Evolutionary dynamics with random payoff matrices

What the paper explicitly states is: in symmetric d‑player two‑strategy games with iid Gaussian payoff entries, E(N)=√(d−1)/2·(1+o(1)), while for asymmetric games the polynomial is Kostlan–Shub–Smale and E(N)=½√(d−1) exactly for all d; furthermore a universality statement with E(N)=√(d−1)/2+O((d−1)^{1/2−c}) is asserted under finite (2+ε)-moment assumptions (and flagged as conjectural only for the symmetric universality), see the summary passages in the survey . Under the standard symmetric modeling actually described (one iid payoff per composition k per strategy), the Bernstein-to-monomial reduction yields coefficients with variance proportional to binom(d−1,k)^2, not binom(d−1,k); applying Edelman–Kostlan’s kernel method then gives a leading positive-zero intensity of √(2(d−1))/(π(1+t^2)) and hence E(N)=½√(2(d−1))·(1+o(1))—a factor √2 larger than the paper’s claim. By contrast, in the asymmetric case the averaging over roles/orderings produces Kostlan weights binom(d−1,k), recovering the exact ½√(d−1) that the paper reports; the stated universality bound likewise matches the Kostlan ensemble. Thus the survey’s symmetric claim is off by √2 unless an unstated variance scaling (Var per k ∝ 1/binom(d−1,k)) is imposed; the model’s solution correctly identifies this and aligns with the paper on the asymmetric and universality parts.