Intrinsic robustness of the price of anarchy

• Authors: Tim Roughgarden
• Venue: Communications of the ACM
• Year: 2012

The price of anarchy, defined as the ratio of the worst-case objective function value of a Nash equilibrium of a game and that of an optimal outcome, quantifies the inefficiency of selfish behavior. Remarkably good bounds on this measure are known for a wide range of application domains. However, such bounds are meaningful only if a game's participants successfully reach a Nash equilibrium. This drawback motivates inefficiency bounds that apply more generally to weaker notions of equilibria, such as mixed Nash equilibria and correlated equilibria, or to sequences of outcomes generated by natural experimentation strategies, such as simultaneous regret-minimization. We prove a general and fundamental connection between the price of anarchy and its seemingly more general relatives. First, we identify a "canonical sufficient condition" for an upper bound on the price of anarchy of pure Nash equilibria, which we call a smoothness argument. Second, we prove an "extension theorem": every bound on the price of anarchy that is derived via a smoothness argument extends automatically, with no quantitative degradation in the bound, to mixed Nash equilibria, correlated equilibria, and the average objective function value of every no-regret sequence of joint repeated play. Third, we prove that in routing games, smoothness arguments are "complete" in a proof-theoretic sense: despite their automatic generality, they are guaranteed to produce an optimal worst-case upper bound on the price of anarchy.