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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^[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 $x$ 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^[2] 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 $\left\{1/4,1/2,1\right\}$ .

References edit

^ M. Sion, P. Wolfe (1957). "On a game with no value". The Annals of Mathematical Studies. 39: 299–306.
^ 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

[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.

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