This article discusses the application of game theory in economics. Game theory is a branch of mathematical economics. It develops recommendations on the rational action of the participants in the process when their interests do not coincide. Game theory helps businesses adopt optimal solution in conditions conflict situation.
- Active operations of commercial banks and their accounting
- Improving the formation of a capital repair fund in apartment buildings
- Legal regulation of the issues of assessing the quality of public (municipal) services provided in Russia
Game theory and economics are inextricably linked, as game theory problem solving methods help determine the best strategy for various economic situations. So how is the concept of "game theory" characterized?
Game theory is a mathematical theory of decision making under conflict conditions. Game theory is an important part of operations research theory that studies the issues of decision making in conflict situations.
Game theory is a branch of mathematical economics. The goal of game theory is to develop recommendations for the rational action of the participants in the process when their interests do not coincide, that is, in a conflict situation. The game is a model of a conflict situation. The players in the economy are the partners who take part in the conflict. The result of conflict is win or loss.
In general, the conflict takes place in different areas of human interest: in economics, sociology, political science, biology, cybernetics, military affairs. Most often, game theory and conflict situations are applied in economics. For each player, there is a specific set of strategies that the player can apply. Intersecting, the strategies of several players create certain situation, where each player receives a certain result (win or loss). When choosing a strategy, it is important to consider not only getting the maximum gain for yourself, but also the possible steps of the enemy, and their impact on the situation as a whole.
In order to improve the quality, as well as the efficiency of economic decisions made in the conditions of market relations and uncertainty, game theory methods can be reasonably applied.
In economic situations, games may have complete information or incomplete information. Most often, economists are faced with complete information for decision making. Therefore, it is necessary to make decisions in conditions of uncertainty, as well as in conditions of a certain risk. When solving economic problems (situations), one usually encounters one-move and multi-move games. The number of strategies can be finite or infinite.
Game theory in economics mainly uses matrix or rectangular games, for which a payoff matrix is compiled (Table 1).
Table 1. The payoff matrix of the game
Should be defined this concept. The payoff matrix of the game is a matrix that shows the payment of one player to another, provided that the first player chooses strategy Ai, the second - Bi.
What is the goal of solving economic problems with the help of game theory? To solve an economic problem is to find the optimal strategy for the first and second players and find the price of the game.
Let's solve the economic problem, compiled by me.
In city G, there are two competing companies (Sladkiy Mir and Sladkoezhka) that are engaged in the production of chocolate. Both companies can produce milk chocolate and dark chocolate. Let's designate the strategy of the "Sweet world" company as Аi, the "Sweet tooth" company - Вi. We calculate the efficiency for all possible combinations of the strategies of the companies "Sweet world" and "Sweet tooth" and build a payment matrix (Table 2).
Table 2. The payoff matrix of the game
This payoff matrix does not have a saddle point, so it is solved in mixed strategies.
U1 \u003d (a22-a21) / (a11 + a22-a21-a12) \u003d (6-3) / (5 + 6-3-4) \u003d 0.75.
U2 \u003d (a11-a12) / (a11 + a22-a21-a12) \u003d (5-4) / (5 + 6-3-4) \u003d 0.25.
Z1 \u003d (a22-a12) / (a11 + a22-a21-a12) \u003d (6-4) / (5 + 6-3-4) \u003d 0.4.
Z2 \u003d (a11-a21) / (a11 + a22-a21-a12) \u003d (5-3) / (5 + 6-3-4) \u003d 0.6.
Game price = (a11*a22-a12*a21) / (a11+a22-a21-a12) = (5*6-4*3) / (5+6-3-4) = 4.5.
We can say that the company "Sladkiy Mir" should distribute the production of chocolate as follows: 75% of the total production should be given to the production of milk chocolate, and 25% to the production of dark chocolate. The Sladkoezhka company should produce 40% milk chocolate and 60% bitter chocolate.
Game theory deals with decision-making in conflict situations by two or more reasonable opponents, each of which seeks to optimize their decisions at the expense of others.
Thus, in this article, the application of game theory in economics was considered. In economics, there are often moments when it is necessary to make an optimal decision, and there are several options for making decisions. Game theory helps to make decisions in a conflict situation. Game theory in economics can help define optimal release products for the enterprise, optimal payment of insurance premiums, etc.
Bibliography
- Belolipetsky, A. A. Economic and mathematical methods [Text]: textbook for students. Higher Proc. Institutions / A. A. Belolipetsky, V. A. Gorelik. - M.: Publishing Center "Academy", 2010. - 368 p.
- Luginin, O. E. Economic and mathematical methods and models: theory and practice with problem solving [Text]: tutorial/ O. E. Luginin, V. N. Fomishina. - Rostov n / D: Phoenix, 2009. - 440 p.
- Nevezhin, V.P. Game Theory. Examples and tasks [Text]: textbook / V. P. Nevezhin. – M.: FORUM, 2012. – 128 p.
- Sliva, I. I. Application of the game theory method for solving economic problems [Text] / I. I. Sliva // Proceedings of the Moscow State Technical University MAMI. - 2013. - No. 1. - S. 154-162.
As a result of studying this chapter, the student should:
know
Concepts of games based on the principle of dominance, Nash equilibrium, what is backward induction, etc.; conceptual approaches to solving the game, the meaning of the concept of rationality and equilibrium in the framework of the interaction strategy;
be able to
Distinguish games in strategic and expanded forms, build a "game tree"; formulate game models of competition for various types of markets;
own
Methods for determining the outcome of the game.
