The saddle point of the payment matrix is a point. Types of criterion functions in games with nature

Games can be classified according to the number of players, the number of strategies, the nature of interaction between players, the nature of winning, the number of moves, the state of information, etc.

Depending on the number of players, games are divided into two and n players. The first of them are the most studied. Games of three or more players are less studied due to the fundamental difficulties encountered and technical capabilities obtaining a decision. How more players- the more problems.

Based on the number of strategies, games are divided into finite and infinite. If all players in a game have a finite number of possible strategies, then it is called ultimate. If at least one of the players has an infinite number of possible strategies, the game is called endless.

Based on the nature of interaction, games are divided into:

non-coalition: players do not have the right to enter into agreements or form coalitions;

coalition(cooperative) – can join coalitions.

In cooperative games, coalitions are predetermined.

According to the nature of winnings, games are divided into: games with zero sum(the total capital of all players does not change, but is redistributed between players; the sum of winnings of all players is zero) and games with non-zero sum.

Based on the type of winning functions, games are divided into: matrix, bimatrix, continuous, convex, separable, duel type, etc.

Definition . If in a game with a matrix A = (the lower net price is equal to the upper net price), then this game is said to have saddle point in pure strategies and net price games==.

Saddle point is a pair of pure strategies (i O , j O ) respectively, players 1 and 2, at which equality is achieved =. If one of the players adheres to the strategy corresponding to the saddle point, then the other player cannot do better than to adhere to the strategy corresponding to the saddle point. Mathematically, this can be written another way: , Where i, j– any pure strategies of players 1 and 2, respectively; (i O , j O ) –strategies that form a saddle point.

Thus, the saddle element is the minimum in the i o -th row and the maximum in the j o -th column in matrix A. Finding the saddle point of matrix A occurs as follows: in matrix A sequentially in each line find the minimum element and check whether this element is the maximum in its column. If yes, then it is a saddle element, and the pair of strategies corresponding to it forms a saddle point. A couple of pure strategies (i O , j O ) players 1 and 2, forming a saddle point and a saddle element is called game solution . Wherein i O And j O are called optimal clean strategies respectively players 1 and 2.

Properties of saddle points:

1. Equivalence. If there are several saddle points in the game, then the values of the payoff function in them are the same.

2. Interchangeability of optimal strategies. Players can replace their optimal strategies with other optimal strategies, while the equilibrium will not be disrupted, and the gain (loss) will remain unchanged.

13.Definition of a mixed strategy. Solution of the 2*2 game in mixed strategies.

A mixed strategy is when, instead of applying any one specific strategy, the participants in the game can randomly alternate (mix) their strategies in accordance with a specially designed scheme that provides the desired frequency, or probability, of the implementation of each of the strategies.

For each player you can set the following components:

Pia – probability of using the i-th strategy by A.

If you choose such a set Pia , which provides biggest win regardless of the actions of the second party, then this set of probabilities (p 1 a, p 2 a, ..., p ma) = S A and will be called a mixed strategy.

S * A = (p * 1 a, p * 2 a, …, p * ma) – optimal mixed strategy.

( S A ) – a set of mixed strategies on the part of A, from which the optimal one must be selected.

Game 2*2 in mixed strategies.

If at least one party has only 2 actions, we apply the graph-analytical solution method. Let's define the game in the form of the following matrix:

For this game, the following can be considered: the player plays against some pure strategy of the other side each time. In this case, he can choose a probability ratio that will give him guaranteed win, the size of the price of the game.

Definition 4. A paired game in which the sum of the players' winnings is zero (the first player's winnings are equal to the second player's loss) is called zero-sum doubles game or antagonistic game .

Comment. Any zero-sum paired game can always be completely specified by the payoff matrix of one of the players. As a rule, the payment matrix of the first player is specified. It is assumed that all players are equally intelligent. The task of each player is that each player strives to ensure the maximum possible win for any actions of the other player.

Let the payoff matrix of the first player in a game of two players, who have strategies respectively, have the form . Then the payoff matrix of the second player is .

Let us construct optimal strategies for players in an antagonistic game.

The optimal strategy of the first player. The first player wants to maximize his own winnings. At the same time, he assumes that in any case the second player will choose a strategy that minimizes the payoff of the first player. The first player's task is to get some guaranteed win.

Let us denote the minimum value of the first player's payoff for each of his strategies (in each row of the matrix): , . Knowing the minimum payoffs for various strategies of the second player, the first player will choose the strategy with the maximum. Let's denote this value. Then .

