Abstract: Game theory and its practical application. Mathematical game theory

Preface

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 and will consider a short history game theory, get acquainted with the basic principles of game theory, including the main types of games and forms of their representation. The article will touch upon the classical problem and the fundamental problem of game theory. The final section of the article is devoted to consideration of the problems of applying game theory to the adoption of management decisions and practical application of game theory in management.

Introduction.

21 century. The age of information, rapidly developing information technologies, innovations and technological innovations. But why the information age? Why does information play a key role in almost all processes occurring 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 no secret that in life you often have to deal with tasks in which you need to make decisions in conditions of uncertainty, in the absence of information about responses to your actions, i.e. situations arise in which two (or more) parties pursue different goals, and the results of any action of each party 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 games are mainly entertaining in nature, by their nature they relate to conflict situations in which the conflict is already inherent in the goal of the game - the winning of one of the partners. At the same time, the result of each player’s move depends on the opponent’s response move. In economics, conflict situations occur very often and are of a diverse nature, and their number is so large that it is impossible to count all the conflict situations that arise in the market in at least one day. Conflict situations in the economy include, for example, relationships between supplier and consumer, buyer and seller, bank and client. In all of the above examples, the conflict situation is generated by the difference in interests of the partners and the desire of each of them to make optimal decisions that realize their goals to the greatest extent. At the same time, everyone has to take into account not only their own goals, but also the goals of their partner, and take into account the decisions unknown in advance that these partners will make. To competently solve problems in conflict situations, scientifically based methods are required. Such methods are developed by the mathematical theory of conflict situations, which is called game theory.

What is game theory?

Game theory is a complex, multi-dimensional concept, so it seems impossible to interpret game theory using only one definition. Let's look at three approaches to defining game theory.

1.Game theory is a mathematical method for studying optimal strategies in games. A game is 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 that can lead to winning or losing - depending on the behavior of other players. Game theory helps you choose best strategies taking into account ideas about other participants, their resources and their possible actions.

2. Game theory is a branch of applied mathematics, or more precisely, operations research. Most often, game theory methods are used in economics, a little less often in others. 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. Very important game theory has implications for artificial intelligence and cybernetics.

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

History of game theory

Optimal solutions or strategies in mathematical modeling were proposed back in the 18th century. The problems of production and pricing under oligopoly conditions, which later became textbook examples of game theory, were considered in the 19th century. A. Cournot and J. Bertrand. At the beginning of the 20th century. E. Lasker, E. Zermelo, E. Borel put forward the idea of ​​a mathematical theory of conflict of interest.

Mathematical game theory originates from neoclassical economics. The mathematical aspects and applications of the theory were first outlined in the classic 1944 book by John von Neumann and Oscar Morgenstern, Game Theory and Economic Behavior.

John Nash after graduation Polytechnic Institute Carnegie with two degrees - a bachelor's and a 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 zero-sum games, where there are losers and winners at their expense. Nash develops methods of analysis in which everyone involved either wins or loses. These situations are called “Nash equilibrium” or “non-cooperative equilibrium”; in the 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 serious contribution to the development of game theory, and the mathematical tools of economic modeling were revised. John Nash shows that A. Smith's classic approach to competition, where everyone is for himself, is suboptimal. More optimal strategies are when everyone tries to do better for themselves while doing better for others. In 1949, John Nash wrote a dissertation on game theory, and 45 years later he received Nobel Prize in economics.

Although game theory originally dealt with economic models, it remained a formal theory within mathematics until the 1950s. But already since the 1950s. attempts are beginning to apply game theory methods not only in economics, but in biology, cybernetics, technology, and anthropology. During World War II and immediately after it, the military became seriously interested in game theory, who saw in it a powerful tool for studying strategic decisions.

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

A major contribution to the application of game theory was the work of Thomas Schelling, Nobel laureate in Economics 2005 “Strategy of Conflict.” T. Schelling considers various “strategies” of behavior of the participants in the conflict. These strategies coincide with conflict management tactics and principles of conflict analysis in conflictology and organizational conflict management.

Basic principles of game theory

Let's get acquainted with the basic concepts of game theory. The mathematical model of a conflict situation is called game, parties involved in the conflict - players. To describe a game, you must first identify its participants (players). This condition is easily fulfilled when we're talking about about ordinary games like chess, etc. The situation is different with “market games”. Here it is not always easy to recognize all the players, i.e. current or potential competitors. Practice shows that it is not necessary to identify all players; it is necessary to discover the most important ones. Games typically span several periods during which players take sequential or simultaneous actions. The choice and implementation of one of the actions provided for by the rules is called progress player. Moves can be personal and random. Personal move- this 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 selected action (for example, choosing a card from a shuffled deck). Actions may be related to prices, sales volumes, research and development costs, etc. The periods during which players make their moves are called stages games. The moves chosen at each stage ultimately determine "payments"(win or loss) of each player, which can be expressed in material assets or money. Another concept in this theory is player strategy. Strategy A player is a set of rules that determine the choice of his action at 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, it is in principle 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 specific strategy, which can be specified as a list of rules or a program. (This way you can play the game using a computer.) In other words, strategy refers to possible actions that allow the player at each stage of the game to choose from a certain number of alternative options the 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 during the course of a given game. The game is called steam room, if it involves two players, 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 their partners; 3) the gain that each set of actions leads to. Typically, winning (or losing) can be quantified; for example, you can value a loss as zero, a win 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. complete task game, it is enough to indicate the size of one of them. If we designate A- winnings of one of the players, b- the other's winnings, then for a zero-sum game b = -a, therefore it is enough to consider, for example A. The game is called ultimate, if each player has a finite number of strategies, and endless- otherwise. In order to decide game, or find game solution, you should choose a strategy for each player that satisfies the condition optimality, those. one of the players must receive maximum win when the second one sticks to his strategy. At the same time, the second player must have minimum loss, if the first one sticks to his strategy. Such strategies are called optimal. Optimal strategies must also satisfy the condition sustainability, i.e., it must be disadvantageous for any of the players to abandon their strategy in this game. If the game is repeated quite a few times, then players may be interested not in winning and losing in each specific game, but in average win (loss) in all batches. Purpose game theory is to determine the optimal strategies for each player. When choosing an optimal strategy, it is natural to assume that both players behave reasonably in terms of their interests.

Cooperative and non-cooperative

The game is called cooperative, or coalition, if players can unite in groups, taking on some obligations to other players and coordinating their actions. This differs from non-cooperative games in which everyone must play for themselves. Entertainment games are rarely cooperative, but such mechanisms are not uncommon in everyday life.

It is often assumed that what makes cooperative games different is the ability for players to communicate with each other. IN general case 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 great detail and produce more accurate results. Cooperatives consider the game process as a whole.

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.

