Parthasarathy's Theorem - Theorem

Theorem

Suppose that is bounded on the unit square ; further suppose that is continuous except possibly on a finite number of curves of the form (with ) where the are continuous functions.

Further suppose

k(\mu,\lambda)=\int_{y=0}^1\int_{x=0}^1 k(y,x)\,d\mu(x)\,d\lambda(y)= \int_{x=0}^1\int_{y=0}^1 k(x,y)\,d\lambda(y)\,d\mu(x).

Then

\max_{\mu\in{\mathcal M}_X}\,\inf_{\lambda\in A_Y}k(\mu,\lambda)= \inf_{\lambda\in A_Y}\,\max_{\mu\in{\mathcal M}_X} k(\mu,\lambda).

This is equivalent to the statement that the game induced by has a value. Note that one player (WLOG ) is forbidden from using a pure strategy.

Parthasarathy goes on to exhibit a game in which

\max_{\mu\in{\mathcal M}_X}\,\inf_{\lambda\in{\mathcal M}_Y}k(\mu,\lambda)\neq \inf_{\lambda\in{\mathcal M}_Y}\,\max_{\mu\in{\mathcal M}_X} k(\mu,\lambda)

which thus has no value. There is no contradiction because in this case neither player is restricted to absolutely continuous distributions (and the demonstration that the game has no value requires both players to use pure strategies).

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