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. Imp...