Beating Price of Anarchy and Gradient Descent without Regret in Potential Games

Arguably one of the thorniest problems in game theory is that of equilibrium selection. Specifically, in the presence of multiple equilibria do self-interested learning dynamics typically select the socially optimal ones? We study a rich class of continuous-time no-regret dynamics in potential games (PGs). Our class of dynamics, *Q-Replicator Dynamics* (QRD), include gradient descent (GD), log-barrier and replicator dynamics (RD) as special cases. We start by establishing *pointwise convergence* of all QRD to Nash equilibria in almost all PGs. In the case of GD, we show a tight average case performance within a factor of two of optimal, for a class of symmetric $2\times2$ potential games with unbounded Price of Anarchy (PoA). Despite this positive result, we show that GD is not always the optimal choice even in this restricted setting. Specifically, GD outperforms RD, if and only if *risk-* and *payoff-dominance* equilibria coincide. Finally, we experimentally show how these insights extend to all QRD dynamics and that unbounded gaps between average case performance and PoA analysis are common even in larger settings.