应用经济学报告系列 (1213-1)

Abstract: Repeated fixed-partnership (FP) Prisoners Dilemma has been the predominant paradigm for investigation of the cooperation problem under direct reciprocity. While any level of efficiency can be achieved via a simple class of tit-for-tat or grim-trigger strategies, also called the folk theorem, generous TFT and win-stay-lose-shift strategies that statically are equally efficient as TFT prove to have differentiated evolutionary advantages when noises are introduced. In comparison, 100% cooperation can never be sustained in equilibrium in the more realistic setup where unilateral and voluntary separation (VS) from the partnership in any period is introduced. Gradual trust building and/or exogenous costs of breakup like in the efficiency-wage hypothesis are featured in equilibrium. Theoretically, however, there has been a glaring gap on results regarding the maximal level of efficiency achievable in this paradigm.

In this study, we start with an experiment with two treatments that simulate the repeated PD paradigms of both FP and VS. For both FP and VS, we observe similar responses dependent on the outcomes in the previous period of an on-going partnership. The pattern is close to what we call TFT with partial forgiveness, where to mend the relationship the “wrongdoer” from the previous period is to show his sincere remorse by giving the wronged partner a chance to “even things out” with commitment to cooperation, while the latter may or may not forgive him. Moreover, within VS, we observe significantly less decision for breakup than initially expected from the familiar grim-trigger strategies. Subjects behaved as if following the conventional wisdom of “Pasteur is not always greener elsewhere.” And both this strong inclination to stay in a relationship through adversities and that to show forgiveness proved to be paying off in terms of average payoff expected from the remaining span of life.

Motivated by these observations, we prove first a result that breakup is often not needed even in the grim-trigger types of equilibria in VS-PD. More importantly, we show that a variation of TFT with partial forgiveness constitutes SPE that for a wide range of game parameters can achieve the upper bound of sustainable efficiency among all SPEs if eternal mutual cooperation is the desired outcome for the partnership. In comparison, other known equilibrium strategies generically are unable to do so.