The Query Complexity of Correlated Equilibria

TL;DR

This paper investigates the query complexity of finding correlated equilibria in n-player games, showing that both randomness and approximation are essential for efficiency, with deterministic algorithms failing even approximately and randomized algorithms failing exactly.

Contribution

It establishes lower bounds on the number of payoff queries needed, proving that no efficient deterministic algorithm can find approximate correlated equilibria and no efficient randomized algorithm can find exact ones.

Findings

Deterministic algorithms cannot efficiently find approximate correlated equilibria.

Randomized algorithms cannot efficiently find exact correlated equilibria.

A lower bound on the number of payoff queries required for these tasks.

Abstract

We consider the complexity of finding a correlated equilibrium of an $n$-player game in a model that allows the algorithm to make queries on players' payoffs at pure strategy profiles. Randomized regret-based dynamics are known to yield an approximate correlated equilibrium efficiently, namely, in time that is polynomial in the number of players $n$. Here we show that both randomization and approximation are necessary: no efficient deterministic algorithm can reach even an approximate correlated equilibrium, and no efficient randomized algorithm can reach an exact correlated equilibrium. The results are obtained by bounding from below the number of payoff queries that are needed.