTION
In practice, it often becomes necessary to coordinate the actions of firms, associations, ministries and other project participants in cases where their interests do not coincide. In such situations, game theory allows you to find the best solution for the behavior of participants who are obliged to coordinate actions in the event of a conflict of interests. Game theory is increasingly penetrating the practice of economic decisions and research. It can be viewed as a tool to help improve the efficiency of planning and management decisions. This is of great importance when solving problems in industry, agriculture, transport, trade, especially when concluding contracts with foreign partners at any level. Thus, it is possible to determine scientifically based levels of retail price reduction and the optimal level of commodity stocks, solve the problems of excursion services and the selection of new lines of urban transport, the task of planning the procedure for organizing the exploitation of mineral deposits in the country, etc. The task of choosing land plots for agricultural crops has become a classic. The method of game theory can be used in sample surveys of finite populations, in testing statistical hypotheses.
Game theory is a mathematical method for studying optimal strategies in games. The game is understood as a process in which two or more parties participate, fighting for the realization of their interests. Each side has its own goal and uses some strategy, which can lead to a win or a loss - depending on the behavior of other players. Game theory helps to choose the best strategies, taking into account ideas about other participants, their resources and their possible actions.
Game theory is a branch of applied mathematics, more precisely, operations research. Most often, the methods of game theory are used in economics, a little less often in other social sciences - sociology, political science, psychology, ethics and others. Since the 1970s, it has been adopted by biologists to study animal behavior and the theory of evolution. It is very important for artificial intelligence and cybernetics, especially with the manifestation of interest in intelligent agents.
Game theory has its origins in neoclassical economics. The mathematical aspects and applications of the theory were first presented in the classic 1944 book Theory of Games and Economic Behavior by John von Neumann and Oscar Morgenstern.
This area of mathematics has found some reflection in public culture. In 1998, the American writer and journalist Sylvia Nazar published a book about the fate of John Nash, a Nobel laureate in economics and a scientist in the field of game theory; and in 2001, based on the book, the film A Beautiful Mind was made. Some American television shows, such as "Friend or Foe", "Alias" or "NUMB3RS", periodically refer to the theory in their episodes.
A non-mathematical version of game theory is presented in the works of Thomas Schelling, the 2005 Nobel laureate in economics.
Nobel laureates in economics for achievements in the field of game theory are: Robert Aumann, Reinhard Zelten, John Nash, John Harsanyi, Thomas Schelling.
2. BASIC CONCEPTS OF GAME THEORY
Let's get acquainted with the basic concepts of game theory. The mathematical model of a conflict situation is called a game, the parties involved in the conflict are called players, and the outcome of the conflict is called a win. For each formalized game, rules are introduced, i.e. a system of conditions that determines: 1) options for the players' actions; 2) the volume of information of each player about the behavior of partners; 3) the payoff to which each set of actions leads. Typically, gain (or loss) can be quantified; for example, you can evaluate a loss by zero, a win by one, and a draw by ½.
A game is called a pair game if two players participate in it, and multiple if the number of players is more than two.
A game is called a zero-sum game, or antagonistic, if the gain of one of the players is equal to the loss of the other, i.e., to complete the task of the game, it is enough to indicate the value of one of them. If we denote a - the payoff of one of the players, b - the payoff of the other, then for a zero-sum game b = -a, so it suffices to consider, for example, a.
The choice and implementation of one of the actions provided for by the rules is called the player's move. Moves can be personal and random. A personal move is a conscious choice by a player of one of the possible actions (for example, a move in a chess game). A random move is a randomly chosen action (for example, choosing a card from a shuffled deck). In what follows, we will consider only the personal moves of the players.
A player's strategy is a set of rules that determine the choice of his action for each personal move, depending on the situation. Usually during the game, at each personal move, the player makes a choice depending on the specific situation. However, in principle it is possible that all decisions are made by the player in advance (in response to any given situation). This means that the player has chosen a certain strategy, which can be given in the form of a list of rules or a program. (So you can play the game using a computer). A game is said to be finite if each player has a finite number of strategies, and infinite otherwise.
In order to solve the game or find a solution to the game, one should choose for each player a strategy that satisfies the optimality condition, i.e. one of the players should get the maximum payoff when the other sticks to his strategy. At the same time, the second player should have a minimum loss if the first one sticks to his strategy. Such strategies are called optimal. Optimal strategies must also satisfy the stability condition, i.e., it must be unprofitable for any of the players to abandon their strategy in this game.
If the game is repeated enough times, then the players may not be interested in winning and losing in each particular game, but in the average win (loss) in all games.
The goal of game theory is to determine the optimal strategy for each player. When choosing the optimal strategy, it is natural to assume that both players behave reasonably from the point of view of their interests. The most important limitation of game theory is the naturalness of payoff as a measure of efficiency, while in most real economic problems there is more than one measure of efficiency. In addition, in the economy, as a rule, there are tasks in which the interests of partners are not necessarily antagonistic.
3. Presentation of games
Games are strictly defined mathematical objects. The game is formed by the players, a set of strategies for each player, and an indication of the payoffs, or payoffs, of the players for each combination of strategies. Most cooperative games are described by a characteristic function, while for other types, the normal or extensive form is more often used.
Extensive form
The game "Ultimatum" in extensive form
Games in the extensive or extended form are represented as a directed tree, where each vertex corresponds to a situation where the player chooses his strategy. Each player is assigned a whole level of vertices. Payments are recorded at the bottom of the tree, under each leaf vertex.
The picture on the left is a game for two players. Player 1 goes first and chooses strategy F or U. Player 2 analyzes his position and decides whether to choose strategy A or R. Most likely, the first player will choose U, and the second - A (for each of them these are optimal strategies); then they will receive respectively 8 and 2 points.
The extensive form is very illustrative, it is especially convenient to represent games with more than two players and games with consecutive moves. If the participants make simultaneous moves, then the corresponding vertices are either connected by a dotted line or outlined by a solid line.
normal form
|
Player 2 |
Player 2 |
|
|
Player 1 |
4 , 3 |
–1 , –1 |
|
Player 1 |
0 , 0 |
3 , 4 |
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Normal form for a game with 2 players, each with 2 strategies. |
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Players chose strategies with the maximum result for themselves, but lost, due to ignorance of the other player's move. Usually, normal form represents games in which the moves are made simultaneously, or at least it is assumed that all players do not know what the other participants are doing. Such games with incomplete information will be considered below.
Characteristic formula
In cooperative games with transferable utility, that is, the ability to transfer funds from one player to another, it is impossible to apply the concept of individual payments. Instead, the so-called characteristic function is used, which determines the payoff of each coalition of players. It is assumed that the payoff of the empty coalition is zero.
