Menus: A Framework for Learning Against Strategic Opponents

Algorithms increasingly mediate repeated strategic interactions in marketplaces, from automated pricing to auction bidding. When one party commits to a learning algorithm, the other party can respond strategically over time by steering the algorithm's internal state toward a favorable long-run outcome. This note surveys a line of work that studies this "learning-as-commitment" perspective via a geometric object we call a menu : the convex set of long-run outcomes an opponent can induce against a fixed learning rule. Menus provide a common language for (i) comparing learning algorithms against strategic opponents, (ii) optimizing over learning rules under uncertainty about opponent objectives, and (iii) characterizing when an opponent can manipulate learning dynamics beyond what they could achieve with a static strategy. Using this machinery, we converge upon no-swap-regret algorithms as an "optimal" commitment strategy for robust learning against a strategic opponent. We also identify principled generalizations of no-swap-regret beyond normal-form games that preserve the same strategic guarantees while remaining computationally tractable.