User:Cretog8/Scratchpad2

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



This article gives an example of a game on the unit square 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 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 $$x$$ and $$y$$ respectively, with $$0\leq x,y\leq 1$$; the payoff to I is


 * $$K(x,y)=

\begin{cases} -1 & \mbox{if } x<y<x+1/2 \\ 0 & \mbox{if } x=y \mbox{ or } y=x+1/2\\ 1 & \mbox{otherwise} \end{cases}$$

If $$(x,y)$$ 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) $$f$$; player II chooses from $$g$$. 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



\sup_{f}\inf_{g}\int\int K\,df\,dg=\frac{1}{3} $$

but



\inf_{g}\sup_{f}\int\int K\,df\,dg=\frac{3}{7}. $$

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

The $$\sup$$ and $$\inf$$ 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 $$3/7$$ if he knows player II's strategy; and player II can hold the payoff down to $$1/3$$ if he knows player I's strategy.

There is clearly no epsilon equilibrium for sufficiently small $$\epsilon$$. Dasgupta and Maskin assert that the game values are achieved if player I puts probability weight only on the set $$\left\{0,1/2,1\right\}$$ and player II puts weight only on $$\left\{1/4,1/2,1\right\}$$.

See also Glicksberg's theorem.