The grounds for this approach can be found in the book of von Neumann and Morgenstern. Studying the normal form for coalition games, they reasoned that if a coalition C is formed in a game with two sides, then the coalition N \ C opposes it. A game for two players is formed, as it were. But since there are many variants of possible coalitions (namely, 2N, where N is the number of players), the payoff for C will be some characteristic value depending on the composition of the coalition. Formally, a game in this form (also called a TU game) is represented by a pair (N, v), where N is the set of all players and v: 2N → R is the characteristic function.
This form of presentation can be applied to all games, including those without transferable utility. Currently, there are ways to convert any game from normal to characteristic form, but the transformation in the opposite direction is not possible in all cases.
4. Types of games
cooperative and non-cooperative.
The game is called cooperative, or coalition, if the players can unite in groups, taking on some obligations to other players and coordinating their actions. In this it differs from non-cooperative games in which everyone is obliged to play for themselves. Entertaining games are rarely cooperative, but such mechanisms are not uncommon in everyday life.
It is often assumed that cooperative games differ precisely in the ability of players to communicate with each other. In general, this is not true. There are games where communication is allowed, but players pursue personal goals, and vice versa.
Of the two types of games, non-cooperative ones describe situations in great detail and produce more accurate results. Cooperatives consider the process of the game as a whole. Attempts to combine the two approaches have yielded considerable results. The so-called Nash program has already found solutions to some cooperative games as equilibrium situations for non-cooperative games.
Hybrid games include elements of cooperative and non-cooperative games. For example, players can form groups, but the game will be played in a non-cooperative style. This means that each player will pursue the interests of his group, while at the same time trying to achieve personal gain.
In recent years, the importance of game theory has increased significantly in many areas of economic and social sciences. In economics, it is applicable not only to solve general business problems, but also to analyze the strategic problems of enterprises, develop organizational structures and incentive systems.
Already at the time of its inception, which is considered the publication in 1944 of the monograph by J. Neumann and O. Morgenstern "Game Theory and Economic Behavior", many predicted a revolution in economic sciences through the use of a new approach. These predictions could not be considered too bold, since from the very beginning this theory claimed to describe rational decision-making behavior in interrelated situations, which is typical for most current problems in economic and social sciences. Thematic areas such as strategic behavior, competition, cooperation, risk and uncertainty are key in game theory and are directly related to managerial tasks.
Early work on game theory was characterized by simplistic assumptions and a high degree of formal abstraction, which made them unsuitable for practical use. Over the past 10-15 years, the situation has changed dramatically. Rapid progress in the industrial economy has shown the fruitfulness of game methods in the applied field.
Recently, these methods have penetrated into management practice. It is likely that game theory, along with the theories of transaction costs and “patron-agent”, will be perceived as the most economically justified element of organization theory. It should be noted that already in the 80s, M. Porter introduced some key concepts of the theory, in particular, such as “strategic move” and “player”. True, an explicit analysis associated with the concept of equilibrium was still absent in this case.
Fundamentals of game theory
To describe a game, you must first identify its participants. This condition is easily fulfilled when it comes to ordinary games such as chess, canasta, etc. The situation is different with “market games”. Here it is not always easy to recognize all the players, i.e. existing or potential competitors. Practice shows that it is not necessary to identify all the players, it is necessary to identify the most important ones.
Games cover, as a rule, several periods during which players take consecutive or simultaneous actions. These actions are denoted by the term "move". Actions can be related to prices, sales volumes, research and development costs, and so on. The periods during which the players make their moves are called game stages. The moves chosen at each stage ultimately determine the “payoff” (win or loss) of each player, which can be expressed in wealth or money (predominantly discounted profits).
Another basic concept of this theory is the player's strategy. It is understood as possible actions that allow the player at each stage of the game to choose from a certain number of alternative options such a move that seems to him to be the “best answer” to the actions of other players. Regarding the concept of strategy, it should be noted that the player determines his actions not only for the stages that a particular game has actually reached, but also for all situations, including those that may not occur in the course of this game.
The form in which the game is presented is also important. Usually, a normal, or matrix, form and an expanded one, given in the form of a tree, are distinguished. These forms for a simple game are shown in Fig. 1a and 1b.
To establish the first connection with the sphere of control, the game can be described as follows. Two enterprises producing homogeneous products are faced with a choice. In one case, they can gain a foothold in the market by setting a high price, which will provide them with an average cartel profit P K . When entering into a tough competition, both make a profit П W . If one of the competitors sets a high price, and the second sets a low price, then the latter realizes monopoly profit P M , while the other incurs losses P G . A similar situation can, for example, arise when both firms have to announce their price, which cannot subsequently be revised.
In the absence of stringent conditions, it is beneficial for both enterprises to charge a low price. The “low price” strategy is dominant for any firm: no matter what price a competing firm chooses, it is always preferable to set a low price itself. But in this case, firms face a dilemma, since profit P K (which for both players is higher than profit P W) is not achieved.
The strategic combination “low prices/low prices” with the corresponding payoffs is a Nash equilibrium, in which it is unprofitable for any of the players to deviate separately from the chosen strategy. Such a concept of equilibrium is fundamental in resolving strategic situations, but under certain circumstances it still needs to be improved.
As for the above dilemma, its solution depends, in particular, on the originality of the players' moves. If the enterprise has the opportunity to revise its strategic variables (in this case, the price), then a cooperative solution to the problem can be found even without a rigid agreement between the players. Intuition suggests that with repeated contacts of players, there are opportunities to achieve acceptable “compensation”. Thus, under certain circumstances, it is inappropriate to seek short-term high profits through price dumping if a “price war” may arise in the future.
As noted, both figures characterize the same game. Presenting the game in normal form usually reflects “synchronicity”. However, this does not mean “simultaneity” of events, but indicates that the choice of strategy by the player is carried out in the absence of knowledge about the choice of strategy by the opponent. With an expanded form, such a situation is expressed through an oval space (information field). In the absence of this space, the game situation acquires a different character: first, one player should make the decision, and the other could do it after him.
Application of game theory for making strategic management decisions
Examples here are decisions regarding the implementation of a principled pricing policy, entry into new markets, cooperation and the creation of joint ventures, identifying leaders and performers in the field of innovation, vertical integration, etc. The provisions of this theory, in principle, can be used for all types of decisions, if their adoption is influenced by other actors. These persons, or players, need not be market competitors; their role can be sub-suppliers, leading customers, employees of organizations, as well as colleagues at work.
quadrants 1 And 2 characterize a situation where the reaction of competitors does not have a significant impact on the company's payments. This happens when the competitor has no motivation (field 1 ) or opportunities (field 2 ) strike back. Therefore, there is no need for a detailed analysis of the strategy of motivated actions of competitors.
A similar conclusion follows, although for a different reason, for the situation reflected by the quadrant 3 . Here, the reaction of competitors could have a significant effect on the firm, but since its own actions cannot greatly affect the payments of a competitor, one should not be afraid of his reaction. Niche entry decisions can be cited as an example: under certain circumstances, large competitors have no reason to react to such a decision of a small firm.
