Algorithms against Anarchy: Understanding Non-Truthful Mechanisms

The algorithmic requirements for dominant strategy incentive compatibility, or truthfulness, are well understood. Is there a similar characterization of algorithms that when combined with a suitable payment rule yield near-optimal welfare in all equilibria? We address this question by providing a tight characterization of a (possibly randomized) mechanism's Price of Anarchy provable via smoothness, for single-parameter settings. The characterization assigns a unique value to each allocation algorithm; this value provides an upper and a matching lower bound on the Price of Anarchy of a derived mechanism provable via smoothness. The characterization also applies to the sequential or simultaneous composition of single-parameter mechanisms. Importantly, the factor that we identify is typically not in one-to-one correspondence to the approximation guarantee of the algorithm. Rather, it is usually the product of the approximation guarantee and the degree to which the mechanism is loser independent. We apply our characterization to show the optimality of greedy mechanisms for single-minded combinatorial auctions, whether these mechanisms are polynomial-time computable or not. We also use it to establish the optimality of a non-greedy, randomized mechanism for independent set in interval graphs and show that it is strictly better than any other deterministic mechanism.