Parthasarathy's theorem

In mathematics – and in particular the study of games on the unit square – Parthasarathy's theorem is a generalization of Von Neumann's minimax theorem. It states that a particular class of games has a mixed value, provided that at least one of the players has a strategy that is restricted to absolutely continuous distributions with respect to the Lebesgue measure.
The theorem is attributed to Thiruvenkatachari Parthasarathy.

Theorem

Let and stand for the unit interval ; denote the set of probability distributions on ; and denote the set of absolutely continuous distributions on .
Suppose that is bounded on the unit square and that is continuous except possibly on a finite number of curves of the form where the are continuous functions. For, define
Then
This is equivalent to the statement that the game induced by has a value. Note that one player is forbidden from using a pure strategy.
Parthasarathy goes on to exhibit a game in which
which thus has no value. There is no contradiction because in this case neither player is restricted to absolutely continuous distributions.