When Can Limited Randomness Be Used in Repeated Games?

TL;DR

This paper investigates the necessity of randomness in repeated games, showing that for many games, linear randomness is required for approximate equilibria, but some games admit equilibria with minimal randomness, especially under cryptographic assumptions.

Contribution

It characterizes when limited randomness suffices for equilibria in repeated games, highlighting differences between classes of games and the impact of cryptographic assumptions.

Findings

Approximate equilibria require linear randomness in many games.

Some games have exact equilibria with constant randomness.

Cryptographic assumptions can reduce randomness needs in repeated games.

Abstract

The central result of classical game theory states that every finite normal form game has a Nash equilibrium, provided that players are allowed to use randomized (mixed) strategies. However, in practice, humans are known to be bad at generating random-like sequences, and true random bits may be unavailable. Even if the players have access to enough random bits for a single instance of the game their randomness might be insufficient if the game is played many times. In this work, we ask whether randomness is necessary for equilibria to exist in finitely repeated games. We show that for a large class of games containing arbitrary two-player zero-sum games, approximate Nash equilibria of the $n$-stage repeated version of the game exist if and only if both players have $\Omega(n)$ random bits. In contrast, we show that there exists a class of games for which no equilibrium exists in pure strategies, yet the $n$-stage repeated version of the game has an exact Nash equilibrium in which each player uses only a constant number of random bits. When the players are assumed to be computationally bounded, if cryptographic pseudorandom generators (or, equivalently, one-way functions) exist, then the players can base their strategies on "random-like" sequences derived from only a small number of truly random bits. We show that, in contrast, in repeated two-player zero-sum games, if pseudorandom generators \emph{do not} exist, then $\Omega(n)$ random bits remain necessary for equilibria to exist.