Penalty-regulated dynamics and robust learning procedures in games

TL;DR

This paper introduces a new class of penalty-regulated continuous-time learning dynamics in games, which converge to Nash equilibria and are robust to noise, with applications to traffic engineering.

Contribution

It proposes a novel penalty-regulated dynamics framework and a payoff-based discrete algorithm that ensures convergence and robustness without synchronization.

Findings

Dynamics converge to Nash equilibria in potential games.

The discrete algorithm is robust to stochastic perturbations.

Players only need to observe their payoffs, not strategies.

Abstract

Starting from a heuristic learning scheme for N-person games, we derive a new class of continuous-time learning dynamics consisting of a replicator-like drift adjusted by a penalty term that renders the boundary of the game's strategy space repelling. These penalty-regulated dynamics are equivalent to players keeping an exponentially discounted aggregate of their on-going payoffs and then using a smooth best response to pick an action based on these performance scores. Owing to this inherent duality, the proposed dynamics satisfy a variant of the folk theorem of evolutionary game theory and they converge to (arbitrarily precise) approximations of Nash equilibria in potential games. Motivated by applications to traffic engineering, we exploit this duality further to design a discrete-time, payoff-based learning algorithm which retains these convergence properties and only requires players to observe their in-game payoffs: moreover, the algorithm remains robust in the presence of stochastic perturbations and observation errors, and it does not require any synchronization between players.