Example of a game without a value

This article gives an example of a game on the unit square that has no value. It is due to Sion and Wolfe [cite journal | author=M. Sion, P. Wolfe | title=On a game with no value | journal= The Annals of Mathematical Studies| year=1957 | volume=39 | pages=299–306] .

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 $0leq\; x,yleq\; 1$; the payoff to I is

:

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

:$sup\_\{f\}inf\_\{g\}intint\; K,df,dg=frac\{1\}\{3\}$

but

:$inf\_\{g\}sup\_\{f\}intint\; K,df,dg=frac\{3\}\{7\}.$

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 $epsilon$ (specifically, if ). Dasgupta and Maskin [cite journal | author=P. Dasgupta and E. Maskin | title=The Existence of Equilibrium in Discontinuous Economic Games, I: Theory | journal= Review of Economic Studies| year=1986 | volume=53 | number=1 | pages=1-26] assert that the game values are achieved if player I puts probability weight only on the set $left\{0,1/2,1\; ight\}$ and player II puts weight only on $left\{1/4,1/2,1\; ight\}$.

See also Glicksberg's theorem.

References