  We consider normal-form games with $n$ players and two strategies for each
player, where the payoffs are i.i.d. random variables with some distribution
$F$ and we consider issues related to the pure equilibria in the game as the
number of players diverges. It is well-known that, if the distribution $F$ has
no atoms, the random number of pure equilibria is asymptotically Poisson$(1)$.
In the presence of atoms, it diverges. For each strategy profile, we consider
the (random) average payoff of the players, called Average Social Utility
(ASU). In particular, we examine the asymptotic behavior of the optimum ASU and
the one associated to the best and worst pure Nash equilibria and we show that,
although these quantities are random, they converge, as $n\to\infty$ to some
deterministic quantities.
