From Behavior to Sparse Graphical Games: Efficient Recovery of Equilibria

In this paper we study the problem of exact recovery of the pure-strategy Nash equilibria (PSNE) set of a graphical game from noisy observations of joint actions of the players alone. We consider sparse linear influence games --- a parametric class of graphical games with linear payoffs, and represented by directed graphs of n nodes (players) and in-degree of at most k. We present an $\ell_1$-regularized logistic regression based algorithm for recovering the PSNE set exactly, that is both computationally efficient --- i.e. runs in polynomial time --- and statistically efficient --- i.e. has logarithmic sample complexity. Specifically, we show that the sufficient number of samples required for exact PSNE recovery scales as $\mathcal{O}(\mathrm{poly}(k) \log n)$. We also validate our theoretical results using synthetic experiments.