Playing games with bounded entropy

We study a two-player zero-sum game in which one of the players is restricted to mixed strategies with limited randomness. More precisely, we consider the maximum payoff that the maximizer (Alice) can secure with limited randomness h. This problem finds an operational interpretation in the context of repeated games with non-ideal sources of randomness. The computational aspects of this problem has not received much attention in the game theory literature. We begin by observing the equivalence of this problem with entropy minimization problems in other scientific contexts. Next, we provide two explicit lower bounds on the entropy-payoff tradeoff curve. To do this, we provide and utilize new results for the set of distribution that guarantee a certain payoff for Alice (mixed strategies corresponding to a security level for Alice). In particular, we study how this set of distribution shrinks as we increase the security level. While the use of total variation distance is common in game theory, our derivation indicates the suitability of utilizing the Renyi-divergence of order two.