Only the situation shown in the quadrant 4 (the possibility of retaliatory steps of market partners), requires the use of the provisions of game theory. However, only the necessary but not sufficient conditions are reflected here to justify the application of the base of game theory to the fight against competitors. There are situations when one strategy unquestionably dominates all others, no matter what actions the competitor takes. If we take the drug market, for example, it is often important for a firm to be the first to introduce a new product to the market: the profit of the “pioneer” turns out to be so significant that all other “players” just have to step up innovation activity faster.
The same game situation can also be represented in normal form (Fig. 4). Two states are designated here – “entry/friendly reaction” and “non-entry/aggressive reaction”. It is obvious that the second equilibrium is untenable. It follows from the detailed form that it is inappropriate for a company already established in the market to react aggressively to the emergence of a new competitor: with aggressive behavior, the current monopolist receives 1 (payment), and with friendly behavior - 3. The outsider company also knows that it is not rational for the monopolist start actions to oust it, and therefore it decides to enter the market. The outsider company will not suffer the threatened losses in the amount of (-1).
Such a rational balance is characteristic of a "partially improved" game, which deliberately excludes absurd moves. Such equilibrium states are, in principle, fairly easy to find in practice. Equilibrium configurations can be identified using a special algorithm from the field of operations research for any finite game. The decision maker proceeds as follows: first, the “best” move in the last stage of the game is chosen, then the “best” move in the previous stage is selected, taking into account the choice in the last stage, and so on, until the initial node of the tree is reached games.
How can companies benefit from game theory-based analysis? There is, for example, a case of a conflict of interests between IBM and Telex. In connection with the announcement of the latter's preparatory plans to enter the market, a "crisis" meeting of the IBM management was held, at which measures were analyzed aimed at forcing the new competitor to abandon its intention to penetrate the new market.
Telex apparently became aware of these events. Game theory based analysis showed that the threats of IBM due to high costs are unfounded.
This shows that it is useful for companies to explicitly consider the possible reactions of their partners in the game. Isolated economic calculations, even based on the theory of decision-making, are often, as in the situation described, limited. For example, an outsider company might choose the “no-entry” move if preliminary analysis convinced it that market penetration would provoke an aggressive response from the monopolist. In this case, in accordance with the criterion of the expected cost, it is reasonable to choose the “non-entry” move with the probability of an aggressive response being 0.5.
For the enterprise 1 it would be best if the company 2 abandoned competition. His benefit in this case would be 3 (payments). It is highly likely that the company 2 would win the competition when the enterprise 1 would accept a cut investment program, and the enterprise 2 - wider. This position is reflected in the upper right quadrant of the matrix.
An analysis of the situation shows that equilibrium occurs at high costs for research and development of the enterprise 2 and low enterprises 1 . In any other scenario, one of the competitors has a reason to deviate from the strategic combination: for example, for the enterprise 1 a reduced budget is preferable if the business 2 refuse to participate in the competition; at the same time the enterprise 2 It is known that at low costs of a competitor it is profitable for him to invest in R&D.
An enterprise with a technological advantage may resort to situation analysis based on game theory in order to ultimately achieve an optimal result for itself. By means of a certain signal, it must show that it is ready to carry out large expenditures on R&D. If such a signal is not received, then for the enterprise 2 it is clear that the company 1 chooses the low cost option.
The reliability of the signal should be evidenced by the obligations of the enterprise. In this case, it may be the decision of the enterprise 1 about purchasing new laboratories or hiring additional research staff.
From the point of view of game theory, such obligations are tantamount to changing the course of the game: the situation of simultaneous decision-making is replaced by the situation of successive moves. Company 1 firmly demonstrates the intention to make large expenditures, the enterprise 2 registers this step and has no more reason to participate in the rivalry. The new equilibrium follows from the scenario “non-participation of the enterprise 2 ” and “high costs for research and development of the enterprise 1 ”.
An important contribution to the use of game theory is made by experimental work. Many theoretical calculations are worked out in the laboratory, and the results obtained serve as an impulse for practitioners. Theoretically, it was found out under what conditions it is expedient for two selfish partners to cooperate and achieve better results for themselves.
This knowledge can be used in the practice of enterprises to help two firms achieve a win-win situation. Today, gaming-trained consultants quickly and unambiguously identify opportunities that businesses can take advantage of to secure stable and long-term contracts with customers, sub-suppliers, development partners, and more.
Problems of practical application
in management
However, it should also be pointed out that there are certain limits to the application of the analytical tools of game theory. In the following cases, it can only be used if additional information is obtained.
Firstly, this is the case when enterprises have different ideas about the game they are participating in, or when they are not sufficiently informed about each other's capabilities. For example, there may be unclear information about a competitor's payments (cost structure). If not too complex information is characterized by incompleteness, then it is possible to operate with a comparison of similar cases, taking into account certain differences.
Second, game theory is difficult to apply to many equilibria. This problem can arise even during simple games with simultaneous choice of strategic decisions.
Thirdly, if the situation of making strategic decisions is very complex, then players often cannot choose the best options for themselves. It is easy to imagine a more complex market penetration situation than the one discussed above. For example, several enterprises may enter the market at different times, or the reaction of enterprises already operating there may be more complex than aggressive or friendly.
It has been experimentally proven that when the game is expanded to ten or more stages, the players are no longer able to use the appropriate algorithms and continue the game with equilibrium strategies.
Nor is the principle underlying assumption of the so-called “common knowledge” underlying game theory, by any means. It says: the game with all the rules is known to the players and each of them knows that all players are aware of what the other partners in the game know. And this situation remains until the end of the game.
But in order for an enterprise to make a decision that is preferable for itself in a particular case, this condition is not always required. Less rigid assumptions, such as “mutual knowledge” or “rationalizable strategies”, are often sufficient for this.
In conclusion, it should be emphasized that game theory is a very complex field of knowledge. When referring to it, one must observe certain caution and clearly know the limits of application. Too simple interpretations, adopted by the firm itself or with the help of consultants, are fraught with hidden danger. Because of their complexity, game theory-based analysis and consultations are only recommended for critical problem areas. The experience of firms shows that the use of appropriate tools is preferable when making one-time, fundamentally important planned strategic decisions, including when preparing large cooperation agreements.
experimental economics
And other methods of analysis
Like any other science that is not entirely conventional, institutional economics uses different methods of analysis. These include traditional microeconomic tools, econometric methods, analysis of statistical information, etc. In this section, we briefly consider the application of game theory, experimental economics, and other methods adapted to institutional analysis.
