  The price of anarchy (PoA) is a popular metric for analyzing the inefficiency
of self-interested decision making. Although its study is widespread,
characterizing the PoA can be challenging. A commonly employed approach is
based on the smoothness framework, which provides tight PoA values under the
assumption that the system objective consists in the sum of the agents'
individual welfares. Unfortunately, several important classes of problems do
not satisfy this requirement (e.g., taxation in congestion games), and our
first result demonstrates that the smoothness framework does *not* tightly
characterize the PoA for such settings. Motivated by this observation, this
work develops a framework that achieves two chief objectives: i) to tightly
characterize the PoA for such scenarios, and ii) to do so through a tractable
approach. As a direct consequence, the proposed framework recovers and
generalizes many existing PoA results, and enables efficient computation of
incentives that optimize the PoA. We conclude by highlighting the applicability
of our contributions to incentive design in congestion games and utility design
in distributed welfare games.
