Passivity Tools for Hybrid Learning Rules in Large Populations

TL;DR

This paper extends passivity-based analysis to hybrid and discontinuous learning rules in large populations, enabling stability assessment of more complex strategic behaviors beyond canonical models.

Contribution

It introduces a generalized $ ext{delta}$-passivity framework for hybrid and discontinuous learning rules, broadening the scope of stability analysis in large population dynamics.

Findings

Established $ ext{delta}$-passivity for broad classes of hybrid learning rules.

Proved that merging certain convex cones preserves $ ext{delta}$-passivity.

Validated theoretical results with numerical examples.

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 $\delta$-passivity. In this paper, we consider that agents exhibit learning behaviors that do not align with a canonical class. Specifically, we focus on characterizing $\delta$-passivity for hybrid learning rules that combine elements from canonical classes. Our analysis also introduces and uses a more general version of $\delta$-passivity, which, for the first time, can handle discontinuous learning rules, including those showing best-response behaviors. We state and prove theorems establishing $\delta$-passivity for two broad convex cones of hybrid learning rules. These cones can merge into a larger one preserving $\delta$-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.