On the Sample Complexity of Learning Graphical Games

TL;DR

This paper investigates the number of behavioral data samples needed to accurately learn the pure-strategy Nash equilibria in graphical games, providing bounds for sparse and dense graph structures.

Contribution

It establishes both sufficient and necessary sample complexity bounds for recovering game equilibria from observed actions, using VC dimension and information-theoretic methods.

Findings

Sample complexity for sparse graphs: O(k n log^2 n)

Sample complexity for dense graphs: O(n^2 log n)

Recovery is impossible below these bounds with high probability

Abstract

We analyze the sample complexity of learning graphical games from purely behavioral data. We assume that we can only observe the players' joint actions and not their payoffs. We analyze the sufficient and necessary number of samples for the correct recovery of the set of pure-strategy Nash equilibria (PSNE) of the true game. Our analysis focuses on directed graphs with $n$ nodes and at most $k$ parents per node. Sparse graphs correspond to ${k \in O(1)}$ with respect to $n$, while dense graphs correspond to ${k \in O(n)}$. By using VC dimension arguments, we show that if the number of samples is greater than ${O(k n \log^2{n})}$ for sparse graphs or ${O(n^2 \log{n})}$ for dense graphs, then maximum likelihood estimation correctly recovers the PSNE with high probability. By using information-theoretic arguments, we show that if the number of samples is less than ${\Omega(k n \log^2{n})}$ for sparse graphs or ${\Omega(n^2 \log{n})}$ for dense graphs, then any conceivable method fails to recover the PSNE with arbitrary probability.