In the 1930s, John and Oscar Morgenstern became the founders of an interesting new branch of mathematics called "game theory". In the 1950s, the young mathematician John Nash became interested in this area. Equilibrium theory became the topic of his dissertation, which he wrote at the age of 21. Thus was born a new one called "Nash Equilibrium", which won the Nobel Prize many years later - in 1994.

The long gap between dissertation writing and general acceptance challenged the mathematician. Genius without recognition resulted in serious mental disorders, but John Nash was able to solve this problem thanks to his excellent logical reason. His theory of "Nash equilibrium" won the Nobel Prize, and his life was adapted in the film "Beautiful mind".

Game theory at a glance

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

Game theory studies the behavior of participants (agents) in terms of interaction with each other like a game, when the outcome depends on the decisions and behavior of several people. The participant makes decisions based on his predictions about the behavior of others, which is called a game strategy.

There is also a dominant strategy in which the participant gets the optimal result for any behavior of the other participants. This is the player's best no-lose strategy.

Prisoner's dilemma and scientific breakthrough

The prisoner's dilemma is a case of play, when the participants are forced to make rational decisions, achieving a common goal in a conflict of alternatives. The question is which of these options he will choose, realizing his personal and general interest, as well as the impossibility of obtaining both. Players seem to be trapped in harsh game conditions, which sometimes makes them think very productively.

This dilemma was explored by an American mathematician. The equilibrium he deduced became revolutionary in its own way. Especially brightly, this new thought influenced the opinion of economists about how market players make their choice, taking into account the interests of others, with close interaction and intersection of interests.

It is best to study game theory with specific examples, since this mathematical discipline itself is not dry-theoretical.

An example of a prisoner's dilemma

For example, two people committed a robbery, fell into the hands of the police and are being interrogated in separate cells. At the same time, the police officers offer each participant favorable conditions under which he will be released if he testifies against his partner. Each of the criminals has the following set of strategies to consider:

1. Both testify at the same time and receive 2.5 years in prison.
2. Both are silent at the same time and receive 1 year each, since in this case the evidence base of their guilt will be small.
3. One gives testimony and gets freedom, while the other is silent and gets 5 years in prison.

Obviously, the outcome of the case depends on the decision of both participants, but they cannot come to an agreement, since they are sitting in different cells. Also clearly visible is the conflict of their personal interests in the struggle for a common interest. Each of the prisoners has two options for action and 4 options for outcomes.

Logical inference chain

So, offender A considers the following options:

1. I am silent and my partner is silent - we both will receive 1 year in prison.
2. I turn in my partner and he turns me over - we both get 2.5 years in prison.
3. I am silent, and my partner turns me over - I will get 5 years in prison, and he is free.
4. I hand over my partner, but he is silent - I get freedom, and he is 5 years in prison.

Here is a matrix of possible solutions and outcomes for clarity.

Table of likely outcomes of the prisoner's dilemma.

The question is, what will each participant choose?

"Silence, you can't speak" or "You can't be silent, you can't speak"

To understand the choice of the participant, you need to go through the chain of his thoughts. Following the reasoning of the offender A: if I remain silent and my partner does not say anything, we will get the minimum term (1 year), but I cannot find out how he will behave. If he testifies against me, then it is also better for me to testify, otherwise I may be imprisoned for 5 years. It is better for me to go to prison for 2.5 years than for 5 years. If he keeps silent, then all the more I need to testify, since this will give me freedom. Participant B argues in the same way.

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

Non-optimal optimal Nash solution

The revolutionary nature of the Nash view is that it is not optimal when considering the individual participant and his personal interest. After all, the best option is to remain silent and be released.

The Nash equilibrium is a point of convergence of interests, where each participant chooses an option that is optimal for him only if other participants choose a certain strategy.

Considering the option when both criminals are silent and receive only 1 year each, we can call it the Pareto-optimal option. However, it is only possible if the criminals could come to an agreement beforehand. But even this would not guarantee this outcome, since the temptation to deviate from the agreement and avoid punishment is great. The lack of complete trust in each other and the danger of getting 5 years makes it necessary to choose the option with recognition. It is simply irrational to speculate that the participants will stick to the silent option, acting in concert. This conclusion can be made if we study the Nash equilibrium. Examples only prove the case.

Selfish or rational

Nash's equilibrium theory yielded startling findings that disproved prior principles. For example, Adam Smith viewed the behavior of each of the participants as completely selfish, which brought the system into balance. This theory was called the "invisible hand of the market."

John Nash saw that if all participants act in pursuit of their own interests, then this will never lead to an optimal group result. Given that rational thinking is inherent in each participant, the choice that the Nash equilibrium strategy offers is more likely.

Purely male experiment

A striking example is the game "blonde paradox", which, although it seems inappropriate, is a vivid illustration of how Nash's theory of games works.

In this game, you need to imagine that a group of free guys came to a bar. Nearby is a group of girls, one of whom is preferable to others, say a blonde. How can guys act to get the best friend for themselves?

So, the guys' reasoning: if everyone starts to get to know a blonde, then most likely nobody will get her, then her friends will not want to meet. Nobody wants to be the second fallback. But if guys choose to avoid the blonde, then the likelihood of each guy finding a good girlfriend among the girls is high.

The Nash equilibrium situation is not optimal for guys, because, pursuing only their own selfish interests, everyone would choose a blonde. It can be seen that the pursuit of only selfish interests will be tantamount to the collapse of group interests. Nash equilibrium will mean that each guy acts in his own interests, which are in contact with the interests of the entire group. This is not an optimal option for everyone personally, but optimal for everyone, based on the overall strategy of success.

Our whole life is a game

Making decisions in real life is very much like a game, when you expect certain rational behavior from other participants. In business, at work, in a team, in a company, and even in relationships with the opposite sex. From big deals to ordinary life situations, everything obeys this or that law.

Of course, the criminals and bar game situations discussed are just great illustrations of the Nash equilibrium. Examples of such dilemmas very often arise in the real market, and this is especially true in cases with two monopolists controlling the market.