Games: basic concepts and principles
The first attempt to create a mathematical theory of games was made in 1921 by E. Borel. As an independent field of science, game theory was first systematically presented in the monograph "Game Theory and Economic Behavior" by J. von Neumann and O. Morgenstern in 1944. Since then, many sections economic theory(for example, the theory of imperfect competition, the theory of economic incentives, etc.) developed in close contact with game theory. Game theory is also successfully applied in the social sciences (for example, the analysis of voting procedures, the search for equilibrium concepts that determine the cooperative and non-cooperative behavior of individuals). As a general rule, voters reject candidates representing extreme points view, but when choosing one of the two candidates offering different compromise solutions, there is a struggle. Even Rousseau's idea of evolution from "natural freedom" to "civil freedom" formally corresponds to the point of view of cooperation from the point of view of game theory.
The game- this is an idealized mathematical model of the collective behavior of several persons (players), whose interests are different, which gives rise to a conflict. The conflict does not necessarily imply the presence of antagonistic contradictions of the parties, but is always associated with a certain kind of disagreement. A conflict situation will be antagonistic if an increase in the payoff of one of the parties by a certain amount leads to a decrease in the payoff of the other side by the same amount and vice versa. Antagonism of interests generates a conflict, and the coincidence of interests reduces the game to coordination of actions (cooperation).
Examples of a conflict situation are situations that develop in the relationship between the buyer and the seller; in the conditions of competition of various firms; in the course of hostilities, etc. Ordinary games are also examples of games: chess, checkers, card games, parlor games, etc. (hence the name "game theory" and its terminology).
In most games arising from analysis financial and economic, managerial situations, the interests of the players (sides) are neither strictly antagonistic nor absolutely coinciding. The buyer and seller agree that it is in their common interest to agree on a sale, but they bargain vigorously to select a specific price within the limits of mutual advantage.
Game theory is a mathematical theory of conflict situations.
The game differs from the real conflict in that it is conducted according to certain rules. These rules establish the sequence of moves, the amount of information each side has about the behavior of the other, and the outcome of the game depending on the situation. The rules also establish the end of the game, when a certain sequence of moves has already been made, and no more moves are allowed.
Game theory, like any mathematical model, has its limitations. One of them is the assumption of complete (ideal) reasonableness of opponents. In a real conflict, often the best strategy is to guess what the enemy is stupid about and use this stupidity to your advantage.
Another disadvantage of game theory is that each of the players must know all the possible actions (strategies) of the opponent, it is only known which of them he will use in a given game. In a real conflict, this is usually not the case: the list of all possible enemy strategies is precisely unknown, and the best solution in a conflict situation will often be to go beyond the strategies known to the enemy, to "stupefy" him with something completely new, unforeseen.
Game theory does not include the element of risk that inevitably accompanies smart decisions in real conflicts. It determines the most cautious, reinsurance behavior of the participants in the conflict.
In addition, in game theory, optimal strategies are found with respect to one indicator (criterion). In practical situations, it is often necessary to take into account not one, but several numerical criteria. A strategy that is optimal in one measure may not be optimal in another.
Being aware of these limitations and therefore not blindly adhering to the recommendations given by game theories, it is still possible to develop a completely acceptable strategy for many real conflict situations.
Currently, scientific research is being carried out aimed at expanding the areas of application of game theory.
The following definitions of the elements that make up the game are found in the literature.
Players- these are the subjects involved in the interaction, represented in the form of a game. In our case, these are households, firms, government. However, in the case of uncertainty of external circumstances, it is quite convenient to represent the random components of the game, which do not depend on the behavior of the players, as actions of "nature".
Rules of the game. The rules of the game are the sets of actions or moves available to the players. In this case, actions can be very diverse: decisions of buyers about the volumes of purchased goods or services; firms - on the volume of output; the level of taxes imposed by the government.
Determining the outcome (result) of the game. For each combination of players' actions, the outcome of the game is set almost mechanically. The result can be: the composition of the consumer basket, the vector of the firm's outputs, or a set of other quantitative indicators.
Winnings. The meaning attached to the concept of winning may differ for different types games. At the same time, it is necessary to clearly distinguish between gains measured on an ordinal scale (for example, the level of utility), and values for which interval comparison makes sense (for example, profit, welfare level).
Information and expectations. Uncertainty and constantly changing information can have an extremely serious impact on the possible outcomes of an interaction. That is why it is necessary to take into account the role of information in the development of the game. In this regard, the concept information set player, i.e. the totality of all information about the state of the game that he possesses in key points time.
When considering players' access to information, the intuitive idea of common knowledge, or publicity, meaning the following: a fact is well known if all players are aware of it and all players know that other players also know about it.
For cases in which the application of the concept of common knowledge is not enough, the concept of individual expectations participants - ideas about how the game situation is on this stage.
In game theory, it is assumed that the game consists of moves, performed by players simultaneously or sequentially.
Moves are personal and random. The move is called personal, if the player consciously chooses it from a set of possible options for action and implements it (for example, any move in a chess game). The move is called random, if his choice is not made by the player, but by some random selection mechanism (for example, based on the results of tossing a coin).
The set of moves taken by the players from the beginning to the end of the game is called party.
One of the basic concepts of game theory is the concept of strategy. strategy player is called a set of rules that determine the choice of a variant of action for each personal move, depending on the situation that has developed during the game. In simple (one-move) games, when a player can make only one move in each game, the concept of strategy and possible option actions match. In this case, the totality of the player's strategies covers all his possible actions, and any possible for the player i action is his strategy. In complex (multi-move) games, the concepts of "variant of possible actions" and "strategy" may differ from each other.
