# Passivity Analysis of Replicator Dynamics and its Variations

## TL;DR

This paper applies passivity theory from control systems to analyze the stability of various replicator dynamics in evolutionary game theory, identifying conditions for convergence and stability.

## Contribution

It introduces passivity-based analysis to assess stability of replicator dynamics and distinguishes between first and second order variants using passivity and negative imaginary properties.

## Key findings

- First order RD satisfies lossless passivity.
- Second order RD does not satisfy standard passivity but has negative imaginary property.
- Certain classes of evolutionary games guarantee stability with RD.

## Abstract

In this paper, we focus on studying the passivity properties of different versions of replicator dynamics (RD). RD is an important class of evolutionary dynamics in evolutionary game theory. Evolutionary dynamics describe how the population composition changes in response to the fitness levels, resulting in a closed-loop feedback system. RD is a deterministic monotone non-linear dynamic that allows incorporation of the distribution of population types through a fitness function. Here, in this paper, we use a tools for control theory, in particular, the passivity theory, to study the stability of the RD when it is in action with evolutionary games. The passivity theory allows us to identify class of evolutionary games in which stability with RD is guaranteed. We show that several variations of the first order RD satisfy the standard loseless passivity property. In contrary, the second order RD do not satisfy the standard passivity property, however, it satisfies a similar dissipativity property known as negative imaginary property. The negative imaginary property of the second order RD allows us to identify the class of games that converge to a stable equilibrium with the second order RD.