Mixed strategies

Often we are involved not in one, but in several games at once. Choosing one of the options for one game, guided by a rational strategy, but you find yourself in another game. After a few rational decisions, you may find that you are not happy with your outcome. What should be done?

Consider two types of strategy:

• A pure strategy is the behavior of a participant that comes from thinking about the possible behavior of other participants.
• A mixed strategy or random strategy is the alternation of pure strategies in a random way or the selection of a pure strategy with a certain probability. This strategy is also called randomized.

By looking at this behavior, we get a new look at the Nash equilibrium. If earlier it was said that the player chooses a strategy once, then another behavior can be imagined. It can be assumed that the players choose a strategy at random with a certain probability. Games that cannot find Nash equilibria in pure strategies always have them in mixed ones.

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

Penalties and mixed strategy

An example of a mixed strategy can be found in the game of soccer. The best illustration of mixed strategy is perhaps the penalty shootout. So, we have a goalkeeper who can only jump to one corner, and a player who will take the penalty.

So, if the first time a player chooses a strategy to kick into the left corner, and the goalkeeper also falls into this corner and catches the ball, then how can events unfold the second time? If a player kicks in the opposite corner, it is probably too obvious, but a kick in the same corner is no less obvious. Therefore, both the goalkeeper and the batter have no choice but to rely on a random choice.

So, alternating random choices with a certain clean strategy, the player and the goalkeeper try to get the maximum result.

From the popular American blog Cracked.

Game theory is concerned with exploring ways to make the best move and, as a result, get as much of the winning pie as possible by chopping off some of it from other players. It teaches you to analyze many factors and make logical conclusions. I think it should be studied after the numbers and before the alphabet. Simply because too many people make important decisions based on intuition, secret prophecies, the location of the stars, and the like. I have studied game theory thoroughly, and now I want to tell you about its basics. Perhaps this will add common sense to your life.

1. The prisoner's dilemma

Berto and Robert were arrested for robbing a bank after failing to properly use a stolen car to escape. The police cannot prove that they were the ones who robbed the bank, but caught them red-handed in a stolen car. They were taken to different rooms and each was offered a deal: to hand over an accomplice and send him to jail for 10 years, and to be released himself. But if they both pass each other, then each will receive 7 years. If no one says anything, then both will be imprisoned for 2 years just for stealing a car.

It turns out that if Berto is silent, but Robert surrenders him, Berto goes to prison for 10 years, and Robert is released.

Each prisoner is a player, and the benefit of each can be represented in the form of a "formula" (what they both get, what the other gets). For example, if I hit you, my winning scheme looks like this (I get a rough win, you are in great pain). Since each prisoner has two options, we can present the results in a table.

Practical Application: Identifying Sociopaths

Here we see the main application of game theory: identifying sociopaths who think only of themselves. True game theory is a powerful analytical tool, and amateurism often serves as a red flag, with the head betraying a person devoid of honor. People who do the calculations intuitively think that it is better to act inappropriately, because it will lead to a shorter prison term no matter what the other player does. Technically, this is correct, but only if you are a shortsighted person who puts numbers above human lives. This is why the game theory is so popular in finance.

The real problem with the prisoner's dilemma is that it ignores data. For example, it does not consider the possibility of you meeting with friends, relatives, or even creditors of the person whom you have imprisoned for 10 years.

Worst of all, everyone involved in the prisoner's dilemma acts as if they have never heard it.

And the best move is to remain silent, and two years later, together with a good friend, use the common money.

2. Dominant strategy

This is a situation in which your actions will give you the greatest payoff, regardless of your opponent's actions. Whatever happens, you did everything right. This is why many people in the "prisoner's dilemma" believe that betrayal leads to the "best" result regardless of what the other person does, and the inherent disregard of reality in this method makes everything look super-simple.

Most of the games we play do not have strictly dominant strategies because otherwise they would be just awful. Imagine that you would always do the same thing. There is no dominant strategy in the rock-paper-scissors game. But if you were to play with a man with mitts on his hands and could only show a rock or paper, you would have a dominant strategy: paper. Your paper will wrap his stone or lead to a draw, and you cannot lose because the opponent cannot show the scissors. Now that you have a dominant strategy, you have to be a fool to try something different.

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 choose between ballet and boxing. Anjali loves boxing because she loves it when blood is shed for the delight of a screaming crowd of spectators who consider themselves civilized just because they paid for someone's broken heads.

Borislav wants to watch ballet because he understands that ballerinas go through a huge number of injuries and difficult training, knowing that one injury can put an end to everything. Ballet dancers are the greatest athletes on earth. A ballerina may kick you in the head, but she never will, because her leg is worth much more than your face.

Each of them wants to go to their favorite event, but they do not want to enjoy it alone, so we get a scheme of their gain: the greatest value is to do what they like, the least value is to just be with another person, and zero is to be. alone.

Some people suggest stubbornly balancing on the brink of war: if you, no matter what, do what you want, the other person must adjust to your choice or lose everything. As I already said, simplified 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 dating as a “battle of the sexes,” it won't work. Part so that each of you can find someone he likes. And the second problem is that in this situation the participants are so insecure that they cannot do it.

A really winning strategy for everyone is to do what they want, and after, or the next day, when they are free, go to a cafe together. Or alternate between boxing and ballet until a revolution occurs in the entertainment world 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 a fait accompli. And if we can make it work, game theory will replace the entire philosophical, religious, and financial system on the planet, because "the desire not to burn out" has become a more powerful driving force for humanity than fire.

Let's quickly divide the $ 100. You and I decide how many of a hundred we demand and at the same time announce the amount. If our total is less than one hundred, everyone gets what they want. If the total is more than one hundred, the person who asked for the least amount gets the desired amount, and the more greedy person gets what is left. If we ask for the same amount, everyone gets $ 50. How much do you ask? How will you split the money? There is only one winning move.

