Collusion with (almost) no information

We analyze noncooperative collusion in an infinitely repeated Bertrand game, where each of the n firms receives a privately observed, i.i.d. cost shock in each period and firms only (and privately) observe whether they have "won" the unit mass of consumers. No other information is available and no communication is allowed. We prove that there exist equilibria in private strategies approximating first-best profits when firms are sufficiently patient. In particular productive efficiency obtains in the limit.