CONTINUOUS-TIME CONVERGENCE RATES IN POTENTIAL AND MONOTONE GAMES

The paper’s Theorem 5.4 proves exponential convergence of mirror descent in relatively strongly monotone games: Dh(x*, x(t)) ≤ e^{−(γ η/ε)t} Dh(x*, x0), plus the Euclidean bound under ρ-strong convexity . In the appendix, they use a dual-space Lyapunov Vz = γ^{−1}∑p D_{ψp*}(zp, z*p) and show V̇z ≤ −(γ η/ε) Vz via η-relative strong monotonicity and Bregman identities, then apply Grönwall’s lemma to get the rate; the Euclidean bound follows from Dh ≥ (ρ/2)∥·∥^2 . The setup (ż = γU(x), x = C(z)) and mirror-map assumptions (ψp = εϑp with Assumption 2(i)/2(ii)) are exactly as in the model’s solution , and the Bregman/Fenchel identities used are standard (Lemma 7.3) . The candidate solution derives the same differential inequality but starts with the Fenchel–Bregman coupling expression for V(t) = Dh(x*, x(t)), differentiates to obtain V̇ = (γ/ε)⟨x − x*, U(x)⟩, and then applies the relative strong monotonicity of −U together with the Nash equilibrium condition to conclude V̇ ≤ −(γ η/ε) V(t). This is equivalent in substance to the paper’s Lyapunov approach (they use the dual Bregman divergence; the model uses the primal coupling), and yields the same rates.