Summary: Game theory and its practical application. Game theory in economics

Foreword

The purpose of this article is to familiarize the reader with the basic concepts of game theory. From the article, the reader will learn what game theory is, consider brief history game theory, will get acquainted with the basic provisions of game theory, including the main types of games and forms of their presentation. The article will touch upon the classical problem and the fundamental problem of game theory. The final section of the article is devoted to the consideration of the problems of application of game theory for making management decisions and the practical application of game theory in management.

Introduction.

21 century. The age of information, rapidly developing information technology, innovation and technological innovation. But why exactly the information age? Why does information play a key role in almost all processes in society? Everything is very simple. Information gives us invaluable time, and in some cases even the opportunity to get ahead of it. After all, it's not a secret for anyone that in life you often have to deal with tasks in which it is necessary to make decisions in conditions of uncertainty, in the absence of information about the responses to your actions, that is, situations arise in which two (or more) parties pursue different goals, and the results of any action by each of the parties depend on the activities of the partner. Such situations arise every day. For example, when playing chess, checkers, dominoes and so on. Despite the fact that the games are mainly entertainment in nature, by their nature they relate to conflict situations in which the conflict is already embedded in the goal of the game - to win one of the partners. At the same time, the result of each player's move depends on the opponent's retaliatory move. In the economy, conflict situations occur very often and are of a varied nature, and their number is so great that it is impossible to calculate all the conflict situations that arise in the market at least in one day. Conflict situations in the economy include, for example, the relationship between a supplier and a consumer, a buyer and a seller, a bank and a client. In all of the above examples, a conflict situation is generated by the difference in the interests of partners and the desire of each of them to make optimal decisions that realize the set goals to the greatest extent. At the same time, everyone has to reckon not only with their own goals, but also with the goals of the partner, and take into account the decisions unknown in advance that these partners will make. For competent problem solving in conflict situations, scientifically based methods are required. Such methods were developed by the mathematical theory of conflict situations, which is called game theory.

What is game theory?

Game theory is a complex multidimensional concept, so it seems impossible to give an interpretation of game theory using just one definition. Consider three approaches to defining game theory.

1. Game theory is a mathematical method for studying optimal strategies in games. A game is understood as a process in which two or more parties are involved in the struggle for the realization of their interests. Each side has its own goal and uses some strategy that can lead to a win or a loss, depending on the behavior of other players. Game theory helps you choose best strategies taking into account the ideas about other participants, their resources and their possible actions.

2. Game theory is a branch of applied mathematics, more precisely, operations research. Most often, game theory methods are used in economics, a little less often in others. social sciences ah - 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. Game theory is very important for artificial intelligence and cybernetics.

3.One of the most important variables on which the success of the organization depends - competitiveness. Obviously, the ability to predict the actions of competitors is an advantage for any organization. Game theory is a method of modeling the assessment of the impact of a decision made on competitors.

History of game theory

Optimal solutions or strategies in mathematical modeling were proposed as early as the 18th century. The problems of production and pricing in an oligopoly, which later became textbook examples of game theory, were considered in the 19th century. A. Cournot and J. Bertrand. At the beginning of the XX century. E. Lasker, E. Cermelo, E. Borel put forward an idea mathematical theory conflicts of interest.

Mathematical game theory has its origins in neoclassical economics. For the first time, the mathematical aspects and applications of the theory were presented in classic book 1944 by John von Neumann and Oskar Morgenstern Game Theory and Economic Behavior.

John Nash, after graduating from Carnegie Polytechnic Institute with two degrees - bachelor's and master's - entered Princeton University, where he attended lectures by John von Neumann. In his writings, Nash developed the principles of "managerial dynamics." The first concepts of game theory analyzed antagonistic games, when there are losers and winners at their expense. Nash develops methods of analysis in which all participants either win or fail. These situations are called "Nash equilibrium", or "noncooperative equilibrium", in a situation the parties use the optimal strategy, which leads to the creation of a stable equilibrium. It is beneficial for the players to maintain this balance, since any change will worsen their situation. These works of Nash made a significant contribution to the development of game theory, the mathematical tools of economic modeling were revised. John Nash shows that A. Smith's classical approach to competition, when everyone is for himself, is not optimal. More optimal strategies are when everyone tries to do better for himself, doing better for others. In 1949, John Nash wrote a dissertation on game theory, 45 years later he received Nobel prize on economics.

Although game theory originally looked at economic models until the 1950s, it remained a formal theory within mathematics. But since the 1950s. Attempts began to apply the methods of game theory not only in economics, but in biology, cybernetics, technology, and anthropology. During and immediately after World War II, the military became seriously interested in game theory, who saw it as a powerful apparatus for researching strategic decisions.

1960 - 1970 interest in game theory is waning, despite the significant mathematical results obtained by that time. Since the mid-1980s. begins an active practical use of game theory, especially in economics and management. Over the past 20 - 30 years, the importance of game theory and interest has grown significantly, some areas of modern economic theory cannot be stated without the application of game theory.

A major contribution to the application of game theory was the work of Thomas Schelling, 2005 Nobel laureate in economics, "The Strategy of Conflict." T. Schelling examines various "strategies" of behavior of the parties to the conflict. These strategies coincide with the tactics of conflict management and the principles of conflict analysis in conflict management and in the management of conflicts in the organization.

