Adapting to Conflict: Equilibrium Structure and Adaptive Learning in Harmonic Games

Zero-sum games are the textbook model of strategic competition—yet they can fail to capture conflict even in simple settings. *Harmonic games* provide a more robust framework for opposed interests: they are invariant under strategic equivalence, and naturally extend beyond pairwise competition. This generality comes at a structural cost. We show that, in odd-dimensional harmonic games, the set of totally mixed Nash equilibria is a generically non-convex real algebraic variety of dimension at least one, extending to the boundary of the strategy space. As a result, learning becomes delicate: although finely tuned methods are known, whether adaptive, parameter-agnostic learning is possible has remained open. We resolve this question in the affirmative. We introduce a flexible extrapolation-based variant of follow-the-regularized-leader (FTRL+) that converges to Nash equilibrium while achieving order-optimal regret: $\mathcal{O}(1)$ in self-play and $\mathcal{O}(\sqrt{T})$ against arbitrary opponents. The method is fully adaptive—each player updates from locally observable information alone, without access to global problem parameters or shared signals—yielding the first parameter-agnostic learning dynamics provably convergent in this class.