Repeated triangular trade: Sustaining circular cooperation with observation errors

We introduce a new fundamental problem called triangular trade, which is a natural extension of the well-studied prisoner’s dilemma for three (or more) players where a player cannot directly punish a seemingly defecting player. More specifically, this problem deals with a situation where the power/influence of players is one-way, players would be better off if they maintain circular cooperation, but each player has an incentive to defect. We analyze whether players can sustain such circular cooperation when they repeatedly play this game and each player observes the actions of another player with some observation errors (imperfect private monitoring). We confirm that no simple strategy can constitute an equilibrium within any reasonable parameter settings when there are only two actions: “Cooperate” and “Defect.” Thus, we introduce two additional actions: “Whistle” and “Punish,” which can be considered as a slight modification of “Cooperate.” Then, players can achieve sustainable cooperation using a simple strategy called Remote Punishment strategy (RP), which constitutes an equilibrium for a wide range of parameters. Furthermore, we show the payoff obtained by a variant of RP is optimal within a very general class of strategies that covers virtually all meaningful strategies.