Calling $ 51 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 get $ 50. And if he asks for less than $ 50, you get $ 51. Either way, there is no other option that will make you more money than this one. Nash Equilibrium is a situation in which we both choose $ 51.

Practical Application: Think First

This is the whole point of game theory. It is not necessary to win, let alone harm other players, but it is imperative to make the best move for yourself, no matter what others prepare for you. And it's even better if this move is beneficial for other players as well. This is a kind of mathematics that could change society.

An interesting variation on this idea is drinking alcohol, which can be called the 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 not doing otherwise.

5. The toss game

The toss is played by Player 1 and Player 2. Each player simultaneously chooses heads or tails. If they guess correctly, Player 1 receives Player 2's penny. If not, Player 2 receives Player 1's coin.

The winning matrix is ​​simple ...

… Optimal strategy: play completely at random. This is more difficult than you think, because the choice must be completely random. If you have a preference for heads or tails, the enemy can use it to take your money.

Of course, the real problem here is that it would be much better if they were just tossing 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 is the worst game ever. And this is the perfect model for a penalty shootout.

Practical Application: Penalty Shootout

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 players in full shape would be able to do the wheel, because that would at least be a measure 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 left or right direction and hope that his choice coincides with the choice of the opponent shooting on goal. This has something to do with playing with a coin.

Note, however, that this is not a perfect example of the similarity to heads and tails, because even with the right 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 with a multiball method, where every minute the players are given an extra ball / puck one-on-one until one of the sides gets a certain result, which was an indicator of the players' real skill, and not a spectacular coincidence.

After all, game theory should be used to make the game smarter. Which means better.

What are free radicals?

Why, if you mix all the colors, you get brown, and not white, because white contains all the colors?

7 unexpected facts about the world around us

Amazing world

10 startling facts about canine thinking

A dog is a friend of man and often takes something from him.

As a result of studying this chapter, the student must:

know

Game concepts based on the principle of dominance, Nash equilibrium, what is reverse induction, etc .; conceptual approaches to solving the game, the meaning of the concept of rationality and balance in the framework of the interaction strategy;

be able to

Distinguish between games in strategic and expanded forms, build a "game tree"; formulate game models of competition for different types of markets;

own

Methods for determining the outcomes 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 for the first time, game theory was systematized in the monograph by J. von Neumann and O. Morgenstern "Game theory and economic behavior" in 1944.Since then, many sections of 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 social sciences (for example, analysis of voting procedures, search for equilibrium concepts that determine cooperative and noncooperative behavior of individuals). As a rule, voters reject candidates representing extreme points of view, but there is a struggle when choosing one of the two candidates offering different compromise solutions. Even Rousseau's idea of ​​evolution from "natural freedom" to "civil freedom" formally corresponds from the standpoint of game theory to the point of view of cooperation.

The game Is an idealized mathematical model of the collective behavior of several persons (players), whose interests are different, which gives rise to a conflict. A conflict does not necessarily imply the presence of antagonistic contradictions between the parties, but is always associated with a certain kind of disagreement. A conflict situation will be antagonistic if an increase in the gain of one of the parties by a certain amount leads to a decrease in the gain of the other party by the same amount and vice versa. The antagonism of interests creates a conflict, and the coincidence of interests reduces the game to the coordination of actions (cooperation).

Examples of a conflict situation are situations that develop in the relationship between the buyer and the seller; in a competitive environment of different firms; in the course of hostilities, etc. Examples of games are ordinary games: chess, checkers, cards, saloon, etc. (hence the name "game theory" and its terminology).

In most games arising from the analysis of financial, economic, managerial situations, the interests of the players (parties) are not strictly antagonistic or absolutely coincident. The buyer and seller agree that it is in their common interest to agree on the sale and purchase, but they bargain vigorously when choosing a specific price within the limits of mutual advantage.

Game theory Is a mathematical theory of conflict situations.

The game differs from a real conflict in that it is played 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 the complete (ideal) intelligence of opponents. In a real conflict, often the best strategy is to guess where the enemy is stupid and use that stupidity to your advantage.

Another drawback of game theory is that each of the players must know all the possible actions (strategies) of the opponent, it is only unknown 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 strategies of the enemy is precisely unknown, and the best solution in a conflict situation will often be to go beyond the limits of the strategies known to the enemy, "dumbfounded" him with something completely new, unforeseen.

Game theory does not include the elements of risk that inevitably accompany intelligent decisions in real-world conflicts. It defines the most cautious, reinsurance behavior of the parties to the conflict.

In addition, in game theory, optimal strategies are found for 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 for one indicator may not be optimal for others.

Recognizing these limitations and therefore not blindly adhering to the recommendations of the given game theories, it is still possible to develop a completely acceptable strategy for many real-life conflict situations.

Research is currently underway to expand the scope of game theory.

In the literature, there are the following definitions of the elements that make up the game.

Players- these are the subjects involved in interaction, represented in the form of a game. In our case, these are households, firms, government. However, in the case of uncertainty in 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 volume of purchased goods or services; firms - on the volume of production; the level of taxes imposed by the government.

Determination of the outcome (result) of the game. For each combination of player actions, the outcome of the game is established 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 of the concept of winning may differ for different types of games. In this case, it is necessary to clearly distinguish between the benefits measured on an ordinal scale (for example, the level of utility), and values ​​for which interval comparison also makes sense (for example, profit, the level of welfare).

Information and expectations. Uncertainty and constant changes in information can be extremely serious in 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 of information set player, i.e. the aggregate of all information about the state of the game that he possesses at key moments in time.

An intuitive idea of ​​shared knowledge is very helpful when considering players' access to information, or common knowledge, meaning the following: a fact is generally 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 at this stage.

In game theory, it is assumed that the game consists of moves, performed by players simultaneously or sequentially.

The moves are personal and random. The move is called personal, if the player consciously chooses it from the set of possible options for actions and carries out it (for example, any move in a chess game). The move is called random, if his choice is made not by the player, but by some random selection mechanism (for example, based on the results of a coin toss).