The player's strategy is called optimal, if it provides a given player with the maximum possible average gain or minimum possible average loss, regardless of what strategies the opponent uses, when the game is repeated many times. Other optimality criteria can also be used.
It is possible that the strategy that provides the maximum payoff does not have another important representation of optimality, such as the stability (equilibrium) of the solution. The solution of the game is sustainable(equilibrium) if the strategies corresponding to this decision form a situation that none of the players is interested in changing.
We repeat that the task of game theory is to find optimal strategies.
The classification of games is shown in fig. 8.1.
- 1. Depending on the types of moves, games are divided into strategic and gambling. gambling games consist only of random moves, which game theory does not deal with. If, along with random moves, there are personal moves or all moves are personal, then such games are called strategic.
- 2. Depending on the number of players, games are divided into doubles and multiples. AT doubles game the number of participants is two multiple- more than two.
- 3. Participants in the multiple game may form coalitions, either permanent or temporary. According to the nature of the relationship between the players, the games are divided into non-cooperative, coalition and cooperative.
Non-coalition called games in which players do not have the right to enter into agreements, form coalitions, and the goal of each player is to obtain the greatest possible individual gain.
Games in which the actions of the players are aimed at maximizing the payoffs of collectives (coalitions) without their subsequent division between the players are called coalition.
Rice. 8.1.
Exodus cooperative game is the division of the coalition's payoff, which arises not as a result of certain actions of the players, but as a result of their predetermined agreements.
In accordance with this, in cooperative games, not situations are compared in terms of preference, as is the case in non-cooperative games, but divisions; and the comparison is not limited to consideration of individual gains, but is more complex.
- 4. According to the number of strategies for each player, games are divided into final(the number of strategies for each player is finite) and endless(the set of strategies for each player is infinite).
- 5. According to the amount of information available to the players regarding past moves, games are divided into games with complete information(all information about previous moves is available) and incomplete information. Examples of games with complete information are chess, checkers, and the like.
- 6. According to the type of description, games are divided into positional games (or games in expanded form) and games in normal form. Positional games are given in the form of a game tree. But any positional game can be reduced to normal form, in which each player makes only one independent move. In positional games, moves are made in discrete moments time. Exist differential games, in which moves are made continuously. These games study the problems of pursuit of a controlled object by another controlled object, taking into account the dynamics of their behavior, which is described by differential equations.
There are also reflective games, which consider situations with regard to the mental reproduction of the possible course of action and behavior of the enemy.
7. If any possible game of some game has zero sum of payoffs of all N players(), then talk about zero sum game. Otherwise, the games are called non-zero sum games.
Clearly, the zero-sum pair game is antagonistic since the gain of one player is equal to the loss of the second, and, consequently, the goals of these players are directly opposite.
A finite pairwise zero-sum game is called matrix game. Such a game is described by a payoff matrix in which the payoffs of the first player are given. The row number of the matrix corresponds to the number of the applied strategy of the first player, the column corresponds to the number of the applied strategy of the second player; at the intersection of the row and column is the corresponding gain of the first player (loss of the second player).
A finite pair game with a non-zero sum is called bimatrix game. Such a game is described by two payoff matrices, each for the corresponding player.
Let's take the following example. Game "Offset". Let player 1 be a student preparing for the test, and player 2 be the teacher taking the test. Let's assume that a student has two strategies: A1 - prepare well for the test; A 2 - do not prepare. The teacher also has two strategies: B1 - put a test; B 2 - do not set off. The estimation of players' payoff values can be based, for example, on the following considerations reflected in the payoff matrices:
This game, in accordance with the above classification, is strategic, paired, non-cooperative, finite, described in normal form, with a non-zero sum. More briefly, this game can be called bimatrix.
The task is to determine the optimal strategies for the student and for the teacher.
Another example of the well-known bimatrix game Prisoner's Dilemma.
Each of the two players has two strategies: A 2 and B 2 – aggressive behavior strategies, a A i and B i - peaceful behavior. Suppose that "peace" (both players are peaceful) is better for both players than "war". The case when one player is aggressive and the other is peaceful is more profitable for the aggressor. Let the payoff matrices of players 1 and 2 in this bimatrix game have the form
For both players, the aggressive strategies A2 and B2 dominate the peaceful strategies Ax and B v Thus, the only equilibrium in dominating strategies has the form (A2, B 2), i.e. it is postulated that the result of non-cooperative behavior is war. At the same time, the outcome (A1, B1) (world) gives a larger payoff for both players. Thus, non-cooperative egoistic behavior comes into conflict with collective interests. Collective interests dictate the choice of peaceful strategies. At the same time, if the players do not exchange information, war is the most likely outcome.
In this case, the situation (A1, B1) is Pareto optimal. However, this situation is unstable, which leads to the possibility of violation of the established agreement by the players. Indeed, if the first player violates the agreement, and the second does not, then the payoff of the first player will increase to three, and the second one will drop to zero, and vice versa. Moreover, each player who does not violate the agreement loses more if the second player violates the agreement than if they both violate the agreement.
There are two main forms of play. game in extensive form represented as a decision-making "tree" diagram, with the "root" corresponding to the starting point of the game, and the beginning of each new "branch", called node,- the state reached at this stage with given actions already taken by the players. Each end node - each end point of the game - is assigned a payoff vector, one component for each player.
strategic, otherwise called normal, form The game representation corresponds to a multidimensional matrix, with each dimension (rows and columns in the two-dimensional case) including a set of possible actions for one agent.
A separate cell of the matrix contains a vector of payoffs corresponding to a given combination of player strategies.