Definition 5. Value – guaranteed maximum win, which the first player can secure for himself is called the lowest price of the game or maximin .

Thus, the formally optimal strategy of the first player consists in choosing a row and a matrix element in it: . This strategy is called maximin from the first player. If the first player adheres to the maximin strategy, then his payoff in any case will be no less than the maximin value:

Optimal strategy of the second player. The second player wants to minimize his own loss. At the same time, he assumes that in any case the first player will choose a strategy that maximizes his own payoff. The second player’s task is to lose no more than a certain guaranteed amount.

Let us denote the maximum value of the first player's payoff for each strategy of the second player (in each column of the matrix): , . Knowing the maximum payoffs of the first player for various strategies of the second player, the second player will choose the strategy with the minimum. Let's denote this value. Then .

Definition 6. The value - the minimum loss that the second player can secure for himself - is called top price of the game or minimax .

Thus, the formally optimal strategy of the second player consists in choosing a column and a matrix element in it: . This strategy is called minimax from the second player. If the second player adheres to the minimax strategy, then in any case he will lose no more than the minimax value:

Theorem 1. For an arbitrary rectangular matrix the inequality always holds or

Definition 7. If the upper price of the game is equal to the lower price, that is, then they say that the game has a saddle point(is being decided)in pure strategies .

Definition 8. The value is called at the cost of the game(at the pure price of the game) .

Definition 9. The matrix element is called saddle element of the matrix .

Comment. The saddle element of the matrix is simultaneously minimal in its row and maximal in its column, that is

Definition 10. A pair of pure strategies and , corresponding to and , is called saddle point games .

Comment. Strategies and , forming a saddle point, are optimal. The existence of a saddle point in the payment matrix corresponds to the presence of an equilibrium state in a given matrix game.

Definition 11. Troika is called game solution .

Example 2.4. The game involves two players. Each of them can write down numbers 1, 2 and 3 independently of the other. If the difference between the numbers written down by the players is positive, then the first player wins a number of points equal to the difference between the numbers. If the difference is negative, then the second player wins. If the difference is zero, then the game ends in a draw. Create a payment matrix and find the price of the game.

Solution. Player A has 3 strategies:

Player B also has three strategies:

This game is a paired game with opposing interests (antagonistic), therefore, for its formal description it is enough to set the payment matrix of the first player.

Let's calculate possible winnings first player:

Then the payment matrix of the first player will take the form:

Let's find the optimal strategy first player. To do this, we find the minimum payoff for each of its strategies (the minimum element in each line):

Let's find the lower price of the game (select the largest from the minimum elements):

Thus, the optimal strategy of the first player is:

Let's find the optimal strategy second player. To do this, we find the maximum payoff of the first player for each strategy of the second player (the maximum element in each column):

Let's find the upper price of the game (select the smallest from the maximum elements):

Thus, the optimal strategy of the second player is:

The first player's deviation from the optimal strategy reduces his payoff. The second player's deviation from the optimal strategy increases his loss.

Consider the game m×n with matrix P = (a ij ), i = 1, 2, ..., m; j = 1, 2, ..., n and determine the best among strategies A 1 , A 2 , ..., A m . Choosing a strategy A i player A must expect that the player IN will answer it using one of the strategies B j , for which the payoff for the player A minimal (player IN seeks to "harm" the player A ). Let us denote by α i , the player's smallest winnings A when choosing a strategy A i for all possible player strategies IN (smallest number in i th row of the payment matrix), i.e.

Among all the numbers α i (i = 1, 2, ..., m) Let's choose the largest: . Let's call α the lowest price of the game, or maximum winnings (maximin). This is a guaranteed win for the player A for any player strategy IN . Hence,

13. Saddle point.

A saddle point is the largest element of a column of the game matrix, which is also the smallest element of the corresponding row (in a two-person zero-sum game). At this point, therefore, the maximin of one player is equal to the minimax of the other; S. t. is a point of equilibrium.

The concept of a saddle point

If in a game with matrix A the lower and upper, the net prices of the game coincide, i.e. , then this game is said to have a saddle point, in pure terms, and a net game value:

A saddle point is a pair of pure strategies (i 0 , j 0) of the first and second players, at which equality is achieved.

The concept of a saddle point has the following meaning:

if one player is pursuing a saddle point strategy, then the other player can do no better than to pursue a saddle point strategy.

The deviation of the first player from the saddle point can only lead to a decrease in his winnings.

The deviation of the second player from the saddle point can lead to an increase in his loss.