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 a player is a set of rules that determine the choice of a variant of action for each personal move, depending on the situation that has developed in the course of the game. In simple (one-move) games, when in each game the player can make only one move, the concept of strategy and possible course of action coincide. 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 "option of possible actions" and "strategy" may differ from each other.

The player's strategy is called optimal, if it provides a given player with a multiple repetition of the game the maximum possible average gain or the minimum possible average loss, regardless of what strategies the opponent uses. Other criteria of optimality can also be used.

It is possible that the strategy that provides the maximum payoff does not have another important concept of optimality, such as the stability (equilibrium) of the solution. The solution to the game is sustainable(equilibrium) if the strategies corresponding to this solution 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. V doubles game the number of participants is two, in multiple- more than two.
• 3. Participants of a multiple game can form coalitions, both permanent and temporary. By the nature of the relationship between the players, games are divided into non-coalition, coalition and cooperative.

Coalition-free are 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.

The outcome cooperative game is the division of the coalition's winnings, 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 the situation is compared in terms of preference, as is the case in non-cooperative games, but the divisions; and this comparison is not limited to the consideration of individual winnings, but is more complex in nature.

• 4. According to the number of strategies of each player, games are divided into final(the number of strategies of each player is finite) and endless(the set of strategies for each player is infinite).
• 5. According to the amount of information available to players regarding past moves, games are subdivided into games with complete information(all information about previous moves is available) and incomplete information. Examples of games with complete information are chess, checkers, etc.
• 6. According to the type of description, games are subdivided into positional games (or games in expanded form) and games in normal form. Positional games are set in the form of a game tree. But any positional play can be reduced to normal form, in which each of the players makes only one independent move. In positional games, moves are made at discrete times. Exists differential games, in which the moves are made continuously. These games study the problems of pursuing 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 in terms of mental reproduction of the possible course of action and behavior of the adversary.

7. If any possible game of a certain game has a zero sum of the winnings of all N players (), then they talk about a zero-sum game. Otherwise the games are called games with non-zero sum.

Obviously, the zero-sum doubles game is antagonistic, since the gain of one player is equal to the loss of the second, and therefore, the goals of these players are directly opposite.

The final zero-sum doubles game is called matrix game. Such a game is described by a payoff matrix, in which the first player's payoffs are set. The row number of the matrix corresponds to the number of the applied strategy of the first player, the column - to the number of the applied strategy of the second player; at the intersection of a row and a column is the corresponding gain of the first player (loss of the second player).

A finite pair game with nonzero sum is called bimatrix game. Such a game is described by two pay matrices, each for the respective player.

Let's give the following example. Game "Test". Let player 1 be the student preparing for the test, and player 2 the teacher taking the test. We will assume that the student has two strategies: A1 - prepare well for the test; A 2 - do not prepare. The teacher also has two strategies: B1 - to give credit; B 2 - no credit. The estimation of the values ​​of the payoffs of the players can be based, for example, on the following considerations reflected in the payoff matrices:

This game, in accordance with the above classification, is strategic, doubles, coalition-free, final, described in normal form, with a non-zero sum. More briefly, this game can be called bimatrix.

The challenge 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 - strategies of aggressive behavior, a A i and B i - peaceful behavior. Suppose “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 beneficial to the aggressor. Let the payoff matrices of players 1 and 2 in a given bimatrix game have the form

For both players, the aggressive strategies A2 and B2 are dominated by the peaceful strategies Ax and B v Thus, the only equilibrium in dominant strategies is (A2, B 2), i.e. it is postulated that war is the result of non-cooperative behavior. At the same time, the outcome (A1, B1) (peace) gives a greater payoff for both players. Thus, non-cooperative selfish behavior conflicts with collective interests. Collective interests dictate the choice of peaceful strategies. At the same time, if 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 players violating the established agreement. Indeed, if the first player violates the agreement, and the second does not violate, then the first player's payoff will increase to three, and the second will fall 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. Playing in extensive form is represented as a diagram of the "tree" type of decision making, with the "root" corresponding to the point of the beginning of the game, and the beginning of each new "branch", called knot,- the state achieved at this stage with the given actions already taken by the players. Each end node - each end point of the game - is assigned a vector of winnings, one component for each player.

Strategic, otherwise called normal, shape representation of the game corresponds to a multidimensional matrix, and each dimension (in the two-dimensional case, rows and columns) includes 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 players' strategies.

In fig. 8.2 shows an extensive form of the game, and in table. 8.1 - strategic form.

Rice. 8.2.

Table 8.1. A game with simultaneous decision-making in a strategic manner

There is a fairly detailed classification of the constituent parts of 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.

• Vorobiev Η. N. Game theory for cyber-ekoiomists. Moscow: Nauka, 1985.
• Wentzel E.S. Operations research. Moscow: Nauka, 1980.

The mathematical theory of games that emerged in the forties of the XX century is most often used in economics. But how can we model the behavior of people in society using the concept of games? Why do economists study in which corner football players are more likely to shoot penalties, and how to win in Rock, Scissors, Paper, Danil Fedorovykh, senior lecturer at the Department of Microeconomic Analysis at the Higher School of Economics, told in his lecture.

John Nash and the blonde at the bar

A game is any situation in which the agent's profit depends not only on his own actions, but also on the behavior of other participants. If you play solitaire at home, from the standpoint of an economist and game theory, it is not a game. It implies the mandatory presence of a conflict of interest.

A Beautiful Mind, about John Nash, a Nobel laureate in economics, features a scene with a blonde in a bar. It shows the idea for which the scientist received the award - the idea of ​​Nash equilibrium, which he himself called control dynamics.

The game- any situation in which the payoffs of agents depend on each other.

Strategy - a description of the player's actions in all possible situations.

The outcome is a combination of the chosen strategies.

So, from the point of view of theory, the players in this situation are only men, that is, those who make the decision. Their preferences are simple: a blonde is better than a brunette, and a brunette is better than nothing. You can act in two ways: go to the blonde or to "your" brunette. The game consists of a single move, decisions are made simultaneously (that is, you cannot see where the others went, and then walk yourself). If a girl rejects a man, the game ends: it is impossible to return to her or choose another.