On fig. 8.2 presents an extensive form of the game, and in table. 8.1 - strategic form.
Rice. 8.2.
Table 8.1. Game with simultaneous decision-making in a strategic form
There is enough detailed classification constituent parts game theory. One of the most general criteria for such a classification is the division of game theory into the theory of non-cooperative games, in which the subjects of decision-making are the individuals themselves, and the theory of cooperative games, in which the subjects of decision-making are groups or coalitions of individuals.
Non-cooperative games are usually presented in normal (strategic) and expanded (extensive) forms.
- Vorobyov N. N. Game theory for eco-yomists-cyberists. Moscow: Nauka, 1985.
- Wentzel E. S. Operations research. Moscow: Nauka, 1980.
BELARUSIAN STATE UNIVERSITY
FACULTY OF ECONOMICS
CHAIR…
Game theory and its application in economics
course project
2nd year student
departments "Management"
scientific adviser
Minsk, 2010
1. Introduction. page 3
2. Basic concepts of game theory p.4
3. Presentation of games page 7
4. Types of games p.9
5. Application of game theory in economics p.14
6. Problems of practical application in management p.21
7. Conclusion p.23
List of references page 24
1. INTRODUCTION
In practice, it often becomes necessary to coordinate the actions of firms, associations, ministries and other project participants in cases where their interests do not coincide. In such situations, game theory makes it possible to find the best solution for the behavior of participants who are obliged to coordinate actions in the event of a conflict of interests. Game theory is increasingly penetrating the practice of economic decisions and research. It can be viewed as a tool to help improve the efficiency of planning and management decisions. This is of great importance when solving problems in industry, agriculture, transport, trade, especially when concluding contracts with foreign partners at any level. Thus, it is possible to determine scientifically based levels of retail price reduction and optimal level commodity stocks, solve the problems of excursion services and the selection of new lines of urban transport, the task of planning the procedure for organizing the exploitation of mineral deposits in the country, etc. The task of choosing land for agricultural crops has become a classic. The method of game theory can be used in sample surveys of finite populations, in testing statistical hypotheses.
Game theory is a mathematical method for studying optimal strategies in games. The game is understood as a process in which two or more parties participate, fighting for the realization of their interests. Each side has its own goal and uses some strategy, which can lead to a win or a loss - depending on the behavior of other players. Game theory helps to choose best strategies taking into account ideas about other participants, their resources and their possible actions.
Game theory is a branch of applied mathematics, more precisely, operations research. Most often, the methods of game theory are used in economics, a little less in other areas. social sciences- sociology, political science, psychology, ethics and others. Since the 1970s, it has been adopted by biologists to study animal behavior and the theory of evolution. It is very important for artificial intelligence and cybernetics, especially with the manifestation of interest in intelligent agents.
Game theory has its origins in neoclassical economics. The mathematical aspects and applications of the theory were first presented in the classic 1944 book Theory of Games and Economic Behavior by John von Neumann and Oscar Morgenstern.
This area of mathematics has found some reflection in public culture. In 1998, the American writer and journalist Sylvia Nazar published a book about the fate of John Nash, 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 achievements in the field of game theory are: Robert Aumann, Reinhard Zelten, John Nash, John Harsanyi, Thomas Schelling.
2. BASIC CONCEPTS OF GAME THEORY
Let's get acquainted with the basic concepts of game theory. The mathematical model of a conflict situation is called a game, the parties involved in the conflict are called players, and the outcome of the conflict is called a win. For each formalized game, rules are introduced, i.e. a system of conditions that determines: 1) options for the players' actions; 2) the volume of information of each player about the behavior of partners; 3) the payoff to which each set of actions leads. Typically, gain (or loss) can be quantified; for example, you can evaluate a loss by zero, a win by one, and a draw by ½.
A game is called a pair game if two players participate in it, and multiple if the number of players is more than two.
The game is called a zero-sum game, or antagonistic, if the gain of one of the players is equal to the loss of the other, i.e. for complete task game, it is enough to indicate the value of one of them. If we denote a - the payoff of one of the players, b - the payoff of the other, then for a zero-sum game b = -a, so it suffices to consider, for example, a.
The choice and implementation of one of the actions provided for by the rules is called the player's move. Moves can be personal and random. A personal move is a conscious choice by a player of one of the possible actions (for example, a move in a chess game). A random move is a randomly chosen action (for example, choosing a card from a shuffled deck). In what follows, we will consider only the personal moves of the players.
A player's strategy is a set of rules that determine the choice of his action for each personal move, depending on the situation. Usually during the game, at each personal move, the player makes a choice depending on the specific situation. However, in principle it is possible that all decisions are made by the player in advance (in response to any given situation). This means that the player has chosen a certain strategy, which can be given in the form of a list of rules or a program. (So you can play the game using a computer). A game is said to be finite if each player has a finite number of strategies, and infinite otherwise.
In order to solve the game or find a solution to the game, one should choose for each player a strategy that satisfies the optimality condition, i.e. one of the players should get the maximum payoff when the other sticks to his strategy. At the same time, the second player should have a minimum loss if the first one sticks to his strategy. Such strategies are called optimal. Optimal strategies must also satisfy the stability condition, i.e., it must be unprofitable for any of the players to abandon their strategy in this game.
If the game is repeated enough times, then the players may not be interested in winning and losing in each particular game, but in the average win (loss) in all games.
The goal of game theory is to determine the optimal strategy for each player. When choosing the optimal strategy, it is natural to assume that both players behave reasonably from the point of view of their interests. The most important limitation of game theory is the naturalness of payoff as a measure of efficiency, while in most real economic problems there is more than one measure of efficiency. In addition, in the economy, as a rule, there are tasks in which the interests of partners are not necessarily antagonistic.
