Passivity Tools for Hybrid Learning Rules in Large Populations
Jair Certorio, Kevin Chang, Nuno C. Martins, Pierluigi Nuzzo, Yasser Shoukry
TL;DR
This work advances the stability analysis of large-population learning dynamics by extending system-theoretic δ-passivity to hybrid, potentially discontinuous learning rules. It introduces a generalized δ-passivity framework using Filippov solutions to accommodate best-response components and defines two large convex cones of hybrid rules (BR with IPC, and BR with SEPT/ND) that preserve δ-passivity. For two-strategy cases, the results extend to a larger cone, showing that a broad class of hybrids remains δ-passive, which enables Lyapunov-based design and global convergence guarantees in contractive payoff environments. Theoretical results are complemented by simulations in a weighted congestion game, illustrating convergence to Nash equilibria under various hybrid and canonical learning rules. Overall, the paper provides a principled approach to designing and analyzing dynamic payoff mechanisms that ensure stable Nash-equilibrium seeking in complex, heterogeneous agent populations, even when agents follow discontinuous or mixed learning rules.
Abstract
Recent work has pioneered the use of system-theoretic passivity to study equilibrium stability for the dynamics of noncooperative strategic interactions in large populations of learning agents. In this and related works, the stability analysis leverages knowledge that certain ``canonical'' classes of learning rules used to model the agents' strategic behaviors satisfy a passivity condition known as $δ$-passivity. In this paper, we consider that agents exhibit learning behaviors that do not align with a canonical class. Specifically, we focus on characterizing $δ$-passivity for hybrid learning rules that combine elements from canonical classes. Our analysis also introduces and uses a more general version of $δ$-passivity, which, for the first time, can handle discontinuous learning rules, including those showing best-response behaviors. We state and prove theorems establishing $δ$-passivity for two broad convex cones of hybrid learning rules. These cones can merge into a larger one preserving $δ$-passivity in scenarios limited to two strategies. In our proofs, we establish intermediate facts that are significant on their own and could potentially be used to further generalize our work. We illustrate the applicability of our results through numerical examples.