On the Robust Price of Anarchy of Altruistic Games

We study the inefficiency of equilibria for several 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 cost-sharing games, utility games, and linear congestion games. Our main contribution is an adaptation of Roughgarden’s smoothness notion to altruistic extensions of games. We show that this extension captures the essential properties to determine the robust price of anarchy of these games, and use it to derive mostly tight bounds. For congestion games and cost-sharing games, the worst-case robust price of anarchy increases with increasing altruism, while for 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, the pure price of anarchy decreases both under increasing uniform altruism and as the fraction of entirely altruistic individuals increases. ∗Department of Computer Science, University of Southern California, USA. Email: {poanchen, dkempe}@usc.edu. †Algorithms, Combinatorics and Optimization, CWI Amsterdam, The Netherlands. Email: [email protected]. ‡Algorithms, Combinatorics and Optimization, CWI Amsterdam and Department for Econometrics and Operations Research, VU University Amsterdam, The Netherlands. Email: [email protected].