Fundamentals of game theory

Let's get acquainted with the basic concepts of game theory. Mathematical model conflict situation called game, parties to the conflict - players... To describe the game, you must first identify its participants (players). This condition is easily met when it comes about ordinary games like chess, etc. The situation is different with "market games". It is not always easy to recognize all the players here, i.e. current or potential competitors. Practice shows that it is not necessary to identify all the players, it is necessary to find the most important ones. Games usually cover several periods during which players take consecutive or simultaneous actions. Selection and implementation of one of stipulated by the rules action is called move player. The moves can be personal or random. Personal move is a conscious choice by the player of one of the possible actions (for example, a move to chess game). Random move is a randomly chosen action (for example, choosing a card from a shuffled deck). Activities can be related to prices, sales volumes, research and development costs, etc. The periods during which the players make their moves are called stages games. The moves chosen at each stage ultimately determine "payments"(gain or loss) of each player, which can be expressed in material values ​​or money. Another concept of this theory is the player's strategy. Strategy a player is a set of rules that determine the choice of his action for each personal move, depending on the current situation. Usually, during the game, with 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 situation that arises). This means that the player has chosen a certain strategy, which can be set in the form of a list of rules or a program. (This is how you can play the game with a computer). In other words, a strategy is understood as possible actions that allow a player at each stage of the game to choose from a certain number of alternative options such a move that seems to him the "best response" 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 arise in the course of this game. The game is called steam room if two players participate in it, and multiple if the number of players is more than two. For each formalized game, rules are introduced, i.e. a system of conditions that determines: 1) options for players' actions; 2) the amount of information each player has about the behavior of partners; 3) the gain to which each set of actions leads. Typically, the gain (or loss) can be quantified; for example, you can estimate a loss as zero, a gain as one, and a draw as ½. 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., for a complete task of the game, it is enough to indicate the value of one of them. If we denote a- winnings of one of the players, b- the other's payoff, then for a zero-sum game b = -а, therefore it suffices to consider, for example a. The game is called ultimate, if each player has a finite number of strategies, and endless- otherwise. To decide game, or find game solution, one should choose a strategy for each player that satisfies the condition optimality, those. one of the players must receive maximum win when the other sticks to his strategy. At the same time, the second player must have minimal loss if the former sticks to his strategy. Such strategy are called optimal... Optimal strategies must also satisfy the condition sustainability, that is, it should be unprofitable for any of the players to abandon their strategy in this game. If the game is repeated many times, then the players may not be interested in winning and losing in each particular game, but average gain (loss) in all parties. The purpose game theory is to determine the optimal strategies 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.

Cooperative and non-cooperative

The game is called cooperative, or coalition, if players can form groups, taking on some obligations to other players and coordinating their actions. This differs from non-cooperative games in which everyone is obliged to play for themselves. Entertaining games 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 the players pursue personal goals, and vice versa.

Of the two types of games, non-cooperative ones describe situations in the smallest details and provide more accurate results. Cooperatives consider the process of the game as a whole.

Hybrid games include elements of co-op and non-co-op 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.

Symmetrical and asymmetrical

Asymmetrical play

The game will be symmetrical when the players have equal strategies, that is, they have the same payments. In other words, if the players can change places and their winnings for the same moves will not change. Many of the two-player games under study are symmetrical. In particular, these are: "Prisoner's Dilemma", "Deer Hunt". In the example on the right, the game at first glance may seem symmetrical due to similar strategies, but this is not so - after all, the second player's payoff with strategy profiles (A, A) and (B, B) will be greater than that of the first.

Zero-sum and non-zero-sum

Zero-sum games are a special kind of fixed-sum games, that is, those where players cannot increase or decrease the available resources or the fund of the game. In this case, the sum of all winnings is equal to the sum of all losses on any move. Look to the right - the numbers represent payments to the players - and their total in each cell is zero. Examples of such games are poker, where one wins all the bets of others; reverse, where the opponent's pieces are captured; or banal theft.

Many games studied by mathematicians, including the already mentioned "Prisoner's Dilemma", are of a different kind: non-zero-sum games the gain of one player does not necessarily mean the loss of another, and vice versa. The outcome of such a game can be less than or greater than zero. Such games can be converted to zero sum - this is done by introducing fictitious player, which "appropriates" the surplus or makes up for the lack of funds.

Another game with a nonzero sum is trade where every member benefits. This also includes checkers and chess; in the last two, the player can turn his ordinary piece into a stronger one, gaining an advantage. In all these cases, the amount of the game increases. A well-known example where it decreases is war.

Parallel and sequential

In parallel games, the players move at the same time, or at least they are not aware of the choices of others until all will not make their move. In sequential, or dynamic In games, participants can make moves in a predetermined or random order, but at the same time they receive some information about the previous actions of others. This information may even be not quite complete, for example, a player can find out that his opponent from his ten strategies definitely didn't choose fifth, knowing nothing about others.

The differences in the presentation of parallel and sequential games were discussed above. The former are usually presented in normal form, and the latter in extensive.

With complete or incomplete information

Games with complete information constitute an important subset of sequential games. In such a game, the participants know all the moves made up to the current moment, as well as the possible strategies of the opponents, which allows them to predict to some extent the subsequent development of the game. Complete information is not available in parallel games, since they do not know the current moves of the opponents. Most of the games studied in mathematics are with incomplete information. For example, all the "salt" Prisoner's dilemmas lies in its incompleteness.

Examples of games with complete information: chess, checkers and others.

Often the concept of complete information is confused with something similar - perfect information... For the latter, only knowledge of all strategies available to opponents is sufficient, knowledge of all their moves is not necessary.

Games with an infinite number of steps

Real-world games or games studied in economics tend to last the final number of moves. Mathematics is not so limited, and in particular, set theory deals with games that can go on indefinitely. Moreover, the winner and his winnings are not determined until the end of all moves.

The problem that is usually posed in this case is not to find an optimal solution, but to find at least a winning strategy.

Discrete and continuous games

Most of the games studied discrete: they have a finite number of players, moves, events, outcomes, etc. However, these components can be extended to a set of real numbers. Games that include these elements are often referred to as differential games. They are associated with some kind of material scale (usually a time scale), although the events occurring in them may be discrete in nature. Differential games are used in engineering and technology, physics.

Metagames

These are games that result in a set of rules for another game (called target or game object). The purpose of metagames is to increase the usefulness of the set of rules produced.

Game presentation form

In game theory, along with the classification of games, the form of game presentation plays a huge role. Usually, a normal or matrix form is distinguished and an expanded one, given in the form of a tree. These forms for a simple game are shown in Fig. 1a and 1b.