Symmetrical and asymmetrical

Asymmetrical game

The game will be symmetrical when the corresponding strategies of the players are equal, that is, they have the same payments. In other words, if players can change places and their winnings for the same moves will not change. Many two-player games studied 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 payoff of the second player 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 type of games with constant amount, that is, those where players cannot increase or decrease the available resources, or the game fund. In this case, the sum of all wins is equal to the sum of all losses for any move. Look to the right - the numbers represent payments to the players - and their sum in each cell is zero. Examples of such games include poker, where one wins all the others' bets; reversi, where enemy pieces are captured; or banal theft.

Many games studied by mathematicians, including the already mentioned “Prisoner’s Dilemma”, are of a different kind: in non-zero sum games One player's win does not necessarily mean another's loss, and vice versa. The outcome of such a game can be less or more 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 non-zero sum is trade, where every participant 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 game amount increases. A well-known example where it decreases is war.

Parallel and serial

In parallel games, players move simultaneously, or at least they are not aware of others' choices until All won't 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 ten of his strategies definitely didn't choose fifth, without learning anything about the 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 form.

With complete or incomplete information

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

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

The concept of complete information is often confused with the similar one - perfect information. For the latter, it is enough just to know all the strategies available to opponents; knowledge of all their moves is not necessary.

Games with an infinite number of steps

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

The task 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 many real numbers. Games that include such elements are often called differential games. They are associated with some kind of material scale (usually a time scale), although the events occurring in them can be discrete in nature. Differential games find their application 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 goal of metagames is to increase the usefulness of the given ruleset.

Game presentation form

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

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

In the absence of strict conditions, it is beneficial for both enterprises to assign low price. The “low price” strategy is the dominant one for any firm: no matter what price a competing firm chooses, it is always preferable to set a low price. 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 of “low prices/low prices” with corresponding payments represents a Nash equilibrium, in which it is disadvantageous for either player 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 resolution depends, in particular, on the originality of the players’ moves. If the enterprise has the opportunity to reconsider its strategic variables (in this case price), then a cooperative solution to the problem can be found even without a rigid agreement between the players. Intuition suggests that with repeated contact between players, opportunities arise 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 pictures characterize the same game. Presenting the game in normal form in the normal case reflects "synchronicity". However, this does not mean the “simultaneity” of events, but indicates that the player’s choice of strategy is carried out in ignorance of the opponent’s choice of strategy. In an expanded form, this situation is expressed through an oval space (information field). In the absence of this space, the game situation takes on a different character: first, one player would have to make a decision, and the other could do it after him.

Classic problem in game theory

Let's consider a classic problem in game theory. Deer hunting is a cooperative symmetric game from game theory that describes the conflict between personal interests and public interests. The game was first described by Jean-Jacques Rousseau in 1755:

"If they were hunting a deer, then everyone understood that for this he was obliged to remain at his post; but if a hare ran near one of the hunters, then there was no doubt that this hunter, without a twinge of conscience, would set off after him and, having overtaken the prey , very few will lament that in this way he deprived his comrades of prey."

Deer hunting is a classic example of the challenge of providing a public good while tempting man to give in to self-interest. Should the hunter remain with his comrades and bet on a less favorable opportunity to deliver large prey to the whole tribe, or should he leave his comrades and entrust himself to a more reliable opportunity that promises his own family a hare?

Fundamental problem in game theory

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

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

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

While behaving individually rationally, together the participants come to an irrational decision: if both betray, they will receive a smaller payoff in total than if they cooperated (the only equilibrium in this game does not lead to Pareto-optimal decision, i.e. a decision that cannot be improved without worsening the situation of other elements.). Therein lies the dilemma.

In a repeated prisoner's dilemma, the game occurs periodically, and each player can "punish" the other for not cooperating earlier. In such a game, cooperation can become an equilibrium, and the incentive to betray can be outweighed by the threat of punishment.

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 - “the bandit's dilemma”).

The classic formulation of the prisoner's dilemma is:

Two criminals, A and B, were caught at about the same time for similar crimes. There is reason to believe that they acted in conspiracy, and the police, isolating them from each other, offer 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 receives the maximum sentence imprisonment (10 years) (20 years). If both are silent, their act is charged under a lighter article, and they are sentenced to 6 months (1 year). If both testify against each other, they receive a minimum sentence of 2 years (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 will happen?

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

The dilemma arises if we assume that both are only concerned with minimizing their own prison term.

Let's imagine the reasoning of one of the prisoners. If your partner is silent, then it is better to betray him and go free (otherwise - six months in prison). If the partner testifies, then it is better to also testify against him in order to get 2 years (otherwise - 10 years). The “testify” strategy strictly dominates the “keep silent” 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 get six months each, as this will reduce the total prison term. Any other solution will be less profitable.

Generalized form

1. The game consists of two players and a banker. Each player holds 2 cards: one says “cooperate”, the other says “defect” (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 anyone else's decision, although knowing someone else's decision does not affect dominance analysis). The banker opens the cards and gives out the winnings.
2. If both choose to cooperate, both receive C. If one chose “to betray”, the other “to cooperate” - the first receives D, second With. If both chose “betray”, both receive d.
3. The values ​​of the variables C, D, c, d can be of any sign (in the example above, all are less than or equal to 0). The inequality D > C > d > c must be satisfied for the game to be a Prisoner's Dilemma (PD).
4. If the game is repeated, that is, played more than 1 time in a row, the total payoff from cooperation must be greater than the total payoff 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 the canonical description of the typical prisoner's dilemma.

Similar but different game

Hofstadter suggested that people understand problems like the prisoner's dilemma more easily if they are presented as a separate game or trading process. One example is “ exchange of closed bags»:

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 what was agreed upon in the bag, or deceive the partner by giving an empty bag.

In this game, cheating will always be the best solution, which also means that rational players will never play the game and that there will be no market for trading closed bags.

Application of game theory to make strategic management decisions

Examples include 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 principles of game theory can in principle be used for all types of decisions if they are influenced by other actors. These individuals, or players, do not necessarily have to be market competitors; their role may be subsuppliers, leading customers, employees of organizations, as well as work colleagues.

 It is especially advisable to use the tools of game theory when there are important dependencies in the field of payments. The situation with possible competitors shown in Fig. 2.

 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 in cases where the competitor has no motivation (field 1 ) or capabilities (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, and for the situation reflected by the quadrant 3 . Here, the reaction of competitors could have a significant impact on the company, but since its own actions cannot greatly affect the payments of a competitor, then one should not be afraid of its reaction. An example is decisions 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 retaliatory steps by market partners) requires the use of game theory provisions. However, these are only necessary but not sufficient conditions to justify the use of a game theory framework to combat competitors. There are situations when one strategy will undoubtedly dominate all others, regardless of what actions the competitor takes. If we take, for example, the market medicines, 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 innovation activities.

 A trivial example of a “dominant strategy” from the standpoint of game theory is the decision regarding penetration on new market. Let's take an enterprise that acts as a monopolist in any market (for example, IBM in the personal computer market in the early 80s). Another enterprise, operating, for example, in the market of computer peripheral equipment, is considering the issue of penetrating the personal computer market by reconfiguring its production. An outsider company may decide to enter or not to enter the market. A monopolist company can react aggressively or friendly to the emergence of a new competitor. Both companies enter into a two-stage game in which the outsider company makes the first move. The game situation indicating 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 indicated 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 that has already established a foothold in the market, it is inappropriate 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 begin actions to displace it, and therefore it decides to enter the market. The outsider company will not bear the threatened losses of (-1).

