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

Lots of text

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

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

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

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

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

3.4.1. Basic concepts of game theory

Currently, many solutions to problems in production, economic or commercial activities depend on the subjective qualities of the decision maker. When choosing decisions under conditions of uncertainty, an element of arbitrariness, and therefore risk, is always inevitable.

The theory of games and statistical decisions deals with problems of decision-making under conditions of complete or partial uncertainty. Uncertainty can take the form of opposition from the other party, which pursues opposite goals, interferes with certain actions or states of the external environment. In such cases, it is necessary to take into account possible options for the behavior of the opposite party.

Possible behavior options for both sides and their outcomes for each combination of alternatives and states can be represented in the form mathematical model called a game. Both sides of the conflict cannot accurately predict mutual actions. Despite such uncertainty, each side of the conflict has to make decisions.

Game theory- this is a mathematical theory conflict situations. The main limitations of this theory are the assumption of the complete (“ideal”) rationality of the enemy and the adoption of the most cautious “reinsurance” decision when resolving the conflict.

The conflicting parties are called players, one implementation of the game – party, outcome of the game - winning or losing.

On the move in game theory is the choice of one of the actions provided for by the rules and its implementation.

Personally called conscious choice player of one of possible options actions and their implementation.

Random move call the choice by a player, carried out not by the player’s volitional decision, but by some mechanism of random selection (tossing a coin, dealing cards, etc.) of one of the possible options for action and its implementation.

Player strategy is a set of rules that determine the choice of action for each personal move of this player, depending on the situation that arises during the game

Optimal strategy player is a strategy that, when repeated many times in a game containing personal and random moves, provides the player with the maximum possible average winnings (or, what is the same, the minimum possible average loss).

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

- Combinatorial games in which the rules, in principle, allow each player to analyze all the various options for behavior and, by comparing these options, choose the best one. The uncertainty here is too large quantities options that need to be analyzed.

- Gambling games in which the outcome is uncertain due to the influence of random factors.

- Strategic games in which the uncertainty of the outcome is caused by the fact that each of the players, when making a decision, does not know what strategy the other participants in the game will follow, since there is no information about the subsequent actions of the opponent (partner).

- The game is called doubles, if the game involves two players.

- The game is called multiple, if there are more than two players in the game.

- The game is called zero sum, if each player wins at the expense of the others, and the sum of the winnings and losses of one side is equal to the other.

- Zero-sum doubles game called antagonistic game.

- The game is called finite, if each player has only a finite number of strategies. Otherwise it's a game endless.

- One step games when the player chooses one of the strategies and makes one move.

- In multi-step games Players make a series of moves to achieve their goals, which may be limited by the rules of the game or may continue until one of the players has no resources left to continue the game.

- Business games imitate organizational and economic interactions in various organizations and enterprises. The advantages of a game simulation over a real object are:

Visibility of the aftereffects of decisions made;

Variable time scale;

Repetition of existing experience with changes in settings;

Variable coverage of phenomena and objects.

Elements of the game model are:

- Participants of the game.

- Rules of the game.

- Information array, reflecting the state and movement of the modeled system.

Carrying out classification and grouping of games allows for similar games to find common methods for searching for alternatives in decision making, and to develop recommendations on the most rational course of action during the development of conflict situations in various fields of activity.

3.4.2. Setting game objectives

Consider a finite zero-sum pairs game. Player A has m strategies (A 1 A 2 A m), and player B has n strategies (B 1, B 2 Bn). Such a game is called a game of dimension m x n. Let a ij be the payoff of player A in a situation where player A chose strategy A i, and player B chose strategy B j. The player's payoff in this situation will be denoted by b ij . A zero-sum game, therefore, a ij = - b ij . To carry out the analysis, it is enough to know the payoff of only one of the players, say A.

If the game consists only of personal moves, then the choice of strategy (A i, B j) uniquely determines the outcome of the game. If the game also contains random moves, then the expected win is the average value (mathematical expectation).

Let us assume that the values ​​of a ij are known for each pair of strategies (A i, B j). Let's create a rectangular table, the rows of which correspond to the strategies of player A, and the columns correspond to the strategies of player B. This table is called payment matrix.