To establish the first connection to the realm of control, the game can be described as follows. Two factories 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 competitive struggle, both receive profit P W. If one of the competitors sets a high price, and the other sets a low price, then the latter realizes the monopoly profit P M, while the other incurs losses P G. A similar situation may, for example, arise when both firms must announce their prices, which cannot be subsequently revised.

In the absence of strict conditions, it is beneficial for both enterprises to set a low price. The "low price" strategy is dominant for any firm: no matter what price the competing firm chooses, it is always preferable to set a low price. But in this case, the firms face a dilemma, since the profit P K (which for both players is higher than the profit P W) is not achieved.

The strategic combination of "low prices / low prices" with the corresponding payments is a Nash equilibrium, in which it is not profitable for any of the players to separately deviate from the chosen strategy. This concept of equilibrium is fundamental in resolving strategic situations, but under certain circumstances it still requires improvement.

As for the above dilemma, its solution depends, in particular, on the originality of the players' moves. If an enterprise has the ability to revise its strategic variables (in this case, price), then it can be found cooperative solution problems even without a rigid agreement between the players. Intuition dictates that with multiple contacts of players, there are opportunities to achieve acceptable "compensation". Thus, under certain circumstances, it is inappropriate to strive for short-term high profits through price dumping if a "price war" may arise in the future.

As noted, both figures characterize the same game. The rendering of the game in normal form normally reflects "synchronicity". However, this does not mean the "simultaneity" of events, but indicates that the choice of strategy by the player is carried out in conditions of ignorance about the choice of strategy by the opponent. In the expanded form, this situation is expressed through the oval space (information field). In the absence of this space, the game situation takes on a different character: first, one player should make a decision, and the other could do it after him.

A classic problem in game theory

Consider classic problem in game theory. Deer hunting is a cooperative symmetric game from game theory that describes the conflict between self-interest and public interest. The game was first described by Jean-Jacques Rousseau in 1755:

"If they hunted a deer, then everyone understood that for this he must remain at his post; but if a hare ran near any of the hunters, then there was no doubt that this hunter, without a twinge of conscience, would chase after him and, having overtaken the prey , very little will lament that in this way he robbed his comrades of prey. "

Deer hunting is a classic example of a supply problem public good when a person is tempted to give in to self-interest. Should the hunter stay with his comrades and bet on a less favorable opportunity to deliver large prey to the entire tribe, or leave his comrades and entrust himself to a more reliable case that promises his own hare family?

A fundamental problem in game theory

Consider a fundamental problem in game theory called the Prisoner's Dilemma.

The Prisoner's Dilemma is a fundamental problem in game theory that players will not always cooperate with each other, even if it is in their best interest to do so. It is assumed that the player (the “prisoner”) maximizes his own gain without caring about the benefit of others. The core of the problem was formulated by Merrill Flood and Melvin Drescher in 1950. The name of the dilemma was given by the mathematician Albert Tucker.

In the prisoner's dilemma, betrayal strictly dominates over cooperation, so the only possible balance is the betrayal of both participants. Simply put, no matter what the other player does, everyone will win more if they betray. Since betrayal is more profitable in any situation than cooperation, all rational players will choose betrayal.

Behaving separately rationally, together the participants come to an irrational decision: if both betray, they will receive in total less gain than if they cooperated (the only equilibrium in this game does not lead to Pareto-optimal solution, i.e. solution that cannot be improved without worsening the position of other elements.). This is the dilemma.

In the repetitive prisoner's dilemma, play occurs intermittently, and each player can “punish” the other for not cooperating earlier. In such a game, cooperation can become a balance, and the incentive to betray can be outweighed by the threat of punishment.

The classic prisoner's dilemma

In all judicial systems the punishment for banditry (committing crimes as part of an organized group) is much heavier than for the same crimes committed alone (hence the alternative name - "bandit's dilemma").

The classic formulation of the prisoner's dilemma is:

Two criminals, A and B, were caught at about the same time on similar crimes. There is reason to believe that they acted in collusion, and the police, isolating them from each other, offers them the same deal: if one testifies against the other, and he remains silent, then the first is released for helping the investigation, and the second gets the maximum sentence imprisonment (10 years) (20 years). If both are silent, their act is subject to a lighter article, and they are sentenced to 6 months (1 year). If both testify against each other, they receive a minimum sentence (2 years each) (5 years). Each prisoner chooses whether to remain silent or testify against the other. However, neither of them knows exactly what the other will do. What's going to happen?

The game can be represented in the form of the following table:

The dilemma arises if we assume that both care only about minimizing their own terms of imprisonment.

Let's present the reasoning of one of the prisoners. If the partner is silent, then it is better to betray him and be released (otherwise - six months in prison). If the partner testifies, then it is better to testify against him too in order to get 2 years (otherwise - 10 years). The “testify” strategy strictly dominates the “keep quiet” strategy. Similarly, another prisoner comes to the same conclusion.

From the point of view of the group (these two prisoners), it is best to cooperate with each other, remain silent and receive six months each, as this will reduce the total term of imprisonment. Any other solution will be less beneficial.

Generalized form

1. The game has two players and a banker. Each player holds 2 cards: one says “cooperate”, the other says “betray” (this is the standard terminology of the game). Each player places one card face down in front of the banker (that is, no one knows the other’s decision, although knowledge of the other’s decision does not affect the dominance analysis). The banker opens the cards and gives out the winnings.
2. If both choose to “cooperate,” both receive C... If one chooses to "betray", the other "to cooperate" - the first one gets D, second with... If both chose to "betray" - both get d.
3. The values ​​of the variables C, D, c, d can be of any sign (in the example above, everything is less than or equal to 0). The inequality D> C> d> c must be observed in order for the game to represent the Prisoner's Dilemma (DZ).
4. If the game is repeated, that is, it is played more than 1 time in a row, the total gain from cooperation should be greater than the total gain in a situation where one betrays and the other does not, that is, 2C> D + c.

These rules were established by Douglas Hofstadter and form a canonical description of the typical prisoner's dilemma.

