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arXiv · 2608.15258

Self-fictitious-play for Potential Monotone Ergodic Mean-field Games

Yupeng Bai·Mathieu Laurière·Zhenjie Ren·Songbo Wang
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Abstract

We investigate long-time learning in ergodic, potential, monotone mean-field games (MFGs) via a self-fictitious-play (SFP) dynamics coupling an optimally controlled diffusion with a slowly evolving belief. At each time, the state follows the optimal feedback associated with the current belief, while the belief is updated using the player's own empirical occupation measure rather than the population distribution. For ergodic monotone potential MFGs on the torus, we prove that the SFP dynamics is contractive and admits a unique invariant law. Moreover, we show that this invariant law is quantitatively close to the MFG Nash equilibrium, with an error of order equal to the square root of the belief-update rate. The proof combines uniform-in-time regularity estimates for the ergodic Hamilton-Jacobi-Bellman equation with an energy argument based on the Lasry-Lions divergence. The linear-quadratic example shows that this rate is sharp, and the numerical experiments illustrate the predicted scaling.

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Yupeng Bai, Mathieu Laurière, Zhenjie Ren, Songbo Wang. 2026-08-15. Self-fictitious-play for Potential Monotone Ergodic Mean-field Games. https://arxiv.org/abs/2608.15258

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