Yoram Bachrach, Vasilis Syrgkanis, Eva Tardos, and Milan Vojnovic
We introduce a framework for studying the effect of cooperation on the quality of outcomes in utility games. Our framework is a coalitional analog of the smoothness framework of non-cooperative games. Coalitional smoothness implies bounds on the strong price of anarchy, the loss of quality of coalitionally stable outcomes, as well as bounds on coalitional versions of coarse correlated equilibria and sink equilibria, which we define as out-of-equilibrium myopic behavior as determined by a natural coalitional version of best-response dynamics.
Our coalitional smoothness framework captures existing results bounding the strong price of anarchy of network design games. We show that in any monotone utility-maximization game, if each player's utility is at least his marginal contribution to the welfare, then the strong price of anarchy is at most 2. This captures a broad class of games, including games with a very high price of anarchy. Additionally, we show that in potential games the strong price of anarchy is close to the price of stability, the quality of the best Nash equilibrium.