Robust Price of Anarchy for Atomic Games with Altruistic Players

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-\beta_i times his direct cost and \beta_i times the social cost. Tuning the parameters \beta_i 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 reveal 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. We also show that the increase in price of anarchy is not a universal phenomenon: for symmetric singleton linear congestion games we derive a bound on the price of anarchy for pure Nash equilibria 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.