What is the likely ending of this game situation? That is, what is its stable configuration, from which everyone will understand that they have made the best choice? First, as Nash correctly observes, if everyone goes to the blonde, it won't end well. Therefore, the scientist further suggests that everyone needs to go to brunettes. But then, if it is known that everyone will go to brunettes, he should go to the blonde, because she is better.

This is the real balance - the outcome, in which one goes to the blonde, and the rest to the brunettes. It may seem like this is unfair. But in a situation of equilibrium, no one can regret their choice: those who go to brunettes understand that they still wouldn't get anything from a blonde. Thus, the Nash equilibrium is a configuration in which no one individually wants to change the strategy chosen by everyone. That is, reflecting at the end of the game, each participant realizes that even knowing how others are like, he would have done the same. In another way, you can call it an outcome, where each participant responds in an optimal way to the actions of the others.

"Rock Paper Scissors"

Consider other balance games. For example, in "Rock, Scissors, Paper" there is no Nash equilibrium: in all its probable outcomes there is no option in which both participants would be happy with their choice. However, there is the World Championship and the World Rock Paper Scissors Society collecting game statistics. Obviously, you can increase your chances of winning if you know something about the usual behavior of people in this game.

A pure strategy in a game is one in which a person always plays the same way, choosing the same moves.

According to the World RPS Society, stone is the most played move (37.8%). Paper is favored by 32.6%, scissors by 29.6%. Now you know to choose paper. However, if you are playing with someone who also knows this, you no longer have to choose paper, because the same is expected of you. There is a famous case: in 2005, two auction houses Sotheby "s and Christie" s were deciding who would get a very large lot - a collection of Picasso and Van Gogh with a starting price of $ 20 million. The owner invited them to play Rock, Scissors, Paper, and the representatives of the houses sent him their options by e-mail. Sotheby's, as they later said, did not hesitate to choose paper. Won Christie ”s. Making a decision, they turned to an expert - the 11-year-old daughter of one of the top managers. She said, “The stone seems to be the strongest, so most people choose it. But if we are not playing with a completely stupid beginner, he will not throw a stone, he will expect us to do it, and he will throw out the paper himself. But we will think ahead and discard the scissors. "

Thus, you can think ahead, but this does not necessarily lead you to victory, because you may not be aware of the competence of your opponent. Therefore, sometimes instead of pure strategies, it is more correct to choose mixed ones, that is, to make decisions at random. So, in "Rock, Scissors, Paper" the balance, which we have not found before, is precisely in mixed strategies: to choose each of the three options for a move with a probability of one third. If you choose a stone more often, your opponent will adjust his choice. Knowing this, you will correct yours, and balance will not come out. But none of you will begin to change behavior if everyone simply chooses rock, scissors, or paper with equal probability. This is because in mixed strategies based on previous actions it is impossible to predict your next move.

Mixed strategies and sports

There are many more serious examples of mixed strategies. For example, where to serve in tennis or to hit / take a penalty kick in football. If you don't know anything about your opponent, or you are just constantly playing against different ones, the best strategy is to act more or less randomly. Professor of the London School of Economics Ignacio Palacios-Huerta published a paper in the American Economic Review in 2003, the essence of which was to find the Nash equilibrium in mixed strategies. Palacios-Huerta chose football as the subject of his research and, as a result, watched over 1400 penalty kicks. Of course, in sports everything is more cunning than in "Rock, Scissors, Paper": it takes into account the athlete's strong leg, hitting different angles when hitting with all his might, and the like. Nash equilibrium here consists in calculating options, that is, for example, determining the angles of the goal at which you need to shoot in order to win with a greater probability, knowing your strengths and weaknesses. The statistics for each footballer and the equilibrium found in it in mixed strategies showed that footballers do something like economists predict. It hardly needs to be argued that the people who take penalties have read game theory textbooks and have been doing some pretty tricky math. Most likely, there are different ways to learn how to behave optimally: you can be a brilliant football player and feel what to do, or you can be an economist and seek balance in mixed strategies.

In 2008, Professor Ignacio Palacios-Huerta met Abraham Grant, the Chelsea coach who was then playing in the Champions League final in Moscow. The scientist wrote a note to the coach with penalty shootout recommendations regarding the behavior of the opponent's goalkeeper, Edwin van der Sar of Manchester United. For example, according to statistics, he almost always hit the middle level and more often rushed to the natural side for a penalty kick. As we defined above, it is more correct to randomize your behavior taking into account knowledge about your opponent. When the penalty was already 6-5, Chelsea striker Nicolas Anelka had to score. Pointing to the right corner before hitting, van der Sar seemed to ask Anelk if he was going to hit there.

The bottom line is that all of Chelsea's previous shots were to the right of the kicker. We do not know exactly why, perhaps because of the advice of an economist, to beat in an unnatural direction for them, because according to statistics, van der Sar is less ready for this. Most of Chelsea's players were right-handed: hitting an unnatural right-hand corner, all but Terry scored. Apparently, the strategy was for Anelka to shoot in the same direction. But van der Sar seems to have figured it out. He acted brilliantly: he pointed to the left corner saying, "Are you going to hit there?" At the last moment, he decided to act differently, hitting his natural side, which was what van der Sar needed, who took this blow and ensured Manchester's victory. This situation teaches a random choice, because otherwise your decision can be calculated and you will lose.

The Prisoner's Dilemma

Perhaps the most famous game that kicks off college game theory courses is Prisoner's Dilemma. According to legend, the two suspects in a serious crime were caught and locked in different cells. There is evidence that they kept weapons, and this allows them to be imprisoned for a short period of time. However, there is no evidence that they committed this terrible crime. The investigator tells each individual about the conditions of the game. If both criminals confess, both will be imprisoned for three years. If one confesses, and the accomplice is silent, the confessed one will leave immediately, and the other will be imprisoned for five years. If, on the contrary, the first does not confess, and the second surrenders him, the first will sit for five years, and the second will leave immediately. If no one confesses, both will go to prison for a year for possession of weapons.