A similar but different game

Hofstadter suggested that people more easily understand tasks as a prisoner's dilemma task when presented as a separate game or trading process. One example is “ closed bags exchange»:

Two people meet and exchange closed bags, realizing that one of them contains money, the other contains goods. Each player can respect the deal and put in the bag what they agreed on, or cheat the partner by giving an empty bag.

In this game, cheating will always be the best solution, meaning also that rational players will never play it and that there will be no closed-bag exchange market.

Application of game theory to strategic management decisions

Examples include decisions about pursuing a principled pricing policy, entering new markets, cooperating and setting up joint ventures, identifying leaders and performers in innovation, vertical integration, etc. Game theory can, in principle, be used for all kinds of decisions if they are influenced by others. characters... These individuals, or players, do not have to be market competitors; they can be subsuppliers, lead clients, organization employees, and work colleagues.

Game theory tools are especially useful when there are important dependencies between the participants in the process. in the field of payments... The situation with possible competitors is shown in Fig. 2.

 Quadrants 1 and 2 characterize a situation where the reaction of competitors does not significantly affect the payments of the firm. This happens when the competitor has no motivation (field 1 ) or opportunity (field 2 ) strike back. Therefore, there is no need for a detailed analysis of the strategy of competitors' motivated actions.

A similar conclusion follows, albeit for a different reason, and for the situation reflected by the quadrant 3 ... This is where a competitor’s reaction could have a significant impact on the firm, but since its own actions cannot greatly affect a competitor’s payments, its reaction should not be feared. An example is the decision to enter a market niche: under certain circumstances, large competitors have no reason to react to such a decision of a small company.

Only the situation shown in the quadrant 4 (the possibility of reciprocal steps of market partners), requires the use of the provisions of game theory. However, it reflects only the necessary but insufficient conditions to justify the application of the base of game theory to combat competitors. There are situations where one strategy will definitely dominate all others, no matter what the competitor will take. If we take, for example, the drug market, then it is often important for a company to be the first to announce a new product on the market: the profit of the “pioneer” turns out to be so significant that all other “players” can only quickly intensify their innovation activity.

 A trivial example of a "dominant strategy" from a game theory standpoint is a decision regarding penetration into a new market. Take an enterprise that acts as a monopolist in some market (for example, IBM in the personal computer market in the early 1980s). Another enterprise, operating, for example, in the market of peripheral equipment for computers, is considering the issue of penetrating the market of personal computers with a changeover of its production. An outsider company may decide to enter or not enter the market. A monopoly company may react aggressively or amiably to the emergence of a new competitor. Both businesses enter a two-stage game in which the outsider company makes the first move. The game situation with the indication of payments is shown in the form of a tree in Fig. 3.

 The same game situation can be presented in normal form (fig. 4).

There are two states designated here - "entry / friendly reaction" and "non-entry / aggressive reaction". Obviously, the second equilibrium is untenable. From the expanded form it follows that for a company already entrenched in the market it is inappropriate to react aggressively to the emergence of a new competitor: in case of aggressive behavior, the current monopolist receives 1 (payment), and if it is friendly - 3. The outsider company also knows that it is not rational for a monopolist initiate actions to oust it, and therefore it decides to enter the market. The outsider company will not suffer threatened losses in the amount of (-1).

Such a rational equilibrium is characteristic of a "partially improved" game, which deliberately excludes absurd moves. In practice, such equilibrium states are, in principle, quite easy to find. 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 choice of the "best" move at the last stage of the game is made, then the "best" move at the previous stage is chosen, taking into account the choice at the last stage, and so on, until the starting node of the tree is reached games.

How can companies benefit from game theory analysis? There is, for example, a case of collision of interests between IBM and Telex. In connection with the announcement of the preparatory plans of the latter to enter the market, a "crisis" meeting of the IBM management was held, at which the measures aimed at making the new competitor abandon their intention to enter the new market were analyzed. Telex apparently became aware of these events. Analysis based on game theory has shown that the high costs of IBM's threats are unreasonable. This shows that it is helpful for companies to consider the possible reactions of their game partners. Isolated business calculations, even based on the theory of decision-making, are often, as in the situation described, limited in nature. Thus, an outsider company could have chosen the "non-entry" course if preliminary analysis had convinced it that market penetration would provoke an aggressive reaction from the monopolist. In this case, in accordance with the expected value criterion, it is reasonable to choose the "non-entry" move with a probability of an aggressive response of 0.5.

 The next example is related to rivalry between companies in the field of technological leadership. The initial situation is when the enterprise 1 previously had technological superiority, but now has fewer financial resources to scientific research and development (R&D) than its competitor. Both enterprises must decide whether to try to achieve a dominant position in the world market in the relevant technological area with the help of large investments. If both competitors invest large funds in the business, then the prospects for success of the enterprise 1 will be better, although it will bear large financial expenses(like the enterprise 2 ). In fig. 5 this situation is represented by payments with negative values.

For the enterprise 1 it would be best if the enterprise 2 abandoned competition. His benefit would then be 3 (payments). Most likely the enterprise 2 the rivalry would win when the enterprise 1 would accept a curtailed investment program, and the company 2 - wider. This position is reflected in the upper right quadrant of the matrix.

Analysis of the situation shows that equilibrium occurs at high costs of 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 an enterprise 1 a reduced budget is preferable if the company 2 refuses to participate in the rivalry; at the same time the enterprise 2 it is known that at low costs for a competitor, it is profitable for him to invest in R&D.

An enterprise with technological advantage, can resort to the analysis of the situation on the basis of game theory in order to ultimately achieve the optimal result for himself. With the help of a certain signal, it must show that it is ready to carry out large expenditures on research and development. If such a signal is not received, then for the enterprise 2 it is clear that the enterprise 1 chooses the low cost option.

The reliability of the signal must be evidenced by the commitment of the enterprise. In this case, it may be a decision of an enterprise 1 on the purchase of new laboratories or the recruitment of additional research personnel.

