A probabilistic-numerical approximation for an obstacle problem arising in game theory

In 1967 Aumann and Maschler presented their celebrated model for games with incomplete information, see [1] and references therein. The game they consider consists in a set of, say I, standard discrete time two person zero-sum games. At the beginning one of these zero-sum games is picked at random according to a probability p. The information which game was picked is transmitted to Player 1 only, while Player 2 just knows p. It is assumed that both players observe the actions of the other one, so Player 2 might infer from the actions of his opponent which game is actually played. It turns out that it is optimal for the informed player to play with an additional randomness. Namely in a such a way, that he optimally manipulates the beliefs of the uninformed player.