E. J. Green (1980, J. Econ. Theory 22, 155-182) and H. Sabourian (1990, J.
Econ. Theory 51, 92-110) studied repeated games where a player's payoff dep
ends on his actions and an anonymous aggregate outcome, and show that long-
run players behave myopically in any equilibrium of such games. In this pap
er we extend these results to games where the aggregate outcome is not nece
ssarily an anonymous function of players' actions, and where players' strat
egies may depend nonanonymously on signals of other players' behavior. Our
argument also provides a conceptually simpler proof of Green and Sabourian'
s results, showing how their analysis is driven by general bounds on the nu
mber of pivotal players in a game. (C) 2001 Academic Press.