What is the theory of games. Mathematical theory of games

Game theory - A combination of mathematical methods for solving conflict situations (collisions of interests). In the theory of games, the game is called Mathematical model of a conflict situation. The subject of special interest of the game theory is the study of decision-making strategies for the participants of the game in conditions of uncertainty. Uncertainty is related to the fact that two or more sides pursue opposite goals, and the results of any action of each part depends on the partner's moves. At the same time, each of the parties seeks to make optimal solutions that implement the goals raised to the greatest extent.

The most consistent game theory is applied in the economy where conflict situations arise, for example, in relations between the supplier and the consumer, the buyer and the seller, the bank and the client. The use of game theory can be found in politics, sociology, biology, military art.

From the history of game theory

History of the theory of games As an independent discipline begins in 1944, when John von Neuman and Oscar Morgenstern published a book "Theory of Games and Economic Behavior" ("Theory of Games and Economic Behavior"). Although the examples of the game theory were even before: the Treatise of the Babylonian Talmud about the division of the deceased husband between his wives, card games in the 18th century, the development of a chess game theory at the beginning of the 20th century, proof of the minimax theorem of the same John von Neuman in 1928 year, without which there would be no game theory.

In the 50s of the 20th century, Melvin Dresher and Meryl Flod from Rand Corporation. The first experimentally applied the file of the prisoner, John Nash in the work on the state of equilibrium in the games of two persons developed the concept of nash equilibrium.

Reinhard Salten in 1965 published the book "Processing Oligopoly in the Theory of Games on Requirements" ("Spieltheoretische Behandlung Eines Oligomodells Mit Nachfrageträgheit"), with which the use of game theory in the economy received a new driving force. A step forward in the evolution of the game theory is associated with the work of John Mainard Smith "Evolutionary Stable Strategy" ("Evolutionary Stable Strategy", 1974). The dilemma of the prisoner was popularized in the book of Robert Axelrod "The Evolution of Cooperation" ("The Evolution of Cooperation") published in 1984. In 1994, it was for the contribution to the theory of Nobel Prize Games, John Nash, John Harsania and Reinhard Salten.

Theory of games in life and business

Let us dwell on the essence of the coffee situation (collision of interests) in the sense, as it is understood in the game theory for further modeling of 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 in relation to these outcomes. But although it can to some extent to control the variable factors determining the outcome, it does not have complete power over them. Sometimes the control is in the hands of several individuals, which, like him, have some preferences in relation to possible outcomes, but in the general, the interests of these individuals are not consistent. In other cases, the final outcome may depend on both randomness (which in legal sciences are sometimes referred to as natural disasters) and from other individuals. The theory of games systematizes observations of such situations and the wording of general principles for the leadership of reasonable actions in such situations.

In some respects, the name "Theory of Games" is unsuccessful, as it suggests that the theory of games is considering only not having a social value of the clashes occurring in the salon games, but still this theory is significantly more widely.

The following economic situation can give an idea of \u200b\u200bthe use of game theory. Let there be several entrepreneurs, each of which seeks to get a maximum profit, while having only limited power over the variables that determine this profits. The entrepreneur does not have the power over the variables, which the other entrepreneur disposes, but which can greatly affect the income of the first. Interpretation of this situation as games can cause the following objection. The game model assumes that each entrepreneur makes one choice from the area of \u200b\u200bpossible elections and these single elections are determined by profits. Obviously, this can almost not be in reality, since in this industry there would be no complex managerial apparatus. There are simply a number of solutions and modifications of these solutions, which depend on the elections committed by other participants in the economic system (players). But in principle, you can imagine that any administrator will foresee all possible randomness and describes in detail the action that needs to be taken in each case, instead of solving each task as it occurs.

Military coflict, by definition, there is a clash of interests in which none of the parties dispose of completely variables that determine the outcome, which is solved by a number of battles. You can simply consider the outcome of the winning or loss and attribute them numerical values \u200b\u200b1 and 0.

One of the most simple conflict situations that can be recorded and solved in the game theory - a duel, which is a conflict of two players 1 and 2, respectively p. and q. Shots. For each player, there is a function indicating the likelihood that player shot i. At the time of time t. will give a falling, which will be fatal.

As a result, the theory of games comes to such a wording of some class of collisions of interest: n. Players, and everyone needs to choose one possibility from a certain set, and when choosing a choice from the Player, there are no information about the elections of other players. The area of \u200b\u200bthe player's possible elections may contain elements such as the "TUI PIC", "production of tanks instead of cars," or in a general sense, a strategy that defines all the actions to be done in all possible circumstances. Before each player is a challenge: what choice does he have to do that its private influence on the outcome brought to him as a larger win?

Mathematical model in the theory of games and formalization of tasks

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

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

Interested in playing the game are called players Each of them can take at least two actions (if there is only one action at the player's disposal, then it does not actually participate in the game, since it is known in advance that he will take). The outcome of the game is called winning .

The real conflict situation is not always, but the game (in the concept of the theory of games) - always - proceeds by defined rules which accurately determine:

1. options for players;
2. the volume of information of each player on the behavior of a partner;
3. winning to which every combination of actions leads.

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

But in the economy, the model of behavior of the players arises, for example, when several firms seek to take a better place in the market, several persons are trying to share any benefit (resources, finance) so that everyone gets more opportunity. Players in conflict situations in the economy, which can be modeling in the form of the game, are firms, banks, individuals and other economic agents. In turn, in the conditions of the war, the game model is used, for example, in the choice of better weapons (of the existing or potentially possible) to defeat the enemy or protect against the attack.

For the game is characterized by the uncertainty of the result . Causes of uncertainty can be distributed in the following groups:

1. combinatorial (both in chess);
2. the effect of random factors (as in the game "Eagle or Rush", bones, card games);
3. strategic (the player does not know what action the enemy will take).

Player strategy There is a set of rules that determine its actions at each time depending on the current situation.

The purpose of the theory of games is the definition of an optimal strategy for each player. Determine this strategy - it means to solve the game. Strategy optimality It is achieved when one of the players should get the maximum winnings, despite the fact that the second adheres to its strategy. And the second player must have a minimum loss if the first adheres to its strategy.

