  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 and is invariant under positive linear
transformations of the payoff functions. Also, it naturally applies to other
solution concepts and other forms of games.
  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.
  In particular, the selfishness level of finite ordinal potential games is
finite, while that of weakly acyclic games can be infinite. We derive explicit
bounds on the selfishness level of fair cost sharing games and linear
congestion games, which depend on specific parameters of the underlying game
but are independent of the number of players. Further, we show that 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, that
of the $n$-players public goods game is $(1-\frac{c}{n})/(c-1)$, where $c$ is
the public good multiplier, and that of the Traveler's Dilemma game is
$\frac{1}{2}(b-1)$, where $b$ is the bonus. Finally, the selfishness level of
Cournot competition (an example of an infinite ordinal potential game, Tragedy
of the Commons, and Bertrand competition is infinite.
