A Game of Martingales
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We consider a two player dynamic game played over T ≤∞ periods. In each period each player chooses any probability distribution with support on [0, 1] with a given mean, where the mean is the realized value of the draw from the previous period. The player with the highest realization in the period achieves a payoff of 1, and the other player, 0; and each player seeks to maximize the discounted sum of his per-period payoffs over the whole time horizon. We solve for the unique subgame perfect equilibrium of this game, and establish properties of the equilibrium strategies and payoffs in the limit.
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