bimatrix pareto game

A game is an idealized mathematical model of collective behavior: several individuals (participants, players) influence the situation (the outcome of the game), and their interests (their payoffs for various possible situations) are different. The antagonism of interests creates conflict, while the coincidence of interests reduces the game to pure coordination, for which the only reasonable behavior is cooperation. In most games emerging from the analysis of socio-economic situations, interests are neither strictly antagonistic nor exactly coinciding. The seller and the buyer agree that it is in their common interest to agree on the sale, of course, provided that the transaction is beneficial to both. However, they are vigorously traded when choosing a specific price within the limits determined by the terms of the mutual advantage of the transaction. Likewise, grassroots voters are generally willing to reject candidates representing extreme views.

However, when choosing one of two candidates offering different compromise solutions, a fierce struggle ensues. One cannot but agree that most game-like conflict situations in social life give rise to both conflict and cooperative behavior. Therefore, it can be concluded that game theory is a useful logical apparatus for analyzing the motives of the behavior of participants in such situations. It has a whole arsenal of formalized scenarios of behavior, from non-cooperative behavior to cooperative agreements using mutual threats. For each game in normal form, using different cooperative and non-cooperative equilibrium concepts tends to lead to different outcomes. Their comparison is the main principle of game-theoretic analysis and, apparently, the source of rigorous and at the same time meaningful reasoning about the incentive motives of behavior arising only from the structure of the game in normal form.

In many social sciences available a large number of models, in the analysis of which it is required to study the ways of choosing strategies. Applications of game theory are predominantly developed in connection with the study of economics.

This corresponds to the principles of the founders of game theory von Neumann and Morgenstern. However, the strong reputation of the game-theoretic approach was established only after the Debray-Scarf theorem, which allows us to consider competitive equilibrium as a result of cooperative actions. Since then, entire sections economic theory(such as the theory of imperfect competition or the theory of economic incentives) are developed in close contact with game theory.

The search for equilibrium concepts, which are the idealization of a whole range of non-cooperative and cooperative patterns of behavior, is closely related to the foundations of sociology. In modern sociological research, formal game-theoretic models are very rare and elementary from a mathematical point of view. And yet, the influence of game theory seems to us already irreversible, at least at the stage of learning.

Mathematical theory offers game theory to solve the set problems, defined as a branch of mathematics focused on the construction of formal models for making optimal decisions in a situation of competitive interaction. This definition The main task of game theory is the sequence of actions of effective behavior in conditions of competition, conflict.).

In game theory, participants in a competitive interaction are called players, each of them has a non-empty set of admissible actions performed by him during the game, which are called moves or choices. The set of all possible moves one from each player's list of possible moves (participating in pairs, triplets, etc. moves) is called a strategy. Properly constructed strategies mutually exclude each other, i.e. mutually exhaust all modes of behavior of the players. The outcome of the game is the realization by the player of the chosen strategy. Each outcome of the game corresponds to the value of utility (winning) determined by the players, called the payoff.

The classification of games can be carried out: by the number of players, the number of strategies, the nature of the interaction of the players, the nature of the payoff, the number of moves, the availability of information, etc.

• 1. Depending on the number of players, pair games and games of n players are distinguished. The mathematical apparatus for the implementation of paired games is the most developed. Games of three or more players are more difficult to study because of the difficulties in the technical implementation of the solution algorithms.
• 2. According to the number of strategies, games are finite and infinite. A game with a finite number of possible strategies for the players is said to be finite. If at least one of the players has an infinite number of possible strategies, then the game is called infinite.
• 3. According to the nature of the interaction, games are divided into:
  • non-cooperative: players do not have the right to enter into agreements, form coalitions;
  • · coalition (cooperative) - players can join coalitions.

IN cooperative games coalitions are hard-coded at the task setting stage and cannot be changed during the game.

• 4. According to the nature of the winnings, the games are divided into:
  • Zero-sum games (the total capital of all players does not change, but is redistributed between the players; the sum of the winnings of all players is zero);
  • non-zero sum games.
• 5. According to the type of payoff functions, games are divided into: matrix, bimatrix, continuous, convex, separable, duels, etc.

A matrix game is a zero-sum final pair game of two players, in which the payoff of player 1 is given in the form of a matrix (the row of the matrix corresponds to the number of the applied strategy of player 2, the column - to the number of the applied strategy of player 2; at the intersection of the row and column of the matrix is ​​the payoff of player 1 corresponding to the applied strategies).

For matrix games, it is proved that any of them has a solution and it can be easily found by reducing the game to a linear programming problem.

The bimatrix game is end game two players with a non-zero sum, in which the payoffs of each player are given by matrices separately for the corresponding player (in each matrix, the row corresponds to the strategy of player 1, the column corresponds to the strategy of player 2, at the intersection of the row and column in the first matrix is ​​the payoff of player 1, in the second matrix - player 2 wins.)

For bimatrix games, the theory of optimal behavior of players has also been developed, but solving such games is more difficult than conventional matrix games.

