Community enforcement when players observe partners' past play

I investigate whether a large community can sustain cooperation in the repeated prisoner's dilemma by having cheaters punished not by their victims but by third parties. In the setting where players can observe their partners' past play only, I show that cooperation can be sustained by an equilibrium that has the following two properties: players choose their actions independently of their own past play, and they are indifferent between cooperation and defection at all histories. This equilibrium carries over to the finite-population setting and is robust to noise in the process of choosing actions or of recording past play.