Such 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, a choice is made of the “best” move on last stage game, then selects the “best” move at the previous stage, taking into account the choice at the last stage, and so on, until the starting node of the game tree is reached.

How can companies benefit from game theory-based analysis? For example, there is a well-known case of conflict of interests between IBM and Telex. In connection with the announcement of the latter's preparatory plans for entering the market, a “crisis” meeting of IBM management was held, at which measures aimed at forcing the new competitor to abandon its intention to penetrate the new market were analyzed. Telex apparently became aware of these events. An analysis based on game theory showed that threats to IBM due to high costs are unfounded. This suggests that it is useful for companies to consider the possible reactions of their gaming partners. Isolated economic calculations, even those based on decision-making theory, are often, as in the situation described, limited in nature. Thus, an outsider company could choose the “non-entry” move if a preliminary analysis convinced it that market penetration would cause an aggressive reaction from the monopolist. In this case, in accordance with the expected value criterion, it is reasonable to choose the “non-intervention” move with a probability of an aggressive response of 0.5.

 The following example is related to the rivalry of companies in the field technological leadership. The starting situation is when the enterprise 1 previously had technological superiority, but now has less financial resources For scientific research and development (R&D) than its competitor. Both companies must decide whether to try to achieve global market dominance in their respective technology area through large capital investments. If both competitors invest large amounts of money in the business, then the prospects for success of the enterprise 1 will be better, although it will incur large financial expenses (like the enterprise 2 ). In Fig. 5 this situation is represented by payments with negative values.

For enterprise 1 it would be best if the enterprise 2 refused to compete. His benefit in this case would be 3 (payments). Most likely the enterprise 2 would win the competition when the enterprise 1 would accept a reduced investment program, and the enterprise 2 - wider. This position is reflected in the upper right quadrant of the matrix.

Analysis of the situation shows that equilibrium occurs at high R&D costs 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 enterprise 2 will refuse to participate in competition; at the same time to the enterprise 2 It is known that when a competitor’s costs are low, it is profitable for him to invest in research and development.

An enterprise with a technological advantage can resort to analyzing the situation based on game theory in order to ultimately achieve the optimal result for itself. With the help of a certain signal, it must show that it is ready to make 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 enterprise's obligations. In this case, it may be the decision of the enterprise 1 on the purchase of new laboratories or the hiring of additional research personnel.

From the point of view of game theory, such obligations are equivalent to changing the course of the game: the situation of simultaneous decision-making is replaced by a situation of sequential moves. Company 1 firmly demonstrates the intention to make large expenditures, the enterprise 2 registers this step and he no longer has any reason to participate in the rivalry. The new equilibrium follows from the scenario “non-participation of the enterprise 2 " and "high costs of research and development of the enterprise 1 ".

 Well-known areas of application of game theory methods also include pricing strategy, creation of joint ventures, timing of new product development.

Important contributions to the use of game theory come from experimental work. Many theoretical calculations are tested in laboratory conditions, and the results obtained serve as an impetus for practitioners. Theoretically, it was clarified under what conditions it is advisable for two selfishly minded partners to cooperate and achieve better results for themselves.

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

Problems of practical application in management

Of course, it should be pointed out that there are certain limits to the application of the analytical tools of game theory. IN following cases it can only be used if additional information is obtained.

Firstly, this is the case when businesses have different ideas about the game they are playing, 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 information that is not too complex is characterized by incompleteness, then one can operate by comparing similar 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 the simultaneous selection of strategic decisions.

Third, If the strategic decision-making situation is very complex, then players often cannot choose the best options for themselves. It's easy to imagine more difficult situation market penetration than that 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 being aggressive or friendly.

It has been experimentally proven that when the game expands to ten or more stages, 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, the situation real world are often very complex and change so quickly that it is impossible to accurately predict how competitors will react to a change in the firm's tactics. However, game theory is useful when it comes to identifying the most important factors to consider in a competitive decision-making situation. This information is important because it allows management to consider additional variables or factors that may affect the situation, thereby increasing the effectiveness of the decision.

In conclusion, it should be especially emphasized that game theory is a very complex field of knowledge. When handling it, you must be careful and clearly know the limits of its use. Too much simple interpretations, adopted by the company independently or with the help of consultants, are fraught with hidden danger. Due to their complexity, game theory analysis and consultation are recommended only for particularly important 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.

Bibliography

1. Game theory and economic behavior, von Neumann J., Morgenstern O., Science publishing house, 1970

2. Petrosyan L.A., Zenkevich N.A., Semina E.A. Game theory: Textbook. manual for universities - M.: Higher. school, Book House "University", 1998

3. Dubina I. N. Fundamentals of theory economic games: textbook. - M.: KNORUS, 2010

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

5. Game theory in management organizational systems. 2nd edition., Gubko M.V., Novikov D.A. 2005

- J. J. Rousseau. Reasoning about the origin and foundations of inequality between people // Treatises / Trans. from French A. Khayutina - M.: Nauka, 1969. - P. 75.

A funny example of the application of game theory is in the fantasy book “The Brave Golem” by Anthony Pearce.

Lots of text

“The point of what I’m about to demonstrate to you all,” Grundy began, “is to set required quantity points. The scores can be very different - it all depends on the combination of decisions made by the participants in the game. For example, suppose each participant testifies against his fellow player. In this case, each participant can be awarded one point!
- One point! – said the Sea Witch, showing unexpected interest in the game. Obviously, the sorceress wanted to make sure that the golem had no chance of making the demon Xanth happy with him.
– Now let’s assume that each of the participants in the game does not testify against his friend! – Grundy continued. – In this case, each person can be awarded three points. I want to especially note that as long as all participants act in the same way, they are awarded the same number of points. No one has any advantage over another.
- Three points! - said the second witch.
– But now we have the right to suggest that one of the players began to testify against the second, but the second is still silent! - said Grundy. - In this case, the one who gives this testimony receives five points at once, and the one who is silent does not receive a single point!
- Yeah! – both witches exclaimed in one voice, licking their lips predatorily. It was clear that both of them were clearly going to get five points.
– I kept losing my glasses! – the demon exclaimed. – But you have only outlined the situation, and have not yet presented a way to resolve it! So what is your strategy? No need to waste time!
- Wait, now I’ll explain everything! - Grundy exclaimed. “Each of us four—there are two of us golems and two witches—will fight against our opponents. Of course, the witches will try not to yield to anyone in anything...
- Certainly! – both witches exclaimed again in unison. They perfectly understood the golem at a glance!
“And the second golem will follow my tactics,” Grundy continued calmly. He looked at his double. - Of course, you know?
- Yes, sure! I'm your copy! I understand perfectly well what you think!
- That is great! In that case, let's make the first move so that the demon can see everything for himself. Each fight will have several rounds to allow the full strategy to play out and make an impression. whole system. Perhaps I should start.

