Selfishness Level of Strategic Games

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 added to the players' payoffs to ensure that a Nash equilibrium of the resulting game is also its social optimum. This notion is unrelated to that of price of stability. We compute the selfishness level for some selected games. In particular, the selfishness level of finite ordinal potential games is finite, while that of a Cournot competition oligopoly game and Tragedy of the Commons game is infinite. We also provide an estimate on the selfishness level of linear congestion games and fair cost sharing games.