From behavior to sparse graphical games: Efficient recovery of equilibria

Abstract

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 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 O (poly(k) log n). We also validate our theoretical results using synthetic experiments.

Provable Sample Complexity Guarantees for Learning of Continuous-Action Graphical Games with Nonparametric Utilities

“…We would like to prove that β ijk " 0, @j R S i and β ijk ‰ 0, @j P S i . This gives us a straight forward way of picking in-neighbors of player i by solving optimization problem (10). We use an auxiliary variable w " ř j‰i ř r k"0 |β ijk | and prove the following lemma to get Karush-Kuhn-Tucker (KKT) conditions at the optimum.…”

“…However, their method runs in exponential time and the authors assumed a specific observation model for the strategy profiles. For the same specific observation model, [10] proposed a polynomial time algorithm, based on 1 -regularized logistic regression, for learning linear influence games. Their strategy profiles (or joint actions) were drawn from a mixture of uniform distributions: one over the pure-strategy Nash equilibria (PSNE) set, and the other over its complement.…”

On the sample complexity of learning graphical games

“…The above results pertain to a specific class of payoff functions with a particular parametric representation, that allows for a logistic regression approach. The results in [8] also assume strict positivity of the payoffs in the PSNE set. Thus, it is unclear how these results can be extended to general discrete actions.…”

“…The binary-action models considered in [8,9,10] are a restricted subclass of the models that we consider here. The results in [8,9,10]…”

Provable Computational and Statistical Guarantees for Efficient Learning of Continuous-Action Graphical Games

“…However, their method runs in exponential time and the authors assumed a specific observation model for the strategy profiles. For the same specific observation model, [15] proposed a polynomial time algorithm, based on ℓ 1 -regularized logistic regression, for learning linear influence games. Their strategy profiles (or joint actions) were drawn from a mixture of uniform distributions: one over the pure-strategy Nash equilibria (PSNE) set, and the other over its complement.…”