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 from 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 (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 non-cooperative 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 trust-building strategies. For example, in small scale transactions in virtual malls, starting honest transactions immediately with some risk of being betrayed is socially better than asking everyone to delay transactions and build up some trust. Our result indicates that behavioral diversity can be stable and beneficial for the society, even if players are free to escape from personalized punishments