From a game theory point of view, such commitments 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 go to large costs, the enterprise 2 registers this step and he no longer has a reason to participate in the rivalry. The new equilibrium arises from the alignment of "non-participation of the enterprise 2 "and" high costs of research and development of the enterprise 1 ".

 Known areas of application of game theory methods include pricing strategy, creation of joint ventures, timing of new product development.

An important contribution to the use of game theory is made by experimental work... Many theoretical calculations are worked out in laboratory conditions, and the results obtained serve as an impetus for practitioners. Theoretically, it was found out under what conditions it is advisable for two selfish partners to cooperate and achieve the best results for themselves.

This knowledge can be used in enterprise practice to help two firms achieve a win / win situation. Today, game-trained consultants quickly and unambiguously identify opportunities that businesses can take advantage of to enter into stable and long-term contracts with customers, sub-suppliers, development partners, and the like.

Problems of practical application in management

Of course, one should also point out the existence of certain limits of application of the analytical toolkit of game theory. In the following cases, it can be used only on condition of additional information.

At first, this is the case when businesses have different ideas about the game in which they participate, 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 by comparing such cases, taking into account certain differences.

Secondly, game theory is difficult to apply to many equilibrium situations. This problem can occur even during simple games with a simultaneous choice of strategic decisions.

Thirdly, if the situation of making strategic decisions is very difficult, then players often cannot choose the best options for themselves. It is easy to imagine more difficult situation market penetration 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 difficult 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.

Game theory is not used very often. Unfortunately, real-world situations are often very complex and change so quickly that it is impossible to accurately predict how competitors will react to changes in firm tactics. Nevertheless, game theory is useful when it is necessary to determine the most important and requiring consideration of factors in a competitive decision-making situation. This information is important because it allows management to consider additional variables or factors that might affect the situation, and thereby improves the effectiveness of the decision.

In conclusion, it should be emphasized that game theory is a very complex area of ​​knowledge. When referring to it, one must observe a certain caution and clearly know the limits of application. Too simple interpretations, adopted by the firm on its own or with the help of consultants, are fraught with hidden dangers. Because of its complexity, game theory analysis and advice is 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 planning strategic decisions, including when preparing large cooperation agreements.

Bibliography

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4. Archive of the journal "Problems of Theory and Practice of Management"., Rainer Felker

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- J. J. Rousseau. Discourse on the origin and foundations of inequality between people // Treatises / Per. with French A. Khayutina - Moscow: Nauka, 1969 .-- P. 75.

In practice, it is often necessary to make decisions in the face of opposition from the other side, which may pursue opposite or different goals, hinder the achievement of the intended goal by certain actions or states of the external environment. Moreover, these impacts of the opposite side can be passive or active. In such cases, it is necessary to take into account the possible options for the behavior of the opposite side, the response actions and their possible consequences.

Possible options for the behavior of both parties and their outcomes for each combination of options and conditions are often presented in the form of a mathematical model, which is called a game .

If the inactive, passive side acts as the opposing side, which deliberately does not oppose the achievement of the intended goal, then this game is called playing with "nature". Nature is usually understood as a set of circumstances in which decisions have to be made (unclear weather conditions, uncertainty about the behavior of customers in commercial activities, uncertainty about the reaction of the population to new types of goods and services, etc.)

In other situations, the opposite side actively, consciously opposes the achievement of the intended goal. In such cases, there is a clash of opposing interests, opinions, ideas. Such situations called conflict , and decision-making in a conflict situation is hampered by the uncertainty of the enemy's behavior. It is known that the adversary deliberately seeks to take the action that is least beneficial to you in order to ensure himself the greatest success. It is not known to what extent the enemy is able to assess the situation and possible consequences, how he assesses your capabilities and intentions. Both sides cannot predict reciprocal actions. Despite such uncertainty, it is up to each side of the conflict to make a decision.

In the economy, conflict situations are very common and diverse in nature. These include, for example, the relationship between a supplier and a consumer, a buyer and a seller, a bank and a client, etc. In all these examples, a conflict situation is generated by the difference in interests of partners and the desire of each of them to make optimal decisions. In this case, everyone has to reckon not only with their goals, but also with the goals of the partner and take into account his possible actions unknown in advance.

Need for justification optimal solutions in conflict situations led to the emergence game theory.

Game theory - it is a mathematical theory of conflict situations... The starting points of this theory are the assumption of the complete “ideal” rationality of the adversary and the adoption of the most cautious decision in resolving the conflict.

The conflicting parties are called players , one implementation of the game - party , the outcome of the game is win or lose ... Any action possible for the player (within the framework of the given rules of the game) is called his strategy .

The meaning of the game is that each of the players, within the framework of the given rules of the game, seeks to apply the strategy that is optimal for him, that is, the strategy that will lead to the best outcome for him. One of the principles of optimal (expedient) behavior is the achievement of an equilibrium situation, in violation of which none of the players is interested.

It is a situation of equilibrium that can be the subject of stable agreements between players. In addition, equilibrium situations are beneficial for each player: in an equilibrium situation, each player receives the greatest payoff, insofar as it depends on him.

Mathematical model of a conflict situation called a game , the parties to the conflict, are called players.

For each formalized game, rules are introduced. In the general case, the rules of the game establish the options for the players' actions; the amount of information each player has about the behavior of partners; the gain to which each set of actions results.

The development of the game in time occurs sequentially, in stages or moves. A move in game theory is called selection of one of the actions provided for by the rules of the game and its implementation. The moves are personal and random. By personal move is called a conscious choice by the player of one of the possible options for action and its implementation. By a random move is called a choice made not by a volitional decision of the player, but by some mechanism of random selection (tossing a coin, passing, dealing cards, etc.).

Depending on the reasons causing the uncertainty of the outcomes, games can be divided into the following main groups:

Combined games, in which the rules give, in principle, the opportunity for each player to analyze all the various options for their behavior and, comparing these options, choose the one that leads to the best outcome for this player. The uncertainty of the outcome is usually associated with the fact that the number of possible options for behavior (moves) is too large and the player is practically unable to enumerate and analyze them all.

