Indirect reciprocity, golden opportunities for defection, and inclusive reputation

In evolutionary models of indirect reciprocity, reputation mechanisms can stabilize cooperation even in severe cooperation problems like the prisoner's dilemma. Under certain circumstances, conditionally cooperative strategies, which cooperate iff their partner has a good reputation, cannot be invaded by any other strategy that conditions behavior only on own and partner reputation. The first point of this paper is to show that an evolutionary version of backward induction can lead to a breakdown of this kind of indirectly reciprocal cooperation. Backward induction, however, requires strategies that count and then cease to cooperate in the last, last but one, last but two, ... game they play. These strategies are unlikely to exist in natural settings. We then present two new findings. (1) Surprisingly, the same kind of breakdown is also possible without counting. Strategies using rare golden opportunities for defection can invade conditional cooperators. This can create further golden opportunities, inviting the next wave of opportunists, and so on, until cooperation breaks down completely. (2) Cooperation can be stabilized against these opportunists, by letting an individual's initial reputation be inherited from that individual's parent. This "inclusive reputation" mechanism can cope with any observably opportunistic strategy. Offspring of opportunists who successfully exploited a conditional cooperator cannot repeat their parents' success because they inherit a bad reputation, which forewarns conditional cooperators in later generations.