## Von Neumann's Minimax Theorem

Before going into the actual game theory, let us begin with an example. Let us refer bact to the sticks game we discussed in 2 person zero sum finite games of perfect information. Previously, the person showing the sticks did not care whether the other person won or lost, but let us assume he wants the player to lose. Now the player's strategy shifts. Since the opponent is attempting to defeat him, he cannot chose the tallest stick after the first two. Instead, he must chose to select one with a given chance of success. The person who shows the sticks also cannot randomize. Instead, he must show sticks with a given probability of getting it past the player. These values shrink with the number of sticks at the rate Pn = 1/(logen), where Pn is the probability of success for the player. These probabilities are given below.

 Chances of Success Position 1 2 3 4 5 Opponent 12/37 12/25 12/19 4/5 1 Player 12/37 12/25 12/19 4/5 1

This illustrates some of the principles of a two person game of imperfect information, and brings us to perhaps the most important theorem in the realm of game theory: Von Neumann's minimax theorem. The minimax theorem states that one can assign every two person finite zero-sum game a value V that is the average payoff each player can expect to win from the other while playing sensibly. Von Neumann reasoned that this was true for three reasons:

1. Player A will never settle for less than an average win of V. This is because there is a strategy for player A that player B cannot circumvent to prevent a gain of V.
2. Player A can be prevented from winning more than V. This is because though a winning of V is allowed by (1), there is also a strategy for Player B that guarantees a loss no greater than V.
3. Player B wants to limit the winnings of A to the minimum possible, which is V. Since by assumption the game is zero sum, the A's loss is B's gain and vice versa, and B wants to minimize his losses. This is critical because in non-zero-sum games, B is not guaranteed to use this capability whenever possible.

Note that the minimax theorem brings certainty to the world of probabilistic game theory. Without it, there would be no way to chose a reasonable direction. With it and the mixed strategies, we can treat all two person zero-sum games as games with equilibrium points, which provides the assurance that in a given state of things, it is impossible to do better as long as the opponent is as good as you are.

It is important to know however that minimax only predict security, which, like a complete description of all chess moves, is hard to come by. It is a matter of preference to not chose the minimax strategy, and with some luck, it is possible to do better. With the real world intervening, minimax often provides the lowest secure limit. Beyond that level of completely rational interchange, it is invalid and possibly misleading.

1. Dauben, Joseph W "Game Theory." Microsoft Encarta. Microsoft Corp, 1999
2. Davis, Morton D. "The General, Finite, Two-Person, Zero-Sum Game" Game Theory, A Nontechnical Introduction. New York: Basic Books, Inc., 1930, pp 19-48
3. McCain, Roger A. "Game Theory: An Introductory Sketch." Hypertext. 1999 (August 1999)