Dynamics of a Stratified Population of Optimum Seeking Agents on a Network -- Part II: Steady State Analysis

TL;DR

This paper analyzes the steady state behavior and social utility of a stratified population of agents seeking optimal strategies on a network, providing bounds and conditions for Nash equilibrium and social utility.

Contribution

It introduces new sufficient conditions for the existence of a unique Nash equilibrium and bounds on social utility in networked populations under three dynamics.

Findings

Existence of a unique Nash equilibrium under certain network conditions.

Upper bounds on steady state social utility derived from flow graph reduction.

Lower bounds on social utility based on graph partitioning and sufficient conditions.

Abstract

In this second part of our work, we study the steady state of the population and the social utility for the three dynamics SSD, NBRD and NRPM; which were introduced in the first part. We provide sufficient conditions on the network based on a maximum payoff density parameter of each node under which there exists a unique Nash equilibrium. We then utilize positive correlation properties of the dynamics to reduce the flow graph in order to provide an upper bound on the steady state social utility. Finally we extend the idea behind the sufficient condition for the existence of a unique Nash equilibrium to partition the graph appropriately in order to provide a lower bound on the steady state social utility. We also illustrate interesting cases as well as our results using simulations.