Equilibria and learning dynamics in mixed network coordination/anti-coordination games

The paper proves (i) existence of a pure Nash equilibrium with the coordinators at consensus under r‑cohesiveness or (1−r)‑cohesiveness (Theorem 3), by fixing the coordinators at the corresponding consensus and using that the S‑restricted game is a potential game, and (ii) global reachability of that set under the added r‑indecomposability assumption (Theorem 4), via a two‑step best‑response path argument on the R‑ and S‑restricted potential games . The candidate solution establishes the same conclusions: for (i) it fixes y=1_R or y=0_R and constructs an explicit exact potential for the S‑subgame; for (ii) it provides a constructive, monotone best‑response path on R using indecomposability to guarantee a strictly dominated move in any non‑consensus split, then finishes on S via the S‑restricted potential. This differs slightly from the paper’s Step‑I/Step‑II potential‑based path (and its fallback pass), but is logically consistent with the paper’s definitions of cohesiveness and indecomposability and with the potential structure of the restricted games (Lemma 1) . No missing hypotheses are required beyond the paper’s (undirected graph, symmetric weights, r∈(0,1)).