On the Sample Complexity of Learning Sparse Graphical Games

We analyze the sample complexity of learning sparse graphical games from purely behavioral data. That is, 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 sparse directed graphs with $n$ nodes and at most $k$ parents per node. By using VC dimension arguments, we show that if the number of samples is greater than $O(k n \log^2{n})$, 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 $\Omega(k n \log^2{n})$, then any conceivable method fails to recover the PSNE with arbitrary probability.