Classification of games

1. Classification by the number of players (game of two or more persons). The games of the two persons occupy a central place in the whole theory of games. The main concept of the game theory for the game of two persons is a generalization of a very essential idea of \u200b\u200bequilibrium, which naturally appears in the games of two persons. As for the games n. Persons, then one part of the game theory is devoted to games in which the cooperation between the players is prohibited. In another part of the game theory n. Persons are assumed that players can cooperate for mutual benefit (see below in this paragraph about non-opoperative and cooperative games).
2. Classification by the number of players and their strategies (The number of strategies of at least two may be infinity).
3. Classification by the number of information Regarding past moves: games with full information and incomplete information. Let there be a player 1 - buyer and player 2 - seller. If the player 1 does not have complete information about the player's actions 2, then the player 1 may not distinguish between two alternatives, between which he has to make a choice. For example, choosing between two species of some product and not knowing that for some signs A. Worse goods B., Player 1 may not see the differences between alternatives.
4. Classification based on winning principles : Cooperative, coalition on the one hand and non-opoperative, impalmed on the other side. IN neooperative game , or otherwise - infaluction game , Players choose strategies at the same time, not knowing which strategy will choose the second player. Communication between players is impossible. IN cooperative game , or otherwise - coalition game The players can unite in the coalition and take collective actions to increase their winnings.
5. The ultimate game of two persons with zero amount Or an antogonistic game is a strategic game with full information in which parties are involved with opposite interests. Anatagonistic games are matrix Games .

A classic example from the theory of games - the file of the prisoner

Two suspects take into custody and isolate apart from each other. The district prosecutor is convinced that they made a serious crime, but does not have sufficient evidence to impose a court charge. He says to each of the prisoners that he has two alternatives: to confess to a crime, which, by conviction, he committed, or not admit. If both are not recognized, the district prosecutor will charge them in any minor crime, for example, small theft or illegal possession of weapons, and they both will receive a small punishment. If they both confess, they will be subject to judicial liability, but it will not require the most strict sentence. If one is recognized, and the other is not, then the sentence will be relaxed for the issuance of the accomplice, while persisting will receive "on the full coil."

If this strategic task is to formulate in the deadlines, it comes down to the following:

Thus, if both prisoners are not recognized, they will receive each year each. If both are recognized, everyone will receive for 8 years. And if one admits, the other is not recognized, then the one who admitted is separated by the three months of the conclusion, and the one that is not recognized will receive 10 years. The resulting matrix correctly reflects the file dilemma: before each there is a question - to admit or not admit. The game that the district prosecutor offers prisoners is neooperative game or otherwise infalliac game . If both prisoners had the opportunity to cooperate (that is, the game would be cooperative or otherwise coalition game ), both would not admit and received the prison each year.

Examples of using mathematical means of game theory

Now we go to the consideration of solutions for examples of common classes of games, for which the methods of research and solutions exist in the theory of games.

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

In the previous paragraph, we have already considered an example of a non-optical (infallional) game (prisoner dilemma). Let's fix our skills. For this, the classic story is also suitable for the "adventures of Sherlock Holmes" Arthur Conan Doyle. You can, of course, argue: an example is not from life, but from literature, but because Conan Doyle has not proven himself as a science fiction writer! Classic is also because the task is made by Oscar Morgettern, as we have already been installed - one of the founders of the theory of games.

Example 1. A shortened presentation of the fragment of one of Sherlock Holmes will be given. According to the well-known concepts of the game theory, make a model of a conflict situation and formally record the game.

Sherlock Holmes intends to go from London to Dover with further go to the continent (European) to escape from Professor Moriarty, who pursues him. Seed in the train, he saw at the vocational platform of Professor Moriarty. Sherlock Holmes admits that Moriarty can choose a special train and overtake it. Sherlock Holmes has two alternatives: continue the trip to Dover or go away at Canterberry Station, which is the only intermediate station on its route. We accept that his opponent is quite intelligent to determine the capabilities of Holmes, so there are two alternatives before him. Both enemies must choose a station to get away on it from the train, not knowing what decision will take each of them. If, as a result of the decision-making, both will be on the same station, you can unambiguously assume that Sherlock Holmes will be killed by Professor Moriarty. If Sherlock Holmes safely get to Dover, he will be saved.

Decision. Heroes Conan Doyle can be viewed as participants in the game, that is, players. At the disposal of each player i. (i.\u003d 1,2) Two net strategies:

• cut in Dover (strategy s.i1 ( i.=1,2) );
• get off at the intermediate station (strategy s.i2 ( i.=1,2) )

Depending on which of the two strategies, each of two players chooses, a special combination of strategies as a couple will be created. s. = (s.1 , s.2 ) .

Each combination can be put in line with an event - the outcome of the attempt to kill Sherlock Holmes by Professor Moriarty. We make a matrix of this game with possible events.

Under each of the events, an index is indicated that means the acquisition of Professor Moriarty, and calculated depending on the salvation of Holmes. Both heroes choose the strategy at the same time, not knowing that he would choose the enemy. Thus, the game is neooperative, since, firstly, players are in different trains, and secondly, they have opposite interests.

An example of formalization and solutions of cooperative (coalition) game n. persons

In this paragraph, the practical part, that is, the decision of the example of the task will be presented the theoretical part in which we will meet the concepts of game theory for solving cooperative (infallusive) games. For this task, the theory of games offers:

• a characteristic function (if it is simplistic, it reflects the magnitude of the benefit of combining the players in the coalition);
• the concept of additivity (the properties of the values \u200b\u200bthat the value of the value corresponding to the whole object is equal to the sum of the values \u200b\u200bof the values \u200b\u200bcorresponding to its parts, in a certain class of partitioning of the object on the part) and superadditivity (the value of the value corresponding to an entire object, more than the amount of values \u200b\u200bof the values, The corresponding parts) of the characteristic function.

Superditivity characteristic function suggests that the association in coalition is beneficial to players, since in this case the value of the coalition win increases with an increase in the number of players.

To formalize the game, we need to introduce formal designations of the above-mentioned concepts.

For Game n. Denote many of all its players as N. \u003d (1.2, ..., n) any non-empty subset of the set N. Denote as T. (including Sam N. and all subsets consisting of one element). The site has a lesson " Sets and sets on sets ", Which when the link clicks, opens in a new window.

The characteristic function is indicated as v. and its definition area consists of possible subsets of the set N.. v.(T.) - The value of the characteristic function for one or another subset, for example, the income obtained by the coalition, including possibly consisting of one player. This is important for the reason that the theory of games requires checking the presence of superadditivity for the values \u200b\u200bof the characteristic function of all inhabited coalitions.

For two non-empty coalitions from subsets T.1 and T.2 The additivity of the characteristic function of the cooperative (coalition) game is written as follows:

And superadditivity so:

Example 2. Three students of the music school work in different clubs, they receive their revenue from club visitors. Install, whether they are beneficial to combine their forces (if so, with what conditions), using the concepts of game theory for solving cooperative games n. Persons under the following source data.

