The Price of Anarchy in Games of Incomplete Information

Abstract

We define smooth games of incomplete information. We prove an ’’extension theorem” for such games:price of anarchy bounds for pure Nash equilibria for all induced full-information games extendautomatically, without quantitative degradation, to all mixed-strategy Bayes-Nash equilibria withrespect to a product prior distribution over players’ preferences. We also note that, for Bayes-Nashequilibria in games with correlated player preferences, there is no general extension theorem forsmooth games. We give several applications of our definition and extension theorem. First, we show that many gamesof incomplete information for which the price of anarchy has been studied are smooth in our sense.Our extension theorem unifies much of the known work on the price of anarchy in games of incompleteinformation. Second, we use our extension theorem to prove new bounds on the price of anarchy ofBayes-Nash equilibria in routing games with incomplete information.