In this talk, we present a unified geometric perspective for studying the convergence behavior of leader-based learning algorithms (e.g., Fictitious Play, Follow-the-Regularized-Leader, Online Mirror Descent, and their optimistic variants) in two-player zero-sum games. By analyzing geometric properties of these dynamics in the dual space of payoff vectors, we gain a more intuitive characterization of their behavior in the primal space, including more precise insights into the effects of using optimism. We use this perspective to establish a variety of new (positive and negative) convergence guarantees in time-average, last-iterate, and best-iterate. We conclude by discussing several open questions.

We develop an axiomatic theory for Automated Market Makers (AMMs) in local energy sharing markets and analyze the Markov Perfect Equilibrium of the resulting economy with a Mean-Field Game. In this game, heterogeneous prosumers solve a Bellman equation to optimize energy consumption, storage, and exchanges. Our axioms identify a class of mechanisms with linear, Lipschitz continuous payment functions, where prices decrease with the aggregate supply-to-demand ratio of energy. We prove that implementing batch execution and concentrated liquidity allows standard design conditions from decentralized finance—quasi-concavity, monotonicity, and homotheticity—to construct AMMs that satisfy our axioms. The resulting AMMs are budget-balanced and achieve ex-ante efficiency, contrasting with the strategy-proof, expost optimal VCG mechanism. Since the AMM implements a Potential Game, we solve its equilibrium by first computing the social planner’s optimum and then decentralizing the allocation. Numerical experiments using data from the Paris administrative region suggest that the prosumer community can achieve gains from trade up to 40% relative to the grid-only benchmark.