– Now each of us must mark our own pieces of paper! – the golem turned to the witch. – First you should draw a smiling face. This will mean that we will not testify against a fellow prisoner. You can also draw a frowning face, which means that we think only about ourselves and are giving the necessary evidence against our comrade. We both realize that it would be better if no one turned out to be that same frowning face, but, on the other hand, a frowning face receives certain advantages over a smiling one! But the point is that each of us does not know what the other will choose! We won't know until our playing partner reveals his drawing!
- Get started, you bastard! – the witch cursed. She, as always, could not do without abusive epithets!
- Ready! - Grundy exclaimed, drawing a big smiling face on his piece of paper so that the witch could not see what he had drawn there. The witch made her move, also making a face. One must think that she certainly put on an unkind face!
“Well, now all we have to do is show each other our drawings,” Grundy announced. Turning back, he opened the drawing to the public and showed it in all directions so that everyone could see the drawing. Grumbling something displeased, the Sea Witch did the same.
As Grundy had expected, an angry, dissatisfied face looked out from the witch’s drawing.
“Now you, dear spectators,” Grundy said solemnly, “see that the witch chose to testify against me.” I'm not going to do that. Thus, the Sea Witch scores five points. And, accordingly, I don’t get a single point. And here…
A slight noise rang through the rows of spectators again. Everyone clearly sympathized with the golem and passionately wanted the Sea Witch to lose.
But the game has just begun! If only his strategy was correct...
– Now we can move on to the second round! – Grundy announced solemnly. – We must repeat the moves again. Everyone draws the face that is closest to them!
And so they did. Grundy now wore a gloomy, dissatisfied face.
As soon as the players showed their drawings, the audience saw that they were both now making angry faces.
- Two points each! - said Grundy.
- Seven two in my favor! – the witch shouted joyfully. “You won’t get out of here, you bastard!”
- Let's start again! - Grundy exclaimed. They made another drawing and showed them to the public. The same angry faces again.
– Each of us repeated the previous move, behaved selfishly, and therefore, it seems to me, it is better not to award points to anyone! - said the golem.
– But I still lead the game! - said the witch, happily rubbing her hands.
- Okay, don't make noise! - said Grundy. - The game is not over. Let's see what happens! So, dear audience, we are starting the fourth round!
The players made drawings again, showing the audience what they had drawn on their sheets. Both sheets of paper again showed the same evil faces to the audience.
- Eight - three! - the witch screamed, bursting into evil laughter. “You dug your own grave with your stupid strategy, golem!”
- Fifth round! - Grundy shouted. The same thing happened as in the previous rounds - angry faces again, only the score changed - it became nine - four in favor of the sorceress.
– Now the last, sixth round! - Grundy announced. His preliminary calculations showed that this particular round should become fateful. Now the theory had to be confirmed or refuted by practice.
A few quick and nervous movements of the pencil on the paper - and both drawings appeared before the eyes of the public. Again two faces, now even with bared teeth!
– Ten – five in my favor! My game! I won! – the Sea Witch cackled.

“You really won,” Grundy agreed gloomily. The audience was ominously silent.
The demon moved his lips to say something.

- But our competition is not over yet! - Grundy shouted loudly. – This was only the first part of the game.
- Give you an eternity! – the demon Xanth grumbled displeasedly.
- It's right! - Grundy said calmly. – But one round does not solve anything, only methodicality indicates the best result.
The golem now approached the other witch.
– I would like to play this round with another opponent! - he announced. – Each of us will depict faces, as it was the previous time, then we will demonstrate what we have drawn to the public!
So they did. The result was the same as last time - Grundy drew a smiling face, and the witch just a skull. She immediately gained a full five point lead, leaving Grundy behind.
The remaining five rounds ended with the results that could be expected. Once again the score was ten - five in favor of the Sea Witch.
– Golem, I really like your strategy! - the witch laughed.
– So, you have watched two rounds of the game, dear viewers! - Grundy exclaimed. “Thus, I scored ten points, and my rivals scored twenty!”
The audience, who were also counting points, nodded their heads mournfully. Their count matched that of the golem. Only the cloud named Fracto seemed very pleased, although, of course, it did not sympathize with the witch either.
But Rapunzel smiled approvingly at the golem - she continued to believe in him. She might be the only one left who believed him now. Grundy hoped that he would justify this boundless trust.
Now Grundy approached his third opponent - his double. He was to be his last opponent. Quickly scribbling their pencils on the paper, the golems showed the pieces of paper to the public. Everyone saw two laughing faces.
– Please note, dear viewers, each of us chose to be a good cellmate! - Grundy exclaimed. “And therefore none of us received the necessary advantage over our opponents in this game.” So we both get three points and move on to the next round!
The second round has begun. The result was the same as the previous time. Then the remaining rounds. And in each round, both opponents again scored three points! It was simply incredible, but the public was ready to confirm everything that was happening.

Finally, this round came to an end, and Grundy, quickly running his pencil over the paper, began to calculate the result. Finally he announced solemnly:
- Eighteen to eighteen! In total, I scored twenty-eight points, while my opponents scored thirty-eight!
“So you lost,” the Sea Witch announced joyfully. – Thus, one of us will become the winner!
- Maybe! – Grundy responded calmly. Now came another important moment. If everything goes as planned...
– We need to bring this matter to an end! – exclaimed the second golem. “I also still need to fight two Sea Witches!” The game is not over yet!
- Yes, of course, go ahead! - said Grundy. – But just be guided by strategy!
- Yes, sure! – assured his double.
This golem approached one of the witches and the tour began. It ended with the same result with which Grundy himself came out of a similar round - the score was ten to five in favor of the sorceress. The witch actually beamed with inexpressible joy, and the audience fell sullenly silent. Demon Xanth looked somewhat tired, which was not a very good omen.
Now it was time for the final round - one witch had to fight against the second. Each had twenty points, which she was able to get by fighting golems.
“And now, if you allow me to score at least a few extra points...” the Sea Witch whispered conspiratorially to her double.
Grundy tried to remain calm, at least outwardly, although a hurricane of conflicting feelings was raging in his soul. His luck now depended on how correctly he predicted the possible behavior of both witches - after all, their character was, in essence, the same!
Now came perhaps the most critical moment. But what if he was wrong?
- Why on earth should I give in to you! – the second witch croaked to the first. – I myself want to score more points and get out of here!
“Well, if you’re acting so impudent,” the applicant screamed, “then I’ll beat you up so that you won’t be like me anymore!”
The witches, giving each other hateful looks, drew their drawings and showed them to the public. Of course, nothing else but two skulls could have been there! Each scored one point.
The witches, showering each other with curses, began the second round. The result is again the same - again two clumsily drawn skulls. The witches thus scored one more point. The public diligently recorded everything.
This continued in the future. When the round was over, the tired witches discovered that they had each scored six points. Draw again!
– Now let’s calculate the results and compare everything! – Grundy said triumphantly. – Each of the witches scored twenty-six points, and the golems scored twenty-eight points. So what do we have? And we have the result that golems have large quantity points!
A sigh of surprise swept through the rows of spectators. Excited spectators began to write columns of numbers on their pieces of paper, checking the accuracy of the count. During this time, many simply did not count the number of points scored, believing that they already knew the result of the game. Both witches began to growl with indignation, it is unclear who exactly they blamed for what had happened. The eyes of the demon Xant again lit up with a wary fire. His trust was justified!
“I ask you, dear audience, to pay attention to the fact,” Grundy raised his hand, demanding that the audience calm down, “that none of the golems won a single round.” But the final victory will still belong to one of us, the golems. The results will be more telling if the competition continues! I want to say, my dear viewers, that in the eternal duel my strategy will invariably turn out to be winning!
The demon Xanth listened with interest to what Grundy was saying. Finally, emitting clouds of steam, he opened his mouth:
– What exactly is your strategy?
– I call it “Be Firm but Fair”! - Grundy explained. – I start the game honestly, but then I start losing because I come across very specific partners. Therefore, in the first round, when it turns out that the Sea Witch begins to testify against me, I automatically remain the loser in the second round - and this continues until the end. The result may be different if the witch changes her tactics of playing the game. But since this couldn’t even occur to her, we continued to play according to the previous pattern. When I started playing with my double, he treated me well, and I treated him well in the next round of the game. Therefore, our game also went differently and somewhat monotonously, since we did not want to change tactics...
– But you haven’t won a single round! – the demon objected in surprise.
– Yes, and these witches haven’t lost a single round! – Grundy confirmed. – But victory does not automatically go to the one who has the remaining rounds. Victory goes to the one who scores the most points, but this is a completely different matter! I managed to score more points when I played with my double than when I played with the witches. Their selfish attitude brought them a momentary victory, but in the longer term, it turned out that it was because of this that they both lost the entire game. This happens often!

