Example of A Game Without A Value - Game Value

Game Value

Sion and Wolfe show that

\sup_f \inf_g \iint K\,df\,dg=\frac{1}{3}

but

\inf_g \sup_f \iint 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 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 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) to either +1 or −1, making the payoff upper or lower semicontinuous respectively. If this is done, the game then has a value.

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