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Moreover, Boyd and Lorberbaum (1987) showed that no deterministic (pure) strategy is evolutionarily stable in the IPD. Consider the following line of argument: a strategy Se is robust (or "collectively stable", Boyd and Lorberbaum 1987) if for any given strategy Si:

V(Se|Se)>=V(Si|Se) (3)

where V(Sm|Sn) is the payoff of individuals using strategy Sm when interacting with individuals using strategy Sn. Consider a randomly meeting population, consisting of n strategies with the frequencies f1, f2,...,fn. Furthermore, a common strategy Se coexists with n-1 rare strategies, so that fe>> fj (e unequal j). In order to fulfil this assumption, the inequality

eq4 (4)

needs to be satisfied. This is still the case, if for some i, V(Se|Se)=V(Si|Se). There may be even strategies Si, for which V(Si|Si)=V(Se|Si) and (4) still is true. For example, the nice strategy TFT has the same expected fitness as another nice strategy Tit for two tats (TF2T, allowing two consecutive Ds before retaliating) when interacting, because neither ever defects. The relative fitness of TFT and TF2T is dependent upon their interaction with any third strategy. Assume this third strategy is Suspicious tit for tat (STFT, playing D in the first interaction), maintained at a low level in the population through recurrent mutation (for example). For V(TF2T|STFT)>V(TFT|STFT) TF2T can invade TFT without random drift. If (2) holds, this is always true because the more tolerant TF2T invokes cooperation in STFT, whereas the retaliatory TFT defects together with STFT. Accordingly, any deterministic strategy can be invaded and outcompeted by the joint effect of a "twin"-strategy together with another strategy. Boyd and Lorberbaum (1987) proved this general case to be true for all w>min[(T-R)/(T-P), (P-S)/(R-S)].
Lorberbaum (1994) extended the instability proof to stochastic (mixed) strategies as well. Depending on the initial set of assumptions, however, there are different types of evolutionary stability of single strategies (Bender and Swistak 1995) or of groups of strategies (see below) in the IPD. All of which have to be considered before seriously testing the model.

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