Game without a value explained
In the mathematical theory of games, in particular the study of zero-sum continuous games, not every 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.
The game
Players I and II choose numbers
and
respectively, between 0 and 1. The payoff to player I is