Frequential chaos

If Gould and Eldredge's (1993) statement that "maintenance of stability within species must be considered as a major evolutionary problem" is also valid for cornpeting strategies (i.e. genotypes) in the IPD, the findings of Nowak and Sigmund (1993b) deserve special interest. According to their terminology, there is a set of 16 deterministic strategies S₀ to S₁₅, representing the 16 corner-points of the four-dimensional vector space (i.e. S₀ is ALLD, S₉ is Pavlov, S₁₀ is TFT, etc.). As in the last simulation, noise is taken into account by replacing 1 by 1-e and 0 by e in the vectors; with 0 <e<<l, the first round no longer matters (Nowak and Sigmund 1993b).
Nowak and Sigmund computed a large population consisting of S₀ to S₁₅, f_i being the relative frequency of S_i in a given generation, V_i the average payoff for an S_i player and u a tiny number of invaders. Selection was computed as described above. This yields

(5)

where f_i denotes the frequency of S_i after one generation. Eq. (5) exhibits complex dynamics. The frequencies may oscillate periodically or display violently chaotic orbits. For example, around u = 0.0004, S₄, S₆, S₇ and S₁₂ almost perish, while the other strategies oscillate vigorously. Only TFT persists at comparatively high frequency, whereas the other oscillating strategies come very close to f_i=0. But whenever f₁₀ (TFT) is high, it is outcompeted by a more generous strategy S₁₁ (1, 0, 1, 1) and S₉ (Pavlov), which in turn are invaded by the parasitic S₁ (0, 0, 0, I). S₁ is then sacked by ALLD-like (S₀) and Grim-like (S₈) rules which, as expected, can easily be overrun by TFT (S₁₀). However, large e favours the domination of defective strategies and the oscillations disappear.

In this model ALLD and Grim are the only ESSs, but the introduction of GTFT may lead to stable equilibria, where GTFT dominates (Nowak and Sigmund 1993b).