Price of Anarchy⋆

Internet users, auction bidders, and stock buyers are examples of agents who are selfishly trying to maximize their own benefit without necessarily caring the global objective of the underlying game such as the overall load on the network or the overall satisfaction of agents. How does these two points of view (agents vs. global) correlate? How much does the society suffer by the lack of coordination between players? The optimal social utility function happens when we have a single authority who dictates every agent what to do. In contrast, when agents choose their own action, we should study their behavior and compare the obtained social utility with the optimal one. There are currently two main approaches for studying behavior of agents. Koutsoupias and Papadimitriou [KP99] assume that agents play according to a Nash equilibrium; thus they, pessimistically, consider the Nash equilibrium that gives the worst social utility and compare it with the optimal social utility. In contrast, Goemans et al. [GMV05] consider games in which agents are reluctant to choose mixed strategies and repeatedly choose their action by playing their best pure responses even if there is no pure strategy Nash equilibrium in the game. In this note we review these two different approaches: price of anarchy and price of sinking. In section 2 we survey some results related to the price of anarchy. In particular, we discuss previous researches on unsplittable flow problem, congestion games, and a class of games, called valid utility systems, whose price of anarchy is constant. Then, in Section 3, we introduce the price of sinking and its value on unsplittable routing games, congestion games, and valid utility systems. The two approaches, price of sinking and price of anarchy, are compared at the end.