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Abstract We study perfect information games played by an infinite sequence of players, each acting only once in the course of the game. We introduce a class of frequency-based minority games and show that these games have no subgame perfect $$\epsilon $$ ϵ -equilibrium for any $$\epsilon $$ ϵ sufficiently small. Furthermore, we present a number of sufficient conditions to guarantee existence of subgame perfect $$\epsilon $$ ϵ -equilibrium.