Playing repeated games with sublinear randomness

TL;DR

This paper characterizes when repeated games have Nash equilibria with sublinear randomness, showing a clear dichotomy and providing a complete answer to an open problem in the field.

Contribution

It provides a complete characterization of games with $O(1)$ randomness Nash equilibria, resolving an open question and establishing a 0-1 law for randomness in repeated games.

Findings

Games either have $O(1)$-randomness equilibria or require $oldsymbol{ extOmega}(n)$ randomness.

The paper offers a full characterization of when sublinear randomness suffices for Nash equilibria.

Techniques developed are applicable to analyzing bounded capabilities in repeated games.

Abstract

We study the amount of entropy players asymptotically need to play a repeated normal-form game in a Nash equilibrium. Hub\'a\v{c}ek, Naor, and Ullman (SAGT'15, TCSys'16) gave sufficient conditions on a game for the minimal amount of randomness required to be $O(1)$ or $\Omega(n)$ for all players, where $n$ is the number of repetitions. We provide a complete characterization of games in which there exists Nash equilibria of the repeated game using $O(1)$ randomness, closing an open question posed by Budinich and Fortnow (EC'11) and Hub\'a\v{c}ek, Naor, and Ullman. Moreover, we show a 0--1 law for randomness in repeated games, showing that any repeated game either has $O(1)$-randomness Nash equilibria, or all of its Nash equilibria require $\Omega(n)$ randomness. Our techniques are general and naturally characterize the payoff space of sublinear-entropy equilibria, and could be of independent interest to the study of players with other bounded capabilities in repeated games.