The Robust Price of Anarchy of Altruistic Games

We study the inefficiency of equilibria for various classes of games when players are (partially) altruistic. We model altruistic behavior by assuming that player i’s perceived cost is a convex combination of 1 − αi times his direct cost and αi times the social cost. Tuning the parameters αi allows smooth interpolation between purely selfish and purely altruistic behavior. Within this framework, we study altruistic extensions of linear congestion games, fair cost-sharing games and valid utility games. We derive (tight) bounds on the price of anarchy of these games for several solution concepts. Thereto, we suitably adapt the smoothness notion introduced by Roughgarden and show that it captures the essential properties to determine the robust price of anarchy of these games. Our bounds show that for congestion games and cost-sharing games, the worst-case robust price of anarchy increases with increasing altruism, while for valid utility games, it remains constant and is not affected by altruism. However, the increase in the price of anarchy is not a universal phenomenon: for symmetric singleton linear congestion games, we derive a bound on the pure price of anarchy that decreases as the level of altruism increases. Since the bound is also strictly lower than the robust price of anarchy, it exhibits a natural example in which Nash equilibria are more efficient than more permissive notions of equilibrium. Department of Computer Science, University of Southern California, USA. Work done in part while visiting CWI Amsterdam. Email: [email protected]. Algorithms, Combinatorics and Optimization, CWI Amsterdam, The Netherlands. Email: [email protected]. Department of Computer Science, University of Southern California, USA. Email: [email protected]. Algorithms, Combinatorics and Optimization, CWI Amsterdam and Department for Econometrics and Operations Research, VU University Amsterdam, The Netherlands. Email: [email protected].