Chapter Chapter-by-Chapter Anaylsis of The Selfish Gene (1989 edition) - Study Guide by Richard Dawkins

S = 0 Sucker’s pay-off
Defect	T = 5 Temptation to defect	P = 1 Punishment for mutual defection

Imagine player A finds another individual B who always co-operates. If A co-operates too it gets a reward of 3, whereas if it defects it gets 5. Therefore, if B co-operates, it pays A to defect. Now imagine player A discovers that B always defects. If A co-operates it gains nothing (the sucker’s pay-off) whereas if it defects it gets 1. Therefore, if B defects, it pays A to defect. The conclusion is that irrespective of the other player’s choice, in a one-off game it pays to defect even though with both players defecting they get less (1) than they would have got it they had both co-operated (3). In other words, co-operation is not an ESS because in a population of "all co-operators" a mutant who defected would spread. On the other hand, always defecting (ALL D) is an ESS because in a population of "all defect" a mutant co- operator does not gain an advantage.

However, this reasoning does not apply if the players interact an indefinite number of times, which is a more realistic assumption than the idea of single or finite meetings. With an indefinite number of interactions, co-operation can emerge because it is the possibility that players meet again that brings about the possibility of co-operation. Axelrod (1984) ran a computer tournament in which different strategies, submitted by scientists from all over the world, were paired against each other in a round robin tournament to determine how best to act when in a Prisoner’s Dilemma situation. Some of the strategies adopted in this tournament were very complex, using procedures such as a Markov process or Bayesian inference, and others were very unforgiving. However, the winning strategy was the simplest one of them all, called "Tit for Tat". Tit for Tat is the policy of co-operating on the first move and then doing whatever the other player did on the previous move. This policy means that Tit for Tat will defect once after each defection of the other player, and thus is a strategy of co-operation based on reciprocity. In the terms of the title of this chapter, Tit for Tat can be seen to be a "nice" strategy, defined as one that is never the first to defect. Axelrod also pointed out that Tit for Tat is a forgiving strategy, defined as one that, although it may retaliate, has a short memory and so swiftly overlooks old misdeeds. Throughout several sequential rounds of the tournament, the robustness of Tit for Tat was demonstrated in that it continued to win and it was subsequently discovered that in the long run, Tit for Tat continued to do well while the less successful strategies were displaced. Therefore, Axelrod identified that two characteristics of winning strategies are niceness and forgivingness.

From this, Dawkins points out that Tit for Tat cannot be invaded by any nasty strategy and so one is tempted to conclude that it is an ESS. However, Tit for Tat can be invaded by another nice strategy such as Always Cooperate. When these two strategies meet, they will always cooperate with each other, thus looking and behaving exactly like the other. In this way, although Always Cooperate does not enjoy a positive selective advantage over Tit for Tat, it can still drift into the population with out being noticed and thus Tit for Tat is not strictly an ESS because it can be invaded by other nice strategies. Therefore, Axelrod coined the phrase "collectively stable" to describe it.