Game theory - a set of mathematical methods for resolving conflict situations (conflicts of interests). In game theory, a game is called mathematical model of a conflict situation. The subject of particular interest in game theory is the study of decision-making strategies of game participants under conditions of uncertainty. Uncertainty stems from the fact that two or more parties pursue opposing goals, and the results of any action of each party depend on the moves of the partner. At the same time, each party strives to make optimal decisions that realize the set goals to the greatest extent.

Game theory is most consistently applied in economics, where conflict situations arise, for example, in the relationship between supplier and consumer, buyer and seller, bank and client. The application of game theory can also be found in politics, sociology, biology, and military art.

From the history of game theory

History of game theory as an independent discipline began in 1944, when John von Neumann and Oscar Morgenstern published the book “The Theory of Games and Economic Behavior”. Although examples of game theory have been encountered before: the treatise of the Babylonian Talmud on the division of the property of a deceased husband between his wives, card games in the 18th century, the development of the theory of chess at the beginning of the 20th century, the proof of the minimax theorem of the same John von Neumann in 1928 year, without which there would be no game theory.

In the 50s of the 20th century, Melvin Drescher and Meryl Flood from Rand Corporation John Nash, the first to experimentally apply the prisoner's dilemma, developed the concept of Nash equilibrium in his works on the state of equilibrium in two-person games.

In 1965, Reinhard Salten published the book "The Treatment of Oligopoly in Game Theory on Demand" ("Spieltheoretische Behandlung eines Oligomodells mit Nachfrageträgheit"), with which the application of game theory in economics received a new driving force. A step forward in the evolution of game theory is associated with the work of John Maynard Smith, “Evolutionary Stable Strategy” (1974). The prisoner's dilemma was popularized in Robert Axelrod's 1984 book The Evolution of Cooperation. In 1994, John Nash, John Harsanyi and Reinhard Selten were awarded the Nobel Prize for their contributions to game theory.

Game theory in life and business

Let us dwell in more detail on the essence of the conflict situation (clash of interests) in the sense as it is understood in game theory for further modeling various situations in life and business. Let an individual be in a position that leads to one of several possible outcomes, and the individual has some personal preferences regarding these outcomes. But although he can to some extent control the variables that determine the outcome, he does not have complete power over them. Sometimes control is in the hands of several individuals who, like him, have some preferences in relation to possible outcomes, but in general the interests of these individuals are not consistent. In other cases, the final outcome may depend both on chance (which in legal sciences is sometimes called natural disasters), and from other individuals. Game theory systematizes observations of such situations and formulations general principles to guide reasonable actions in such situations.

In some respects, the name "game theory" is unfortunate, since it suggests that game theory deals only with socially insignificant encounters that occur in parlor games, but nevertheless the theory has a much broader meaning.

The following economic situation can give an idea of ​​the application of game theory. Suppose there are several entrepreneurs, each of whom strives to obtain maximum profit, while having only limited power over the variables that determine this profit. An entrepreneur has no power over variables that another entrepreneur controls, but which can greatly influence the income of the first. Treating this situation as a game may raise the following objection. In the game model, it is assumed that each entrepreneur makes one choice from the range of possible choices, and these single choices determine profits. Obviously, this almost cannot happen in reality, since in this case complex management apparatuses would not be needed in industry. There are simply a number of decisions and modifications of these decisions that depend on the choices made by other participants in the economic system (players). But in principle one can imagine some administrator anticipating all possible contingencies and detailing the action to be taken in each case, rather than solving each problem as it arises.

A military conflict, by definition, is a clash of interests in which neither side has complete control over the variables that determine the outcome, which is decided by a series of battles. You can simply consider the outcome to be a win or a loss and assign the numerical values ​​1 and 0 to them.

One of the simplest conflict situations that can be written down and resolved in game theory is a duel, which is a conflict between two players 1 and 2, having respectively p And q shots. For each player there is a function indicating the probability that the player's shot i at a point in time t will give a hit that will be fatal.

As a result, game theory comes to the following formulation of a certain class of conflicts of interests: there are n players, and each needs to choose one option from a hundred specific set, and when making a choice, the player has no information about the choices of other players. The player's possible choice area may contain elements such as "playing the ace of spades", "producing tanks instead of cars", or more generally, a strategy that defines all the actions to be taken in all possible circumstances. Each player is faced with a task: what choice should he make so that his private influence on the outcome brings him the greatest possible win?

Mathematical model in game theory and formalization of problems

As we have already noted, the game is a mathematical model of a conflict situation and requires the following components:

1. interested parties;
2. possible actions on each side;
3. interests of the parties.

The parties interested in the game are called players , each of them can take at least two actions (if the player has only one action at his disposal, then he does not actually participate in the game, since it is known in advance what he will take). The outcome of the game is called winning .

A real conflict situation is not always, but the game (in the concept of game theory) always proceeds according to certain rules , which precisely determine:

1. options for players' actions;
2. the amount of information each player has about their partner’s behavior;
3. the payoff that each set of actions leads to.

Examples of formalized games include football, card game, chess.

