  We introduce and study a class of infinite-horizon non-zero-sum
non-cooperative stochastic games with infinitely many interacting agents using
ideas of statistical mechanics. First we show, in the general case of
asymmetric interactions, the existence of a strategy that allows any player to
eliminate losses after a finite random time. In the special case of symmetric
interactions, we also prove that, as time goes to infinity, the game converges
to a Nash equilibrium. Moreover, assuming that all agents adopt the same
strategy, using arguments related to those leading to perfect simulation
algorithms, spatial mixing and ergodicity are proved. In turn, ergodicity
allows us to prove ``fixation'', i.e. that players will adopt a constant
strategy after a finite time. The resulting dynamics is related to
zero-temperature Glauber dynamics on random graphs of possibly infinite volume.
