Example of A Game Without A Value

Example Of A Game Without A Value

In game theory, and in particular the study of zero-sum continuous games, it is commonly assumed that a game has a minimax value. This is the expected value to one of the players when both play a perfect strategy (which is to choose from a particular PDF).

This article gives an example of a zero sum game that has no value. It is due to Sion and Wolfe.

Zero sum games with a finite number of pure strategies are known to have a minimax value (originally proved by John von Neumann) but this is not necessarily the case if the game has an infinite set of strategies. There follows a simple example of a game with no minimax value.

The existence of such zero-sum games is interesting because many of the results of game theory become inapplicable if there is no minimax value.