Institutional Incentives for the Evolution of Committed Cooperation: Ensuring Participation is as Important as Enhancing Compliance

The paper’s Supporting Information derives exactly the same risk-dominance thresholds the model reports: (i) without noise, ACD is risk-dominant against all strategies except ACC if and only if α < 1 − (T+P−R−S)/(2u) and ε bounds equal 2(u+uα+min{T−S,R−P}) for reward, 2(2uα+min{T−S,R−P}) for punishment; under α=0 these reduce to the familiar thresholds. (ii) For the donation game, these specialize to α < 1 − c/u and the corresponding ε-bounds. (iii) With noisy acceptance (A↔N flips with probability ω), ACD becomes risk-dominant against ACC for any ω>0, and the remaining RD conditions match the paper’s ω-dependent bounds. The model follows the same Δ(X,Y) risk-dominance calculus as the SI, adds compatible pairwise decompositions, and is consistent throughout.