Player A's goal is to maximize his winnings, and player B's goal is to minimize his loss.

Thus, the payment matrix looks like:

The task is to determine:

1) The best (optimal) strategy of player A from the strategies A 1 A 2 A m;

2) The best (optimal) strategy of player B from strategies B 1, B 2 Bn.

To solve the problem, the principle is applied according to which the participants in the game are equally intelligent and each of them does everything to achieve their goal.

3.4.3. Methods for solving game problems

Minimax principle

Let us analyze sequentially each strategy of player A. If player A chooses strategy A 1, then player B can choose such strategy B j, in which the payoff of player A will be equal to the smallest of the numbers a 1j. Let's denote it a 1:

that is, a 1 is the minimum value of all the numbers in the first line.

This can be extended to all rows. Therefore, player A must choose the strategy for which the number a i is the maximum.

Value a is a guaranteed win that player a can secure for himself for any behavior of player B. Value a is called the lower price of the game.

Player B is interested in reducing his loss, that is, reducing player A's winnings to a minimum. To choose the optimal strategy, he must find maximum value winnings in each column and choose the smallest among them.

Let's denote by b j the maximum value in each column:

Lowest value b j denote by b.

b = min max a ij

b is called the upper bound of the game. The principle that dictates that players choose appropriate strategies is called the minimax principle.

There are matrix games for which the lower price of the game is equal to the upper price; such games are called saddle point games. In this case, g=a=b is called the net price of the game, and strategies A * i, B * j, allowing to achieve this value are called optimal. The pair (A * i, B * j) is called the saddle point of the matrix, since the element a ij .= g is simultaneously the minimum in the i-row and the maximum in the j-column. Optimal Strategies A*i, B*j, and net price are a solution to the game in pure strategies, i.e., without involving a random selection mechanism.

Example 1.

Let a payment matrix be given. Find a solution to the game, i.e. determine the lower and upper prices of the game and minimax strategies.

Here a 1 =min a 1 j =min(5,3,8,2) =2

a =max min a ij = max(2,1,4) =4

b = min max a ij =min(9,6,8,7) =6

Thus, the lower price of the game (a=4) corresponds to strategy A 3. By choosing this strategy, player A will achieve a payoff of at least 4 for any behavior of player B. The upper price of the game (b=6) corresponds to the strategy of player B. These strategies are minimax . If both sides follow these strategies, the payoff will be 4 (a 33).

Example 2.

The payment matrix is ​​given. Find the lower and upper prices of the game.

a =max min a ij = max(1,2,3) =3

b = min max a ij =min(5,6,3) =3

Therefore, a =b=g=3. The saddle point is the pair (A * 3, B * 3). If a matrix game contains a saddle point, then its solution is found using the minimax principle.

Solving games in mixed strategies

If the payment matrix does not contain a saddle point (a mixed strategy.

To use mixed strategies, the following conditions are required:

1) There is no saddle point in the game.

2) Players use a random mixture of pure strategies with corresponding probabilities.

3) The game is repeated many times under the same conditions.

4) During each move, the player is not informed about the choice of strategy by the other player.

5) Averaging of game results is allowed.

It is proven in game theory that every zero-sum paired game has at least one mixed strategy solution, which implies that every finite game has a cost g. g - average winning per game, satisfying condition a<=g<=b . Оптимальное решение игры в смешанных стратегиях обладает следующим свойством: каждый из игроков не заинтересован в отходе от своей оптимальной смешанной стратегии.

The players' strategies in their optimal mixed strategies are called active.

Theorem on active strategies.

The application of an optimal mixed strategy provides a player with a maximum average win (or minimum average loss) equal to the cost of the game g, regardless of what actions the other player takes, as long as he does not go beyond the limits of his active strategies.