A saddle element is the minimum element in a row and the maximum element in a column.

To determine the saddle element, it is necessary to sequentially determine the minimum element at each point, and then check whether it is the maximum element of the column, and if so, then the saddle point is found in this way - the price of the game, the optimal strategies of the first and second player:

14. Optimal strategies.

In a matrix game, each player chooses his own strategies without knowing about the actions of the other player. Let's find out what the best guaranteed winnings are for them. The first player, having chosen some strategy i, can receive as a payoff one of two elements ai1, ai2 of matrix A, depending on which strategy the second player uses. IN worst case he must count on the minimum winnings, i.e.

At the same time, when good choice strategy i = i* the first player can get the maximum payoff from the minimum:

The second player argues in a similar way. When choosing strategy j, its maximum loss from two possible a1 j, a2 j is equal to

If the choice of strategy j = j* was successful, then he can count on the minimum loss out of the maximum:

Formulas (5.2), (5.3) determine the best guaranteed winnings for players. If they match, then they general meaning can be considered an acceptable compromise for players, and the corresponding strategies i*, j* can be considered optimal strategies.

Direct calculations using formulas (5.2), (5.3) using (5.1) give

Here the best guaranteed wins are not equal and there are no optimal strategies.

The reason for the lack of optimal strategies obviously lies in their definition. Let's try to change the definition of optimal strategies without losing sight of the game meaning of the task and the goals of the players.

Depending on the number of players, games are divided into two and n players. The first of them are the most studied. Games of three and more players are less explored due to the fundamental difficulties encountered and the technical possibilities of obtaining a solution. The more players there are, the more problems there are.

Based on the nature of interaction, games are divided into:

1) non-coalition: players do not have the right to enter into agreements or form coalitions;

2) coalition(cooperative) - can join coalitions.

IN cooperative games coalitions are determined in advance.

Based on the type of winning functions, games are divided into: matrix, bimatrix, continuous, convex, separable, duel type, etc.

Definition. If in a game with a matrix A = (the lower net price is equal to the upper net price), then this game is said to have saddle point in pure strategies and net price games u = = .

Saddle point is a pair of pure strategies (i o, j o) respectively, players 1 and 2, at which equality is achieved = . If one of the players adheres to the strategy corresponding to the saddle point, then the other player cannot do better than to adhere to the strategy corresponding to the saddle point. Mathematically, this can be written differently: , where i, j– any pure strategies respectively players 1 and 2; (i o, j o)– strategies that form a saddle point.

Thus, the saddle element is minimal in the i o -th row and maximum in the j o -th column in matrix A. Finding the saddle point of matrix A occurs as follows: in matrix A sequentially in each line find the minimum element and check whether this element is the maximum in its column. If yes, then it is a saddle element, and the pair of strategies corresponding to it forms a saddle point. A couple of pure strategies (i o, j o) players 1 and 2, forming a saddle point and a saddle element, is called game solution . Wherein i o And j o are called optimal clean strategies respectively players 1 and 2.

Properties of saddle points:

1. Equivalence. If there are several saddle points in the game, then the values of the payoff function in them are the same.

13.Definition of a mixed strategy. Solution of the 2*2 game in mixed strategies.

For each player you can set the following components:

Pia– probability of using the i-th strategy by A.

If you choose such a set Pia, which provides the greatest gain regardless of the actions of the second party, then this set of probabilities (p 1 a, p 2 a, ..., p ma) = S A and will be called a mixed strategy.

S * A = (p * 1 a, p * 2 a, …, p * ma) – optimal mixed strategy.

( S A ) – a set of mixed strategies on the part of A, from which the optimal one must be selected.

Game 2*2 in mixed strategies.

If at least one party has only 2 actions, we apply the graph-analytical solution method. Let's define the game in the form of the following matrix.

A zero-sum game in which each player has a finite set of strategies at his disposal. The rules of the matrix game are determined by the payment matrix, the elements of which are the winnings of the first player, which are also the losses of the second player.

Matrix game is an antagonistic game. The first player receives the maximum guaranteed (independent of the behavior of the second player) winnings, equal to the price of the game; similarly, the second player achieves the minimum guaranteed loss.

Under strategy is understood as a set of rules (principles) that determine the choice of action for each personal move of the player, depending on the current situation.

Now about everything in order and in detail.

Payment matrix, pure strategies, game price

IN matrix game its rules are determined payment matrix .