3. Presentation of games
Games are strictly defined mathematical objects. The game is formed by the players, a set of strategies for each player, and an indication of the payoffs, or payoffs, of the players for each combination of strategies. Most cooperative games are described by a characteristic function, while for other types, the normal or extensive form is more often used.
Extensive form
The game "Ultimatum" in extensive form
Games in the extensive or extended form are represented as a directed tree, where each vertex corresponds to a situation where the player chooses his strategy. Each player is assigned a whole level of vertices. Payments are recorded at the bottom of the tree, under each leaf vertex.
The picture on the left is a game for two players. Player 1 moves first and chooses strategy F or U. Player 2 analyzes his position and decides whether to choose strategy A or R. Most likely, the first player will choose U, and the second - A (for each of them these are optimal strategies); then they will receive respectively 8 and 2 points.
The extensive form is very illustrative, it is especially convenient to represent games with more than two players and games with consecutive moves. If the participants make simultaneous moves, then the corresponding vertices are either connected by a dotted line or outlined by a solid line.
normal form
|
Player 2 |
Player 2 |
|
|
Player 1 |
4 , 3 |
–1 , –1 |
|
Player 1 |
0 , 0 |
3 , 4 |
|
Normal form for a game with 2 players, each with 2 strategies. |
||
Players chose strategies with the maximum result for themselves, but lost, due to ignorance of the other player's move. Usually, normal form represents games in which the moves are made simultaneously, or at least it is assumed that all players do not know what the other participants are doing. Such games with incomplete information will be considered below.
Characteristic formula
In cooperative games with transferable utility, that is, the ability to transfer funds from one player to another, it is impossible to apply the concept of individual payments. Instead, the so-called characteristic function is used, which determines the payoff of each coalition of players. It is assumed that the payoff of the empty coalition is zero.
The grounds for this approach can be found in the book of von Neumann and Morgenstern. Studying the normal form for coalition games, they reasoned that if a coalition C is formed in a game with two sides, then the coalition N \ C opposes it. A game for two players is formed, as it were. But since there are many variants of possible coalitions (namely, 2N, where N is the number of players), the payoff for C will be some characteristic value depending on the composition of the coalition. Formally, a game in this form (also called a TU game) is represented by a pair (N, v), where N is the set of all players and v: 2N → R is the characteristic function.
This form of presentation can be applied to all games, including those without transferable utility. Currently, there are ways to convert any game from normal to characteristic form, but the transformation in the opposite direction is not possible in all cases.
4. Types of games
cooperative and non-cooperative.
The game is called cooperative, or coalition, if the players can unite in groups, taking on some obligations to other players and coordinating their actions. In this it differs from non-cooperative games in which everyone is obliged to play for themselves. Entertainment games are rarely cooperative, but such mechanisms are not uncommon in everyday life.
It is often assumed that cooperative games differ precisely in the ability of players to communicate with each other. In general, this is not true. There are games where communication is allowed, but players pursue personal goals, and vice versa.
Of the two types of games, non-cooperative ones describe situations in great detail and produce more accurate results. Cooperatives consider the process of the game as a whole. Attempts to combine the two approaches have yielded considerable results. The so-called Nash program has already found solutions to some cooperative games as equilibrium situations for non-cooperative games.
Hybrid games include elements of cooperative and non-cooperative games. For example, players can form groups, but the game will be played in a non-cooperative style. This means that each player will pursue the interests of his group, while at the same time trying to achieve personal gain.
AT last years the importance of game theory has increased significantly in many areas of economic and social sciences. In economics, it is applicable not only to solve general business problems, but also to analyze the strategic problems of enterprises, developments organizational structures and incentive systems.
Already at the time of its inception, which is considered the publication in 1944 of the monograph by J. Neumann and O. Morgenstern "Game Theory and Economic Behavior", many predicted a revolution in economic sciences through the use of a new approach. These predictions could not be considered too bold, since from the very beginning this theory claimed to describe rational decision-making behavior in interrelated situations, which is typical for most current problems in economic and social sciences. Thematic areas such as strategic behavior, competition, cooperation, risk and uncertainty are key in game theory and are directly related to managerial tasks.
Early work on game theory was characterized by simplistic assumptions and a high degree of formal abstraction, which made them unsuitable for practical use. Over the past 10-15 years, the situation has changed dramatically. Rapid progress in the industrial economy has shown the fruitfulness of game methods in the applied field.
AT recent times these methods have penetrated into management practice. It is likely that game theory, along with the theories of transaction costs and “patron-agent”, will be perceived as the most economically justified element of organization theory. It should be noted that already in the 80s, M. Porter introduced some key concepts theories, in particular such as "strategic move" and "player". True, an explicit analysis associated with the concept of equilibrium was still absent in this case.
Fundamentals of game theory
To describe a game, you must first identify its participants. This condition is easily fulfilled when it comes to ordinary games such as chess, canasta, etc. The situation is different with “market games”. Here it is not always easy to recognize all the players, i.e. existing or potential competitors. Practice shows that it is not necessary to identify all the players, it is necessary to identify the most important ones.
Games cover, as a rule, several periods during which players take consecutive or simultaneous actions. These actions are denoted by the term "move". Actions can be related to prices, sales volumes, research and development costs, and so on. The periods during which the players make their moves are called game stages. The moves chosen at each stage ultimately determine the “payoff” (win or loss) of each player, which can be expressed in wealth or money (predominantly discounted profits).
