MATH2901 Operations Research I Game Theory p.1

Transcription

1 MATH2901 Operations Research Game Theory p1 GAME THEORY A game represents a competitive or conflicting situation between two or more players Each player has a number of choices, called moves (or pure strategies) A player selects his moves without any knowledge of the moves chosen by the other players The simultaneous choices of all players lead to the respective payoffs of the game f the sum of the payoffs to all players is zero, then it is called a zero-sum game A game played by n persons is called an n-person game We shall consider only two-person zero-sum games, which are also called matrix games because the gain of one player signifies an equal loss to the other that it suffices to express the outcomes (as a result of the selection of moves) in terms of the payoffs to one player (player ) in a matrix A strategy for a given player ( or ) is a plan that specifies which of the available choices he should made with what probabilities Hence a strategy for player is the specification of x i (x i 0, x i = 1) and for player, y j (y j 0, y j = 1) A strategy is generally a mixed strategy to distinguish it from a pure strategy in which x i (or y j ) is 1 for some i (or j) and all other x k (or y k ) being 0, k i, j Furthermore, we explicitly adopt the rationality assumptions about the players, who are then assumed to be intelligent players that act so as to maximize his expected payoff (or minimize his expected loss) Game theory deals with the determination of the optimal strategies for each player n view of the conflicting nature and lack of information about the specific strategies selected by each player, optimality is based on a rather conservative criterion, namely, each player selects his strategy (mixed or pure) which guarantees a payoff that can never be worsened by the selections of his opponent This criterion is known as the minimax criterion (or maximin criterion ) 1 Stable game Consider the following (two-person zero-sum) game matrix which represents the payoff to player Row minimum * Column maximum

2 MATH2901 Operations Research Game Theory p2 Player, by playing his first (pure) strategy guarantees a gain of at least 2 = Min {8, 2, 9, 5} Similarly the second strategy guarantees at least 5 = Min {6, 5, 7, 8}, and the third, 4 = Min {7, 3, 4, 7} Thus the row minimum is the value guaranteed for each pure strategy Player, if he selects strategy 2, is maximizing his minimum (guaranteed) gain Hence this selection is called the maximin strategy and the corresponding gain (= 5) is called the maximin (or lower) value of the game A completely analogous consideration for player indicates that he will be interested in the column maximum and that he seeks to minimize the column minimum by using the minimax strategy leading to a minimax (or upper) value of the game The selections made by and are based on the so-called maximin (minimax) criterion The criterion expresses a conservative attitude which guarantees the best of the worst results For any matrix game, the minimax (upper) value is greater than or equal to the maximin (lower) value n the case when equality holds, that is, minimax value = maximin value, the corresponding pure strategies are called the optimal strategies and the game is said to be a stable game, which possesses a saddle point that equals to this common value called the value of the game [Exercise: Prove that the saddle point is highest in its column and lowest in its row] 2 Unstable game n general, the value of the game must satisfy the inequality, maximin value value of the game minmax value The existance of a saddle point immediately yields the optimal pure strategies for the game However, there are unstable games where such a saddle point does not exist Consider the very simple game of coin-matching Each player selects either a head (H) or a tail (T ) f the outcomes match (ie H, H or T, T ), wins $1 form, otherwise wins $1 from The matrix of the game is H T Row minimum H T Maximum value = 1 Column maximum Minimax value = +1