On average, their revenues in one evening amounted to:

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

Trying to increase the revenue, students have created various groups for several months. The results showed that, uniting, they can increase their revenue for the evening as follows:

• violinist + guitarist earned 1500 units;
• violinist + singer earned 1,800 units;
• guitarist + singer earned 1900 units;
• violinist + guitarist + singer earned 3000 units.

Decision. In this example, the number of participants of the game n. \u003d 3, therefore, the field of determining the characteristic function of the game consists of 2³ \u003d 8 possible subsets of a plurality of all players. List all possible coalitions T.:

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

Each of the players assign the sequence number:

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

According to the task, we define the characteristic function of the game. v.:

v (t (1)) \u003d 600; V (T (2)) \u003d 700; V (T (3)) \u003d 900; These values \u200b\u200bof the characteristic function are determined based on the winnings of the first, second and third players, respectively, when they are not combined in the coalition;

v (T (1,2)) \u003d 1500; V (T (1,3)) \u003d 1800; V (T (2,3)) \u003d 1900; These values \u200b\u200bof the characteristic function are determined by the revenue of each pair of players who united in the coalition;

v (T (1,2,3)) \u003d 3000; This value of the characteristic function is determined by medium revenue in the case when the players combined in the triple.

Thus, we have listed all the possible coalitions of players, they turned out eight, as it should be, since the area of \u200b\u200bdetermining the characteristic function of the game consists precisely of eight possible subsets of many players. Which requires the theory of games, since we need to check the presence of superadditivity for the values \u200b\u200bof the characteristic function of all inhabitable coalitions.

How are the conditions of superadditivity in this example? We define how players form inhabitant coalitions T.1 and T.2 . If part of the players enter the coalition T.1 All other players enter the coalition T.2 And by definition, this coalition is formed as the difference between the entire set of players and many T.1 . Then, if T.1 - coalition from one player, then in coalition T.2 will be the second and third players if in the coalition T.1 there will be the first and third players, then the coalition T.2 It will consist only from the second player, and so on.

Game theory is a science that studies the principles of decision-making in situations in which several agents interact with each other. Solutions taken by someone influence the decisions of the rest and on the outcome of the interaction in general. The interactions of this type are called strategic.

The word "game" should not be misleading. This concept in the theory of games is extended wider than in everyday life. The situation of strategic interaction can be described as a model, which is called the game. Thus, in the game theory, the game will be considered not only a game of chess, but also a vote in the UN Security Council, and the vendor to the vendor with the buyer in the market.

Strategic interactions are found in almost any sphere of our life. An example from the economy: several companies competing in the market, when making decisions should look at the actions of competitors. If we talk about politics, the candidates competing in the elections, declaring their election platform, naturally, take into account the positions of other candidates relative to this issue. And if we study the interaction of people in society, then with the help of the theory of games you can learn a lot of interesting things about the tendency of people to cooperation.

Representatives of social sciences often use the theory of games as a tool that allows you to solve their tasks. Simplifying, theoretical and gaming modeling can be divided into two stages.

First, by a real life situation, you need to build a formal model. As a rule, in the model you need to reflect the three main characteristics of the life situation: who interacts with each other (such agents in the theory of games are called players), what decisions players can receive and what payments they are as a result of this interaction. The formal model is called the game.

As soon as we built the game, it needs to be solved somehow. At this stage, we fully abstract from reality and we study the solely formal model. How is the model solution arranged? We must fix the concept of the behavior of players in the game, that is, the principles of the decisions they have. As soon as we recorded this concept, we can try to solve the game with it, that is, to make the outcome to end the game.

With the help of various theoretical and game concepts, you can solve different classes of games. One of the most beautiful theoretical results of the game theory proves that in some very wide class of models, it is guaranteed to find a solution. I mean the result of John Nash, received by him in 1950: in any ultimate game in a normal form, you can always find at least one equilibrium in mixed strategies. Chronologically, it was the first universal theoretical and game concept, which allows you to be guaranteed to find a solution in a very wide class of models.

Unlike representatives of social sciences, Mathematics-Games are more interested in the internal properties of games and the concepts of their decision. It is thanks to such theoretical results we can be confident that, building and solving this or that theoretical and game model, we eventually obtain a solution with the necessary properties.

Of course, John Nash is not the sole author of the game theory. The theory of games as an independent science began to develop a little earlier, at the beginning of the twentieth century. The first attempts to formally identify the games, player strategies and the concept of game solutions to rise to the names of Emil Borel and John von Neuman. However, it was Nash who presented the concept of equilibrium that allows you to be guaranteed to find a solution in the ultimate games. In honor of the author of the theorem on the existence of equilibrium in mixed strategies in the ultimate games, this equilibrium began to be called Nash's equilibrium.

In 1994, the first Nobel Prize for results in the field of game theory (John Nashu, Reinhard Zelten and John Harsanka) actually approved the status of the game theory as an independent scientific direction with its tasks and methods of their decisions. The next few Nobel Prizes following this were awarded both for fundamental theoretical and gaming results and for apps the theory of games to one or another side of our life. In leading universities in the world in programs and in economics, and on political sciences, the theory of games is necessarily included in the standard set of courses. Often, psychologists and mathematics are studying it.

Today, if you look at the sections of large conferences and on articles in leading scientific journals on the theory of games, the number of works that use the apparatus of game theory for solving applied tasks is much larger than the number of fundamental theoretical and game results. The current state of discipline can be described as follows: in the theory of games, a fairly powerful core has been formed, knowledge reservoir, which allows to obtain good and interesting results to researchers from related regions.

Nevertheless, new interesting areas of research and the theory of games are always open. So, thanks to the development of computational technologies, new theoretical and gaming concepts appeared, taking into account the possibilities and limitations of computing machines. Thanks to them, they have the opportunity to solve new tasks. The result of 2015 is on equilibrium in one of the versions of poker, obtained by bowling, Berech, Johanson and Tammlin, is a wonderful example of using modern theories and technologies.

experimental economy

And other analysis methods

Like any other not fully conventional science, the institutional economy applies different methods of analysis. These include traditional microeconomic tools, econometric methods, analysis of statistical information, etc. In this section, we will briefly consider the use of game theory, experimental economy and other methods adapted to institutional analysis.

