(i) In a spirit similar to that of Exercise 12.9, now assume that the mutation
probabilities may depend on the prevailing state so that there is some
smooth function φ : [0, 1) × _ → [0, 1), where εω = φ(ε, ω) is the mutation
probability at state ω when the base parameter reflecting the overall
noise of the system is ε. (For simplicity, assume all players mutate with the
same probability.) Further suppose that φ(0, ·) = 0 and, as the counterpart
of (12.62), we have that
for each ω,ω
_ *_. Focus on the context in which the population plays
a bilateral coordination game under global interaction and show that the
results established in the text are not affected by the proposed variation.
(ii) Now consider a situation such as the one described in (i) above but, instead
of (12.63), suppose φ(ε, ω) = ε for all ω _= ωα but
for some r * N. Given n (the population size), characterize the values of
r that lead to the same selection result (i.e., the same stochastically stable
state) as in the text. What happens for other values of r?