Consider a game in which there are two participants: the first player and the second player. Let the first player have at his disposal m pure strategies, and at the disposal of the second player - n pure strategies. Since the game is being considered, it is natural that in this game there are wins and there are losses.

IN payment matrix the elements are numbers expressing the players' wins and losses. Wins and losses can be expressed in points, amount of money or other units.

Let's create a payment matrix:

If the first player chooses i-th pure strategy, and the second player - j th pure strategy, then the payoff of the first player will be aij units, and the loss of the second player is also aij units.

Because aij + (- a ij) = 0, then the described game is a zero-sum matrix game.

The simplest example of a matrix game is coin toss. The rules of the game are as follows. The first and second players throw a coin and the result is either heads or tails. If "heads" and "heads" or "tails" or "tails" are thrown at the same time, then the first player will win one unit, and in other cases he will lose one unit (the second player will win one unit). The same two strategies are at the disposal of the second player. The corresponding payment matrix will be as follows:

The task of game theory is to determine the choice of the first player's strategy, which would guarantee him the maximum average win, as well as the choice of the second player's strategy, which would guarantee him the maximum average loss.

How do you choose a strategy in a matrix game?

Let's look at the payment matrix again:

First, let's determine the amount of winnings for the first player if he uses i th pure strategy. If the first player uses i th pure strategy, then it is logical to assume that the second player will use such a pure strategy due to which the first player’s payoff would be minimal. In turn, the first player will use such a pure strategy that would provide him with the maximum win. Based on these conditions, the winnings of the first player, which we denote as v1 , called maximin winnings or the lowest price of the game .

At for these values, the first player should proceed as follows. From each line, write down the value of the minimum element and select the maximum one from them. Thus, the winnings of the first player will be the maximum of the minimum. Hence the name - maximin winning. The line number of this element will be the number of the pure strategy that the first player chooses.

Now let’s determine the amount of loss for the second player if he uses j th strategy. In this case, the first player uses his own pure strategy in which the loss of the second player would be maximum. The second player must choose a pure strategy in which his loss would be minimal. The loss of the second player, which we denote as v2 , called minimax loss or top price of the game .

At solving problems on the price of the game and determining the strategy To determine these values for the second player, proceed as follows. From each column, write down the value of the maximum element and select the minimum from them. Thus, the loss of the second player will be the minimum of the maximum. Hence the name - minimax win. The column number of this element will be the number of the pure strategy that the second player chooses. If the second player uses "minimax", then regardless of the choice of strategy by the first player, he will lose no more than v2 units.

Example 1.

The largest of the smallest elements of the strings is 2, this is lowest price game, the first line corresponds to it, therefore, the maximin strategy of the first player is the first. The smallest of the largest elements of the columns is 5, this is the upper price of the game, the second column corresponds to it, therefore, the minimax strategy of the second player is the second.

Now that we have learned to find the lower and upper price of the game, the maximin and minimax strategies, it’s time to learn how to formally define these concepts.

So, the guaranteed win for the first player is:

The first player must choose a pure strategy that would provide him with the maximum of the minimum winnings. This gain (maximin) is denoted as follows:

The first player uses his pure strategy so that the loss of the second player is maximum. This loss is indicated as follows:

The second player must choose his pure strategy so that his loss is minimal. This loss (minimax) is indicated as follows:

Another example from the same series.

Example 2. Given a matrix game with a payoff matrix

Determine the maximin strategy of the first player, the minimax strategy of the second player, the lower and upper price of the game.

Solution. To the right of the payment matrix, we will write out the smallest elements in its rows and note the maximum of them, and below the matrix - the largest elements in the columns and select the minimum of them:

The largest of the smallest elements of the lines is 3, this is the lower price of the game, the second line corresponds to it, therefore, the maximin strategy of the first player is the second. The smallest of the largest elements of the columns is 5, this is the upper price of the game, the first column corresponds to it, therefore, the minimax strategy of the second player is the first.

Saddle point in matrix games

If the upper and lower prices of the game are the same, then the matrix game is considered to have a saddle point. The converse is also true: if a matrix game has a saddle point, then the upper and lower prices of the matrix game are the same. The corresponding element is both the smallest in the row and the largest in the column and is equal to the price of the game.

Thus, if , then is the optimal pure strategy of the first player, and is the optimal pure strategy of the second player. That is, equal lower and upper game prices are achieved using the same pair of strategies.

In this case matrix game has a solution in pure strategies .

