By Ye. S. Venttsel
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Extra resources for Elements Of Game Theory
1 a(x,y) x Fig. 2 The value of a (x, y) at that point is precisely the value of the game, v: 60 The presence of a saddle point means that the given infinite game has a solution in the range of pure strategies; xo, Yo are optimum pure strategies for A and B. In the general case where rJ. i= ~ the game may have a solution (possibly not a unique one) only in the range of mixed strategies. A mixed strategy for infinite games is some probability distribution for the strategies x and y regarded as random variables.
To find our opponent's optimal strategy, we can in the general case proceed as follows. We change the sign of the payoff, add to the matrix elements a constant value L to make them nonnegative, and solve the problem for our opponent in the same way as we did for ourselves. The fact that we already know the value of the game, v, to some extent simplifies the problem, however. Besides, in this particular case the problem is further simplified by the fact that the solution comprises only two "utility" strategies for our opponent, B t and B 2 , since the variable Z3 is nonzero and hence when strategy B 3 is played the value of the game is not attained.
Consider a game between two opponents, A and B, each having an infinite (uncountable) set of strategies; for player A these strategies correspond to different values of a continuously varying parameter x, and for B, to those of a parameter y. In this case, instead of being given by matrix II aij II the game is defined by some function of two continuously varying arguments, a(x, y), which we shall call a gain function (notice that the function a (x, y) itself need not be continuous). The function a(x, y) can be represented geometrically by some surface a (x, y) over the range of the arguments (x, y) (Fig.