A. Show that the derived family of Markov chains {(_0, ˆQε)}ε* [0,1)

constructed in Subsection 12.6.4.2 from the original canonical model defines by

itself a proper canonical evolutionary model.

B. Let A *_0 satisfy R(A) > CR(A). Showdirectly (i.e., not merely

invoking Proposition 12.3) that no other set B ⊂ _0\A may satisfy R(B) > CR(B).

Q150A. Let (_, Qε) stand for the perturbed stochastic process modeling reinforcement learning in the prisoner’s dilemma (cf. Subsection 12.7.1). Show that the sufficient condition (12.58) invoked to guarantee the ergodicity of the process holds in this case.

B.Recall the argument spelled out in Subsection 12.7.1 concerning the stochastic stability of ωC in the prisoner’s dilemma when η/2 > ν. This argument was restricted to transitions across the symmetric states ωC and ωD. Extend it to the transitions involving the other two limit states (C, 0, D, η) and (D, η,C, 0).