Let us introduce the following notation:

P 1 P 2 ... P m - the probability of player A using strategies A 1 A 2 ..... A m ;

Q 1 Q 2 …Q n the probability of player B using strategies B 1, B 2….. Bn

We write the mixed strategy of player A in the form:

A 1 A 2…. A m

Р 1 Р 2 … Р m

We write the mixed strategy of player B in the form:

B 1 B 2…. Bn

Knowing the payment matrix A, you can determine the average winnings (mathematical expectation) M(A,P,Q):

M(A,P,Q)=S Sa ij P i Q j

Player A's average winnings:

a =max minM(A,P,Q)

Player B's average loss:

b = min maxM(A,P,Q)

Let us denote by P A * and Q B * the vectors corresponding to optimal mixed strategies under which:

max minM(A,P,Q) = min maxM(A,P,Q)= M(A,P A * ,Q B *)

In this case, the following condition is satisfied:

maxM(A,P,Q B *)<=maxМ(А,P А * ,Q В *)<= maxМ(А,P А * ,Q)

Solving a game means finding the price of the game and optimal strategies.

Geometric method for determining game prices and optimal strategies

(For the game 2X2)

A segment of length 1 is plotted on the abscissa axis. The left end of this segment corresponds to strategy A 1, the right end to strategy A 2.

The y-axis shows the winnings a 11 and a 12.

The winnings a 21 and a 22 are plotted along a line parallel to the ordinate axis from point 1.

If player B uses strategy B 1, then we connect points a 11 and a 21, if B 2, then a 12 and a 22.

The average winning is represented by point N, the point of intersection of straight lines B 1 B 1 and B 2 B 2. The abscissa of this point is equal to P 2, and the ordinate of the game price is g.

Compared to the previous technology, the gain is 55%.

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

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

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

In other situations, the opposite party actively, consciously opposes the achievement of the intended goal. In such cases, there is a clash of opposing interests, opinions, and ideas. Such situations are called conflict , and decision-making in a conflict situation is difficult due to the uncertainty of the enemy’s behavior. It is known that the enemy deliberately seeks to take the least beneficial actions for you in order to ensure the greatest success. It is unknown to what extent the enemy knows how to assess the situation and possible consequences, how he assesses your capabilities and intentions. Both sides cannot predict mutual actions. Despite such uncertainty, each side of the conflict has to make a decision

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

The need to justify optimal decisions in conflict situations has led to the emergence game theory.

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

The conflicting parties are called players , one implementation of the game – party , the outcome of the game is winning or losing . Any possible action for a player (within the given rules of the game) is called his strategy .

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

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

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

For each formalized game, rules are introduced. In general, the rules of the game establish the players' options for action; the amount of information each player has about the behavior of their partners; the payoff that each set of actions leads to.

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

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

Combined games, in which the rules provide, in principle, the opportunity for each player to analyze all the various options for his behavior and, having compared these options, choose the one that leads to the best outcome for this player. The uncertainty of the outcome is usually due to the fact that the number of possible behavior options (moves) is too large and the player is practically unable to sort through and analyze them all.

Gambling , in which the outcome is uncertain due to the influence of various random factors. Gambling games consist only of random moves, the analysis of which uses the theory of probability. Mathematical game theory does not deal with gambling.

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

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

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

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

Optimal strategy A player is called a strategy that, when repeated many times in a game containing personal and random moves, provides the player with the maximum possible average win or minimum possible loss, regardless of the opponent’s behavior.

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

In multi-move game theory problems, the concepts of “strategy” and “option of possible actions” are significantly different from each other. In simple (one-move) game problems, when in each game each player can make one move, these concepts coincide, and, therefore, the set of player strategies covers all possible actions that he can take in any possible situation and under any possible actual situation. information.

Games are also differentiated by the amount of winnings. The game is called game with zero sum th, if each player wins at the expense of the others, and the amount of winning of one side is equal to the amount of loss of the other. In a zero-sum doubles game, the interests of the players are directly opposed. A zero-sum pairs game is called Iantagonistic game .

Games in which one player's gain and another's loss are not equal are callednon-zero sum games .

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

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

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

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

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

Game theory concept

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

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

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

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

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

Classification of games

There are separate classes of non-cooperative games:

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

and others also refer to non-cooperative games.

Mathematical apparatus

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

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

Optimality and solutions

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

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

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

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

Story

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

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