Understanding Emergent Behaviours in Multi-Agent Systems with Evolutionary Game Theory

The paper explicitly states the two core facts under i.i.d. Gaussian payoffs: (i) for n=2 and large d, E(2,d) grows on the order of sqrt(d−1) up to a logarithmic factor, i.e., sqrt(d−1) ≤ E(2,d) ≤ sqrt(d−1) ln(d−1), implying lim_{d→∞} ln E(2,d)/ln(d−1)=1/2; and (ii) for d=2, E(n,2)=2^{−(n−1)}. These are presented in Section 5 as headline results of Duong–Han’s program connecting equilibria to real zeros of random polynomials via Edelman–Kostlan. The candidate solution derives the same two conclusions using the same backbone (Edelman–Kostlan + Bernstein basis/weights and an orthant-symmetry argument). The only caveat is that the candidate informally sketches a sharper constant-level asymptotic E(2,d)~sqrt{(d−1)/2} without fully justifying uniform tail control; but the paper only claims the order bounds, so there is no conflict. Overall, the statements agree and the proof strategies are substantially aligned with the paper’s approach. See the paper’s summary bullets for E(2,d) bounds and E(n,2)=2^{−(n−1)} in Section 5 .