copy of page Example of a game without a value September 10, 2008, just in case it gets deleted

Game square for a game with no value, due to Sion and Wolfe

This article gives an example of a game on the unit square that has no value. It is due to Sion and Wolfe[1].

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 for games with an infinite set of strategies. There follows a simple example of a game with no value.

Players I and II choose numbers and respectively, with ; the payoff to I is

If is interpreted as a point on the unit square, the figure shows the payoff to player I. Now suppose that player I adopts a mixed strategy: choosing a number from probability density function (pdf) ; player II chooses from . Player I seeks to maximize the payoff, player I to minimize the payoff, in the knowledge that the adversary plays likewise.

Sion and Wolfe show that

but

These are the maximal and minimal expectations of the game's value of player I and II respectively.

The and respectively take the supremum and infimum over pdf's on the unit interval (actually Probability Borel measures). These represent player I and player II's (mixed) strategies. Thus, player I can assure himself of a payoff of at least if he knows player II's strategy; and player II can hold the payoff down to if he knows player I's strategy.

There is clearly no epsilon equilibrium for sufficiently small . Dasgupta and Maskin[2] assert that the game values are achieved if player I puts probability weight only on the set and player II puts weight only on .


See also Glicksberg's theorem.

References

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  1. ^ M. Sion, P. Wolfe (1957). "On a game with no value". The Annals of Mathematical Studies. 39: 299–306.
  2. ^ P. Dasgupta and E. Maskin (1986). "The Existence of Equilibrium in Discontinuous Economic Games, I: Theory". Review of Economic Studies. 53 (1): 1–26.

Category: game theory