Optimal Collusion with Private Information

Abstract

We consider an infinitely-repeated Bertrand game, in which prices are perfectly observed and each firm receives a privately-observed, i.i.d. cost shock in each period. Productive efficiency is possible only if high-cost firms are willing to relinquish market share. In the most profitable collusive schemes, firms implement productive efficiency, and high-cost firms are favored with higher expected market share in future periods. If types are discrete, there exists a discount factor strictly less than one above which first-best profits can be attained purely through history-dependent reallocation of market share between equally-efficient firms. We provide further characterizations and several computational examples. We next examine different institutional features. We find that firms may find explicit communication (smoke-filled rooms) about costs beneficial after some histories but not others. We show that if firm-level behavior is not publicly observable, the best collusive scheme sacrifices all productive efficiency. Finally, if firms can make explicit side-payments and these entail any inefficiency (e.g., if they are illegal and bear some risk of detection), then optimal collusive equilibria are non-stationary and thus involve the use of future market-share favors.