Game theory. Game theory - Analytical method that has developed after World War II and used to analyze situations in which individuals strategically interact. Chess is a prototype of a strategic game, since the result depends on the behavior of the enemy, as well as from the behavior of the player's actual. Due to the analogies found between the strategic games and the forms of political and economic cooperation, the theory of games is given increased attention in the social sciences. Modern game theory begins with the work of D. Nimanan and O. Morgen Shextern "Theory of Games and Economic Behavior" (1944, Russian Option - 1970). The theory explores the interaction of individual solutions in some assumptions concerning the decision-making in terms of risk, the overall state of the environment, cooperative or non-optherapy behavior of other individuals. Obviously, a rational individual has to make decisions in conditions of uncertainty and interaction. If the winnings of one individual is a loss of another, then this is a zero amount game. When each individuals can win from solving one of them, then there is a game with a nonzero amount. The game can be cooperative when collusion is possible, and the antagonism is prevailing. One of the well-known examples of the game with a nonzero amount is the prisoner (DZ) dilemma. This example shows that, contrary to the allegations of liberalism, the persecution of an individual of one's own interest leads to a solution to a less satisfactory than possible alternatives.

Limit theorem F.I. Ezuorta is seen as an early example of a cooperative game n. participants. Theorem argues that as the number of participants in the economy of pure exchange increases, the conspirass becomes less useful, and many possible equilibrium relative prices (kernel) decrease. If the number of participants tends to infinity, then only one system of relative prices remains, corresponding to the prices of general equilibrium.

The concept of optimal (equilibrium) on the NASH solution is one of the key in the theory of games. It was introduced in 1951 by the American Economist-Mathematics John F. Nast.

In this context, it suffices to consider this concept in relation to the theoretical and game model of two persons 25. In this model, each of the participants has some non-empty multiple strategies. S. i. , i.\u003d 1, 2. The selection of specific strategies from among the available player is carried out in such a way as to maximize the value of its own function of winning (utility) u. i. , i. \u003d 1, 2. The values \u200b\u200bof the winnings function are set on the set of ordered pairs of player strategies S. one S. 2, whose elements are all sorts of combinations of player strategies ( s. 1 , s. 2) (the ordering of pairs of strategies is that in each of the combinations in the first place there is a strategy of the first player, on the second - second), i.e. u. i. = u. i. (s. 1 , s. 2), i. \u003d 1, 2. In other words, the winning of each player depends not only on the strategy chosen by them, but also from the strategy adopted by his opponent.

A pair of strategies is recognized as optimal on Nash. s. 1 *, s. 2 *), s. i. *Î S. i. , i. \u003d 1, 2, which has the following property: Strategy s. 1 * provides a player 1 Maximum winnings when player 2 chooses a strategy s. 2 *, and symmetrically s. 2 * delivers the maximum value of the player winning function 2 When a player 1 Accepted strategy s. one *. A pair of strategies leads to equilibrium on Nash, if the choice made by the player 1 , optimal with this choice of player 2 , and the choice made by the player 2 is optimal with this choice of player 1 . The concept of nash optimality is obviously summarized in case of game n. Persons. It should be noted that the existence of an equilibrium on Nash does not mean its pass-optimality, and the pass-optimal set of strategies does not have to satisfy the equilibrium on Nash. In 1994, J. F. Nashu, R. Oltenu and J. Ch. Kharshani was awarded the memory premium A. Nobel in economics for their contribution to the development of game theory and its app to the economy.

Appeal to this method relies on its explicit force in coverage of the causes and consequences of institutional change. The ability of the game theory to help analyze the consequences of changing the rules is indisputable; Her power in the disclosure of causes is ambiguous. Any theoretical-game analysis should assume the preceding definition of the basic rules of the game. So, O. Morgenishtern in 1968 wrote: "The games are described by determining possible behavior within the rules of the game. Rules are in each case unambiguous; For example, in chess, certain moves are allowed for specific figures, but are prohibited for others. The rules are also impatient. When the social situation is considered as a game, the rules are given to the physical and legal environment, within which there is an individual action "26.

If this point of view is accepted, it is impossible to expect that the theory of games will explain the reason for changes in the fundamental rules for organizing economic, political and social life: the definition of such rules is obviously a prerequisite for carrying out such an analysis.

To understand the values \u200b\u200bof the institutes are used models of the coordination game and the divisions dilemma.

Consider clean and generalized coordination problem. A net coordination game shows that economic agents cannot be guaranteed to implement mutual benefits of cooperation, even if there is no conflict of interest. In other words, there is a multiple equilibrium in the situation "clean" coordination, which is equally preferred to each side. In this case, there is no conflict of interest, but there is no guarantee that everyone will strive for one equilibrium result. The famous example is the choice of the side of the road (right or left), in which people should ride (Fig. 2.1). This game has two nash equilibriums corresponding to the strategy sets (left, left) and (right, right). No one does not object to ride on the right or on the left, but the achievement of a coordinated result with a large number of negotiation participants will require high transaction costs. A institution is needed that would fulfill the function of the focal point, i.e. introduced a coherent solution. In this institution, there may be a result of a general knowledge obtained on the basis of a similar analysis of the situation, and may be a state that interferes in order to introduce the coordination rule and reduce transaction costs. In general, institutions perform a coordination function, reducing uncertainty.

The generalized coordination problem exists if the wins matrix is \u200b\u200bsuch that at any point of equilibrium, none of the players have an incentive to change their behavior with this behavior of other players, but none of the players want some other player to change it. In this case, each would prefer the coordinated result not coordinated, but perhaps everyone wants to choose a special coordinated result (Fig. 2.2). For example, two manufacturer BUT and B. Use various technology X. and Y.But they want to enter the National Standard of the product that will cause network external effects. Manufacturer BUT will win more if the technology becomes the standard H., and producer B. - Technology Y.. The winnings turn out to be distributed asymmetrically. So producer BUT(B.) would prefer in order to become the standard X.(Y.) -Technology, but both will prefer any of the coordinated results not coordinated. Transaction costs in this model will be higher than in the previous one (especially with the participation of a large number of parties), as there is a collision of interest. Replacing private attempts to coordinate with state intervention would reduce transaction costs in the economy. Examples are the government introduction of technological standards, measurement and quality standards, etc. The generalized coordination model illustrates the importance of not only the coordination function of institutions, but also the distribution, on which the method restricts the possible alternatives to the players, and ultimately the effectiveness of interaction.

Dilemma prisoner It is often brought as an example of the problem of establishing cooperation between individuals. Two players are involved in the game, two prisoners who are separated by their warders. Everyone has two choices: to cooperate, i.e. Store silence, or refuse to cooperate, i.e. betraying another. Everyone must act, not knowing that it will take another. Each say that recognition, if the other is silent, leads to freedom. Refusal of recognition in the event of a betrayal of another means death. If both are recognized, they will spend several years in prison. If each of them refuses recognition, it will be arrested for a short time and then released. Assuming that the prison is preferable to death, and freedom is the most desired state, prisoners face a paradox: although they would both prefer not to betray each other and carry out a short time in prison, everyone will be in a better position, betraying another, not believable with the fact that Take another. Analytically the ability of prisoners to establish the connection is in the background, since the incentives for betrayal remain equally strong if there are or without communication. The betrayal remains the dominant strategy.

