Learning in Large Populations With Anticipatory Payoffs: Bregman Meets Willems and Lyapunov
Abstract
We study evolutionary Nash equilibrium learning in large populations whose payoffs are generated by dynamic mechanisms. Counterclockwise dissipativity (CCW) provides a passivity-like framework for establishing learning/convergence under a broad convex cone of so-called well-behaved learning rules satisfying a natural payoff-alignment condition according to which, away from equilibria, the population's state (strategic profile) moves in a direction positively correlated with the payoffs. The cone is closed under conic combinations and encompasses a broad range of noise-free rules, including discontinuous rules such as best response, imitative rules such as the replicator rule that is not δ-passive, and all noise-free δ-passive rules. Existing CCW results require no knowledge of the particular rule, but establish convergence without providing a Lyapunov function or Lyapunov stability. We develop a methodology that closes this gap by constructing rule-independent Lyapunov functions from CCW storage functions of the payoff mechanism. For a nonlinear class of anticipatory mechanisms, in which agents compare current payoffs with those associated with a smoothed population state, we show that a storage can be derived as a Bregman divergence. The resulting Lyapunov function combines a potential gap with this divergence and establishes global asymptotic stability. A congestion game with a nonlinear congestion-growth toll illustrates the results.
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BibTeX
@misc{jeng2026learning,
title = {{Learning in Large Populations With Anticipatory Payoffs: Bregman Meets Willems and Lyapunov}},
author = {Tzuyu Jeng and Nuno C. Martins},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.04086},
eprint = {2610.04086},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.04086},
}
RIS
TY - UNPB
TI - Learning in Large Populations With Anticipatory Payoffs: Bregman Meets Willems and Lyapunov
AU - Tzuyu Jeng
AU - Nuno C. Martins
PY - 2026
DA - 2026/10/02
PB - arXiv
DO - 10.48550/arxiv.2610.04086
UR - https://arxiv.org/abs/2610.04086
AB - We study evolutionary Nash equilibrium learning in large populations whose payoffs are generated by dynamic mechanisms. Counterclockwise dissipativity (CCW) provides a passivity-like framework for establishing learning/convergence under a broad convex cone of so-called well-behaved learning rules satisfying a natural payoff-alignment condition according to which, away from equilibria, the population's state (strategic profile) moves in a direction positively correlated with the payoffs. The cone is closed under conic combinations and encompasses a broad range of noise-free rules, including discontinuous rules such as best response, imitative rules such as the replicator rule that is not δ-passive, and all noise-free δ-passive rules. Existing CCW results require no knowledge of the particular rule, but establish convergence without providing a Lyapunov function or Lyapunov stability. We develop a methodology that closes this gap by constructing rule-independent Lyapunov functions from CCW storage functions of the payoff mechanism. For a nonlinear class of anticipatory mechanisms, in which agents compare current payoffs with those associated with a smoothed population state, we show that a storage can be derived as a Bregman divergence. The resulting Lyapunov function combines a potential gap with this divergence and establishes global asymptotic stability. A congestion game with a nonlinear congestion-growth toll illustrates the results.
ER -