Game theory. Game theory- an analytical method developed after the Second World War and used to analyze situations in which individuals interact strategically. Chess is the prototype of a strategic game, since the result depends on the behavior of the opponent, as well as on the behavior of the player himself. Because of the analogies found between strategic games and forms of political and economic interaction, game theory has received increased attention in the social sciences. Modern game theory begins with the work of D. Neumann and O. Morgenstern "Game Theory and Economic Behavior" (1944, Russian version - 1970). The theory explores the interaction of individual decisions under certain assumptions about decision making under risk, the general state of the environment, the cooperative or non-cooperative behavior of other individuals. Obviously, a rational individual has to make decisions under conditions of uncertainty and interaction. If one individual's gain is another's loss, then it is a zero-sum game. When each of the individuals can benefit from the decision of one of them, then a game with a non-zero sum takes place. The game can be cooperative, when collusion is possible, and non-cooperative, when antagonism prevails. One well-known example of a non-zero-sum game is the Prisoner's Dilemma (PD). This example shows that, contrary to the claims of liberalism, the pursuit of self-interest by the individual leads to a solution less satisfactory than the possible alternatives.
Limit theorem F.I. Edgeworth is seen as an early example of a cooperative game n participants. The theorem states that as the number of participants in a pure exchange economy increases, collusion becomes less useful and the set of possible equilibrium relative prices (the core) decreases. If the number of participants tends to infinity, then there remains only one system of relative prices corresponding to general equilibrium prices.
The concept of an optimal (equilibrium) solution according to Nash is one of the key concepts in game theory. It was introduced in 1951 by the American mathematician John F. Nash.
In this context, it suffices to consider this concept in relation to the game-theoretic model of two persons 25 . In this model, each of the participants has some non-empty set of strategies S i , i= 1, 2. In this case, the choice of specific strategies from among those available to the player is carried out in such a way as to maximize the value of the own payoff function (utility) u i , i= 1, 2. The values of the payoff function are given on the set of ordered pairs of player strategies S one S 2 , whose elements are all possible combinations of the strategies of the players ( s 1 , s 2) (the ordering of the pairs of strategies is that in each of the combinations the strategy of the first player is in the first place, the second - of the second), i.e. u i = u i (s 1 , s 2), i= 1, 2. In other words, the payoff of each player depends not only on the strategy chosen by him, but also on the strategy adopted by his opponent.
The Nash optimal solution is a pair of strategies ( s 1 *, s 2 *), s i *Î S i , i= 1, 2, which has the following property: strategy s 1 * provides the player 1 maximum payoff when player 2 chooses a strategy s 2 *, and symmetrical s 2* delivers the maximum value of the player's payoff function 2 when the player 1 strategy is adopted s one *. A pair of strategies leads to a Nash equilibrium if the choice made by the player 1 , is optimal for the given choice of the player 2 , and the choice made by player 2 is optimal for the given choice of player 1 . The concept of Nash optimality can be generalized in an obvious way to the case of the game n persons. It should be noted that the existence of a Nash equilibrium does not mean that it is Pareto-optimal, and a Pareto-optimal set of strategies does not have to satisfy a Nash equilibrium. In 1994, J. F. Nash, R. Selten, and J. C. Harshani were awarded the A. Nobel Memorial Prize in Economics for their contributions to the development of game theory and its application to economics.
The appeal to this method relies on its sheer power in highlighting the causes and consequences of institutional change. The ability of game theory to help analyze the consequences of changing the rules is undeniable; its power in revealing causes is ambiguous. Any game-theoretic analysis must presuppose a prior definition of the basic rules of the game. Thus, O. Morgenstern wrote in 1968: “Games are described by defining possible behavior within the rules of the game. The rules are in each case unambiguous; for example, in chess certain moves are allowed for specific pieces, but forbidden for others. The rules are also inviolable. When a social situation is viewed as a game, the rules are given by the physical and legal environment within which the actions of individuals take place.
If this view is accepted, then game theory cannot be expected to account for the change in the fundamental rules of the organization of economic, political, and social life: the definition of such rules is obviously a prerequisite for such an analysis.
Coordination game models and prisoners' dilemmas are used to understand the meaning of institutions.
Consider the problem of pure and generalized coordination. The pure coordination game shows that economic agents cannot be guaranteed to realize the mutual benefits of cooperation, even if there is no conflict of interest. In other words, in a situation of "pure" coordination, there is a multiple equilibrium that is equally preferred by each side. In this case, there is no conflict of interest, but there is no guarantee that everyone will strive for the same equilibrium outcome. A well-known example is the choice of which side of the road (right or left) people should drive on (Figure 2.1). This game has two Nash equilibria corresponding to the sets of strategies (left, left) and (right, right). No one has any objections to driving on the right or left in advance, but achieving a coordinated outcome with a large number of negotiators will require high transaction costs. An institution is needed that would fulfill the function of a focal point, i.e. came up with a consensus solution. Such an institution may be the result of common knowledge obtained on the basis of the same type of analysis of the situation, or it may be the state that intervenes to introduce a rule of coordination and reduce transaction costs. In general, institutions perform a coordinating function, reducing uncertainty.
A generalized coordination problem exists if the payoff matrix is such that, at any equilibrium point, none of the players has an incentive to change their behavior given the behavior of other players, but neither player wants any other player to change it. In this case, everyone would prefer a coordinated outcome over an uncoordinated one, but perhaps everyone would want to prefer a particular coordinated outcome (Figure 2.2). For example, two manufacturers BUT And B use different technology X And Y, but want to introduce a national product standard that will cause network externalities. Manufacturer BUT would benefit more if technology became the standard X, and the manufacturer B- technology Y. The payoff is distributed asymmetrically. So the manufacturer BUT(B) would prefer that the standard become X(Y)-technology, but both will prefer any of the coordinated results to the uncoordinated. Transaction costs in this model will be higher than in the previous one (especially with the participation of a large number of parties), since there is a conflict of interests. Replacing private attempts at coordination with government intervention would reduce transaction costs in the economy. Examples are government implementation of technology standards, measurement and quality standards, etc. The generalized coordination model illustrates the importance of not only the coordination function of institutions, but also the distributive one, which determines the way that limits the possible alternatives of the players, and ultimately the effectiveness of interaction.
The Prisoner's Dilemma often cited as an example of the problem of establishing cooperation between individuals. The game is played by two players, two prisoners who are separated by their guards. Everyone has two choices: cooperate, i.e. remain silent, or refuse to cooperate, i.e. betray another. Each must act without knowing what the other will do. Everyone is told that recognition, if the other is silent, leads to freedom. Refusal of recognition in case of betrayal of another means death. If both confess, they will spend several years in prison together. If each of them refuses to confess, he will be briefly arrested and then released. Assuming that prison is preferable to death, and that freedom is the most desired state, the prisoners are faced with a paradox: although they would both prefer not to betray each other and spend a short time in prison, each will be in a better position to betray the other, regardless of the fact that will undertake another. Analytically, the prisoners' ability to connect takes a backseat, as the incentives to betray remain equally strong with or without the connection. Betrayal remains the dominant strategy.
