game with stochastic subgradient strategy updates

Hi,

I would like to seek help on proving convergence for the following
game-theoretical model.

The strategy of each player in the game is the probability of taking a
certain action (e.g., starting a price war). In each time slot, every
player takes the action according to his probability. Each player then
updates his probability for the next time slot based on the observed
actions of other players (e.g., if no one starts a price war, he will
decrease his own probability. Otherwise, he will increase his
probability). It can be shown that the update algorithm is following a
stochastic subgradient update to maximize his own utility function.

Since the game is abstracted from a real world problem, we can not
change the way that the players update their strategies. Is there any
literature on how to prove convergence for this kind of game to its
Nash Equilibrium (i.e., update with stochastic subgradient, partially
observed strategies of others.)

Thanks.

.

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