Heterogeneous Mixed Populations of Coordinating, Anticoordinating, and Imitating Individuals

The paper proves that the equilibrium set X* coincides with the set E of candidate states xr,j1,j1′ that satisfy the strict threshold inequalities for best-responders and the envelope inequalities for imitators (≥ if r>0, ≤ if r<m), which jointly force equality when 0<r<m (Theorem 4.5) . The paper’s construction of Cj1,j1′ and Dj1+1,j1′+1, the threshold conditions (4.1)–(4.2), and the necessity/sufficiency arguments via CM and DM appear explicitly in Lemmas 4.1–4.4 . The candidate solution reproduces the same characterization, uses the same CM/DM envelopes and the same activation-sequence invariance logic, and reaches X*=E with the same conditions, i.e., substantially the same proof as the paper’s Lemmas 4.2–4.4 and Theorem 4.5 .