3 MATH2901 Operations Research Game Theory p3 Optimal solution to such games requires each player to use a mixture of pure strategies, or mixed strategies [Exercise: Prove the inequality Maximin value of game Minimax] 3 Mixed strategies Each player, instead of selecting pure strategies only, may play all his strategies according to a predetermined set of ratios Let x i, i = 1, 2,, m and y j, j = 1, 2,, n be the row and column ratios representing the relative frequencies by which and, respectively, select their pure strategies Then x i 0, y j 0 and x i = y j = 1 y 1 y 2 y n x 1 a 11 a 12 a 1n x m a m1 a m2 a mn We can think of x i and y j as probabilities (generated by some random mechanism) by which and select their i th and j th pure strategies, respectively The solution of the mixed strategy problem is based also on the minimax criterion The only difference is that selects the ratios x i (instead of the pure strategies i) which maximize the minimum expected payoff in a column, while selects the ratios y j (instead of pure strategies j) which minimize the maximum expected payoff in a row Mathematically, player selects x i (x i 0, x i = 1) which will yield { [ m ] } Max Min a i1 x i, a i2 x i,, a in x n, x i and player selects y j (y j 0, y j = 1) which will yield [ n Min y j Max ] a 1j y j, a 2j y j,, a mj y j These values are referred to as the maximin and minimax expected payoffs, respectively As in the pure strategies case, the relationship, Maximin expected payoff Minimax expected payoff,

4 MATH2901 Operations Research Game Theory p4 holds in general When x i and y i correspond to the optimal solution, the above relation holds in equality sense and the resulting expected values become equal to optimal expected value of the game (This is known as the Minimax Theorem in Game Theory, which we shall see the justification using duality of linear programming later) Now if x i and y j are the optimal solutions, then each payoff element a ij will be associated with the probability (x i, y j ) Thus, if v denotes the optimal expected value of the game, then v = a ij x i yj 4 (2 2) unstable game Consider the following (2 2) game in which we assume there is no saddle point y 1 y 2 = 1 y 1 x 1 a 11 a 12 x 2 = 1 x 1 a 21 a 22 Player s expected payoffs corresponding to the pure strategies of are given by s pure strategy s expected payoff 1 x 1 a 11 + x 2 a 21 = (a 11 a 21 )x 1 + a 21 2 x 1 a 12 + x 2 a 22 = (a 12 a 22 )x 1 + a 22 Since the optimal x 1 and x 2 have been chosen to make s mixture of moves optimal against any of the two possible s moves, the two expected payoffs for must be equal if x 1 and x 2 are optimal Hence we have (a 11 a 21 )x 1 + a 21 = (a 12 a 22 )x 1 + a 22, from which x 1 (and x 2 = 1 x 1) can be determined A similar analysis for leads to (a 11 a 12 )y 1 + a 12 = (a 21 a 22 )y 1 + a 22, from which y 1 (and y 2 = 1 y 1) can be determined

5 MATH2901 Operations Research Game Theory p5 5 Graphical solution of (2 n) and (m 2) games Consider a game in which one of the player (say ) has avaiable to him only two pure strategies This is then a (2 n) game as follows: y 1 y 2 y n x 1 a 11 a 12 a 1n x 2 = 1 x 1 a 21 a 22 a 2n t is assumed that the game does not have a saddle point As before, the expected payoffs to corresponding to s pure strategies are 1 (a 11 a 21 )x 1 + a 21 2 (a 12 a 22 )x 1 + a 22 n (a 1n a 2n )x 1 + a 2n According to the minimax criterion, player should select the value of x 1 so as to maximize his minimum expected payoffs This may be done by plotting the straight lines of expected payoffs as functions of x 1 Typically, we have Each line is numbered according to its corresponding s pure strategy The lower envelope of these lines give the minimum expected payoff as a function of x 1 : Lower envelope (x 1 ) = Min[(a 11 a 21 )x 1 + a 21,, (a 1n a 2n )x 1 + a 2n ]

