Consensus as a Nash Equilibrium of a Dynamic Game

TL;DR

This paper models consensus formation in social networks as a dynamic game where individuals minimize a cost function balancing opinion differences and stubbornness, deriving explicit Nash equilibrium solutions and analyzing their behavior under various influence and stubbornness parameters.

Contribution

It introduces a novel dynamic game framework for consensus modeling and derives explicit Nash equilibrium solutions for different network topologies.

Findings

Explicit formulas for opinion trajectories in two network cases

Analysis of opinion dynamics under varying influence and stubbornness

Identification of conditions leading to consensus or persistent disagreement

Abstract

Consensus formation in a social network is modeled by a dynamic game of a prescribed duration played by members of the network. Each member independently minimizes a cost function that represents his/her motive. An integral cost function penalizes a member's differences of opinion from the others as well as from his/her own initial opinion, weighted by influence and stubbornness parameters. Each member uses its rate of change of opinion as a control input. This defines a dynamic non-cooperative game that turns out to have a unique Nash equilibrium. Analytic explicit expressions are derived for the opinion trajectory of each member for two representative cases obtained by suitable assumptions on the graph topology of the network. These trajectories are then examined under different assumptions on the relative sizes of the influence and stubbornness parameters that appear in the cost functions.