But in economics, a model of player behavior arises, for example, when several firms strive to take a more advantageous place in the market, several individuals try to divide some good (resources, finances) among themselves so that everyone gets as much as possible. Players in conflict situations in the economy, which can be modeled as a game, are firms, banks, individuals and other economic agents. In turn, in war conditions, the game model is used, for example, in choosing more best weapons(from existing or potential) to defeat the enemy or protect against attack.

The game is characterized by uncertainty of the outcome . The reasons for uncertainty can be divided into the following groups:

1. combinatorial (as in chess);
2. the influence of random factors (as in the game "heads or tails", dice, card games);
3. strategic (the player does not know what action the enemy will take).

Player strategy is a set of rules that determine his actions at each move depending on the current situation.

The purpose of game theory is to determine the optimal strategy for each player. Determining such a strategy means solving the game. Optimality of strategy is achieved when one of the players should get the maximum win, while the second one sticks to his strategy. And the second player should have a minimal loss if the first one sticks to his strategy.

Classification of games

1. Classification by number of players (game of two or more persons). Two-person games occupy a central place in all game theory. The core concept of game theory for two-person games is a generalization of the very significant idea of ​​equilibrium that naturally appears in two-person games. As for games n individuals, then one part of game theory is devoted to games in which cooperation between players is prohibited. In another part of game theory n individuals are assumed to be able to cooperate for mutual benefit (see later in this paragraph on non-cooperative and cooperative games).
2. Classification by the number of players and their strategies (the number of strategies is at least two, may be infinity).
3. Classification by amount of information relative to past moves: games with complete information and incomplete information. Let there be player 1 - buyer and player 2 - seller. If player 1 does not have complete information about the actions of player 2, then player 1 may not distinguish between the two alternatives between which he must make a choice. For example, choosing between two types of some product and not knowing that, according to some characteristics, the product A worse product B, player 1 may not see the difference between the alternatives.
4. Classification according to the principles of division of winnings : cooperative, coalition on the one hand and non-cooperative, non-coalition on the other hand. IN non-cooperative game , or otherwise - non-cooperative game , players choose strategies simultaneously without knowing which strategy the second player will choose. Communication between players is impossible. IN cooperative game , or otherwise - coalition game , players can form coalitions and take collective actions to increase their winnings.
5. Finite two-person zero-sum game or antagonistic game is a strategic game with complete information, which involves parties with opposing interests. Antagonistic games are matrix games .

A classic example from game theory is the prisoner's dilemma.

The two suspects are taken into custody and separated from each other. The district attorney is convinced that they committed a serious crime, but does not have enough evidence to charge them at trial. He tells each prisoner that he has two alternatives: confess to the crime the police believe he committed, or not confess. If both don't confess, the DA will charge them with some minor crime, such as petty theft or illegal possession of a weapon, and they will both receive a small sentence. If they both confess, they will be subject to prosecution, but he will not demand the harshest sentence. If one confesses and the other does not, then the one who confessed will have his sentence commuted for extraditing an accomplice, while the one who persists will receive “to the fullest.”

If this strategic task is formulated in terms of conclusion, then it boils down to the following:

Thus, if both prisoners do not confess, they will receive 1 year each. If both confess, each will receive 8 years. And if one confesses, the other does not confess, then the one who confessed will get off with three months in prison, and the one who does not confess will receive 10 years. The above matrix correctly reflects the prisoner's dilemma: everyone is faced with the question of whether to confess or not to confess. The game that the district attorney offers to the prisoners is non-cooperative game or otherwise - non-cooperative game . If both prisoners had the opportunity to cooperate (i.e. the game would be co-op or else coalition game ), then both would not confess and would receive a year in prison each.

Examples of using mathematical tools of game theory

We now move on to consider solutions to examples of common classes of games, for which there are research and solution methods in game theory.

An example of formalization of a non-cooperative (non-cooperative) game of two persons

In the previous paragraph, we already looked at an example of a non-cooperative (non-cooperative) game (prisoner's dilemma). Let's strengthen our skills. A classic plot inspired by “The Adventures of Sherlock Holmes” by Arthur Conan Doyle is also suitable for this. One can, of course, object: the example is not from life, but from literature, but Conan Doyle has not established himself as a science fiction writer! Classic also because the task was completed by Oskar Morgenstern, as we have already established, one of the founders of game theory.

Example 1. An abbreviated summary of a fragment of one of “The Adventures of Sherlock Holmes” will be given. According to the well-known concepts of game theory, create a model of a conflict situation and formally write down the game.

Sherlock Holmes intends to travel from London to Dover with the further goal of getting to the continent (European) in order to escape from Professor Moriarty, who is pursuing him. Having boarded the train, he saw Professor Moriarty on the station platform. Sherlock Holmes admits that Moriarty can choose a special train and overtake it. Sherlock Holmes has two alternatives: continue the journey to Dover or get off at Canterbury station, which is the only intermediate station on his route. We accept that his opponent is intelligent enough to determine Holmes' capabilities, so he has the same two alternatives. Both opponents must choose a station to get off the train at, without knowing what decision each of them will make. If, as a result of making a decision, both end up at the same station, then we can definitely assume that Sherlock Holmes will be killed by Professor Moriarty. If Sherlock Holmes reaches Dover safely, he will be saved.

Solution. We can consider Conan Doyle's heroes as participants in the game, that is, players. Available to every player i (i=1,2) two pure strategies:

• get off at Dover (strategy si1 ( i=1,2) );
• get off at an intermediate station (strategy si2 ( i=1,2) )

Depending on which of the two strategies each of the two players chooses, a special combination of strategies will be created as a pair s = (s1 , s 2 ) .

Each combination can be associated with an event - the outcome of the attempted murder of Sherlock Holmes by Professor Moriarty. We create a matrix of this game with possible events.

Under each of the events there is an index indicating the acquisition of Professor Moriarty, and calculated depending on the salvation of Holmes. Both heroes choose a strategy at the same time, not knowing what the enemy will choose. Thus, the game is non-cooperative because, firstly, the players are on different trains, and secondly, they have opposing interests.

An example of formalization and solution of a cooperative (coalition) game n persons

At this point, the practical part, that is, the process of solving an example problem, will be preceded by a theoretical part, in which we will become familiar with the concepts of game theory for solving cooperative (non-cooperative) games. For this task, game theory suggests:

• characteristic function (to put it simply, it reflects the magnitude of the benefit of uniting players into a coalition);
• the concept of additivity (the property of quantities, consisting in the fact that the value of a quantity corresponding to the whole object is equal to the sum of the values ​​of quantities corresponding to its parts in a certain class of partitions of the object into parts) and superadditivity (the value of a quantity corresponding to the whole object is greater than the sum of the values ​​of quantities, corresponding to its parts) of the characteristic function.

The superadditivity of the characteristic function suggests that joining a coalition is beneficial to the players, since in this case the value of the coalition's payoff increases with the number of players.

To formalize the game, we need to introduce formal notations for the above concepts.

For Game n let us denote the set of all its players as N= (1,2,...,n) Any non-empty subset of the set N let's denote it as T(including itself N and all subsets consisting of one element). There is a lesson on the site " Sets and operations on sets", which opens in a new window when you click on the link.

