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According to Nowak and Sigmund (1992, 1993a, b) this weakness is caused by the deterministic nature of the strategies discussed so far. In reducing the probability for C or D specified by the outcome of the previous round from 1 to <1, one takes random errors into account. For the infinitely IPD (the limit case, w=1) the first move is irrelevant since its effect is 'forgotten' in the long run (Nowak and Sigmund 1992). The decision rule then consists of a point (p, q) in the unit square, with the probability p (q) for a C after a cooperative (defective) move of the partner; 0<p,q<1. In this terminology TFT corresponds to (1,0), ALLD to (0,0), ALLC to (1, 1) and so on. In such a scenario, Generous tit for tat (GTFT), a more forgiving strategy with q=min[l -(T-R)/(R-S), (R-P)/(T-P)]= 1/3 is said to provide the highest payoff (Nowak and Sigmund 1992).
Fig. 3. The evolution of a set of strategies is shown. 99 strategies were randomly sampled and a TFT-like (0.99, 0.01) strategy was added. The initial frequencies f were uniformly set to 0.01 and all subsequent frequencies evolved in proportion to the payoff of the previous round. a) Relative frequencies after 0, 20, 100, 150, 200, 1000 generations. Strategies with very low frequencies may disappear from this plot, although they are present in the numerical computations. b) Population averages for p, q and payoff. See text for details. From: Nowak, M. A. and Sigmund, K. 1992. Tit for tat in heterogeneous populations. Nature 355: 251.
Nowak and Sigmund (1992) used a similar approach as Axelrod in his ecological tournament: 100 strategies S1 to S100 (uniformly distributed on the unit square, equal initial relative frequencies) were sampled, frequencies were assessed according to the payoff of the previous round, below-threshold strategies were discarded and the frequencies evolved as Fig. 3 shows: initially, the strategies closest to ALLD (0,0) feed on a large percentage of high q suckers, and increase drastically while the others vanish. If some of the initial strategies are close to TFT (1, 0), however, the strategies near ALLD will never reach fixation: the depletion of suckers reduces the exploiters' fitness (they now gain mostly P) while the TFT-like strategies still obtain R interacting among themselves and P confronting the defecting strategies. Slowly at first but gathering momentum, the cooperators increase in frequency and the defectors wane. Yet, having extinguished the ALLDlike strategies, the rules close to TFT are superseded by a third strategy, very close to the less severe retaliator GTFT. Due to the noise, generated by the stochasticity of the decision rules, the harsh retaliation of TFT proved 'fatal' to TFT. The possibility of forgiveness invading a retaliatorily cooperating community has previously been shown (Boyd and Lorberbaum 1987); however, the simulations conducted by Nowak and Sigmund (1992) clearly demonstrate the 'catalyser'-effect of TFT:
"TFT's strictness is salutary for the community, but harms its own. [...] It needs to be present, initially only in a tiny amount; in the intermediate phase, its concentration is high; but in the end, only a trace remains."
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