Example 3. Given a matrix game with a payoff matrix

The lower price of the game coincides with the upper price of the game. Thus, the price of the game is 5. That is . The price of the game is equal to the value of the saddle point. The first player's maxin strategy is the second pure strategy, and the second player's minimax strategy is the third pure strategy. This matrix game has a solution in pure strategies.

Solve a matrix game problem yourself, and then look at the solution

Example 4. Given a matrix game with a payoff matrix

Find the lower and upper price of the game. Does this matrix game have a saddle point?

Matrix games with optimal mixed strategy

In most cases, a matrix game does not have a saddle point, so the corresponding matrix game has no solutions in pure strategies.

But it has a solution in optimal mixed strategies. To find them, you need to assume that the game is repeated a sufficient number of times so that, based on experience, you can guess which strategy is more preferable. Therefore, the decision is associated with the concept of probability and average (mathematical expectation). In the final solution there is also an analogue of the saddle point (that is, the equality of the lower and top price games), and an analogue of their corresponding strategies.

So, in order for the first player to get the maximum average win and for the second player to have a minimum average loss, pure strategies should be used with a certain probability.

If the first player uses pure strategies with probabilities , then the vector is called a mixed first player strategy. In other words, it is a “mixture” of pure strategies. In this case, the sum of these probabilities is equal to one:

If the second player uses pure strategies with probabilities , then the vector is called a second player mixed strategy. In this case, the sum of these probabilities is equal to one:

If the first player uses a mixed strategy p, and the second player - a mixed strategy q, then it makes sense expected value the first player's win (the second player's loss). To find it, you need to multiply the first player's mixed strategy vector (which will be a one-row matrix), the payoff matrix and the second player's mixed strategy vector (which will be a one-column matrix):

Example 5. Given a matrix game with a payoff matrix

Determine the mathematical expectation of the first player's win (the second player's loss), if the first player's mixed strategy is , and the second player's mixed strategy is .

Solution. According to the formula for the mathematical expectation of the first player’s win (the second player’s loss), it is equal to the product of the first player’s mixed strategy vector, the payment matrix and the second player’s mixed strategy vector:

The first player is called such a mixed strategy that would provide him with the maximum average payoff if the game is repeated a sufficient number of times.

Optimal mixed strategy the second player is called such a mixed strategy that would provide him with a minimum average loss if the game is repeated a sufficient number of times.

By analogy with the maximin and minimax notations, in the case of pure strategies the optimal mixed strategies are designated as follows (and are linked to mathematical expectation, that is, the average of the winnings of the first player and the losses of the second player):

In this case, for the function E there is a saddle point , which means equality.

In order to find optimal mixed strategies and a saddle point, that is, solve a matrix game in mixed strategies , we need to reduce the matrix game to the problem linear programming, that is, to an optimization problem, and solve the corresponding linear programming problem.

Reducing a matrix game to a linear programming problem

In order to solve a matrix game in mixed strategies, you need to construct a straight line linear programming problem And dual task. In the dual problem, the extended matrix, which stores the coefficients of the variables in the system of constraints, free terms and coefficients of the variables in the objective function, is transposed. In this case, the minimum of the goal function of the original problem is matched to the maximum in the dual problem.

Goal function in a direct linear programming problem:

System of constraints in a direct linear programming problem:

The goal function in the dual problem is:

System of restrictions in the dual problem:

The optimal plan for a direct linear programming problem is denoted by

and the optimal plan for the dual problem is denoted by

We denote the linear forms for the corresponding optimal plans by and ,

and they need to be found as sums of the corresponding coordinates of optimal plans.

In accordance with the definitions of the previous paragraph and the coordinates of optimal plans, the following mixed strategies of the first and second players are valid:

Theoretical mathematicians have proven that game price is expressed in the following way through the linear forms of optimal plans:

that is, it is the reciprocal of the sums of coordinates of optimal plans.

We, practitioners, can only use this formula to solve matrix games in mixed strategies. Like formulas for finding optimal mixed strategies the first and second players respectively:

in which the second factors are vectors. Optimal mixed strategies are also, as we already defined in the previous paragraph, vectors. Therefore, multiplying the number (game price) by a vector (with the coordinates of optimal plans) we also obtain a vector.

Example 6. Given a matrix game with a payoff matrix

Find the price of the game V and optimal mixed strategies and .

Solution. We create a linear programming problem corresponding to this matrix game:

We obtain a solution to the direct problem:

We find the linear form of the optimal plans as the sum of the found coordinates.

The saddle point of the payment matrix is ​​a point. Types of criterion functions in games with nature