A game is considered continuous if the payoff function of each player is continuous depending on the strategies. In the theory of mathematics, it has been proved that games of this class have solutions, but so far no practically acceptable methods for finding them have been developed.

The goal of any game is to maximize each player's profit. The meaning of the mathematical theory of games, built on the above classification, is to formalize (simplify) and facilitate the optimal choice. The set of all possible game strategies is a large number, growing the stronger, the more players and the set of moves available to each. So for a pair of players, if the conditions of the game allow each player to make n moves, there are 2n strategies in the game.

A simple enumeration and evaluation (comparison) of such a number of strategies is technically a very difficult task and is unacceptable in practice. The mathematical apparatus can significantly reduce the number of strategies that require analysis and comparison, discarding obviously inefficient ones. When a limited set of equilibrium points, reasonable for analysis, is obtained (the outcomes of the game are equally preferred by all players), based on the analysis of the payoffs of the players, the most rational result is chosen. When choosing an outcome, there are two main approaches that give the name of the final strategy of the game:

• · Minimax strategy (selection from maximum (worst) losses to minimum (best) losses.
• Maximin strategy (selection from the minimum (worst) payoffs to the maximum (best) ones.

The development of game theory using the methods of probabilistic analysis is the mathematical theory of decision making. This theory operates not with a real (actual) solution, but with an average, which is the expected solution of the game during its multiple repetition. This property is relevant for solving legal problems, since the normative nature of law means that it is focused on an indefinite subject and involves multiple repetition of legal relations. In order not to go into deep mathematical calculations, we only note that decision theory offers a system of criteria (for example, the Hurwitz criterion, Hadji-Lehman criterion, the expected value criterion), which, using a probabilistic analysis of the outcomes of games, make it possible to choose the optimal solution under risk and uncertainty .

65. In a graphical method for solving 3 * 3 games to find the optimal strategies of players:
a) two triangles are built (*answer*)
b) one triangle is being built.
c) triangles are not built at all.
66. The graph of the lower envelope for the graphical method of solving games 2 * m represents, in the general case, the function:
a) monotonically decreasing.
b) monotonically increasing.
c) non-motonic.
67. If in an antagonistic game on a segment the payoff function of the 1st player F(x,y) is equal to 2*x+C, then depending on C:
a) there are never saddle points.
b) there are always saddle points (*answer*)
c) other option
68. Than you can set the task of making a decision under conditions of uncertainty on finite sets:
a) two matrices.
b) wins.
c) something else (*answer*)
69. In an antagonistic game of arbitrary dimension, the payoff of the first player is:
a) number.
b) set.
c) a vector, or an ordered set.
d) function (*answer*)
70. In matrix game 3*3 two components of the player's mixed strategy:
a) determine the third (*answer*)
b) not defined.
71. A bimatrix game can be defined:
a) two matrices of the same dimension with arbitrary elements,
b) two matrices not necessarily of the same dimension,
c) one matrix.
72. In the matrix game, the element aij is:
a) the loss of the 2nd player when he uses the j-th strategy, and the 2nd - i-th strategy(*answer*)
b) optimal strategy 2nd player when used adversary i-th or j-th strategy,
c) the payoff of the 1st player when he uses the j-th strategy, and the 2nd - the i-th strategy,
73. Matrix element aij corresponds to a saddle point. The following situations are possible:
a) optimal.
b) clean.
c) there is no clear answer (*answer*)
84. If all columns in the matrix are the same and look like (4 3 0 2), then what strategy is optimal for the 2nd player?
a) first. b) third. c) any (*answer*)
85. What is the maximum number of saddle points in a 3*3 game (the matrix can contain any numbers):
a) 3.
b) 9.
c) 27 (*answer*)
86. Let in the antagonistic game X=(1;5) be the set of strategies of the 1st
player, Y=(2;8) - the set of strategies of the 2nd player. Is a pair (1,2)
be a saddle point in this game:
a) always.
b) sometimes (*answer*)
c) never.
87. Are there exactly 2 equilibrium situations in a 3*3 bimatrix game?
a) Always.
b) sometimes (*answer*)
c) never.
88. Let in a matrix game of dimension 2 * 3 one of mixed strategies 1st player has the form (0.3, 0.7), and one of the mixed strategies of the 2nd player has the form (0.3, x, x). What is the number x?
a) 0.7 b) 0.4 c) something else (*answer*)
89. Matrix game is special case bimatrix, which always holds true:
a) matrix A is equal to matrix B, taken with the opposite sign.
b) matrix A is equal to matrix B.
c) The product of matrices A and B is the identity matrix.
90. In a bimatrix game, the by element is:
a) the payoff of the 2nd player when he uses the i-th strategy, and the 1st - the j-th strategy,
b) the optimal strategy of the 2nd player when the opponent uses the i-th or j-th strategy /
c) something else (*answer*)
91. In a bimatrix game, the element ac corresponds to an equilibrium situation. The following situations are possible:
a) there are elements in the column that are equal to this element (*answer*)
b) this element is less than some in the column.
c) this element is the smallest in the column.
92. In a matrix game, knowing the strategies of each player and the payoff function,
the price of a game in pure strategies can be found:
a) always.
b) sometimes (*answer*)
c) the question is incorrect.

