Deterministic strategies
Let there be a homogenous population with individuals that are able
to recognise a previous interactant and remember some aspects of prior
outcomes. If the average chance to meet a given partner more than THETA
times in a row is w^{THETA} (where 0<w<1),
then a strategy would be a rule, modifying the behaviour (C or D) according
to the history of the interaction so far. Obviously, unconditional defection
(ALLD) is still an ESS against single cooperating mutants. But there may
be other stable strategies as well. In fact, in the Iterated Prisoner's
Dilemma (IPD) no strategy is evolutionarily stable (Boyd and Lorberbaum
1987, Lorberbaum 1994).