The characteristic function is denoted as v and its domain of definition consists of possible subsets of the set N. v(T) - the value of the characteristic function for a particular subset, for example, the income received by a coalition, possibly including one consisting of one player. This is important because game theory requires checking the presence of superadditivity for the values ​​of the characteristic function of all disjoint coalitions.

For two non-empty subset coalitions T1 And T2 The additivity of the characteristic function of a cooperative (coalition) game is written as follows:

And superadditivity is like this:

Example 2. Three music school students work part-time in different clubs; they receive their income from club visitors. Determine whether it is profitable for them to join forces (if so, under what conditions), using the concepts of game theory to solve cooperative games n persons, with the following initial data.

On average, their revenue per evening was:

• the violinist has 600 units;
• the guitarist has 700 units;
• the singer has 900 units.

In an attempt to increase revenue, students created various groups over the course of several months. The results showed that by teaming up, they could increase their evening revenue by:

• violinist + guitarist earned 1500 units;
• violinist + singer earned 1800 units;
• guitarist + singer earned 1900 units;
• violinist + guitarist + singer earned 3000 units.

Solution. In this example, the number of players in the game n= 3, therefore, the domain of definition of the characteristic function of the game consists of 2³ = 8 possible subsets of the set of all players. Let us list all possible coalitions T:

• coalitions of one element, each of which consists of one player - a musician: T{1} , T{2} , T{3} ;
• coalition of two elements: T{1,2} , T{1,3} , T{2,3} ;
• a coalition of three elements: T{1,2,3} .

We will assign a serial number to each player:

• violinist - 1st player;
• guitarist - 2nd player;
• singer - 3rd player.

Based on the problem data, we determine the characteristic function of the game v:

v(T(1)) = 600 ; v(T(2)) = 700 ; v(T(3)) = 900 ; these values ​​of the characteristic function are determined based on the payoffs of the first, second and third players, respectively, when they do not unite in a coalition;

v(T(1,2)) = 1500 ; v(T(1,3)) = 1800 ; v(T(2,3)) = 1900 ; these values ​​of the characteristic function are determined by the revenue of each pair of players united in a coalition;

v(T(1,2,3)) = 3000 ; this value of the characteristic function is determined by the average revenue in the case when the players united in threes.

Thus, we have listed all possible coalitions of players; there are eight of them, as it should be, since the domain of definition of the characteristic function of the game consists of exactly eight possible subsets of the set of all players. This is what game theory requires, since we need to check the presence of superadditivity for the values ​​of the characteristic function of all disjoint coalitions.

How are the superadditivity conditions satisfied in this example? Let's determine how players form disjoint coalitions T1 And T2 . If some players are part of a coalition T1 , then all other players are part of the coalition T2 and by definition, this coalition is formed as the difference of the entire set of players and the set T1 . Then if T1 - a coalition of one player, then in a coalition T2 there will be second and third players if in a coalition T1 there will be the first and third players, then the coalition T2 will consist of only the second player, and so on.

BELARUSIAN STATE UNIVERSITY

FACULTY OF ECONOMICS

DEPARTMENT…

Game theory and its application in economics

Course project

2nd year student

Department "Management"

Scientific director

Minsk, 2010

1. Introduction. p.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 used literature p.24

1. INTRODUCTION

In practice, there is often a need 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 makes it possible to find the best solution for the behavior of participants who are required 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 considered as a tool that helps improve the efficiency of planning and management decisions. It has great importance when solving problems in industry, agriculture, transport, trade, especially when concluding agreements with foreign partners at any level. Thus, it is possible to determine scientifically based levels of reduction in retail prices and the optimal level of inventory, solve the problems of excursion services and the selection of new lines of urban transport, the problem of planning the procedure for organizing the exploitation of mineral deposits in the country, etc. The problem of selecting plots of land for agricultural crops has become a classic one. The game theory method can be used in sample surveys of finite populations and in testing statistical hypotheses.

Game theory is a mathematical method for studying optimal strategies in games. A game is 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 that can lead to winning or losing - 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, or more precisely, operations research. Most often, game theory methods are used in economics, and 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 interest in intelligent agents.

Game theory originates from neoclassical economics. The mathematical aspects and applications of the theory were first outlined in the classic 1944 book by John von Neumann and Oscar Morgenstern, The Theory of Games and Economic Behavior.

This area of ​​mathematics has found some reflection in public culture. In 1998, the American writer and journalist Sylvia Nasar 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, Nobel laureate in economics in 2005.

Nobel laureates in economics for their achievements in the field of game theory were: Robert Aumann, Reinhard Selten, 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 players’ actions; 2) the amount of information each player has about the behavior of their partners; 3) the gain that each set of actions leads to. Typically, winning (or losing) can be quantified; for example, you can evaluate a loss as zero, a win as one, and a draw as ½.

A game is called doubles if it involves two players, and multiple if there are more than two players.

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 game, it is enough to indicate the value of one of them. If we denote a as the gain of one of the players, b as the gain of the other, then for a zero-sum game b = -a, so it is enough 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 player’s conscious choice 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 the future, 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 at 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, it is in principle 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 specific strategy, which can be specified as a list of rules or a program. (This way you can play the game using a computer.) A game is called 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, you should choose a strategy for each player that satisfies the optimality condition, i.e. one of the players should receive the maximum winnings when the other sticks to his strategy. At the same time, the second player should have a minimal 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 player to abandon his strategy in this game.

If the game is repeated quite a few times, then players may not be interested in winning and losing in each specific game, but in the average winning (loss) in all games.

The goal of game theory is to determine the optimal strategy for each player. When choosing an optimal strategy, it is natural to assume that both players behave reasonably in terms of their interests. The most important limitation of game theory is the naturalness of winning as an indicator of efficiency, while in most real economic problems there is more than one indicator of efficiency. In addition, in economics, as a rule, problems arise in which the interests of partners are not necessarily antagonistic.

3. Presentation of games

Games are strictly defined mathematical objects. A game is formed by the players, a set of strategies for each player, and the players' payoffs, or payoffs, 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

Game "Ultimatum" in extensive form

Games in extensive, or expanded, form are represented in the form of an oriented tree, where each vertex corresponds to the situation when the player chooses his strategy. Each player is assigned whole level peaks 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 8 and 2 points respectively.

The extensive form is very visual and is especially useful for representing games with more than two players and games with sequential moves. If participants make simultaneous moves, then the corresponding vertices are either connected by a dotted line or outlined with a solid line.

Normal form

	Player 2 strategy 1	Player 2 strategy 2
Player 1 strategy 1	4 , 3	–1 , –1
Player 1 strategy 2	0 , 0	3 , 4
Normal form for a game with 2 players, each with 2 strategies.

In normal, or strategic, form, the game is described by a payoff matrix. Each side (more precisely, dimension) of the matrix is ​​a player, the rows determine the strategies of the first player, and the columns determine the strategies of the second. At the intersection of the two strategies, you can see the winnings that players will receive. In the example on the right, if player 1 chooses the first strategy, and player 2 chooses the second strategy, then at the intersection we see (−1, −1), which means that as a result of the move, both players lost one point.