The Nash equilibrium here lies in the first combination, when both suspects are not silent and both are imprisoned for three years. The reasoning of each is as follows: “If I speak, I will sit for three years, if I remain silent, for five years. If the other is silent, I would also better say: not sitting down is better than sitting down for a year. " This is the dominant strategy: speaking is beneficial, no matter what the other is doing. However, there is a problem in it - the availability of a better option, because sitting for three years is worse than sitting for a year (if we consider the story only from the point of view of the participants and do not take into account moral issues). But it is impossible to sit down for a year, because, as we understood above, it is unprofitable for both criminals to remain silent.

Pareto improvement

There is a famous metaphor about the invisible hand of the market, which belongs to Adam Smith. He said that if the butcher tries to make money for himself, it will be better for everyone: he will make tasty meat, which the baker will buy with money from the sale of rolls, which he, in turn, will also have to make tasty so that they are sold ... But it turns out that this invisible hand does not always work, and there are a lot of situations when everyone acts for himself, but everyone is bad.

Therefore, sometimes economists and game theorists do not think about the optimal behavior of each player, that is, not about the Nash equilibrium, but about the outcome in which the whole society will be better (in "Dilemma" the society consists of two criminals). From this point of view, the outcome is effective when there is no Pareto improvement in it, that is, it is impossible to make someone better without making others worse. If people just exchange goods and services, this is a Pareto improvement: they do it voluntarily, and it is unlikely that anybody is bad for it. But sometimes, if you just let people interact and not even interfere, then what they come to will not be Pareto optimal. This is what happens in The Prisoner's Dilemma. In it, if we allow everyone to act in a way that suits them, it turns out that everyone is bad from this. It would be better for everyone if everyone did not act optimally for themselves, that is, they were silent.

Community tragedy

The Prisoner's Dilemma is a stylized toy story. You might not expect to find yourself in a similar situation, but similar effects are everywhere around us. Consider the multiplayer Dilemma, sometimes referred to as a community tragedy. For example, there are traffic jams on the roads, and I decide how to go to work: by car or by bus. Others do the same. If I go by car, and everyone decides to do the same, there will be a traffic jam, but we will get there in comfort. If I go by bus, there will still be a traffic jam, but I will be uncomfortable and not particularly fast, so this outcome is even worse. If, on average, everyone travels by bus, then I, having done the same, will get there pretty quickly without a traffic jam. But if, under such conditions, I go by car, I will also get there quickly, but also with comfort. So, the presence of a plug does not depend on my actions. The Nash equilibrium is here - in a situation where everyone chooses to go by car. Whatever the others do, I'd better choose a car, because there will be a traffic jam or not, it is unknown, but in any case I will get there with comfort. It's the dominant strategy, so everyone ends up driving and we have what we have. The task of the state is to make traveling by bus the best option for at least some, so there are paid entrances to the center, parking lots, and so on.

Another classic story is the rational ignorance of the voter. Imagine that you do not know the outcome of an election in advance. You can study the program of all candidates, listen to the debates and then vote for the best. The second strategy is to come to the polling station and vote at random or for someone who is more often shown on TV. What is the best behavior if my vote never determines who wins (and in a country with a population of 140 million, one vote will never decide anything)? Of course, I want the country to have a good president, but I know that no one else will study the candidates' programs carefully. Therefore, not wasting time on this is the dominant strategy of behavior.

When you are invited to come to the Saturday clean-up, it will not depend on anyone separately whether the courtyard becomes clean or not: if I go out alone, I cannot remove everything, or, if everyone goes out, I won’t go out, because everything is without me removed. Another example is the shipping of goods in China, which I learned about in Stephen Landsburg's excellent book The Economist on the Couch. 100-150 years ago, a method of transporting goods was widespread in China: everything was folded into a large body, which was dragged by seven people. Customers paid if the goods were delivered on time. Imagine that you are one of these six. You can make an effort and pull with all your might, and if everyone does that, the load will arrive on time. If someone alone does not do this, everyone will also arrive on time. Everyone thinks: "If everyone else is pulling properly, why should I do it, and if everyone else is not pulling with all their strength, then I cannot change anything." As a result, over the time of delivery, everything was very bad, and the movers themselves found a way out: they began to hire a seventh and pay him money to lash the lazy with a whip. The very presence of such a person forced everyone to work with all their might, because otherwise everyone fell into a bad balance, from which no one individually could profitably get out.

The same example can be observed in nature. A tree growing in a garden differs from that growing in a forest in its crown. In the first case, it surrounds the entire trunk, in the second, it is only at the top. In the forest, this is the Nash equilibrium. If all the trees agreed and grew the same way, they would equally distribute the number of photons, and everyone would be better off. But it is unprofitable for anyone to do so. Therefore, each tree wants to grow a little taller than those around it.

Communication device

In many situations, one of the players in the game may need a tool to convince others that they are not bluffing. It's called a commitment device. For example, the law of some countries prohibits the payment of ransom to kidnappers in order to reduce the motivation of criminals. However, this legislation often does not work. If your relative is captured, and you have the opportunity to save him, bypassing the law, you will. Imagine a situation that the law can be circumvented, but the relatives turned out to be poor and they have nothing to pay the ransom. The perpetrator has two options in this situation: release or kill the victim. He doesn't like to kill, but he doesn't like prison anymore. The released victim, in turn, can either give evidence so that the kidnapper was punished, or remain silent. The best outcome for the offender is to release the victim who will not surrender him. The victim wants to be released and testify.

The balance here is that the terrorist does not want to be caught, which means that the victim dies. But this is not Pareto equilibrium, because there is an option in which everyone is better - the victim at large remains silent. But for this it is necessary to do so that it would be profitable for her to be silent. Somewhere I read an option when she can ask a terrorist to arrange an erotic photo session. If the perpetrator is imprisoned, his accomplices will post photos on the Internet. Now, if the kidnapper remains free, that's bad, but open-access photos are even worse, so a balance is obtained. It's a way for the victim to stay alive.

