John von Neumann’s minimax theory

“Keeping up with him was all but impossible. The feeling was you were on a tricycle chasing a racing car”

Israel Halperin on John von Neumann

John von Neumann made huge contributions to many areas of mathematics and was also one of the founders of game theory.

His mathematical ability shone through from a young age. At the age of six he was able to divide two eight-digit numbers in his head and by the age of eight he had mastered calculus.

One of his major contributions to game theory was the Minimax theorem, which he proved in 1928.


The basic idea applies to two player¬†zero-sum games. As it is a zero-sum game one player’s payoff is the opposite of the other’s payoff. So if one gets a payoff of +10 then the other will get a payoff of -10, the two payoffs add up to zero.

In any game you are trying to maximise your own payoff and your opponent is trying to do the same thing. In a zero-sum game, if you maximise your payoff then that is the same as minimising your opponent’s payoff. There is only a fixed amount to be shared between the players in a zero-sum game so if one wins an amount then the other loses an equal amount.

Given that you know that your opponent is going to play a strategy that will maximise their payoff, you want to play the strategy against them that will minimise their payoff (and therefore maximise yours).

Putting these two things together means that you want to play the strategy that minimises your opponent’ s maximum payoff. This is where the name ‘minimax’ comes from.

In zero-sum games the minimax solution is the same as the Nash equilibrium.

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One Response to John von Neumann’s minimax theory

  1. What else did he find out? Since every player will not always win or loose in a sequence of games with the same strategy, they will mix their strategies. Every player will elaborate a best mix of both strategies, considering his own assets and anticipating the strategy of the opponent. And abracadabra: If both play their best mix, the minimum of the maximum payoff for one, and the maximum of the minimum payoff for the other are the same: Min Max Theorem. A special case of Nash’s equilibrium, where it doesn’t make any sense for both to change their strategy, since they can’t improve their payoffs with a change.

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