When are Non-Anonymous Players Negligible?

We examine games played by a single large player and a large number of opponents who are small, but not anonymous. If the play of the small players is observed with noise, and if the number of actions the large player controls is bounded independently of the number of small players, then as the number of small players grows, the equilibrium set converges to that of the game where there is a continuum of small players. The paper extends previous work on the negligibility of small players by dropping the assumption that small players actions are “anonymous.” That is, we allow each small player’s actions to be observed separately, instead of supposing that the small players’ actions are only observed through their effect on an aggregate statistic.