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 S_{0}
to S_{15}, representing the 16 cornerpoints of the fourdimensional
vector space (i.e. S_{0} is ALLD, S_{9} is Pavlov, S_{10}
is TFT, etc.). As in the last simulation, noise is taken into account by
replacing 1 by 1e 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 S_{0}
to S_{15}, 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, S_{4}, S_{6}, S_{7} and S_{12}
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 f_{10}
(TFT) is high, it is outcompeted by a more generous strategy S_{11}
(1, 0, 1, 1) and S_{9} (Pavlov), which in turn are invaded by the
parasitic S_{1} (0, 0, 0, I). S_{1} is then sacked by ALLDlike
(S_{0}) and Grimlike (S_{8}) rules which, as expected,
can easily be overrun by TFT (S_{10}). 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).