Other examples of games:

Bertrand model

While we're on the subject of economics, let's look at an economic example. In Bertrand's model, two stores sell the same product, buying it from the manufacturer at the same price. If the prices in the stores are the same, then their profit is approximately the same, because then the buyers choose the store by chance. The only Nash equilibrium here is to sell the product at cost. But shops want to make money. Therefore, if one puts the price of 10 rubles, the second will reduce it by a penny, thereby doubling his revenue, since all buyers will go to him. Therefore, it is beneficial for market participants to reduce prices, thereby distributing profits among themselves.

Exit on a narrow road

Let's consider examples of choosing between two possible equilibria. Imagine that Petya and Masha are driving towards each other along a narrow road. The road is so narrow that they both need to pull over to the side. If they decide to turn left or right from themselves, they will simply disperse. If one turns to the right, and the other to the left, or vice versa, an accident will happen. How to choose where to move out? To help find balance in such games, there are, for example, the rules of the road. In Russia, everyone needs to turn right.

In the fun of Chiken, when two people ride at high speed towards each other, there are also two balances. If both swerve to the side of the road, a situation arises, which is called Chiken out, if both do not swerve, then they die in a terrible accident. If I know that my opponent is going straight, it’s advantageous for me to move out in order to survive. If I know that my opponent will move out, then it’s advantageous for me to go straight in order to get $ 100 afterwards. It is difficult to predict what will actually happen, however, each of the players has their own method of winning. Imagine that I secured the steering wheel so that it cannot be turned, and showed this to my opponent. Knowing that I have no choice, the opponent will bounce.

QWERTY effect

Sometimes it can be very difficult to move from one balance to another, even if it means benefits for everyone. The QWERTY layout was created to slow down your typing speed. Because if everyone typed too fast, the typewriter heads that hit the paper would cling to each other. Therefore, Christopher Scholes placed frequently adjacent letters as far away as possible. If you go to the keyboard settings on your computer, you can select the Dvorak layout there and type much faster, since there is no problem with analog presses right now. Dvorak expected the world to move to his keyboard, but we still live with QWERTY. Of course, if we switched to the Dvorak layout, the future generation would be grateful to us. We would all work hard and relearn, and the result would be a balance in which everyone types quickly. Now we are also in balance - in a bad one. But it is not beneficial for anyone to be the only one who will retrain, because it will be inconvenient to work with any computer, except for a personal one.

1. Basic concepts of game theory and their classification .................... 4

1.1. The subject and tasks of game theory ............................................. ....................................... 4

1.2. Terminology and classification of games .............................................. ............................ 7

1.3. Examples of games ................................................ .................................................. ........... 12

Tests ................................................. .................................................. ............................. 15

2. Matrix games .............................................. .................................................. ... sixteen

2.1. Description of the matrix game ............................................... ........................................ sixteen

Game theory is a mathematical theory of conflict situations.

The Purpose of Game Theory - development of recommendations on the reasonable behavior of the parties to the conflict (determination of optimal strategies for the behavior of players).

The game differs from a real conflict in that it is played 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 the complete (“ideal”) intelligence of opponents. In a real conflict, often the best strategy is to guess where the enemy is “stupid” and take advantage of this stupidity.

Another drawback of game theory is that each of the players must know all the possible actions (strategies) of the opponent, it is only unknown 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 strategies of the enemy is precisely unknown, and the best solution in a conflict situation is often going beyond the limits of the strategies known to the enemy, "dumbfounded" him with something completely new, unforeseen.

Game theory does not include the elements of risk that inevitably accompany intelligent decisions in real-world conflicts. It defines the most cautious, “reinsurance” behavior of the parties to the conflict.

In addition, in game theory, optimal strategies are found for 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 for one indicator may not be optimal for others.

Recognizing 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-life conflict situations.

Research is currently underway to expand the scope of game theory.

1.2. Terminology and classification of games

In game theory, it is assumed that the game consists of moves performed by players simultaneously or sequentially.

There are moves personal and random ... The move is called personal if the player deliberately chooses it from the set of possible options for actions and carries out it (for example, any move in a chess game). The move is called random if his choice is made not by the player, but by some mechanism of random selection (for example, by the results of a coin toss).

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 a player is a set of rules that determine the choice of a variant of action for each personal move, depending on the situation that has developed in the course of the game. In simple (one-move) games, when in each game the player can make only one move, the concept of strategy and possible course of action coincide. 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 concept of “option of possible actions” and “strategy” may differ from each other.

A player's strategy is called optimal if it provides a given player with a multiple repetition of the game the maximum possible average gain or the minimum possible average loss, regardless of what strategies the opponent uses. Other criteria of optimality can also be used.

It is possible that the strategy that provides the maximum payoff does not have another important concept of optimality, such as the stability (equilibrium) of the solution. A game solution is stable (equilibrium) if the strategies corresponding to this solution 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. 1.1.

1. Depending from the types of moves games are divided into strategic and gambling. Gambling games consist only of random moves - game theory does not deal with them. If along with random moves there are personal moves, or all moves are personal, then such games are called strategic .

2. Depending of the number of participants games are divided into doubles and multiples. In the steam room the number of participants in the game is two, in multiple - more than two.

3. Participants of a multiple game can form coalitions, both permanent and temporary. The nature The relationship between the players of the game is divided into non-coalition, coalition and cooperative.

Coalition-free are 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 .

https://pandia.ru/text/78/553/images/image002_69.gif "width =" 509 "height =" 75 ">

https://pandia.ru/text/78/553/images/image006_35.gif "width =" 509 "height =" 108 ">

Rice. 1.1. Game classification

The outcome cooperative game is the division of the coalition's winnings, 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 the situation is compared in terms of preference, as is the case in non-cooperative games, but the divisions; and this comparison is not limited to the consideration of individual winnings, but is more complex in nature.