This analysis helps to explain why selfish-maximizing agents cannot rationally come to a cooperative result or maintain it (individual rationality paradox). It is useful in explaining the EX POST of the collapse of the cartel or other cooperative agreement, but does not explain how the cartel or cooperative agreement is formed. If the prisoners are able to achieve an agreement, the problem disappears: they agree not to betray each other and come to maximize joint gains. So, it is enough to join the agreement, which is together preferably, but makes everyone separately more vulnerable to the damage than in the absence of such an agreement. This analysis draws attention to institutions, which, from an individual point of view, can transform such agreements into less risky.

Theoretical literature gives a distinction between the analysis of cooperative and non-optics games. As already described, players are able to conclude their agreements. The guarantor of such agreements is implicit. Many game theorists insist that the deception and rupture of agreements are common features of human relationships, so such behavior should remain inside the strategic space. They are trying to explain the emergence and preservation of cooperation in the model of non-optics games, especially in the model of an infinitely repeated sequence of DZ games. The final sequence of games will not give the result, because from the moment when the dominant strategy in the last game will be clearly apostable, and from the moment it becomes expected, the same will be true for the penultimate game and so on, before the first game. In infinite series of games, a cooperation may appear as an equilibrium strategy under certain assumptions about discounting wins. Thus, uncooperative analysis does not avoid the need to adopt the basic rules of the game as part of the description of the strategic space. He simply assumes an excellent and less restrictive set of rules. In contrast to cooperative analysis of the agreement can be broken at will. On the other hand, the exit from the continuous game is limited. No approach avoids the need to identify the rules of the game before you start the analysis.

One of the most interesting recent achievements in the DZ study was the organization of tournaments between predefined strategies for carrying out of course repetitive DZ games with two participants. The first one was organized by Robert Axelrod (described in 1984) and included the game by a sequence of 200 parties. Experienced in DZ participants were offered computer programs, and which then competed with each other.

R. Axelrod informed the players that strategies will be estimated not by the number of victories, but according to the amount of glasses against all other strategies, and three points each receives for mutual cooperation, one point for mutual apostasy and win 5 to 0 for apostasy / cooperation. As noted earlier, it is analyzically clear that apostasy is the dominant strategy of the last game and, therefore, each previous game.

Consider the wins matrix in DZ, analyzed by R. Axelrod 27 (Fig. 2.3). No matter what the other player does, the betrayal gives a higher remuneration than cooperation. If the first player thinks that another player will be silent, it is more profitable for him to betray ($ 5\u003e $ 3). On the other hand, if the first player thinks that another betrayed, it is still more profitable to betray himself ($ 1 better than nothing). Consequently, the temptation inclines to betrayal. But if both betrayed, both get less than in the cooperation situation ($ 1 + $ 1<$3+$3).

		Second player
			Cooperative
First player
	Cooperative

Fig. 2.3. Matrix winnings in the division of prisoner

The dilemma of the prisoner is the famous problem in the economy - shows: what is rational or optimal for one agent may not be rational or optimal for a group of individuals considered jointly. The selfish behavior of the individual can be harmful or destructive for the group. In repeated games of DZ, the corresponding strategy is not obvious. To find a good strategy, and tournaments were organized. If the winnings were obtained strictly based on the victory-loss, then each participant of the tournament was to offer continuous apostasy. However, the rules of winnings made it clear that the organization of some cooperation could lead to higher general results. To the surprise of many, the simple strategy "tooth for the tooth" won, proposed by A. Rapoporta: the player coexisted in the first step and then makes one move that another player did in the previous step.

Much more players participated in the second tournament, including professionals, as well as those who knew about the results of the first round. The result was another victory of copy strategy ("tooth for the tooth").

Analysis of the results of tournaments revealed four properties, leading to a successful strategy: 1) the desire to avoid unnecessary conflict and coexisted as long as the other; 2) the ability to call in the face of nothing caused by the betrayal of another; 3) forgiveness after answering the call; 4) Clarity of behavior so that another player can recognize and adapt to the image of the first.

R. Axelrod showed that cooperation can begin, develop and stabilize in situations that are otherwise extraordinary, not promising nothing good. It can be acceed with the fact that the strategy "tooth for the tooth" in the analytical sense of irrational in a certainly repeated game, but empirically, obviously, no. If the strategy "tooth for the tooth" competed with other analytical strategies, all of which consisted of continuous apostasses, she could not win the tournament.

The theory of games can be an important tool for studying human interaction in limited rules of circumstances. Thanks to its capabilities, study the consequences of different institutional agreements, it can also be useful in terms of public policy in the design of new institutional agreements. The theory of games was used in the analysis of public goods, oligopoly, cartel and cortex in the markets of goods and labor. With all its merits, the theory of games has relative weaknesses. Some authors expressed doubts about the use of a model of a dilemma of the prisoner in social science. For example, M. Taylor in 1987 suggested that such games correspond to the circumstances of providing public benefits. In 1985, N. Schofield argued that agents should form agreed concepts about the beliefs and desires of other agents, including problems of knowledge and interpretation, which are not simple for modeling 28. Many economists noted that the use of game theory without reservations can reduce economic activity to too static scheme. In particular, the Nobel Laureate R. Stone wrote in 1948: "The main feature, thanks to which the theory of games flows into a contradiction with live validity, is that the object of the study is limited in time - the game has the beginning and end. You can't tell this economic reality. It is in the ability to separate the parti from  game and is the deep discrepancy between the theory with reality, and this discrepancy limits its application "29. However, since then, it is more invaluable for smoothing this discrepancies and expanding the use of the theory of games in the economy.

Experimental economy. Another methodological approach used to verify the postulates of economic theory and related sciences, as well as an explanation of institutional problems is experimental economy. The influence of the design of the institutions on the efficiency of resource accommodation is not always possible to predict EXTE. One of the costs of saving on the costs EX POST is imitation of the work of institutions in laboratory conditions.

In general, the economic experiment is reproduction of an economic phenomenon or process in order to study under the most favorable conditions and further practical changes. Experiments that are carried out in real conditions are called natural, or field, and experiments conducted in artificial conditions - laboratory. The latter often require the use of economic and mathematical methods and models. Natural experiments can be carried out at the micro-level (experiments R. Owen, F. Taylor, on the introduction of Hosrat at the enterprise, etc.) and on the macro level (economic policy options, free economic zones, etc.). Lab experiments are artificially reproduced economic situations, some economic models whose environment (experimental conditions) is controlled by a researcher in the laboratory.

