“Proposition 4. For a nice strategy to be collectively stable, it must be provoked by the very first defection of the other player. The reason is simple enough. If a nice strategy were not provoked by a defection on move n, then it would not be collectively stable because it could be invaded by a rule which defected only on move n.”

这里解释了为什么一定要在背叛后惩罚对方（复仇）来达到stable，就是因为如果有一种策略只在n轮背叛，如果你不惩罚它，那就会被入侵。

“This proposition has important implications for the evolution of cooperation. If one imagines a system starting with individuals who cannot be enticed to cooperate, the collective stability of ALL D implies that no single individual can hope to do any better than go along and be uncooperative as well. A world of “meanies” can resist invasion by anyone using any other strategy—provided that the newcomers arrive one at a time. The problem, of course, is that a single newcomer in such a mean world has no one who will reciprocate any cooperation. If the newcomers arrive in small clusters, however, they will have a chance to get cooperation started.”

“If a TIT FOR TAT has some proportion, p, of its interactions with other TIT FOR TAT players, it will have 1–p with the meanies. So its average score will be 30p + 9(1–p). If this score is more than 10 points, it pays to use a TIT FOR TAT strategy rather than be a meanie like the bulk of the population. And this will be true even if only 5 percent of the interactions of the TIT FOR TAT players are with other TIT FOR TAT players.8 Thus, even a small cluster of TIT FOR TAT players can get a higher average score than the large population of meanies they enter. Because the TIT FOR TAT players do so well when they do meet each other, they do not have to meet each other very often to make their strategy the superior one to use.”

这里说了一个很有意思的事，ALL D：全部背叛的策略是stable的regardless of w （折现率）。这种策略可以抵御任何单个策略的入侵，因为合作得不到任何回报。但是只要有一小群同样的策略，比如TIT FOR TAT，内部能有5%的互动率，那就可以达到比ALL D更好的表现