Fast convergence in semi-anonymous potential games

Abstract

Abstract-The log-linear learning algorithm has been extensively studied in both the game theoretic and distributed control literature. One of the central appeals with regards to log-linear learning for distributed control is that it often guarantees that the agents' behavior will converge in probability to the optimal configuration. However, one of the central issues with log-linear learning for this purpose is that the worst case convergence time can be prohibitively long, e.g., exponential in the number of players. In this work, we formalize a modified log linear learning algorithm that exhibits a worst case convergence time that is roughly linear in the number of players. We prove this characterization in semi-anonymous potential games with limited populations. That is, potential games where the agents' utility functions can be expressed as a function of aggregate behavior within each population.

Fast convergence for time-varying semi-anonymous potential games

“…For example, [19] introduces a variant of log-linear learning and demonstrates that the mixing time grows linearly in the number of players for a class of single-selection congestion games where the agents have homogenous route choices [19]. In [4], the authors extend these mixing time results to the broader class of multi-selection congestion games where the agents could have heterogenous route choices. The primary features of this class of congestion game, or more generally the framework of semi-anonymous potential games [4] are that (i) the agents can be divided into populations according to their capabilities, and (ii) each agent's utility can be evaluated using aggregate information about the actions of agents in each population.…”

“…First we review a modified log-linear learning algorithm first stated in [19] and extended in [4] to semianonymous potential games with multiple populations. Let a(t) ∈ A be the action profile at time t ≥ 0.…”

“…We begin by reviewing the definition of a semianonymous potential game in [4]. Consider a game with agents N = {1, 2, .…”

“…Recent research has shown that one can often attain desirable convergence rates when the underlying game has additional structure, e.g., [4], [9], [16], [19]. For example, [19] introduces a variant of log-linear learning and demonstrates that the mixing time grows linearly in the number of players for a class of single-selection congestion games where the agents have homogenous route choices [19].…”

“…The proof of this result relies on existing bounds relating the Sobolov constant to the underlying mixing times of the process. In extending the static game result of [4] to trajectories of games, the main challenge is bounding the change in distance between the distribution of the underlying Markov process at a given time and its stationary distribution, both of which change as players enter or exit the game. It is important to highlight that the structure of the proof follows closely to the proof in [19] which focused purely on the simplified single-selection congestion games depicted above.…”