This paper studies an infinitely repeated duopoly game with incomplete information and with costly entry decisions. Every period, each player learns her private type and decides whether to pay a cost in order for her to enter or not. If she enters, she plays a game belonging to a class that includes Bertrand duopoly and some auction games as special cases, either as a monopolist or as a duopolist. The players can communicate before they make their entry decisions. We study full collusion (joint profit maximization) in this environment which requires a higher-quality player to solely enter and to choose an action maximizing the stage payoff. We present a condition on the stage game which is both necessary and sufficient in order for full collusion to be an equilibrium outcome for sufficiently patient players. The condition is more likely to hold when the entry cost increases, which signifies that the entry cost is an important factor facilitating full collusion. We also show that under some parameter restrictions, asymmetric equilibria where only one player reveals her type every period sustain full collusion for a wider range of discount factors. These asymmetric equilibria reduce the total amount of communication, which makes it harder for antitrust authorities to detect collusion