This analysis helps explain why selfish-maximizing agents cannot rationally arrive at or maintain a cooperative outcome (the paradox of individual rationality). It is useful in explaining the ex post breakdown of a cartel or other cooperative arrangement, but does not explain how a cartel or cooperative arrangement is formed. If the prisoners are able to reach an agreement, then the problem disappears: they agree not to betray each other and come to a point where they maximize their joint gains. So it suffices to enter into an agreement that is collectively desirable, but leaves each individually potentially more vulnerable to harm than in the absence of such an agreement. This analysis draws attention to the institutions that, from an individual point of view, can make such agreements less risky.
The theoretical literature distinguishes between the analysis of cooperative and non-cooperative games. As already described, players are able to enter into agreements that bind them. The guarantor of such agreements is implicit. Many game theorists insist that cheating and breaching agreements are common features of human relationships, so such behavior should remain within the strategic space. They try to explain the emergence and maintenance of cooperation in the model of non-cooperative games, especially in the model of an infinitely repeating sequence of PD games. The final sequence of games will not produce a result, because from the moment the dominant strategy in the last game becomes clearly defective, and from the moment it becomes expected, the same will be true for the penultimate game, and so on, until the first game. In infinite series of games, under certain assumptions about the discounting of payoffs, cooperation can appear as an equilibrium strategy. Thus, non-cooperative analysis does not avoid the need to accept the ground rules of the game as part of the description of the strategic space. It simply suggests a different and less restrictive set of rules. Unlike cooperative analysis, agreements can be broken at will. On the other hand, exit from continuous play is limited. Neither approach escapes the need to define the rules of the game before starting the analysis.
One of the most interesting recent advances in PD research has been the organization of tournaments between predetermined strategies to play finitely repeated 2-player PD games. The first of these was organized by Robert Axelrod (described in 1984) and included playing in a sequence of 200 games. Experienced in the DZ participants were offered computer programs, and which then competed with each other.
R. Axelrod informed the players that strategies would not be judged by the number of wins, but by the sum of their points against all other strategies, with three points each for mutual cooperation, one point for mutual defection, and a 5 to 0 win for defection/cooperation. As noted earlier, it is analytically clear that apostasy is the dominant strategy of the last game, and therefore every previous game.
Consider the payoff matrix in PD analyzed by R. Axelrod 27 (Fig. 2.3). Regardless of what the other player does, betrayal provides a higher reward than cooperation. If the first player thinks that the other player will remain silent, then it is more profitable for him to betray ($5>$3). On the other hand, if the first player thinks that the other will betray, it is still more profitable for him to betray himself ($1 is better than nothing). Therefore, temptation inclines to betrayal. But if both betray, then both get less than in the cooperation situation ($1+$1<$3+$3).
|
Second player |
|||
|
cooperates |
|||
|
First Player | |||
|
cooperates | |||
Rice. 2.3. The payoff matrix in the prisoner's dilemma
The Prisoner's Dilemma, a famous problem in economics, shows that what is rational or optimal for one agent may not be rational or optimal for a group of individuals considered together. The selfish behavior of an individual can be harmful or destructive to the group. In repeated PD games, the appropriate strategy is not obvious. To find a good strategy, and tournaments were organized. If winnings were to be obtained strictly on a win-lose basis, then each participant in the tournament would have to offer continuous defection. However, the scoring rules made it clear that the organization of some cooperation could lead to higher overall results. To the surprise of many, the simple tit-for-tat strategy proposed by A. Rapoport won: the player cooperates on the first step and then makes the move that the other player made on the previous step.
Many more players participated in the second tournament, including professionals, as well as those who knew about the results of the first round. The result was another victory for the copy strategy (“tit for tat”).
An analysis of the results of tournaments revealed four properties that lead to a successful strategy: 1) the desire to avoid unnecessary conflict and cooperate for as long as the other does; 2) the ability to challenge in the face of the unprovoked betrayal of another; 3) forgiveness after answering the challenge; 4) clarity of behavior so that the other player can recognize and adapt to the mode of action of the first.
R. Axelrod showed that cooperation can start, develop and stabilize in situations that are otherwise extraordinary, promising nothing good. One might agree that the tit-for-tat strategy is analytically irrational in a finitely repeating game, but empirically it is obviously not. If the tit-for-tat strategy competed with other analytic strategies, all of which consisted of continuous backsliding, it could not win the tournament.
Game theory can be an important tool for studying human interaction under rule-bound circumstances. Because of its ability to study the implications of different institutional arrangements, it can also be useful from a public policy perspective when designing new institutional arrangements. Game theory has been used in the analysis of public goods, oligopoly, cartel, and collusion in product and labor markets. For all its strengths, game theory also has relative weaknesses. Some authors have raised doubts about the application of the prisoner's dilemma model in social science. For example, M. Taylor in 1987 suggested that such games correspond to the circumstances of providing public goods. In 1985, N. Schofield argued that agents must form consistent concepts about the beliefs and desires of other agents, including problems of cognition and interpretation, which are not easy to model 28 . Many economists have pointed out that the use of game theory without reservation can reduce economic activity to a too static pattern. In particular, the Nobel laureate R. Stone wrote in 1948: “The main feature, due to which game theory conflicts with living reality, is that the object of study is limited in time - the game has a beginning and an end. You can't say the same about economic reality. It is precisely in the possibility of isolating the party from the game that the deep divergence of theory and reality lies, and this divergence limits its application” 29 . However, since then, invaluable work has been done to smooth this discrepancy and expand the application of game theory in economics.
Experimental economics. Another methodological approach used to test the postulates of economic theory and related sciences, as well as to explain institutional problems, is experimental economics. The impact of designed institutions on the efficiency of resource allocation cannot always be predicted ex ante. One way to save on ex post costs is to simulate the work of institutes in laboratory conditions.
In general, an economic experiment is a reproduction of an economic phenomenon or process with the aim of studying it under the most favorable conditions and further practical change. Experiments that are carried out in real conditions are called natural or field experiments, and experiments carried out in artificial conditions are called laboratory experiments. The latter often require the use of economic and mathematical methods and models. Natural experiments can be carried out at the micro level (experiments by R. Owen, F. Taylor, on the introduction of cost accounting at the enterprise, etc.) and at the macro level (economic policy options, free economic zones, etc.). Laboratory experiments are artificially reproduced economic situations, some economic models, whose environment (experimental conditions) is controlled by the researcher in the laboratory.
American Economist Roth, since the late 70s. working in the field of experimental economics, notes a number of advantages of laboratory experiments over "field" ones 30 . Under laboratory conditions, the experimenter can have complete control over the environment and the behavior of subjects, while in "field" experiments it is possible to control only a limited number of environmental factors and almost impossible - the behavior of economic subjects. It is precisely because of this that laboratory experiments make it possible to more accurately determine the conditions under which the repetition of individual phenomena can be expected. Moreover, natural experiments are costly and, if they fail, affect the lives of many people.
