Second-Order Mirror Descent: Convergence in Games Beyond Averaging and Discounting

The paper proves that MD2 converges to an interior mere variationally stable state under either (i) C2 Legendre + supercoercive regularizers or (ii) compact domain with strongly convex regularizers, for all α,β,γ,ε>0, via a Lyapunov function built from a Fenchel/Bregman coupling plus a quadratic term and LaSalle’s invariance principle; see the MD2 definition and Theorem 1 statement, and the proof details for cases (i) and (ii) . The candidate solution follows the same structure: identical Lyapunov functional (up to constant scaling), the same derivative calculation d/dt F(x*,z)=⟨x−x*,ż⟩, the same use of variational stability to make V̇≤0, and LaSalle to locate the ω-limit set. The model additionally gives a standard Bregman-divergence argument to rule out multiple equilibrium cluster points, which strengthens (but does not contradict) the paper’s sketch. Overall, the model’s solution matches the paper’s main argument and assumptions.