Efficiency may improve when defectors exist

In repeated Prisoner’s Dilemma, the C-trigger strategy played by all players is well known to achieve symmetric efficiency when players are sufficiently patient. By contrast, if players are free to quit a repeated Prisoner’s Dilemma without information flow to new partners, cooperation from the outset of new partnerships cannot be a part of any symmetric equilibrium. Fujiwara-Greve and Okuno-Fujiwara (Rev Econ Stud 76:993–1021, 2009) showed that symmetric trust-building strategies can constitute an equilibrium for sufficiently long initial (D, D) (trust-building) periods. However, trust-building periods create social loss of payoffs, and there is a possibility that an asymmetric equilibrium with some players cooperating immediately, while others defect, may be more efficient. We show that there is a “fundamentally asymmetric” locally stable Nash equilibrium consisting of the most cooperative strategy (C-trigger with ending the partnership when betrayed) and the most noncooperative strategy, which plays D and ends the partnership immediately. When the deviation gain is relatively small, the fundamentally asymmetric equilibrium is neutrally stable against equilibrium entrants within trust-building strategies and is more efficient than any Nash equilibrium consisting of non-degenerate trust-building strategies. Our result indicates that behavioral diversity can be stable and beneficial for the society, even if players are free to escape from personalized punishments.