Cost of the action is f(α)=α⁴. Survivors (fraction 1−ε) keep their action; an ε-fraction are new samplers who best-respond. Population follows the population dynamics map h(α)=(1−ε)α+ε·R(α), with response R(α)=1−Bernₖ(α). The paper’s sufficient condition is ε<1/(1+f′(1))=0.20 — a conservative bound; the dynamics in fact still settle for larger ε, up to an edge that shifts with the sample size k (marked on the bar below).
α₀ is a believed rate, not yet an action profile. In period 0 nobody has a prior action to keep, so everyone decides and the town realises α₁=R(α₀) — the first magenta dot, sitting on the blue curve. From period 1 the ε-replacement dynamic takes over: the 1−ε survivors keep their action rather than re-optimising, so every later step lands on h, and the remaining magenta dots sit on the orange curve.