TF2T and STFT
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 S_{e} is robust (or "collectively
stable", Boyd and Lorberbaum 1987) if for any given strategy S_{i}:
V(S_{e}S_{e})>=V(S_{i}S_{e})
(3)
where V(S_{m}S_{n}) is the payoff of individuals using
strategy Sm when interacting with individuals using strategy S_{n}.
Consider a randomly meeting population, consisting of n strategies with
the frequencies f_{1}, f_{2},...,f_{n}. Furthermore,
a common strategy S_{e} coexists with n1 rare strategies, so that
f_{e}>> f_{j} (e unequal j). In order to fulfil this
assumption, the inequality
(4)
needs to be satisfied. This is still the case, if for some i,
V(S_{e}S_{e})=V(S_{i}S_{e}). There may
be even strategies S_{i}, for which V(S_{i}S_{i})=V(S_{e}S_{i})
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(TF2TSTFT)>V(TFTSTFT) 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[(TR)/(TP), (PS)/(RS)].
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.
