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

In this second part of our work, we study the steady state of the population and the social utility for a general class of dynamics that converge to the set of Nash equilibria and follow a certain positive correlation property. This class of dynamics includes the three dynamics 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 the 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.