Example of a game without a value
- 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
:K(x,y)=egin{cases} -1 & mbox{if } x
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 epsilon < frac{1}{2}left(frac{3}{7}-frac{1}{3}
ight)). 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
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