Insights into the Structured Coordination Game with Neutral Options through Simulation

The paper reports simulation evidence that the consensus-basin probability increases with “connectivity,” relates it to ER connectedness, and formulates Conjecture 2 (a threshold B(n), plausibly at the connectedness scale) without a proof. It also notes an upper bound for disconnected graphs via component counts. The candidate solution rigorously refutes finite-n monotonicity by explicit n=4 counterexamples and shows a non‑monotone Φ4(p); it proves a clean component bound ϕG(Q0) ≤ c^{1−r} and uses ER connectivity to obtain the necessary lower scale Ω(n log n) for any threshold, while (correctly) not claiming sufficiency at that scale. Since the central sufficiency direction (2b) remains unproven in the paper and is not settled by the candidate, the main question is likely open as of the cutoff. This aligns with the paper’s conjectural status and simulations (see Fig. 11–12 and Conjecture 2 in the PDF) .