Performative Mean Field Games

This paper investigates performative prediction in large-population games where the number of players is unknown and effectively infinite. In many real-world settings, such as traders adjusting investment strategies or merchants setting prices, any single player's decision has negligible influence on the overall environment, yet the collective decision distribution of all players significantly shapes the data distributions that each player observes and makes decisions on. Existing performative prediction frameworks, which rely on discrete, finite-player models, fail to capture these dynamics, since a change in any single player's discrete decision does not meaningfully alter the population-level decision distribution and thus induces no perceptible shift in the induced data distributions. We address this challenge by introducing the *performative mean field game* framework. In this framework, a continuum of players interacts through an unknown mapping from the population decision distribution to the resulting data distributions. We define the *performatively stable mean-field equilibrium* as the fixed point where the decision distribution is exactly the aggregate of each player's optimal response to the data distribution induced by that same decision distribution, eliminating the need for further decision updates. Our main technical contributions are threefold. First, we formalize the above game and equilibrium concept. Second, we introduce a sensitivity measure in Sobolev norms that quantifies the impact of distribution shifts on individual decision-making and use it to establish theoretical convergence conditions for performatively stable mean-field equilibria. Third, building on these theoretical results, we design an adaptive algorithm that estimates sensitivity on the fly and provably converges to a performatively stable mean-field equilibrium without requiring prior knowledge of the underlying decision-to-data mapping. Experiments on quantitative trading, recommendation system, and ride-sharing market validate our theoretical results and demonstrate the effectiveness of the proposed algorithm.