# Playing Games with Bounded Entropy: Convergence Rate and Approximate Equilibria

## TL;DR

This paper analyzes the convergence rate of the long-run value in zero-sum repeated games with limited randomness strategies and characterizes the set of approximate Nash equilibria related to this value.

## Contribution

It introduces a new simulation tool for sources based on Rénnyi entropies and characterizes approximate equilibria in games with bounded entropy strategies.

## Key findings

- Convergence rate of $v_n$ to its limit is exponentially bounded.
- Simulation precision depends on Rénnyi entropy difference.
- Set of approximate equilibria closely relates to the long-run max-min value.

## Abstract

We consider zero-sum repeated games in which the players are restricted to strategies that require only a limited amount of randomness. Let $v_n$ be the max-min value of the $n$ stage game; previous works have characterized $\lim_{n\rightarrow\infty}v_n$, i.e., the long-run max-min value. Our first contribution is to study the convergence rate of $v_n$ to its limit. To this end, we provide a new tool for simulation of a source (target source) from another source (coin source). Considering the total variation distance as the measure of precision, this tool offers an upper bound for the precision of simulation, which is vanishing exponentially in the difference of R\'enyi entropies of the coin and target sources. In the second part of paper, we characterize the set of all approximate Nash equilibria achieved in long run. It turns out that this set is in close relation with the long-run max-min value.