(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*?