# Playing Games with Bounded Entropy

## TL;DR

This paper investigates the impact of limited randomness on the outcomes of zero-sum repeated games, introducing new methods from information theory to characterize and compute the maximum guaranteed payoff under entropy constraints.

## Contribution

It introduces a novel approach using random hashing and source simulation to analyze the maxmin payoff in games with bounded entropy, simplifying and generalizing prior results.

## Key findings

- Characterized the maxmin payoff using the function J(h).
- Provided explicit lower bounds on the entropy-payoff trade-off curve.
- Highlighted the effectiveness of Renyi-divergence in game-theoretic analysis.

## Abstract

In this paper, we consider zero-sum repeated games in which the maximizer is restricted to strategies requiring no more than a limited amount of randomness. Particularly, we analyze the maxmin payoff of the maximizer in two models: the first model forces the maximizer to randomize her action in each stage just by conditioning her decision to outcomes of a given sequence of random source, whereas, in the second model, the maximizer is a team of players who are free to privately randomize their corresponding actions but do not have access to any explicit source of shared randomness needed for cooperation. The works of Gossner and Vieille, and Gossner and Tomala adopted the method of types to establish their results; however, we utilize the idea of random hashing which is the core of randomness extractors in the information theory literature. In addition, we adopt the well-studied tool of simulation of a source from another source. By utilizing these tools, we are able to simplify the prior results and generalize them as well. We characterize the maxmin payoff of the maximizer in the repeated games under study. Particularly, the maxmin payoff of the first model is fully described by the function $J(h)$ which is the maximum payoff that the maximizer can secure in a one-shot game by choosing mixed strategies of entropy at most $h$. In the second part of the paper, we study the computational aspects of $J(h)$. We offer three explicit lower bounds on the entropy-payoff trade-off curve. To do this, we provide and utilize new results for the set of distributions that guarantee a certain payoff for Alice. In particular, we study how this set of distributions 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.