American economist EL. Roth, from the end of the 70s. Working in the field of the experimental economy, notes a number of advantages of laboratory experiments before the "field" 30. In the laboratory conditions, full control of the experimenter over the medium and behavior of the subjects is possible, while under the "field" experiments can only be monitored by a limited number of environmental factors and is almost impossible - the behavior of economic entities. It is due to this, laboratory experiments make it possible to more accurately determine the conditions under which you can expect the repetition of individual phenomena. In addition, natural experiments are expensive, and in case of failure, they affect the fate of many people.

The area of \u200b\u200binterests of the experimental economy is quite extensive: the provisions of the game theory, theory of industry markets, a model of rational choice, the phenomenon of market balance, the problems of public goods, etc.

For example, we will focus on the results of the study of the comparative effectiveness of the market institutions that are published by Ch.A. Holt and represented A.E. Shastitko 31. The study compares the findings of theoretical and experimental models of the market, obtained using controlled experiments. The results of the behavior of agents are measured using the coefficient of exhaustion of the amount of the potential rent of the buyer and the seller, which corresponds to the effectiveness of the exchange. The coefficient of exhaustion is the ratio of actually (experimentally) the resulting rent to the highest possible value - varies from 0 to 1. Comparison was carried out in the following market forms: bilateral auction, trade on the basis of price applications of one of the parties, the calculation chamber, decentralized price negotiations, trade on The basis of applications with subsequent negotiations. The most interesting results of experiments were obtained by various groups of researchers in two first market forms (Table 2.1).

A funny example of applying game theory is in the fantasy book Anthony Pier "Brave Golem"

Many text

- The meaning of what I will demonstrate to all now, - the Grande began, - is the set of the required number of points. Points can be the most different - it all depends on the combination of solutions that are accepted by the participants of the game. For example, suppose that each participant testifies against his comrade on the game. In this case, each participant can be awarded one point!
- One point! - said the sea witch, showing unexpected interest to the game. Obviously, the sorceress wanted to make sure that the Golem had no chance that the Demon Xunt was pleased with them.
- And now let's assume that each of the participants of the game does not testify against their comrade! - continued Grande. - In this case, each one can award three points. I want to especially note that as long as all participants act equally, then they are awarded the same number of points. No one has no advantages over to others.
- 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, and the second is still silent! - said Grande. - In this case, the one who gives these testimony, receives five points at once, and the one that is silent does not receive a single point!
- Yeah! - Both witches were exclaimed in one voice, predatory licking lips. It was clear that both of them were clearly going to get five points.
- I have lost my glasses all the time! - exclaimed the demon. - But after all, you just outlined the situation, and the method of its permission has not yet introduced! So what is your strategy? Do not pull the time!
- Wait, now I will explain everything! - exclaimed Grande. - Each of us four - we are here two golems and two witches - will fight against their opponents. Of course, the witches will try to no one in anything ...
- Sure! - exclaimed both witches in unison again. They perfectly understood the golem from the Poluslov!
"And the second golem will follow my tactic," the Grande continued calmly. He looked at his twin. - You, of course, know?
- Yes of course! I'm your copy! I understand everything I understand what you think!
- That's great! In this case, let's make the first move so that the demon can see everything himself. Each fight will be several rounds so that the entire strategy can manifest itself to the end and impress the holistic system. Perhaps, I should start.

- Now each of us should apply marks on their sheets of paper! - He applied to the witch. - First you should draw a smiling face. This will mean that we will not testify to the comrade on the conclusion. You can also draw a purple face that means that we only think about yourself and the necessary testimony on your comrade give. We are both aware that it would be better if no one turned out to be the most populous face, but, on the other hand, a petitioner receives certain advantages over smiling! But the essence is that each of us does not know what the other will choose! We will not know until then, as long as the partner of the game will not open its drawing!
- Start you, bastard! - witch cut out. She, as always, could not do without parangular epithets!
- Ready! - exclaimed Grandi, having drawn a big smiling face on his sheet of paper so that the witch could not see what he portrayed there. Witch made his turn, also by portraying a person. We must think, she certainly portrayed the unkind physiognomy!
"Well, now we can only show our drawings to each other," Grande announced. Wrapped back, he opened the drawing to the public and showed it in all directions so that the drawing could see everything. Something disappeared, the same witch did the same.
As Grande and I expected, with the drawing of the sorceress I watched an evil, displeased face.
"Now you, dear viewers," Grandi said solemnly, "you see that the witch chose to give me a testimony. I'm not going to do that. Thus, the sea witch picks five points. And I, accordingly, do not get a single score. And here…
In the ranks of the spectators, a light slum rolled again. Everyone clearly sympathized with Gola and passionately wanted to lose the sea witch.
But the game just started! If only his strategy was faithful ...
- Now we can go to the second round! - announced Grande solemnly. - We must repeat the moves again. Each paints the face that is closer to him!
So done. Grande depicted now the gloomy, displeased face.
As soon as the players showed their drawings, the public saw that now both of them depicted evil faces.
- Two points to everyone! - said Grande.
- Seven two in my favor! - the witch shouted joyfully. - You will not go anywhere from here, the barefoot!
- We start again! - exclaimed Grande. They did in the next drawing and showed them to the public. Again the same evil faces.
- Each of us repeated the previous move, he behaved selfishly, and therefore it seems to me, it is better not to amend the glasses! - said Golem.
- But I still lead in the game! - said the witch, happily rubbing his hands.
- Okay, not Shumi! - said Grande. - The game is not over. Let's see what will happen! So, dear public, we start the fourth round of Round!
Players again made pictures, showing the public what they were depicted on their sheets. Both leaves again appeared to the audience the same evil physiognomy.
- Eight - three! - the witch shouted, pouring the evil laugh. - You dug up our stupid strategy, Golem!
- Fifth round! - shouted the Grande. Repeated the same thing as in the former rounds - again evil faces, only the expense changed - he began nine - four in favor of the sorcerer.
- Now the last, sixth round! - He announced Grande. His preliminary calculations showed that this row should be fateful. Now the theory was to be confirmed or be refuted by practice.
Several quick and nerve motions of the pencil on paper - and both pictures appeared before the eyes of the public. Again two faces, now even with cracked teeth!
- Ten - five in my favor! My game! I won! - The sea witch burned.

"You really won," Grande agreed in gloomily. The audience was silent.
The demon moved was lips to say something.

