Barriers to Near-Optimal Equilibria

This paper explains when and how communication and computational lower bounds for algorithms for an optimization problem translate to lower bounds on the worst-case quality of equilibria in games derived from the problem. We give three families of lower bounds on the quality of equilibria, each motivated by a different set of problems: congestion, scheduling, and distributed welfare games, welfare-maximization in combinatorial auctions with "black-box" bidder valuations, and welfare-maximization in combinatorial auctions with succinctly described valuations. The most straightforward use of our lower bound framework is to harness an existing computational or communication lower bound to derive a lower bound on the worst-case price of anarchy (POA) in a class of games. This is a new approach to POA lower bounds, which relies on reductions in lieu of explicit constructions. More generally, the POA lower bounds implied by our framework apply to all classes of games that share the same underlying optimization problem, independent of the details of players' utility functions. For this reason, our lower bounds are particularly significant for problems of game design -- ranging from the design of simple combinatorial auctions to the computation of tolls for routing networks -- where the goal is to design a game that has only near-optimal equilibria. For example, our results imply that the simultaneous first-price auction format is optimal among all "simple combinatorial auctions" in several settings.