Another basic concept of this theory is the player's strategy. It refers to the possible actions that allow the player at each stage of the game to choose from a certain number of alternatives a move that seems to him to be the “best response” to the actions of other players. With regard to the concept of strategy, it should be noted that the player determines his actions not only for the stages that a particular game has actually reached, but also for all situations, including those that may not occur in the course of this game.
The form in which the game is presented is also important. Usually, a normal, or matrix, form and an expanded one, given in the form of a tree, are distinguished. These forms for a simple game are shown in Fig. 1a and 1b.
To establish the first connection with the sphere of control, the game can be described as follows. Two enterprises producing homogeneous products are faced with a choice. In one case, they can gain a foothold in the market by setting a high price, which will provide them with an average cartel profit P K . When entering into a tough competition, both make a profit П W . If one of the competitors sets a high price, and the second sets a low price, then the latter realizes monopoly profit P M , while the other incurs losses P G . A similar situation can, for example, arise when both firms have to announce their price, which cannot subsequently be revised.
In the absence of stringent conditions, it is beneficial for both enterprises to charge a low price. The “low price” strategy is dominant for any firm: no matter what price a competing firm chooses, it is always preferable to set a low price itself. But in this case, firms face a dilemma, since profit P K (which for both players is higher than profit P W) is not achieved.
The strategic combination “low prices/low prices” with the corresponding payoffs is a Nash equilibrium, in which it is unprofitable for any of the players to deviate separately from the chosen strategy. Such a concept of equilibrium is fundamental in resolving strategic situations, but under certain circumstances it still needs to be improved.
As for the above dilemma, its resolution depends, in particular, on the originality of the players' moves. If the enterprise has the opportunity to revise its strategic variables (in this case, the price), then a cooperative solution to the problem can be found even without a rigid agreement between the players. Intuition suggests that with repeated contacts of players, there are opportunities to achieve acceptable “compensation”. Thus, under certain circumstances, it is inappropriate to seek short-term high profits through price dumping if a “price war” may arise in the future.
As noted, both figures characterize the same game. Presenting the game in normal form usually reflects “synchronicity”. However, this does not mean “simultaneity” of events, but indicates that the choice of strategy by the player is carried out in conditions of ignorance about the choice of strategy by the opponent. With an expanded form, such a situation is expressed through an oval space (information field). In the absence of this space, the game situation acquires a different character: first, one player should make the decision, and the other could do it after him.
Application of game theory for making strategic management decisions
Examples here are decisions to conduct a principled pricing policy, entry into new markets, cooperation and creation of joint ventures, identifying leaders and performers in the field of innovation, vertical integration, etc. The provisions of this theory can, in principle, be used for all types of decisions, if their adoption is influenced by others. characters. These persons, or players, need not be market competitors; their role may be sub-suppliers, leading customers, employees of organizations, as well as colleagues at work.
quadrants 1 and 2 characterize a situation where the reaction of competitors does not have a significant impact on the company's payments. This happens when the competitor has no motivation (field 1 ) or opportunities (field 2 ) strike back. Therefore, there is no need for a detailed analysis of the strategy of motivated actions of competitors.
A similar conclusion follows, although for a different reason, for the situation reflected by the quadrant 3 . Here, the reaction of competitors could have a great effect on the firm, but since its own actions cannot greatly affect the payments of a competitor, one should not be afraid of his reaction. Niche entry decisions can be cited as an example: under certain circumstances, large competitors have no reason to react to such a decision of a small firm.
Only the situation shown in the quadrant 4 (the possibility of retaliatory steps of market partners), requires the use of the provisions of game theory. However, only the necessary but not sufficient conditions are reflected here to justify the application of the base of game theory to the fight against competitors. There are times when one strategy unquestionably dominates all others, no matter what the competitor does. If we take the drug market, for example, it is often important for a company to be the first to introduce a new product to the market: the profit of the “pioneer” turns out to be so significant that all other “players” just have to step up innovation activity faster.
The same game situation can also be represented in normal form (Fig. 4). Two states are designated here – “entry/friendly reaction” and “non-entry/aggressive reaction”. It is obvious that the second equilibrium is untenable. It follows from the detailed form that it is inappropriate for a company already established in the market to react aggressively to the emergence of a new competitor: with aggressive behavior, the current monopolist receives 1 (payment), and with friendly behavior - 3. The outsider company also knows that it is not rational for the monopolist start actions to oust it, and therefore it decides to enter the market. The outsider company will not suffer the threatened losses in the amount of (-1).
Such a rational balance is characteristic of a "partially improved" game, which deliberately excludes absurd moves. Such equilibrium states are, in principle, fairly easy to find in practice. Equilibrium configurations can be identified using a special algorithm from the field of operations research for any finite game. The decision maker proceeds as follows: first, the “best” move in the last stage of the game is selected, then the “best” move in the previous stage is selected, taking into account the choice in the last stage, and so on, until the initial node of the tree is reached games.
How can companies benefit from game theory-based analysis? There is, for example, a case of a conflict of interests between IBM and Telex. In connection with the announcement of preparatory plans The last to enter the market was a “crisis” meeting of IBM management, which analyzed measures aimed at forcing a new competitor to abandon its intention to penetrate a new market.
Telex apparently became aware of these events. Game theory based analysis showed that the threats of IBM due to high costs are unfounded.
This shows that it is useful for companies to explicitly consider the possible reactions of game partners. Isolated economic calculations, even based on the theory of decision-making, are often, as in the situation described, limited. For example, an outsider company might choose the “no-entry” move if preliminary analysis convinced it that market penetration would provoke an aggressive response from the monopolist. In this case, in accordance with the criterion of the expected cost, it is reasonable to choose the “non-entry” move with the probability of an aggressive response being 0.5.
