1. MinMax Theorem John von Neumann
Zur Theorie der Gesellschaftsspiele (The Theory of games of strategy)
Mathematische Annalen, 100, 1928, pp
John von Neumann
Institute for Advanced Study
Presented by Franson, C.W. Chen
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2. MinMax 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.
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3. A two players zero-sum game.
The value of g(x, y) is being tugged at from two sides, S1 and S2.
Player S1 controls the variable x, and wants to maximize g(x,y); Player S2 controls the variable y, and wants to minimize g(x,y).
In 1928 von Neumann was not able to prove anything about the existence of optimal strategies for the general case. In stead he analyzed the simplest case, namely a game of strategy with only two players S1 and S2. The situation is then, that player S1 chooses a number x ∈ {1, 2, , 1}, and player S2 chooses a number y ∈ {1, 2, , 2} each without knowing what the other player has chosen, and they then receive the amounts g(x, y), −g(x, y) respectively. Von Neumann then gave the following description of the tension in the two-person game:
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4. If S1 chose the number x0 (x0 ∈ {1, 2,
If S1 chose the number x0 (x0 ∈ {1, 2, ..., }), that is the strategy x0, his result g(x0, y) would then also depend on the choice of S2; but no matter which choice (y) S2 comes up with, the following inequality holds:
(1)
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5. Now if we suppose that S2 knew x0, S2 would according to the assumptions in the model choose y = y0 such that
(2)
Facing this situation the best thing for S1 would be to choose x0 such that
(3)
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6. no matter what strategy x, S1 chooses.
According (1) and (3), the conclusion of von Neumann is then that S1 can make
(4)
independently of the choice of S2. The same argument holds for S2, which can make
(5)
no matter what strategy x, S1 chooses.
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7. From this von Neumann concluded that if a pair of strategies x0, y0 can be found for which
(6)
then that would necessary be the choices for S1, and S2 respectively, and would be the value of the game.
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8. Saddle Point
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