Tests for final control

1. Antagonistic game can be set:

a) a set of strategies for both players and a saddle point.

b) the set of strategies of both players and the payoff function of the first player.

2. The price of the game always exists for matrix games in mixed strategies.

a) yes.

3. If all columns in the payoff matrix are the same and look like (4 5 0 1), then what strategy is optimal for the 1st player?

a) first.

b) second.

c) any of the four.

4. Let in the matrix game one of the mixed strategies of the 1st player has the form (0.3, 0.7), and one of the mixed strategies of the 2nd player has the form (0.4, 0, 0.6). What is the dimension of this matrix?

a) 2*3.

c) another dimension.

5. The principle of dominance allows you to remove from the matrix in one step:

a) entire lines.

b) separate numbers.

6. In the graphical method for solving games 2 * m, directly from the graph find:

a) the optimal strategies of both players.

b) the price of the game and the optimal strategies of the 2nd player.

c) the price of the game and the optimal strategies of the 1st player.

7. The graph of the lower envelope for the graphical method of solving games 2 * m is in the general case:

a) broken.

b) straight.

c) a parabola.

8. In the 2*2 matrix game, the two components of the player's mixed strategy are:

a) determine each other's values.

b) are independent.

9. In the matrix game, the element aij is:

a) the payoff of the 1st player when he uses the i-th strategy, and the 2nd - the j-th strategy.

b) the optimal strategy of the 1st player when the opponent uses the i-th or j-th strategy.

c) the loss of the 1st player when he uses the j-th strategy, and the 2nd - the i-th strategy.

10. Matrix element aij corresponds to the saddle point. The following situations are possible:

a) this element is strictly less than all in the string.

b) this element is second in order in the string.

11. In the Brown-Robinson method, each player, when choosing a strategy at the next step, is guided by:

a) opponent's strategies on previous steps.

b) their strategies in the previous steps.

c) something else.

12. According to the criterion of mathematical expectation, each player proceeds from the fact that:

a) the worst situation for him will happen.

c) all or some situations are possible with some given probabilities.

13. Let the matrix game be given by a matrix in which all elements are negative. The value of the game is positive:

b) no.

c) there is no clear answer.

14. The price of the game is:

a) number.

b) vector.

c) matrix.

15. What is the maximum number of saddle points in a 5 * 5 game (matrix can contain any numbers):

16. Let in a matrix game of dimension 2*3 one of the mixed strategies of the 1st player has the form (0.3, 0.7), and one of the mixed strategies of the 2nd player has the form (0.3, x, 0.5). What is the number x?

c) another number.

17. For what dimension of the game matrix does the Wald criterion turn into the Laplace criterion?

c) only in other cases.

18. The upper price of the game is always less than the lower price of the game.

b) no.

b) the question is incorrect.

19. What strategies are there in the matrix game:

a) clean.

b) mixed.

c) both.

20. Can they in some antagonistic game, the values ​​of the payoff function of both players for some values ​​of the variables equal to 1?

a) always.

b) sometimes.

c) never.

21. Let in the matrix game one of the mixed strategies of the 1st player has the form (0.3, 0.7), and one of the mixed strategies of the 2nd player has the form (0.4, 0.1,0.1,0.4). What is the dimension of this matrix?

c) different dimensions.

22. The principle of dominance allows you to remove from the matrix in one step:

a) whole columns

b) separate numbers.

c) smaller submatrices.

23. In a 3*3 matrix game, the two components of the player's mixed strategy are:

a) determine the third.

b) not defined.

24. In the matrix game, the element aij is:

a) the loss of the 2nd player when he uses the j-th strategy, and the 2nd player - the i-th strategy.

b) the optimal strategy of the 2nd player when the opponent uses the i-th or j-th strategy,

c) the payoff of the 1st player when he uses the j-th strategy, and the 2nd - the i-th strategy,

25. The matrix element aij corresponds to the saddle point. The following situations are possible:

a) this element is the largest in the column.

b) this element is strictly greater than all in order in the string.

c) the string contains elements both greater and less than this element.

26. According to the Wald criterion, each player proceeds from the fact that:

a) the worst situation for him will happen.

b) all situations are equally possible.

c) all situations are possible with some given probabilities.

27. The lower price is less than the upper price of the game:

b) not always.

c) never.

28. The sum of the components of a mixed strategy for a matrix game is always:

a) is equal to 1.

b) is non-negative.

c) is positive.

d) not always.

29. Let in a matrix game of dimension 2*3 one of the mixed strategies of the 1st player has the form (0.3, 0.7), and one of the mixed strategies of the 2nd player has the form (0.2, x, x). What is the number x?