Gambling , in which the outcome is uncertain due to the influence of various random factors. Gambling games consist only of random moves, which are analyzed using the theory of probability. Mathematical game theory is not involved in gambling.

Strategy games , in which the complete uncertainty of the choice is justified by the fact that each of the players, when deciding on the choice of the upcoming move, does not know what strategy the other participants in the game will adhere to, and the player's ignorance of the behavior and intentions of the partners is of fundamental nature, since there is no information about the subsequent actions of the enemy (partner).

There are games that combine the properties of combined and gambling, the strategy of games can be combined with combinatoriality, etc.

Depending on the number of participants in the game are subdivided into paired and multiple. In a double game, the number of participants is two, in a multiple game, the number of participants is more than two. Participants in a multiple game can form coalitions. In this case, the games are called coalition ... A multiple game turns into a pair if its participants form two permanent coalitions.

One of the basic concepts of game theory is strategy. Player strategy is a set of rules that determine the choice of a variant of action for each personal move of this player, depending on the situation that has developed in the course of the game.

Optimal strategy A player's strategy is a strategy that, with multiple repetitions of a game containing personal and random moves, provides the player with the maximum possible average gain or the minimum possible loss, regardless of the behavior of the opponent.

The game is called the ultimate if the number of player strategies is finite, and endless if at least one of the players has an infinite number of strategies.

In multi-pass problems of game theory, the concepts of "strategy" and "variant of possible actions" differ significantly from each other. In simple (one-move) game problems, when in each game game each player can make one move, these concepts coincide, and, therefore, the set of player strategies covers all possible actions that he can take in any possible situation and for any possible actual information.

Distinguish between games and the amount of winnings. The game is called game with zero sum th, if each player wins at the expense of the others, and the amount won by one side is equal to the amount lost by the other. In zero-sum doubles play, the interests of the players are exactly the opposite. The zero-sum doubles game is called I amantagonistic game .

Games in which the gain of one player and the loss of another are not equal to each other, are callednon-zero-sum games .

There are two ways to describe games: positional and normal ... The positional method is associated with an expanded form of the game and is reduced to a graph of successive steps (game tree). The normal way is to explicitly represent the set of players' strategies and payment function ... The payment function in the game determines the payoff of each of the parties for each set of strategies chosen by the players.

The use of mathematical methods, including game theory, in analysis economic processes allows you to identify such trends, relationships that remain hidden when using other methods.

In economic reality, at every step there are situations when individuals, firms or entire countries are trying to bypass each other in the struggle for supremacy. Such situations are dealt with by a branch of economic analysis called "game theory".

"Game theory studies the way in which two or more players choose individual actions or entire strategies. The name of this theory sets in a somewhat abstract way, since it is associated with playing chess and bridge or waging wars. In fact, the conclusions of this discipline are very deep Game theory was developed by a native of Hungary, brilliant mathematician John von Neumann (1903-1957) This theory is a relatively young mathematical discipline.

Later, game theory was supplemented by such developments as the Nash equilibrium (named after the mathematician John Nash). A Nash equilibrium occurs when none of the players can improve their position if their opponents do not change their strategies. Each player's strategy is the best answer to his opponent's strategy. Sometimes the Nash equilibrium is also called non-cooperative equilibrium, since the participants make their choice without entering into any agreements with each other and without taking into account any other considerations (the interests of society or the interests of other parties), except for their own benefit.

The equilibrium of a perfectly competitive market is also a Nash equilibrium, or noncooperative equilibrium, in which every firm and every consumer makes decisions based on already existing prices as not depending on his will. We already know that in conditions when each firm seeks to maximize profits, and each consumer - utility, equilibrium arises when prices are equal marginal cost and the profit is zero. "Mamaeva LN Institutional economics: A course of lectures - M .: Publishing and trade corporation" Dashkov and K ", 2012. - 200 p.

Let us recall the concept of the "invisible hand" of Adam Smith: "Pursuing his own interests, he (an individual) often contributes to the prosperity of society to a greater extent than if he consciously aspired to it" Smith A. Research on the nature and causes of the wealth of peoples // Anthology of economic classics ... - M .: Ekonov-Klyuch, 19931. The paradox of the "invisible hand" is that, although everyone acts as an independent force, in the end society remains a winner. At the same time, competitive equilibrium is also a Nash equilibrium in the sense that no one has a reason to change their strategy if everyone else adheres to theirs. In a completely competitive economy, non-cooperative behavior is cost-effective from the point of view of the interests of society.

On the contrary, when members of a certain group decide to cooperate and jointly arrive at a monopoly price, such behavior will be detrimental economic efficiency... The state is forced to create antimonopoly legislation and thereby reason with those who are trying to inflate prices and divide the market. However, disunity in behavior is not always cost effective. Rivalry between firms leads to low prices and a competitive production volume. "Not visible hand"has an almost magical effect on perfectly competitive markets: efficient allocation of resources occurs as a result of the actions of individuals seeking to maximize profits.

However, in many cases, non-cooperative behavior leads to economic inefficiency or even poses a threat to society (for example, an arms race). Non-cooperative behavior on the part of both the United States and the USSR forced both sides to invest huge amounts of money in the military field and led to the creation of an arsenal of almost 100,000 nuclear warheads. There is also concern that America's excessive availability of weapons could trigger a kind of internal arms race. Some people arm themselves against others - and this "race race" can continue indefinitely. Here a completely "visible hand" comes into play, directing this destructive competition and having nothing to do with the "invisible hand" of Adam Smith. Another important economic example- "pollution games" ( environment). Here the object of our attention will be the following view side effects like pollution. If firms never asked anyone what to do, they would rather create pollution than install expensive cleaners. If any firm, out of noble motives, decided to reduce harmful emissions, then the costs, and therefore the prices of its products, would increase, and the demand would fall. It is quite possible that this firm would have simply gone bankrupt. Living in a brutal world of natural selection, firms are more likely to choose to remain in a Nash equilibrium. No firm can improve profits by reducing pollution.