For the enterprise 1 it would be best if the company 2 abandoned competition. His benefit in this case would be 3 (payments). It is highly likely that the company 2 would win the competition when the enterprise 1 would accept a cut investment program, and the enterprise 2 - wider. This position is reflected in the upper right quadrant of the matrix.
An analysis of the situation shows that equilibrium occurs at high costs for research and development of the enterprise 2 and low enterprises 1 . In any other scenario, one of the competitors has a reason to deviate from the strategic combination: for example, for the enterprise 1 a reduced budget is preferable if the business 2 refuse to participate in the competition; at the same time the enterprise 2 It is known that at low costs of a competitor it is profitable for him to invest in R&D.
An enterprise with a technological advantage may resort to situation analysis based on game theory in order to ultimately achieve an optimal result for itself. By means of a certain signal, it must show that it is ready to carry out large expenditures on R&D. If such a signal is not received, then for the enterprise 2 it is clear that the company 1 chooses the low cost option.
The reliability of the signal should be evidenced by the obligations of the enterprise. In this case, it may be the decision of the enterprise 1 about purchasing new laboratories or hiring additional research staff.
From the point of view of game theory, such obligations are tantamount to changing the course of the game: the situation of simultaneous decision-making is replaced by the situation of successive moves. Company 1 firmly demonstrates the intention to make large expenditures, the enterprise 2 registers this step and has no more reason to participate in the rivalry. The new equilibrium follows from the scenario “non-participation of the enterprise 2 ” and “high costs for research and development of the enterprise 1 ”.
An important contribution to the use of game theory is made by experimental work. Many theoretical calculations are worked out in the laboratory, and the results obtained serve as an impulse for practitioners. Theoretically, it was found out under what conditions it is expedient for two selfish partners to cooperate and achieve better results for themselves.
This knowledge can be used in the practice of enterprises to help two firms achieve a win-win situation. Today, gaming-trained consultants quickly and unambiguously identify opportunities that businesses can take advantage of to secure stable, long-term contracts with customers, sub-suppliers, development partners, and more.
Problems of practical application
in management
However, it should also be pointed out that there are certain limits to the application of the analytical tools of game theory. In the following cases, it can only be used if additional information is obtained.
Firstly, this is the case when enterprises have different ideas about the game they are participating in, or when they are not sufficiently informed about each other's capabilities. For example, there may be unclear information about a competitor's payments (cost structure). If not too complex information is characterized by incompleteness, then it is possible to operate with a comparison of similar cases, taking into account certain differences.
Second, game theory is difficult to apply to many equilibria. This problem may occur even during simple games with the simultaneous choice of strategic decisions.
Thirdly, if the situation of making strategic decisions is very complex, then players often cannot choose the best options for themselves. It's easy to imagine more difficult situation market penetration than the one discussed above. For example, to the market in different dates several enterprises may enter, or the reaction of enterprises already operating there may be more complex than aggressive or friendly.
It has been experimentally proven that when the game is expanded to ten or more stages, the players are no longer able to use the appropriate algorithms and continue the game with equilibrium strategies.
The fundamental assumption of the so-called “common knowledge” underlying the theory of games is by no means indisputable. It says: the game with all the rules is known to the players and each of them knows that all players are aware of what the other partners in the game know. And this situation remains until the end of the game.
But in order for an enterprise to make a decision that is preferable for itself in a particular case, this condition not always required. Less rigid assumptions, such as “mutual knowledge” or “rationalizable strategies”, are often sufficient for this.
In conclusion, it should be emphasized that game theory is a very complex field of knowledge. When referring to it, one must observe certain caution and clearly know the limits of application. Too simple interpretations, adopted by the firm itself or with the help of consultants, are fraught with hidden danger. Because of their complexity, game theory-based analysis and consultations are only recommended for critical problem areas. The experience of firms shows that the use of appropriate tools is preferable when making one-time, fundamentally important planned strategic decisions, including when preparing large cooperation agreements.
From the popular American blog Cracked.
Game theory is about learning how to make the best move and, as a result, get the biggest piece of the winning pie possible by chopping off some of it from other players. It teaches you to analyze many factors and draw logically weighted conclusions. I think it should be studied after the numbers and before the alphabet. Just because too many people accept important decisions, based on intuition, secret prophecies, the location of the stars and other such. I have carefully studied game theory, and now I want to tell you about its basics. Perhaps this will add common sense to your life.
1. Prisoner's dilemma
Berto and Robert were arrested for bank robbery after failing to properly use a stolen car to escape. The police can't prove they were the ones who robbed the bank, but caught them red-handed in a stolen car. They were separated by different rooms and everyone was offered a deal: to hand over an accomplice and send him to jail for 10 years, and go free himself. But if they both betray each other, then each will receive 7 years. If no one says anything, then both will sit down for 2 years only for stealing a car.
It turns out that if Berto is silent, but Robert betrays him, Berto goes to prison for 10 years, and Robert goes free. Each prisoner is a player, and the benefit of each can be represented as a "formula" (what they both get, what the other gets). For example, if I hit you, my winning scheme will look like this (I get a rough win, you suffer from severe pain). Since each prisoner has two options, we can present the results in a table.
Practical Application: Spotting Sociopaths
Here we see the main application of game theory: identifying sociopaths who think only about themselves. Real game theory is a powerful analytical tool, and amateurism often serves as a red flag, with a head betraying a person devoid of honor. Intuitive people think it's better to be ugly because it will result in a shorter prison sentence no matter what the other player does. Technically, this is correct, but only if you are a short-sighted person who puts numbers higher human lives. This is why game theory is so popular in finance.