- But our contest has not yet completed! - shouted the Grand Walk. - That was just the first part of the game.
- Yes, you have a whole eternity! - shouted Demon Xant displeased.
- It's right! - said Grandi calmly. - But one tour does not solve anything, only the methodology indicates the best result.
Now the golem went to another witch.
- I would like to play this tour with another opponent! - he announced. - Each of us will depict faces as it was in the previous time, then will demonstrate the published public!
So they did. The result was the same as last time - Grandi drew a smiling face of the face, and the witch is so in general a skull. She immediately gained an advantage in the entire five points, leaving the Grande behind.
The remaining five rounds ended with those results that could be expected. Again the score was ten - five in favor of a sea witch.
- Golem, I really like your strategy! - laughing sordogne.
- So, you viewed two rounds of games, dear viewers! - exclaimed Grande. - So, thus scored ten points, and my rivals - twenty!
The audience, who also led the counting points, thoroughly rinsed his heads. Their counting coincided with the calculations of the golem. Only a cloud named Frakto seemed very pleased, although, of course, it also did not sympathize with the witch.
But Rapunzelia smiled approvingly by Glau - she continued to believe in him. She may have remained the only one who believed him now. Grande hoped he would justify this limitless confidence.
Now Grandi approached his third opponent - his twin. He had to become his last opponent. Quickly chirking pencils on paper, the golems showed the leaves to the public. Everyone saw two laughing faces.
- Note, expensive viewers, each of us chose to be a good ceamer! - exclaimed Grande. - And therefore, none of us received in this game the necessary advantage over the opponent. Thus, we both get three points and proceed to the next round!
The second round began. The result was the same as the previous time. Then the remaining rounds. And in each round, both enemies were gaining three points again! It was just incredible, but the audience was ready to confirm all what was happening.

Finally, this tour came to the end, and Grande, quickly driving her pencil on paper, began to count the result. Finally he announced solemnly:
- Eighteen to eighteen! In total, I scored twenty-eight points, and my rivals scored thirty eight!
"So you lost," the sea witch heard joyfully. - The winner will become, so one of us!
- Maybe! - calmly responded Grande. Now there has been another important point. If everything goes as it was conceived ...
- You need to bring the point to the end! - exclaimed the second golem. - I also need to fight with two marine witch! The game is not finished yet!
- Yes, of course, come on! - said Grande. - But only guided the strategy!
- Yes of course! - His twilight assured him.
This goel went to one of the witches, and the tour began. He ended with the same result, with which the Grande himself came out of such a round was ten-five in favor of the witch. The witch was constantly shone from an inexpressible joy, and the public silently silent. Demon Xant looked somewhat tired, which was not too kindly a foremason.
Now the final round has come - one witch was supposed to fight against the second. Each had in the asset for twenty points she was able to get, fighting with the Golets.
"And now, if you let me gain at least a few extra glasses ..." The maritime witch whispered to his twin.
Grandi tried to preserve calm at least outwardly, although in his soul he was raging the hurricane of contradictory feelings. His luck now depended on how true he predicted the possible behavior of both witches - because their character was, in essence, the same!
Now the most, perhaps, the critical moment. But if he was wrong!
- What are these things I have to give up! - Skrying the second witch first. - I myself want to gain more points and get out of here!
- Well, if you are so cheering yourself, - the contender screamed, - then I will finish you now so that you will no longer be like me!
Witches, giving each other with hated views, draw their drawings and showed them to the public. Of course, nothing else, except for two skulls, there simply could not! Each drew one point.
Witches, showering each other curses, started the second round. The result is again the same - again two coryato drawn skulls. Witch, therefore, they scored another point. The public diligently fixed everything.
So continued in the future. When the tour ended, tired witches found that each of them scored six points. Again a draw!
- Now let's calculate the resulting results and everything will be comparable! - Grandi 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 the golems have more points!
At the ranks of the audience, a sigh of surprise swept. Excited spectators began to write on their sheets of pincions of numbers, checking the accuracy of the counting. Many during this time simply did not consider the number of points scored, believing that the result of the game was already known. Both witches began to growl from indignation, it is not clear who exactly accusing to what happened. The eyes of Demon Xanta again caught fire with a wary fire. His confidence was justified!
"I ask you, dear public, pay attention to the fact," the Grande's hand raised, demanding to calm down from the audience, "that none of the golems won a single round. But the final victory will still be at one of us, from the golems. The results will be more eloquent if the contest continues further! I want to say my dear spectators that in the eternal fight my strategy will always be advantageous!
The Demon Xunt was listened to the fact that he spoke Grande. Finally, he, the emitting clubs of the couple, opened his mouth:
- What exactly is your strategy?
- I call it "to be solid, but honest"! - explained Grande. - I'm starting to play honestly, but then I start to lose, because I come across very specific partners. Therefore, in the first round, when it turns out that the sea witch begins to give testimony against me, I automatically remain the loser and in the second round - and so continues until the end. The result may be different, if the witch will change its playing tactics. But since she could not even come to mind, we continued to play on the previous template. When I started playing with my double, he was well treated for me, and I treated him well in the next round of the game. Therefore, we went to the game too differently and somewhat monotonous as we did not want to change the tactics ...
- But you did not won a single tour! - The demon objected.
- Yes, and these witches did not lose any tour! - confirmed Grandi. - But the victory does not automatically go to the one who remained tours. The victory goes to the one who scored more points, and this is quite another thing! I managed to score more points when we played with my twin than when I played with witch. Their selfish attitude brought them a momentary victory, but in terms of more long-term it turned out that it was because of this that they lost the game entirely. It often happens and that!

In the 1930s, John and Oscar Morgenishtern became the founders of the new interesting destination of mathematics, which was called the "game theory". In the 1950s, a young mathematician John Nash was interested in this direction. The theory of equilibrium became the topic of his thesis, which he wrote, being aged 21 years. This was born a new called "Nesha Equilibrium", who earned the Nobel Prize many years later - in 1994.

Long gap between the writing of the thesis and universal recognition was the test for mathematics. The genius without recognition broke into serious mental violations, but also this task John Nash was able to solve the mind of the wonder. His theory "Equilibrium on Nash" was awarded the nobel award, and his life is an empty in the film "Beautiful Mind" ("Mind Games").

Brief about the theory of games

Since the theory of Nash's equilibrium explains the behavior of people in the conditions of interaction, therefore it is worth considering the basic concepts of the theory of games.

The theory of games is studied by the behavior of participants (agents) in conditions of interaction with each other by the type of game, when the outcome depends on the solution and behavior of several people. The participant makes decisions, guided by its forecasts regarding the behavior of the rest, which is called a gaming strategy.

There is also a dominant strategy at which the participant receives an optimal result for any behavior of other participants. This is the best player noticeable strategy.