In a deadly economic game, every profit-maximizing and uncontrolled steel firm will generate water and air pollution. If a firm tries to clean up its emissions, it will be forced to raise prices and suffer losses. Non-cooperative behavior will establish a Nash equilibrium in a high-emission environment. The government can take steps to shift the balance. In this position, the pollution will be negligible, the profits will remain the same. Mamaeva L.N. Institutional Economics: A course of lectures - M .: Publishing and Trade Corporation "Dashkov and K", 2012. - 203 p.

The pollution games are one of the cases when the mechanism of action of the "invisible hand" does not work. This is a situation where the Nash equilibrium is ineffective. Sometimes these out-of-control games become threatening, and the government can intervene. By establishing a system of fines and emission quotas, the government can induce firms to choose an outcome that is low in pollution. Firms earn exactly the same as before, with large emissions, but the world is becoming somewhat cleaner.

Game theory also applies to macroeconomic policy. Economists and politicians in the United States often scold current monetary and fiscal policies: the federal deficit is too large and reduces national savings, while monetary policy generates interest rates that constrain investment. Moreover, this "fiscal and monetary syndrome" has been a feature of the macroeconomic landscape for over a decade. Why is America so stubbornly pursuing both types of policies, although neither of them is desirable?

One can try to explain this syndrome in terms of game theory. It has become customary in modern economics to separate these types of politics. The Central Bank of America - the Federal Reserve System - determines monetary policy independently of the government by setting interest rates. Fiscal policy, taxes and expenditures are in charge of the legislative and executive power... However, each of these policies has different goals. The central bank aims to limit the growth of the money supply and keep inflation low.

Arthur Berne, an expert on economic cycles and a former head of the Federal Reserve, wrote: “Central bank officials tend to keep prices in check, by tradition, perhaps by personal warehouse. private financial circles ". The authorities in charge of fiscal policy are more concerned with issues such as full employment, their own popularity, keeping taxes low, and upcoming elections.

Fiscal policy makers prefer the lowest possible unemployment, higher government spending coupled with tax cuts, and do not care about inflation and private investment.

In the money-budget game, a cooperative strategy leads to moderate inflation and unemployment, coupled with a large volume of investment that stimulates economic growth. However, the desire to reduce unemployment and implement social programs prompts the country's leadership to resort to increasing the budget deficit, while aversion to inflation forces the central bank to raise interest rates. Noncooperative equilibrium means the smallest possible investment.

They choose the "big budget deficit". On the other hand, the central bank tries to reduce inflation, is unaffected by trade unions and lobbying groups, and chooses "high interest rates." The result is a non-cooperative equilibrium with moderate inflation and unemployment, but low investment.

It is possible that it was thanks to the "fiscal game" that President Clinton put forward an economic program to reduce the budget deficit, lower interest rates and expand investment.

There are different ways to describe games. One of them is that all possible strategies of the players are considered and payments are determined that correspond to any possible combination of strategies of the players. A game described in this way is called playing in normal form.

The normal form of a two-player game consists of two payment matrices showing how much each player will receive for any of the possible pairs of strategies. Usually these matrices are expressed in the form of a single matrix, which is called bimatrix. The elements of the bimatrix are pairs of numbers, the first of which determines the amount of the first player's payoff, and the second, the amount of the second player’s payoff. The first player (state) chooses one of m strategies, with each strategy corresponding to a row of the matrix I (i = 1,…, m). The second player (business) chooses one of n strategies, with each strategy corresponding to a column of the matrix j (j = 1,…, n). A pair of numbers at the intersection of a row and a column, which correspond to the strategies chosen by the players, shows the amount of winnings for each of them. In general, if player I chooses the strategy i and player II is strategy j, then the payoffs of the first and second players are, respectively, and (i = 1,…, m; j = 1,…, n), where m, n is the number of final strategies of players I and II, respectively. It is assumed that each of the players knows all the elements of the payoff bimatrix. In this case, their strategy is called definite and has a finite number of options.

If the player does not know any variants of the opponent's strategies (matrix elements), then the game is called indefinite and may have infinite number options (strategies).

There are other classes of games where players win and lose at the same time.

Antagonistic games of two persons connected with the fact that one of the players wins exactly as much as the other loses. In such games, the interests of its players are directly opposite to each other.

As an example, consider a game in which two players participate, each of them has two strategies. The winnings of each of the players are determined by the following rules: if both players choose strategies with the same numbers (player I -, player II -), then the first player wins, and the second loses (the state raises taxes - business pays them, i.e. the state's gain determines loss of business); if both players choose different strategies (player I - і 1 player II - j 2 then the first loses, and the second wins (the state raises taxes on business - business evades them; the state’s loss is a business gain).

Game theory is theory mathematical models such phenomena in which the participants ("players") have different interests and have more or less freely chosen ways (strategies) to achieve their goals. In most works on game theory, it is assumed that the interests of the participants in the game are quantifiable and are real functions of situations, i.e. a set of strategies obtained when each player chooses some of his own strategies. To obtain results, it is necessary to consider certain classes of games, distinguished by some restrictive assumptions. Such restrictions can be imposed in several ways.

Can be distinguished several ways (ways) of imposing restrictions.

1. Limitations of the possibilities of the players' relationships with each other. The simplest case is when the players are completely disconnected and cannot knowingly help or interfere with each other by action or inaction, information or misinformation. This state of affairs inevitably occurs when only two players (the state and business) participate in the game, having diametrically opposite interests: an increase in the gain of one of them means a decrease in the gain of the other, and, moreover, by the same amount, provided that the gains of both players are expressed in the same units of measurement. Without breaking the generality, we can take the total payoff of both players to be zero and treat the payoff of one of them as the loss of the other.

These games are called antagonistic (or zero-sum games, or zero-sum games). They assume that there can be no relationships between players, no compromises, exchanges of information and other resources by the very nature of things, in the essence of the game, since every message received by the player about the intentions of the other can only increase the payoff of the first player and thereby increase losing to his opponent.

Thus, we conclude that in antagonistic games the players may not have direct relationships and, at the same time, be in a state of play (opposition) in relation to each other.