The real problem with the Prisoner's Dilemma is that it ignores the data. For example, it does not consider the possibility of you meeting with friends, relatives, or even creditors of the person you put in jail for 10 years.
Worst of all, everyone involved in the Prisoner's Dilemma acts like they've never heard it.
And the best move is to remain silent, and two years later, together with good friend use public money.
2. Dominant strategy
This is a situation in which your actions give the greatest gain, regardless of the actions of your opponent. Whatever happens, you did everything right. This is why many people in the Prisoner's Dilemma believe that betrayal leads to the "best" outcome no matter what the other person does, and the ignorance of reality inherent in this method makes everything look super-simple.
Most of the games we play don't have strictly dominant strategies because they would be terrible otherwise. Imagine that you would always do the same thing. There is no dominant strategy in the game of rock-paper-scissors. But if you were playing with a person who had oven gloves on and could only show rock or paper, you would have the dominant strategy: paper. Your paper will wrap his stone or result in a tie and you can't lose because your opponent can't show scissors. Now that you have a dominant strategy, it would take a fool to try anything else.
3. Battle of the sexes
Games are more interesting when they don't have a strictly dominant strategy. For example, the battle of the sexes. Anjali and Borislav go on a date but can't decide between ballet and boxing. Anjali loves boxing because she likes blood being shed for the joy of a screaming crowd of spectators who consider themselves civilized only because they paid for someone's broken heads.
Borislav wants to watch ballet because he understands that ballerinas go through a lot of injuries and the most difficult training, knowing that one injury can end everything. Ballet dancers are the greatest athletes on earth. A ballerina may kick you in the head, but she will never do it, because her leg is worth much more than your face.
They each want to go to their favorite activity, but they don't want to enjoy it alone, so here's their winning scheme: highest value is doing what they enjoy, lowest value is just being with another person, and zero is being lonely.
Some people suggest stubbornly balancing on the brink of war: if you do what you want, no matter what, the other person must conform to your choice or lose everything. As I already said, simplistic game theory is great at spotting fools.
Practical Application: Avoid Sharp Corners
Of course, this strategy also has its significant drawbacks. First of all, if you treat your dates like a "battle of the sexes", it won't work. Separate so that each of you can find a person that he likes. And the second problem is that in this situation, the participants are so unsure of themselves that they cannot do it.
A truly winning strategy for everyone is to do what they want, and after, or the next day, when they are free, go together to a cafe. Or alternate between boxing and ballet until the entertainment world is revolutionized and boxing ballet is invented.
4. Nash equilibrium
A Nash equilibrium is a set of moves where no one wants to do something differently after the fact. And if we can make it work, game theory will replace all the philosophical, religious, and financial system on the planet, because the “desire not to burn out” has become more powerful for humanity driving force than fire.
Let's split the $100 quickly. You and I decide how many of the hundred we demand and at the same time announce the amounts. If our total is less than a hundred, everyone gets what they wanted. If the total is over one hundred, the one who asked for the least amount gets the desired amount, while the greedier person gets what's left. If we ask for the same amount, each gets $50. How much will you ask? How will you split the money? There is only one winning move.
The $51 claim will give you the maximum amount no matter what your opponent chooses. If he asks for more, you will receive $51. If he asks for $50 or $51, you will get $50. And if he asks for less than $50, you will get $51. In any case, there is no other option that will bring you more money than this one. The Nash equilibrium is a situation in which we both choose $51.
Practical Application: Think First
This is the whole point of game theory. You don't have to win, let alone hurt other players, but you do need to make the best move for yourself, no matter what others have in store for you. And even better if this move is beneficial for other players. This is a kind of mathematics that could change society.
An interesting variant of this idea is drinking, which can be called a Nash Equilibrium with a time dependence. When you drink enough, you don't care about other people's actions no matter what they do, but the next day you really regret that you didn't do otherwise.
5. The game of toss
Player 1 and Player 2 participate in the toss. Each player simultaneously chooses heads or tails. If they guess correctly, Player 1 gets Player 2's penny. If they don't, Player 2 gets Player 1's coin.
The winning matrix is simple...
…optimal strategy: play completely at random. It's harder than you think, because the selection must be completely random. If you have a preference for heads or tails, the opponent can use it to take your money.
Of course, the real problem here is that it would be much better if they just threw one penny at each other. As a result, their profits would be the same, and the resulting trauma could help these unfortunate people feel something other than terrible boredom. After all, this worst game ever existing. And this is the perfect model for a penalty shootout.
Practical Application: Penalty
In football, hockey and many other games, extra time is a penalty shootout. And they would be more interesting if they were based on how many times the players full form be able to make a "wheel" because that would at least be an indicator of their physical ability and would be fun to watch. Goalkeepers cannot clearly determine the movement of the ball or puck at the very beginning of their movement, because, unfortunately, robots still do not participate in our sports. The goalkeeper must choose a left or right direction and hope that his choice will coincide with the choice of the opponent kicking at the goal. It has something in common with the game of coin.
However, please note that this is not perfect example resemblance to the game of heads and tails, because even with right choice direction, the goalkeeper may not catch the ball, and the attacker may miss the goal.
So what is our conclusion according to game theory? Ball games should end in a “multi-ball” manner, where an extra ball/puck is given to the players one-on-one every minute, until either side has a certain result that was indicative of the true skill of the players, and not a showy coincidence.
After all, game theory should be used to make the game smarter. And that means better.