The field of interest in experimental economics is quite extensive: the provisions of game theory, the theory of industrial markets, the model of rational choice, the phenomenon of market equilibrium, problems of public goods, etc.
For example, let us dwell on the results of a study of the comparative effectiveness of market institutions, which were published by Ch.A. Holt and presented by A.E. Shastitko 31 . The study compares the conclusions of the theoretical and experimental models of the market obtained through controlled experiments. The outcomes of the agents' behavior are measured by the exhaustion ratio of the sum of the potential rents of the buyer and the seller, which corresponds to the efficiency of the exchange. The exhaustion coefficient - the ratio of the actually (experimentally) received rent to the maximum possible value - varies from 0 to 1. The comparison was carried out according to the following forms of the market: a bilateral auction, trading based on price bids from one of the parties, a clearing house, decentralized price negotiations, trading on based on bids followed by negotiations. The most interesting experimental results were obtained by different groups of researchers on the first two forms of the market (Table 2.1).
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INTRODUCTION
Any person all over the world daily performs some actions, makes a choice for himself in something. In order to perform any actions, a person needs to think about their consequences, choose the most correct, rational of all possible decisions. The choice must be made based on the interests of one's own or group, depending on who the decision refers to (an individual or a group, the organization as a whole).
Institutions are created by people to maintain order and reduce the uncertainty of exchange. They provide predictability of human behavior. Institutions allow us to save our mental abilities, since having learned the rules, we can adapt to the external environment without trying to comprehend and understand it. Petrosyan L.A., Zenkevich N.A., Shevkoplyas E.V.: Game theory: textbook. Publisher: BHV, 2012.-p.18.
Institutions are the "rules of the game" in society, or, more formally, the man-made limits that organize relationships between people. Labsker L.G., Yashchenko N.A.: Game theory in economics. Practice with problem solving. Tutorial. Publisher: Knorus, 2014.-p.21. Institutions appear to solve problems that arise from the repeated interaction of people. At the same time, they not only have to solve the problem, but also minimize the resources spent on solving it.
Game theory is a mathematical method for studying optimal strategies in games. The game is understood as a process in which two or more parties participate, fighting for the implementation of their interests. Each side has its own goal and uses some strategy that can lead to a win or a loss - depending on their behavior and the behavior of other players. Game theory helps to choose the most profitable strategies, taking into account some factors:
1. considerations about other participants;
2. resources of participants;
3. the intended actions of the participants.
In game theory, it is assumed that the payoff functions and the set of strategies available to each of the players are well known, i.e. each player knows his payoff function and the set of strategies available to him, as well as the payoff functions and strategies of all other players, and in accordance with this information forms his behavior.
The relevance of the topic lies in a wide range of applications of game theory in practice (biology, sociology, mathematics, management, etc.). Specifically, in the economy - at such moments when the theoretical foundations of the theory of choice in classical economic theory do not work, which, for example, is that the consumer makes his choice rationally, he is fully aware of the situation in this market and about this particular product.
CHAPTER 1. THEORETICAL FOUNDATIONS OF GAME THEORY
1.1 GAME THEORY CONCEPT
As mentioned above, game theory is a branch of mathematics that studies formal models for making optimal decisions in a conflict. At the same time, conflict is understood as a phenomenon in which various parties participate, endowed with various interests and opportunities to choose actions available to them in accordance with these interests. Each of the parties has its own goal and uses some strategy, which can lead to a win or a loss - depending on the behavior of other players. Game theory helps to choose the best strategies, taking into account ideas about other participants, their resources and their possible actions.
Game theory has its origins in neoclassical economics. The mathematical aspects and applications of the theory were first presented in the classic 1944 book by John von Neumann and Oskar Morgenstern, Game Theory and Economic Behavior.
The game is a simplified formalized model of a real conflict situation. Mathematically, formalization means that certain rules for the actions of the parties in the course of the game have been developed: options for the actions of the parties; the outcome of the game with this variant of action; the amount of information each party has about the behavior of all other parties.
Situations in which the interests of two parties collide and the result of any operation carried out by one of the parties depends on the actions of the other party are called conflict situations.
The player is one of the parties in the game situation. The player's strategy is his rules of action in each of the possible situations of the game. Dominance in game theory is a situation in which one of the strategies of a certain player gives a greater payoff than another, for any actions of his opponents. Protasov I.D. Game theory and operations research: textbook. allowance. - M.: Helios ARV, 2013.-S.121.
The focal point is the equilibrium in the coordination game, chosen by all participants in the interaction on the basis of a common knowledge that helps them coordinate their choice. The focal point concept was introduced by the 2005 Nobel Prize-winning economist Thomas Schelling in a 1957 article that became the third chapter of his famous book The Strategy of Conflict (1960).
If there is a strictly dominant strategy for one of the players, he will use it in any of the Nash equilibria in the game. If all players have strictly dominant strategies, the game has a unique Nash equilibrium. However, this equilibrium will not necessarily be Pareto efficient, i.e. disequilibrium outcomes can provide all players with a greater payoff. A classic example of this situation is the Prisoner's Dilemma game. A Nash equilibrium is a set of strategies (one for each player) such that none of the players has an incentive to deviate from their strategy. A situation will be Pareto efficient if neither player can improve their position without making the other player worse off.
We should also mention the Stackelberg equilibrium. Stackelberg equilibrium is a situation where none of the players can increase their payoff unilaterally, and decisions are made first by one player and become known to the second player. Unlike the dominant strategy equilibrium and the Nash equilibrium, this kind of equilibrium always exists.
Interpretation of game theory can be carried out in two ways: matrix and graphic. The matrix method will be depicted below, where situations leading to the emergence of institutions will be considered.
For an example of a graphic representation, consider the following situation, where there is one pasture for cows to graze. Now let's ask the question: for what number of cows, n, would the use of this pasture be optimal? In accordance with the marginal principle of optimization, which assumes the equation of marginal cost and marginal income, it should be answered that the optimal number of cows will be the one at which the value of the marginal product from grazing the last cow, VMP, will be equal to the cost of one cow, c. In the conditions of private ownership of this pasture, this principle would be observed, since the individual owner would compare the benefits and costs associated with each additional cow, and would stop at the number of them, Ep, in which the possibilities of obtaining a positive rent from grazing cows on the pasture , Rp, would be exhausted, and, accordingly, the maximum of this rent would be reached (Fig. 1). This is summarized in the equation below, which, while respecting the margin principle, maximizes the difference between the value of the total product, VTP, and the total cost, i.e. the cost of a cow times the number of cows.