2. Restrictions or simplifying assumptions on the set of player strategies. In the most simple case these sets of strategies are finite, which eliminates situations associated with possible coincidences (convergences) in sets of strategies, eliminates the need to introduce any technology on the sets.

Games in which the sets of strategies of each of the players are finite are called end games.

3. Suggestions about the internal structure of each strategy, ie. about its content. So, for example, as strategies can be considered functions of time (continuous or discrete), the values ​​of which are the actions of the player at the appropriate moment. These and similar games are usually called dynamic (positional) games.

The limitations of the players' strategies can be their target functions, i.e. determination of the goals to achieve which this or that strategy is directed. It can be assumed that the restrictions on the strategy are also related to the ways of achieving these goals in certain time intervals, for example, the desire of a business to achieve a decrease in the size of the mandatory sales of foreign exchange earnings within the next three months (or one year). If no assumptions are made about the nature of strategies, then they are considered to be some abstract set. Games of this kind in the simplest formulation of the question are called games in normal form.

Final antagonistic games in normal form are called matrix. This name is explained by the possibility of the following interpretation of games of this type. We will understand the strategies of the first player (player I - the state) as rows of some matrix, and the strategies of the second player (player II - business) - as its columns. For brevity, the strategies of the players are not the rows or columns of the matrix themselves, but their numbers. Then the situations of the game are the cells of this matrix, standing at the intersections of each row with each of the columns. Having filled these cells-situations with numbers describing the payoffs of player I in these situations, we complete the task of the game. The resulting matrix is ​​called the payoff matrix of the game, or the matrix of the game. Due to the antagonism of the matrix game, the payoff of player II in each situation is completely determined by the payoff of player I in this situation, differing from him only in sign. Therefore, additional indications about the payoff function of player II in the matrix game are not required.

A matrix with m rows and n columns is called an (m * n) - matrix, and a game with this matrix is ​​called an (m * n) - game.

The process of (m * n) - games with a matrix can be represented as follows:

Player I fixes the number of row i, and player II fixes the number of column j, after which the first player receives from his opponent the sum

The goal of player I in the matrix game is to get the maximum payoff, the goal of player II is to give player I the minimum payoff.

Let player I (state) choose some of its strategies i. Then, in the worst case, he will receive a payoff min. In game theory, players are assumed to be cautious, counting on the least favorable turn of events for themselves.

Such a state of affairs that is least favorable for player I can occur, for example, in the case when strategy i becomes known to player II (business). Anticipating such a possibility, player I must choose his strategy so as to maximize this minimum gain:

min = max min (I)

The value on the right-hand side of the equality is the guaranteed payoff of player I. Player II (business) must choose a strategy such that

max = min max (II)

The value on the right side of the equality is the payoff of player I, more than which he cannot receive with the correct actions of the opponent.

The actual payoff of player I must, with reasonable actions of partners, be in the interval between the payoff values ​​in the first and second cases. If these values ​​are equal, then the payoff of player I is a well-defined number, the games themselves are called quite definite. The payoff of player I is called the value of the game, and it is equal to the element of the matrix.

Players may have additional features- the choice of their strategies randomly and independently of each other (strategies correspond to the rows and columns of the matrix). A player's random choice of his strategies is called mixed country this player's tags. In (m * n) - the game, the mixed strategies of player I are determined by the sets of probabilities: X = (, ...), with which this player chooses his initial, pure strategies.

The theory of matrix games is based on the Neumann theorem for active strategies: “If one of the players adheres to his optimal strategy, then the payoff remains unchanged and equal to the game price regardless of what the other player does, if he does not go beyond his active strategies (i.e. That is, he uses any of them in pure form or mixes them in any proportions "Neumann J. Contributions to the theory of games. 1995 .. - 155 p.). Note that active is the player's pure strategy that is included in his optimal mixed strategy with a nonzero probability.

The main goal of the game is finding the optimal strategy for both players, if not with the maximum gain for one of them, then with the minimum loss for both. The method of finding optimal strategies often gives more than is necessary for practical purposes. In a matrix game, it is not necessary for the player to know all of his optimal structures, since they are all interchangeable and the player only needs to know one of them for a successful game. Therefore, in relation to matrix games, the question of finding at least one optimal strategy for each of the players is relevant.

The main theorem on matrix games states the existence of game values ​​and optimal mixed strategies for both players. The optimal strategy does not have to be one-off. This is a very important conclusion drawn from game theory.

For the gamer matrix game subject are characteristic the following quality:

matrix elements interpreted as cash payments and, accordingly, their winnings and losing evaluated in monetary form;

each of players applies the function to these elements usefulness;

in the game, each player acts as if his opponent's utility function had exactly the same effect on the matrix, i.e. everyone looks at the game "with their own bell towers ".

These assumptions lead to zero-sum games, in which relations of cooperation, bargaining and other type of interactions arise between players as before games, and in its process. Mamaeva L.N. Institutional Economics: A course of lectures - M .: Publishing and Trade Corporation "Dashkov and K", 2012. - 210 - 211s.

Generalization of game theory to include others analysis capabilities, leads to interesting, but rather difficult tasks. In the development of game theory, it is necessary to apply the utility function not only to monetary outcomes, but also to the amounts with the expected future outcomes. These the assumptions are controversial, but they exist. In this case, we proceed from the assumption that this assumption about a similar operation It has likeness with behavior players in certain decision-making situations and admits the possibility that the method playing the game this player depends on the state of his capital during conducting them games.

Consider this in the following example. Let be the first player at the start of the game G has capital x dollars. Then his capital at the end games will is equal to + x, where is the actual winnings he gets from the game. The usefulness he attributes to such the outcome, is equal to f (+ x), where f is the utility function.

These few examples illustrate only a fraction of the vast variety of results that can be obtained using game theory. This section of economic theory is an extremely useful (for economists and other social scientists) tool for analyzing situations in which a small number of people are well informed and try to outwit each other in the markets, in politics or in hostilities.