# Game without a value

*https://en.wikipedia.org/wiki/Game_without_a_value*

In the mathematical theory of games, in particular the study of zero-sum continuous games, not every game has a minimax value. This is the expected value to one of the players when both play a perfect strategy (which is to choose from a particular PDF).

This article gives an example of a
zero-sum game 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 if the game has an infinite set of strategies. There follows a simple example of a game with no minimax value.

The existence of such zero-sum games is interesting because many of the results of game theory become inapplicable if there is no minimax value.

## The game

Players I and II each choose a number, and respectively, with ; the payoff to I is

(i.e. player II pays to player I;the game is
zero-sum). Sometimes player I is referred to as the *maximizing player* and player II the *minimizing player*.

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 II to minimize the payoff. Note that each player is aware of the other's objective.

## Game value

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 Borel probability 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 , specifically, if . 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 .

Glicksberg's theorem shows that any zero-sum game with
upper or
lower semicontinuous payoff function has a value (in this context, an upper (lower) semicontinuous function *K* is one in which the set (resp ) is
open for any
real *c*).

Observe that the payoff function of Sion and Wolfe's example is clearly not semicontinuous. However, it may be made so by changing the value of *K*(*x*, *x*) and *K*(*x*, *x* + 1/2) [i.e. the payoff along the two discontinuities] to either +1 or −1, making the payoff upper or lower semicontinuous respectively. If this is done, the game then has a value.

## Generalizations

Subsequent work by Heuer ^{
[3]} discusses a class of games in which the unit square is divided into three regions, the payoff function being constant in each of the regions.

## References

**^**Sion, Maurice; Wolfe, Phillip (1957), "On a game without a value", in Dresher, M.; Tucker, A. W.; Wolfe, P. (eds.),*Contributions to the Theory of Games III*, Annals of Mathematics Studies 39, Princeton University Press, pp. 299–306, ISBN 9780691079363**^**P. Dasgupta and E. Maskin (1986). "The Existence of Equilibrium in Discontinuous Economic Games, I: Theory".*Review of Economic Studies*.**53**(1): 1–26. doi: 10.2307/2297588. JSTOR 2297588.**^**G. A. Heuer (2001). "Three-part partition games on rectangles".*Theoretical Computer Science*.**259**: 639–661. doi: 10.1016/S0304-3975(00)00404-7.