Min Max Theorem Zur Theorie der Gesellschaftsspiele The

Min. Max Theorem Zur Theorie der Gesellschaftsspiele (The Theory of games of strategy) Mathematische Annalen, 100, 1928, pp. 295 -320 John von Neumann Institute for Advanced Study Presented by Franson, C. W. Chen 2021/6/6 1

Min. Max Theorem In any finite, two-player, zero-sum game, in any Nash equilibrium each player receives a payoff that is equal to both his maxmin value and his minmax value. 2021/6/6 2

1. A two players zero-sum game. 2. The value of g(x, y) is being tugged at from two sides, S 1 and S 2. 3. Player S 1 controls the variable x, and wants to maximize g(x, y); Player S 2 controls the variable y, and wants to minimize g(x, y). 2021/6/6 3

If S 1 chose the number x 0 (x 0 ∈ {1, 2, . . . , }), that is the strategy x 0, his result g(x 0, y) would then also depend on the choice of S 2; but no matter which choice (y) S 2 comes up with, the following inequality holds: (1) 2021/6/6 4

Now if we suppose that S 2 knew x 0, S 2 would according to the assumptions in the model choose y = y 0 such that (2) Facing this situation the best thing for S 1 would be to choose x 0 such that (3) 2021/6/6 5

According (1) and (3), the conclusion of von Neumann is then that S 1 can make (4) independently of the choice of S 2. The same argument holds for S 2, which can make (5) no matter what strategy x, S 1 chooses. 2021/6/6 6

From this von Neumann concluded that if a pair of strategies x 0, y 0 can be found for which (6) then that would necessary be the choices for S 1, and S 2 respectively, and would be the value of the game. 2021/6/6 7

Saddle Point 2021/6/6 8