Example of A Game Without A Value - The Game

The Game

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

K(x,y)= \begin{cases} -1 & \text{if } x<y<x+1/2 \\ 0 & \text{if } x=y \text{ or } y=x+1/2\\ 1 & \text{otherwise} \end{cases}

(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.

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