4. By the number of strategies of each player, the games are subdivided into finite (the number of strategies of each player is finite) and endless (the set of strategies for each player is infinite).

5. By the amount of information available to players relative to past moves, games are subdivided into games with complete information (all information about previous moves is available) and incomplete information ... Examples of games with complete information are chess, checkers, etc.

6. By description type games are classified into positional games (or games in expanded form) and games in normal form. Positional games are set in the form of a game tree. But any positional play can be reversed to normal form , in which each player makes only one independent move. In positional In games, moves are made at discrete moments in time. Exists differential games in which moves are made continuously. These games study the problems of pursuing 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 that consider situations taking into account the mental reproduction of the possible course of action and behavior of the enemy.

7. If any possible game of some game has zero winnings f i, https://pandia.ru/text/78/553/images/image009_21.gif "width =" 60 height = 45 "height =" 45 ">), then they talk about the game zero sum ... Otherwise the games are called games with non-zero sum .

Obviously, the zero-sum doubles game is antagonistic , since the gain of one player is equal to the loss of the second, and therefore the goals of these players are directly opposite.

The final zero-sum doubles game is called matrix game. Such a game is described by a payoff matrix, in which the first player's payoffs are set. The row number of the matrix corresponds to the number of the applied strategy of the first player, the column - to the number of the applied strategy of the second player; at the intersection of a row and a column is the corresponding gain of the first player (loss of the second player).

A finite pair game with nonzero sum is called bimatrix game. Such a game is described by two pay matrices, each for the respective player.

1.3. Examples of games

Game 1. Test

Let player 1 be the student preparing for the test, and player 2 the teacher taking the test. We will assume that the student has two strategies: A1 - prepare well for the test; A2 - do not prepare. The teacher also has two strategies: B1 - to give credit; B2 - no credit. The estimation of the values ​​of the payoffs of the players can be based, for example, on the following considerations reflected in the payoff matrices

	(appreciated)				(everything is fine)	(was unfair)
	(managed to catch it)	(got what he deserved)			(let himself be fooled)	(student will come again)
Student winnings	Teacher benefits

This game, in accordance with the above classification, is strategic, doubles, coalition-free, final, described in normal form, with a non-zero sum. More briefly, this game can be called bimatrix.

The challenge is to determine the optimal strategies for the student and for the teacher.

Game 2. Morra

The game "morra" is a game of any number of faces, in which all players simultaneously show ("throw") a certain number of fingers. Each situation is attributed to the winnings that the players in this situation receive from the "bank". For example, each player wins the number of fingers shown to him if all other players show a different number; he gains nothing in all other cases. In accordance with the above classification, this game is strategic; in the general case, multiple (in this case, the game can be non-coalitional, coalitional, and cooperative) finite.

In the special case, when the game is doubles, it will be a matrix game (matrix game is always antagonistic).

Have two players "throw" one, two, or three fingers at the same time. If the sum is even, the first player wins, if the sum is odd, the second. The winning is equal to the sum of thrown fingers. Thus, in this case, each of the players has three strategies, and the matrix of the first player's payoffs (the second player’s losses) has the form:

where A i- the first player's strategy of "throwing out" i fingers;

V j- the strategy of the second player, which consists in "throwing out" j fingers.

What should each of the players do to ensure that they receive the maximum winnings?

Game 3. Fight for the markets

A certain firm A, having at its disposal 5 conventional monetary units, is trying to keep two equal sales markets. Its competitor (firm B), having an amount equal to 4 conventional monetary units, is trying to oust firm A from one of the markets. Each of the competitors can allocate a whole number of units of their funds to protect and conquer the respective market. It is believed that if firm A allocates less funds than firm B to protect at least one of the markets, then it loses, and in all other cases it wins. Let the gain of firm A be equal to 1, and the loss is equal to (-1), then the game is reduced to a matrix game, for which the matrix of the gains of firm A (losses of firm B) has the form:

Here A i- the strategy of firm A, which consists in highlighting i conventional monetary units for the protection of the first market; V j- the strategy of firm B, which consists in highlighting j conventional monetary units to conquer the first market.

If firms could devote any amount of funds available to defend or conquer markets, the game would be endless.

TESTS

(B - True, N - False)

1. Any conflict situation is antagonistic.

2. Any antagonistic situation is a conflict.

4. The disadvantage of game theory is the assumption that the opponents are completely reasonable.

5. In game theory, it is assumed that not all of the opponent's possible strategies are known.

6. Game theory includes elements of risk that inevitably accompany intelligent decisions in real-world conflicts.

7. In game theory, finding the optimal strategy is carried out according to many criteria.

8. Strategy games consist only of personal moves.

9. In a doubles game, the number of strategies of each participant is equal to two.

10. Games in which the actions of the players are aimed at maximizing the payoffs of the coalitions without their subsequent division between the players are called coalition games.

11. The outcome of the cooperative game is the division of the coalition's winnings, which arises not as a result of certain actions of the players, but as a result of their predetermined agreements.

12. According to the type of description, games are divided into games with complete information or games with incomplete information.

13. A finite multiple zero-sum game is called a matrix game.

14. The final zero-sum doubles game is called a bimatrix game.

(Answers: 1-H; 2-B; 3-B; 4-B; 5-H; 6-H; 7-H; 8-H; 9-H; 10-B; 11-B; 12-H ; 13-H; 14-H.)

2. MATRIX GAMES

2.1. Description of the matrix game

The most developed in game theory is a finite pair zero-sum game (an antagonistic game of two persons or two coalitions) called a matrix game.

Consider a game like this G, in which two players participate A and V having antagonistic interests: the gain of one player is equal to the loss of the second. Since the player's winnings A equal to the player's payoff V with the opposite sign, we can only be interested in winning a player A... Naturally the player A wants to maximize a and the player V- minimize a... For the prostate, we identify ourselves mentally with one of the players (let it be the player A), then we will call the player V- “enemy” (of course, some real advantages for A does not follow from this).