VMP (n*) = c maxn VTP (n) - cn (1)
Figure 1. Graph of the value of the marginal and average grazing of cows
However, in the conditions of free access to the pasture, i.e., the absence of exclusive rights to it, the marginal optimization principle will not be observed and the number of cows on the pasture will exceed the optimal value, Ep, and reach the point of equality of the value of the average cow grazing product, VAP, and the cost of a cow . As a result, there will be a new equilibrium number of cows in free access conditions, Ec. In this case, the positive rent, Rp, created by grazing cows until reaching their optimal number, Еp, will be spent on additional cows and, when the point Еc is reached, will become equal to zero as a result of the accumulation of negative rent equal to it in absolute value. This is summarized in the equations below:
VTP(n")/n"=c?VTP(n")-cn"=0;
1.2 VARIETY OF SITUATIONS AND AREAS OF HUMAN LIFE IN WHICH GAME THEORY IS APPLICABLE
In life, there are many examples of a clash of opposite sides, taking the form of a conflict with two acting parties pursuing opposing interests.
Such situations arise, for example, when it comes to trust. Compliance of the counterparty's actions with expectations becomes especially important in situations where the risk of decisions made by the individual is determined by the actions of the counterparty. Game theory models are the best illustration of what has been said: the choice of a particular strategy by a player depends on the actions of another player. Trust consists in "the expectation of certain actions of others that influence the choice of the individual, when the individual must begin to act before the actions of others become known." Let us emphasize the connection of transactions in the market with trust in a depersonalized form (trust as a norm governing relations between individuals), since the circle of participants in transactions should not be limited to personally familiar people. The following model helps to make sure that trust exists in a depersonalized form for the implementation of the simplest market transaction using an advance payment (Fig. 2).
Figure 2
Let's assume that the buyer is opposed by many sellers and he knows from his previous business experience the probability of deception (1 - p). Let's calculate the value of p so that the transaction takes place, i.e., "make an advance payment" is an evolutionarily stable strategy.
EU (make an advance payment) = 10p - 5 (1 - p) = 15p - 5,
EU(do not prepay) = 0.15p - -5 > 0, p>1/3.
In other words, when the level of buyer's confidence in sellers is less than 33.3%, transactions with prepayment under the given conditions become impossible. In other words, p=1/3 is the critical, minimum required level of confidence.
To generalize the results, we replace the specific values of the buyer's gain (10) and loss (--5) with the symbols G and L. Then, with the previous structure of the game, the transaction will take place at
the higher the loss relative to the gain, the higher should be the level of trust between the participants in the transaction. James Coleman depicted the dependence of the need for trust on the terms of the deal being concluded as follows (Fig. 3).
Figure 3
Estimated data on the minimum required level of confidence are confirmed empirically. Thus, the level of depersonalized trust in countries with developed market economies, measured by answering the question: “Based on your personal experience, do you think that people around you can be trusted? ”, was 94% in Denmark 24, 90 in Germany, 88 in Great Britain, 84 in France, 72 in northern Italy and 65% in the south. The low level of trust in the south of Italy, where the mafia is traditionally strong, is indicative. It is no coincidence that one of the mafia researchers, D. Gambetta, explains its emergence by a critically low level of trust in the southern regions of Italy and, consequently, by the need for a substitute for trust, which takes the form of the intervention of a “third party” trusted by both participants in the transaction.
Another striking example of game theory is contracts between an investor and the state for the development of mineral deposits.
To illustrate this example, let's take a contract for the sale of chairs, given that the presence of wired treasures in them is in question. We will depict an example taking into account the fact that, within the framework of game theory, factors external to the intentions of the parties to the contract are taken into account by introducing a third player, “nature”, into the game with two participants (Fig. 4).
Figure 4
As follows from the presentation of the game in expanded form, instead of four outcomes, there are six of them in the game. And if the problem of the dependence of Ostap's payoff on the actions of the stage engineer finds its solution in the presence of any non-zero level of Ostap's confidence, then the problem of the dependence of Ostap's payoff on the presence of treasures in the chairs remains unsolvable, which, however, confirms the ending of the novel.
1.3 POSSIBLE STRATEGIES IN REPEATED GAMES
1. Mixed strategies. When players find themselves in a certain choice situation repeatedly, their interaction becomes significantly more complicated. They can afford to combine strategies to maximize the overall payoff. We will show this with the help of a model that describes the relationship between the Central Bank (CB) and an economic agent in connection with the monetary policy pursued by the Central Bank.
The Central Bank focuses either on a tight monetary policy, seeking to maintain inflation at a fixed level (р0), or on emission and, consequently, an increase in inflation rates (р1). In turn, the economic agent acts on the basis of its inflation expectations re (sets prices for its products, decides on the purchase of goods and services, etc.), which can either be confirmed or not confirmed as a result of the policy pursued by the Central Bank. If p1 > pe, the Central Bank receives profit from seigniorage and inflation tax. If pe = p1, then both the Central Bank losers because of the reduction in seigniorage revenues, and economic agents that continue to bear the burden of the inflation tax. If pe = p0, then the status quo is maintained and no one loses. Finally, if pe > p0, then only economic agents lose: producers - due to the loss of demand for products that have risen unreasonably in price, consumers - due to the creation of unjustified stocks.
In the proposed model, with a single interaction, agents do not have dominant strategies, and there is no Nash equilibrium. With repeated repeated interactions, and it is precisely such interactions that are characteristic of real situations, both participants can use either one or the other strategy at their disposal. Does alternating strategies in a certain sequence allow players to maximize their utility, i.e., achieve Nash equilibrium in mixed strategies: an outcome in which no player can increase his payoff by unilaterally changing his strategy? Assume that the Central Bank pursues a tight monetary policy with probability Р1 (in P1 % of cases), and with probability (1 - Р1) - an inflationary policy. Then, when an economic agent chooses non-inflationary expectations (pe = p0), the Central Bank can expect to receive a gain equal to
theory game strategy
EU(CB) = Р1 0+,
1 (1 - P1) = 1- -P1
In the case of inflationary expectations of an economic agent, the gain of the Central Bank will be
EU(CB) = Р10 + (1 - Р1)(-2) = 2Р1 - 2.
Now let's assume that an economic agent has non-inflationary expectations with probability P2 (in P2% of cases), and inflationary expectations with probability (1 - P2). Hence, the expected utility of the Central Bank will be
EU(CB) = Р2(1 - Р1) + (1 - Р2)(2Р1-2) = =ЗР2-ЗР1 Р2+2Р1 - 2 (Fig. 5).
Figure 5
Similar calculations for an economic agent will give
EU (e.a.) = P1(P2-1) + (1 - P1)(-P2-2) = 2P1P2 + P1-P2-2.
If we rewrite these expressions in the following form
EU(CB) = Pl(2-3P2) + ЗР2-2
EU(e.a.)==P2(2P1-1) + P1-2,
it is easy to see that when
the gain of the Central Bank does not depend on its own policy, and when
the economic agent's payoff does not depend on his expectations.
