The prisoners’ dilemma and the mini-max strategy by Charles Warner The Prisoners’ Dilemma

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Figure 1

Figure 2

The decision tree and the payoff matrix show the different consequences or outcome of each decision in years in the Gulag. The saddle point in the payoff matrix is that decision in which both sides get the same payoff – equilibrium. The saddle point, or equilibrium, is not the maximum benefit for either side, but is the optimum benefit for both sides.
The Mini-max Theorem
“The theorem states that in zero-sum games in which the players’ interests are strictly opposed (one’s gain is the other other’s loss), one player should attempt to minimize his opponent’s maximum payoff while his opponent attempts to maximize his own minimum payoff. When they do so, the surprising conclusion is that the minimum of the maximum (mini-max) payoffs equals the maximum of the minimum (maxi-min) payoffs. Neither player can improve his position, and so these strategies form an equilibrium of the game.”ii

“The general proof of the mini-max theorem is quite complicated, but the result is useful and worth remembering. If all you want to know is the gain of one player or the loss of the other when both play their best mixes, you need only compute the best mix for one of them and determine its result.”iii

“To put it in plain language, the mini-max theorem says that there is always a rational solution to a precisely defined conflict between two people whose interests are completely opposite. It is a rational solution in that both parties can convince themselves that they cannot expect to do any better, given the nature of the conflict.”iv

In other words, “You know that the best you can expect is to avoid the worst.”v

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