Selfishness Level of Strategic Games

Krzysztof R. Apt ( CWI and University of Amsterdam )

We introduce a new measure of the discrepancy in strategic games between the social welfare in a Nash equilibrium and in a social optimum, that we call selfishness level. It is the smallest fraction of the social welfare that needs to be offered to each player to achieve that a social optimum is realized in a pure Nash equilibrium. The selfishness level is unrelated to the price of stability and the price of anarchy. We study the selfishness level of several well-known strategic games. This allows us to quantify the implicit tension within a game between players' individual interests and the impact of their decisions on the society as a whole. Our analyses reveal that the selfishness level often provides a deeper understanding of the characteristics of the underlying game that influence the players' willingness to cooperate. For instance, the selfishness level of the n-players Prisoner's Dilemma is c/(b(n-1)-c), where b and c are the benefit and cost for cooperation, respectively, and that of the n-players public goods game is (1-cn)/(c-1), where c is the public good multiplier. In turn, the selfishness level of Cournot competition, Tragedy of the Commons, and Bertrand competition is infinite.

This is a joint work with Guido Schäfer.