# Dynamics of a Population of Optimum Seeking Agents on a Network

### Nirabhra Mandal

**Introduction**

In this project, we consider a population composed of a continuum of agents that seek to maximize a payoff function by moving on a network. The nodes in the network may represent physical locations or they may be choices in a more abstract sense. The proposed framework finds applications in a variety of problems where the evolution of a population’s choices and nudging the behavior at a population level, over a period of time, to more pro-social choices is of interest. Examples include fleet distribution of ride sharing services like Ola and Uber; transportation mode choices of a population; opinion dynamics; human and insect swarm migrations etc. Problem Description Let V be a set of nodes, E ⊂ V × V be a set of edges and G := (V, E) be an undirected graph that does not contain any self loop. In the model, each node i ∈ V represents the aforementioned choice available to the infinitesimal agents constituting the population. We consider xi ∈ [0, 1] to be the fraction of the population present in node i, or equivalently making the choice i (the total population is normalized to 1). The vector x represents the population configuration. The payoff that each agent in node i receives is given by the function pi(xi) (which we assume to be twice continuously differentiable and strictly concave). We study the dynamics of the optimum seeking agents with different levels of coordination. To model agents of selfish nature we consider a well known dynamics in the area of population games and evolutionary dynamics described as follows.

**Smith Dynamics**

Here each agent selfishly revises its choice at independent and random time instants. An agent in node i takes a decision to revise its choice by comparing its payoff with the payoff of a neighboring agent. We propose two dynamics which are described as follows. ** Nodal Best Response Dynamics (NBRD)**

Here we assume that the agents have coordination at the nodal level and each agent revises its choice, among its neighbor set, at independent and random time instants. Each agent in node i decides to revise or retain its choice by computing the best response of the fraction xi (to maximize the payoff that xi receives) to the current population configuration x. **Network Restricted Payoff Maximization (NRPM)**

Here we assume that the agents have coordination across the whole population and evolve under a centralized decision scheme. They do so in order to maximize the overall social utility of the entire population (which is defined as the sum of the payoffs received by every agent in the population). For both cases we analyse properties such as existence and uniqueness of solutions and convergence. We also compare the converging population configuration of the two dynamics. We plan to compare NBRD and NRPM with Smith Dynamics in the future. **Results Obtained**

We have modeled the evolution of the population configuration under NBRD and NRPM as continuous-time systems x˙ = fNBRD(x) and x˙ = fNRPM(x), respectively. The dynamics in both cases arises by solving optimization problems and introduces the possibility of state dependent switches. For both NBRD and NRPM, we have shown existence and uniqueness of solutions ∀t ≥ 0. We have characterized the equilibrium set X and shown that for all feasible initial conditions, both dynamics asymptotically converge to some point in X . If ∀i ∈ V, pi(xi) = − 1 2 xi 2 − aixi , we have established similarities of the model with a system of water in a network of connected tanks. Here xi represents the volume of water in tank i and ai represents the height of the base of tank i. The evolution of the population is similar to water trying to minimize its potential by moving across the network of tanks given by G. We have derived sufficient conditions on G such that NBRD and NRPM converge to the same population configuration. In cases where the two dynamics may not converge to the same point, we have provided bounds on the difference between the converging configurations.

**List of Publications**

N. Mandal and P. Tallapragada, “Evolution of a Population of Selfish Agents on a Network”, Accepted for 21st IFAC World Congress, 2020. N. Mandal and P. Tallapragada, “Dynamics of a Population of Optimum Seeking Agents on a Network”, (journal) to be submitted.