In other words, the Nash equilibrium in mixed strategies will be the formation by an economic agent of non-inflationary expectations in 2/3 of the cases and the implementation of the Central Bank in half of the cases of a tight monetary policy. The found equilibrium is achievable provided that economic agents form expectations in a rational way, and not on the basis of inflationary expectations in the previous period, adjusted for the forecast error of the previous period8. Consequently, changes in the policy of the Central Bank affect the behavior of economic agents only to the extent that they are unexpected and unpredictable. The strategy of the Central Bank in 50% of cases to pursue a tight monetary policy, and in 50% - soft is the best way to create an atmosphere of unpredictability.
2. Evolutionary-stable strategy. An evolutionarily stable strategy is one such that if the majority of individuals use it, then no alternative strategy can supplant it through natural selection, even if the latter is more Pareto efficient.
A kind of repetitive games are situations when an individual repeatedly finds himself in a certain situation of choice, but his counterparty is not constant, and in each period the individual interacts with a new counterpart. Therefore, the probability of a counterparty choosing one or another strategy will depend not so much on the configuration of the mixed strategy, but on the preferences of each of the counterparties. In particular, it is assumed that out of the total number N of potential counterparties, n (n/N%) always choose strategy A, and m (m/N%) - strategy B. This creates the prerequisites for achieving a new type of equilibrium, evolutionarily stable strategies. An Evolutionary Stable Strategy (ESS) is one in which if all members of a particular population use it, then no alternative strategy can displace it through the mechanism of natural selection. Consider, as an example, the simplest variant of the coordination problem: two cars passing on a narrow road. It is assumed that in a given area, left-hand and right-hand traffic standards are equal (or the Rules of the Road are simply not always followed). Car A is moving towards several cars with which it needs to pass. If both cars take to the left, entering the left side of the road in the direction of travel, then they part without problems. The same thing happens if both cars take to the right. When one car takes to the right, and the second - to the left and vice versa, they will not be able to part (Fig. 6).
Figure 6
So, motorist A knows the approximate percentage of motorists B who systematically take to the left (P), and the percentage of motorists B who take to the right (1 - P). The condition for the “take right” strategy to become evolutionarily stable for motorist A is formulated as follows: EU(right) > EU(left), or
0P+ 1(1 - P) > 1P+ 0(1 - P),
whence R< 1/2. Таким образом, при превышении доли автомобилистов во встречном потоке, принимающих вправо, уровня 50% эволюционно-стабильной стратегией становится «принять вправо» -- сворачивать на правую обочину при каждом разъезде.
In general, the requirements for an evolutionarily stable strategy are written as follows. Strategy I, used by counterparties with probability p, is evolutionarily stable for the player if and only if the following conditions are satisfied
EU(I, p) > EU(J, p),
which is identical
pU(I, I) + (l -p)U(I,J)>pU(J,I) + (1 - p)U(J,J) (3)
From what it follows:
U(I, I)> U(J, I)
U(I, I) = U(J, I)
U(I, J) > U(J, J),
where -- U(I, I) is the player's payoff when choosing strategy I, if the counterparty chooses strategy I; U(J, I) is the player's payoff when choosing strategy J, if the counterparty chooses strategy I, etc.
Figure 7
These conditions can also be represented graphically. Let us plot the expected utility of choosing one or another strategy along the vertical axis, and the proportion of individuals in the total population of players choosing both strategies along the horizontal axis. Then we will get the following graph (the values are taken from the model of two cars passing), shown in Fig. 7.
It follows from the figure that both “take left” and “take right” have an equal chance of becoming an evolutionarily stable strategy as long as neither of them covers more than half of the “population” of drivers. If the strategy crosses this threshold, then it will gradually but inevitably crowd out the other strategy and cover the entire population of drivers. The fact is that if the strategy crosses the 50% threshold, it becomes profitable for any driver to use it in maneuvers, which, in turn, further increases the attractiveness of this strategy for other drivers. In strict form, this statement will look like this:
dp/dt = G, G">0 (4)
The main result of the analysis of repeated games is an increase in the number of equilibrium points and, on this basis, the solution of the problems of coordination, cooperation, compatibility and justice. Even in the Prisoner's Dilemma, the transition to repetitive interaction makes it possible to achieve the Pareto optimal result ("deny guilt"), without going beyond the norm of rationality and the prohibition on the exchange of information between players. This is the meaning of the “general theorem”: any outcome that suits an individual individually can become an equilibrium in the transition to the structure of a repeated game. In the situation of the prisoners' dilemma, the equilibrium outcome under certain conditions can be both a simple strategy "not to recognize", and a set of mixed strategies. Among the mixed and evolutionary strategies, we note the following: Tit-For-Two-Tats - start with a denial of guilt and admit guilt only if the counterparty admitted guilt in the previous two periods in a row; DOWING is a strategy based on the assumption that the counterparty is equally likely to use the "deny" and "admit" strategies at the very beginning of the game. Further, each denial of guilt on the part of the counterparty is encouraged, and each confession is punished by the choice of the “admit guilt” strategy in the next period; TESTER - start with an admission of guilt, and if the counterparty also admits guilt, then deny guilt in the next period.
CONCLUSION
At the end of the essay, we can conclude that it is necessary to use game theory in modern economic conditions.
In conditions of alternative (choice) it is very often not easy to make a decision and choose this or that strategy. Operations research allows using appropriate mathematical methods to make an informed decision on the appropriateness of a particular strategy. Game theory, which has an arsenal of methods for solving matrix games, allows you to effectively solve these problems by several methods and choose the most effective from their set, as well as simplify the original game matrices.
In the essay, the practical application of the main strategies of game theory was illustrated and the corresponding conclusions were drawn, the most used and frequently used strategies and basic concepts were studied.
LIST OF USED LITERATURE
1. Petrosyan L.A., Zenkevich N.A., Shevkoplyas E.V.: Game theory: textbook. Publisher: BHV, 2012.-212p.
2. Labsker L.G., Yashchenko N.A.: Game theory in economics. Practice with problem solving. Tutorial. Publisher: Knorus, 2014.-125p.
3. Nailbuff, Dixit: Game Theory. The art of strategic thinking in business and life. Publisher: Mann, Ivanov and Ferber, 2015 .- 99p.
4. Oleinik A.N. Institutional economics. Textbook, Moscow INFRA-M, 2013.-78s.
5. Protasov I.D. Game theory and operations research: textbook. allowance. - M.: Helios ARV, 2013.-100s.
6. Samarov K.L. Maths. Teaching aid for the section "Elements of game theory", Resolventa LLC, 2011.-211p.
7. Shikin E.V. Mathematical methods and models in management: textbook. allowance for students ex. specialist. universities. - M.: Delo, 2014.-201s.
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