The players chose strategies with the maximum result for themselves, but lost due to ignorance of the other player’s move. Typically, normal form represents games in which moves are made simultaneously, or at least in which all players are assumed to be unaware of what the other participants are doing. Such games with incomplete information will be discussed 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, a so-called characteristic function is used, which determines the payoff of each coalition of players. It is assumed that the gain of the empty coalition is zero.

The basis 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 options for possible coalitions (namely 2N, where N is the number of players), the gain 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 representation can be used for all games, including those without transferable utility. There are currently ways to convert any game from normal form to characteristic form, but the reverse transformation is not possible in all cases.

4. Types of games

Cooperative and non-cooperative.

A game is called cooperative or coalition if players can form groups, taking on certain obligations to other players and coordinating their actions. This differs from non-cooperative games in which everyone must play for themselves. Entertainment games are rarely cooperative, but such mechanisms are not uncommon in everyday life.

It is often assumed that what makes cooperative games different is the ability for 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 great detail and produce more accurate results. Cooperatives consider the game process 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 of 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.

Game theory- theory of mathematical models for making optimal decisions in conflict conditions. Since the parties involved in most conflicts are interested in hiding their intentions from the enemy, decision-making in conflict situations usually occurs under conditions of uncertainty. On the contrary, the uncertainty factor can be interpreted as an opponent of the subject making the decision (thus, decision-making under conditions of uncertainty can be understood as decision-making under conditions of conflict). In particular, many statements of mathematical statistics are naturally formulated as game-theoretic ones.

Game theory is a branch of applied mathematics that is used in the social sciences (mostly economics), biology, political science, computer science (mainly for artificial intelligence) and philosophy. Game theory attempts to mathematically capture behavior in strategic situations, in which the success of the subject making the choice depends on the choices of other participants. If at first the analysis of games in which one of the opponents wins at the expense of others (zero-sum games) developed, then subsequently they began to consider a wide class of interactions that were classified according to certain criteria. Today, “game theory is something like an umbrella or universal theory for the rational side of the social sciences, where social can be understood broadly, including both human and non-human players (computers, animals, plants)” (Robert Aumann, 1987)

This branch of mathematics has received some reflection in popular culture. In 1998, American writer and journalist Sylvia Nasar published a book about the life of John Nash, a Nobel laureate in economics for his achievements in game theory, and in 2001, the film A Beautiful Mind was based on the book. (Thus, game theory is one of the few branches of mathematics in which you can receive a Nobel Prize). Some American television shows, e.g. Friend or Foe, Alias or NUMBERS periodically use game theory in their releases.

John Nash is a mathematician and Nobel laureate known to the general public thanks to the film A Beautiful Mind.

Game theory concept

The logical basis of game theory is the formalization of three concepts included in its definition and which are fundamental to the entire theory:

• Conflict,
• Making decisions in conflict
• Optimality of the decision made.

These concepts are considered in game theory in the broadest sense. Their formalizations respond with a meaningful idea of ​​the corresponding objects.

If we name the participants in the conflict action coalitions(denoting their set as D, the possible actions of each of the action coalitions are its strategies(the set of all action coalition strategies K denoted as S), the results of the conflict - situations(the set of all situations is denoted as S; it is believed that each situation develops as a result of the choice of each of the coalitions to act on some of its strategies, so that ), parties concerned - coalitions of interests(there are many of them - I) and, finally, talk about possible advantages for each coalition of interests K one situation s" in front of another s"(this fact is denoted as ), then the conflict as a whole can be described as a system

.

Such a system representing conflict is called game. Specification of the components that define the game leads to different classes of games.

Classification of games

There are separate classes of non-cooperative games:

• zero-sum games, including matrix games and unit square games.
• dynamic games, including differential games,
• recursive games,
• survival games

and others also refer to non-cooperative games.

Mathematical apparatus

Game theory widely uses various mathematical methods and results from probability theory, classical analysis, functional analysis (fixed point theorems are especially important), combinatorial topology, the theory of differential and integral equations, and others. The specifics of game theory contribute to the development of various mathematical areas (for example, the theory of convex sets, linear programming, etc.).

Decision making in game theory is considered to be the choice of an action by a coalition, or, in particular, the choice by a player of some of its strategies. This choice can be imagined as a one-time action and can be formally raised to the selection of an element from a set. Games with such an understanding of the choice of strategies are called games in normal form. They are contrasted with dynamic games, in which the choice of strategy is a process that occurs over a period of time, which is accompanied by the expansion and contraction of possibilities, the acquisition and loss of information about current state affairs, etc. Formally, a strategy in such a game is a function defined on the set of all information states of the decision-maker. The uncritical use of “freedom of choice” strategies can lead to paradoxical phenomena.

Optimality and solutions

The question of formalizing the concept of optimality is very complex. There is no single idea of ​​optimality in game theory, so we have to consider several principles of optimality. The scope of application of each of the optimality principles used in game theory is limited to relatively narrow classes of games, or concerns limited aspects of their consideration.

Each of these principles is based on certain intuitive ideas about the optimum, as something “sustainable” or “fair”. The formalization of these ideas gives the requirements for the optimum and have the nature of axioms.

Among these requirements there may be those that contradict each other (for example, it is possible to show conflicts in which the parties are forced to be content with small gains, because big wins can only be achieved in uncertain situations); Therefore, a single principle of optimality cannot be formulated in game theory.

Situations (or sets of situations) that satisfy certain optimality requirements in a certain game are called decisions this game. Since the idea of ​​optimality is not unambiguous, there were outcomes of games in different meanings. Creating definitions of game solutions, establishing their existence, and developing ways to actually find them are the three main issues of modern game theory. Close to them are questions about the uniqueness of solutions to games, about the existence in certain classes of games of solutions that have certain predetermined properties.

Story

As a mathematical discipline, game theory originated at the same time as probability theory in the 17th century, but saw little development for nearly 300 years. The first significant work on game theory should be considered the article by J. von Neumann “Towards the Theory of Strategic Games” (1928), and with the publication of the monograph by American mathematicians J. von Neumann and O. Morgenstern “Game Theory and Economic Behavior” (1944), game theory emerged as an independent mathematical discipline. Unlike other branches of mathematics, which have a predominantly physical or physical-technological origin, game theory from the very beginning of its development was aimed at solving problems arising in economics (namely, in a competitive economy).

Subsequently, the ideas, methods and results of game theory began to be applied in other areas of knowledge dealing with conflicts: in military affairs, in matters of morality, in the study of mass behavior of individuals with different interests (for example, in issues of population migration, or when considering biological control for existence). Game-theoretic methods for making optimal decisions under conditions of uncertainty can be widely used in medicine, economics and social planning and forecasting, in a number of issues of science and technology. Sometimes game theory is referred to as the mathematical apparatus of cybernetics, or the theory of operations research.