The dilemma of the prisoner and scientific breakthrough

The prisoner's dilemma is the case of the game when the participants are forced to make rational decisions, reaching a common goal in the condition of the conflict of alternatives. The question is which of these options it will choose, aware of personal and common interest, as well as the inability to get both. Players are as if concluded in hard gaming conditions, which sometimes makes them think very productive.

This dilemma was investigated by an American mathematician equilibrium, which he brought, became revolutionary in its kind. This new thought was particularly bright on the opinion of economists about how the marketing market players make, given the interests of others, with dense interaction and interacting interests.

It is best to study the theory of games on concrete examples, since this mathematical discipline itself is not a dry-theoretical.

An example of the prisoner of the prisoner

An example, two people made a robbery, hit the hands of the police and pass interrogation in individual cameras. At the same time, police ministers offer each participant favorable conditions under which he will be released in the event of testimony against his partner. Each criminals have the following set of strategies that it will consider:

1. Both simultaneously give indications and receive 2.5 years in prison.
2. Both are sometimes silent and get to 1 year, because in this case the evidence base of their guilt will be small.
3. One gives testimony and gets freedom, and the other is silent and gets 5 years in prison.

Obviously, the outcome of the case depends on the solution of both participants, but they cannot consist, because they are sitting in different cameras. Also brightly visible conflict of their personal interests in the struggle for common interest. Each of the prisoners have two options for action and 4 outcomes.

Chain logical conclusion

So, the criminal and considers the following options:

1. I am silent and silent my partner - we both get a prison for 1 year.
2. I rent a partner and he dresses me - we both get 2,5 years in prison.
3. I am silent, and the partner drops me - I will get 5 years in prison, and he is freedom.
4. I rent a partner, and he silent - I get freedom, and he is 5 years in prison.

We give the matrix of possible solutions and outcomes for clarity.

Table of probable outcomes of the dilemma of the prisoner.

The question is that each participant will choose?

"Silent, you can not talk" or "you can not be silent, talk"

To understand the choice of the participant, you need to go through the chain of his reflection. Following the reasoning of the criminal A: if I keep silent and silent my partner, we will receive a minimum of the term (1 year), but I can't learn how he behaves. If he gives testimony against me, then I also have a better testimony, otherwise I can sit for 5 years. It is better for me to sit for 2.5 years than for 5 years. If he is silent, then I need to testify, since I get freedom. Similarly, a member of B is also arguing.

It is not difficult to understand that the dominant strategy for each of the criminals is a dacha test. The optimal point of this game comes when both criminals give testimony and receive their "prize" - 2.5 years in prison. Nash game theory calls it to equilibrium.

Non-optimal nonsense solution

The revolutionary of the Nashevsky view is not optimal if we consider a separate participant and his personal interest. After all, the best option is to keep silent and get free.

Equilibrium on Nash is the point of contact of interests, where each participant chooses this option, which is optimal for him only on the condition that other participants choose a specific strategy.

Considering the option when both criminal silent and get only for 1 year, you can call it a pass-optimal option. However, it is possible only if the criminals could consist in advance. But even this would not guarantee this outcome, as the temptation to retreat from the persuasion and avoid punishment is great. The lack of complete trust in each other and the danger get 5 years forcing the choice of an option with recognition. To reflect on the fact that participants will adhere to an option with silence, acting consistently, is simply irrational. This conclusion can be done if you study the balance of Nash. Examples only prove the correctness.

Egoistic or rational

Nash's equilibrium theory gave stunning conclusions that refuted the existing principles before. For example, Adam Smith examined the behavior of each of the participants as absolutely selfish, which led the system to balance. This theory was called the "invisible hand hand."

John Nash saw that if all participants would act, pursuing only their interests, it would never lead to an optimal group result. Considering that the rational thinking is inherent to each participant, the choice is more likely, which offers Nash's equilibrium strategy.

Purely male experiment

A bright example is the game "Blonde Paradox", which, although it seems inappropriate, but is a bright illustration showing how the theory of Nash games works.

In this game you need to imagine that the company of free guys came to the bar. Nearby turns out to be a company, one of which is preferable to others, say the blonde. How guys to behave to get the best girlfriend for yourself?

So, the reasoning guys: if everyone is started to get acquainted with blonde, then, most likely, she will not get anyone, then her friends will not want dating. No one wants to be a second spare option. But if the guys choose to avoid blonde, then the likelihood of each of the guys to find a good girlfriend is high.

The situation of equilibrium on Nash is non-optimal for guys, because, pursuing only its egoistic interests, everyone would choose exactly blonde. It can be seen that the persecution of only selfish interests will be equivalent to collapse of group interests. Nash equilibrium will mean that each guy acts in their personal interests that come into contact with the interests of the whole group. This is a non-optimal option for each personally, but optimal for everyone, based on the overall success strategy.

Our whole life is a game

Decision making in real conditions is very similar to the game when you expect certain rational behavior from other participants. In business, in work, in the team, in the company and even in relations with the opposite sex. From large transactions and to ordinary life situations, everything obeys anyone or other law.

Of course, the survey situations with criminals and the bar are just excellent illustrations showing Nash's equilibrium. Examples of such dilemmas are very often arising in the real market, and especially it works in cases with two monopolists controlling the market.

Mixed strategies

Often we are not involved in one, but immediately in several games. Choosing one of the options for one game, guided by a rational strategy, but fall into another game. After several rational solutions, you may find that your result does not suit you. What to do?

Consider two types of strategy:

• The net strategy is the behavior of the participant, which comes from reflection over the possible behavior of other participants.
• A mixed strategy or random strategy is an alternation of net strategies randomly or choosing a pure strategy with a certain probability. This strategy is also called the Randomized.

Considering this behavior, we get a new look at the equilibrium on nose. If it was previously stated that the player chooses a strategy once, then you can imagine another behavior. You can allow the option that players choose a strategy randomly with a certain probability. Games in which Nash's equilibrium cannot be found in pure strategies, always have them in mixed.

Nash equilibrium in mixed strategies is called mixed equilibrium. This is an equilibrium, where each participant chooses the optimal frequency of choosing its strategies, provided that other participants choose their strategies at a given frequency.

Penalty and mixed strategy

An example of a mixed strategy can be brought in a football game. The best illustration of a mixed strategy is, perhaps, a penalty series. So, we have a goalkeeper who can only jump in one corner, and a player who will beat the penalty.

So, if for the first time the player chooses the strategy to make a blow to the left corner, and the goalkeeper will also fall into this angle and is lying the ball, how can events develop a second time? If a player beat in the opposite corner, it is most likely too obvious, but also a blow to the same corner is no less obvious. Therefore, goalkeeper, and having nothing remains, how to rely on a random choice.

So, alternating a random choice with a specific pure strategy, a player and goalkeeper try to get the maximum result.