6 MATH2901 Operations Research Game Theory p6 The highest point in this lower envelope then gives the maximum of the minimum expected payoff and hence the optimal value of x 1 (= x 1), with optimal value of the game: v = Max x 1 { Min [ (a11 a 21 )x 1 + a 21,, (a 1n a 2n )x 1 + a 2n ]} The optimal y j for can be obtained by observing that y j have been chosen to make s mixture of moves optimal against any of the possible strategies of Hence v = yj { (a1j a 2j )x } 1 + a 2j We claim that all lines {(a 1j a 2j )x 1 + a 2j } that do not pass through the maximin point must have their corresponding y j = 0 (Why? Hint: y j = 1, y j 0) Because the maximin point is determined by the intersection of two straight lines (if more than two, can be any two with opposite slopes), we have the important result that any (2 n) game is basically equivalent to a (2 2) game because only two of the n moves are effective For notational simplicity, assume the first two pure strategies of are effective Now, the expected payoffs (loss) to corresponding to s pure strategies are 1 (a 11 a 12 )y 1 + a 12 2 (a 21 a 22 )y 1 + a 22 Typically, the situation is as follows: Of course, algebraically, this intersection can also be obtained by solving the two linear equations for y 1 The (m 2) games are treated similarly as in the (2 n) games except that s optimal strategies yj, j = 1, 2, are first determined using the minimax criterion This

7 MATH2901 Operations Research Game Theory p7 automatically determines the two effective strategies for, from which the application of maximin criterion determines the optimal x i, i = 1, 2,, m (Note the close connection of having only two (positive) effective strategies with the number of positive entries for a BFS in an linear program) [Exercise: Give the (similar) analysis of an (m 2) game] 6 Numerical example [Exercise: Verify that the solutions to the following (2 4) game, are: x 1 = x 2 = 1/2; y 1 = y 4 = 0, y 2 = y 3 = 1/2; v = 5/2] Remark: t is sometimes possible to reduce the size of the game matrix by using the dominance property This occurs (as in the numerical example above) when one or more of the pure strategies of either player can be deleted because they are inferior to at least one of the remaining, or to a weighted average of two or more other pure strategies [Exercise: Verify that in the above, strategy 1 of is dominated by strategy 2; and that in the following game matrix, the third strategy of is dominated by a weighted average of s first and second pure strategies What are the weights?] Upon deletion of s pure strategy 3 (domainated), we have reduced the game to a (2 3), or effectively a (2 2) game [Exercise: Solve this game by the graphical procedure]

8 MATH2901 Operations Research Game Theory p8 7 (m n) games and linear programming For any (m n) game, player selects his optimal mixed strategies which yield { [ m ] } Max Min a i1 x i,, a in x i x i subject to the constraints m x i = 1, x i 0, i = 1, 2,, m This problem can be put into a linear program as follows: Let v = Min [ i a i1 x i,, i a in x i ], then we have Max subject to v a ij x i v, j = 1, 2,, n x i = 1 x i 0, i = 1, 2,, m Clearly, v represents the value of the game in this case Dividing all constraints by v [Exercise: What modification to the original game matrix may be necessary and legitimate to ensure v > 0?] Defining X i = x i /v, i = 1, 2,, m, we get, since max v yields the same solutions x i as min 1/v, Min 1/v = Min m x i /v = Min m X i Finally, we have Min X 0 = subject to X i a ij X i 1 x i 0, i = 1, 2,, m After the optimal X i follow are obtianed using the simplex method (say), the original x i easily Play s problem, on the other hand, is given by [ n Min y j Max a ij y j,, ] a mj y j subject to the constraints y j = 1, y j 0, j = 1, 2,, n Putting in a linear

9 MATH2901 Operations Research Game Theory p9 programming form gives [Exercise: Verify!], Max Y 0 = Y j subject to a ij Y j 1 Y j 0, i = 1, 2,, n where Y 0 1/v, Y j y j /v, j = 1, 2,, n Notice that s and s problems together is actually a dual pair! Duality theorem then leads to an easy conclusion that the optimal solution of one problem will automatically yield the optimal solution to the other one, and Minimax expected payoff = Maximin expected payoff (n fact, Dantzig states that when non Neumann, father of game theory, was first introduced to the simplex method of linear programming in 1947 immediately recognized this relationship and further pinpointed and stressed the concept of duality in linear programming!)