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We study a static game played by a finite number of agents, in which agentsare assigned independent and identically distributed random types and eachagent minimizes its objective function by choosing from a set of admissibleactions that depends on its type. The game is anonymous in the sense that theobjective function of each agent depends on the actions of other agents onlythrough the empirical distribution of their type-action pairs. We study theasymptotic behavior of Nash equilibria, as the number of agents tends toinfinity, first by deriving laws of large numbers characterizes almost surelimit points of Nash equilibria in terms of so-called Cournot-Nash equilibriaof an associated nonatomic game. Our main results are large deviationprinciples that characterize the probability of rare Nash equilibria andassociated conditional limit theorems describing the behavior of equilibriaconditioned on a rare event. The results cover situations when neither thefinite-player game nor the associated nonatomic game has a unique equilibrium.In addition, we study the asymptotic behavior of the price of anarchy,complementing existing worst-case bounds with new probabilistic bounds in thecontext of congestion games, which are used to model traffic routing innetworks.

Rare Nash Equilibria and the Price of Anarchy in Large Static Games