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Differential Game Strategies for Social Networks with Self-Interested Individuals

Hossein B. Jond

TL;DR

This work develops a differential-game framework for continuous-time HK opinion dynamics with time-delayed input on graphs, deriving explicit open-loop Nash equilibria for both non-stubborn and stubborn agents. It shows how local neighbor information suffices to compute equilibrium actions and extends to receding-horizon (MPC) implementations to realize feedback-like strategies under fixed, complete, and second-neighborhood graph structures. Simulations on Zachary's Karate Club demonstrate how the interplay of delay, stubbornness, and graph topology yields consensus, polarization, or disagreement. The results provide a rigorous, distributed mechanism for strategic opinion formation with potential applications in networked social systems and control of collective behavior.

Abstract

A social network population engages in collective actions as a direct result of forming a particular opinion. The strategic interactions among the individuals acting independently and selfishly naturally portray a noncooperative game. Nash equilibrium allows for self-enforcing strategic interactions between selfish and self-interested individuals. This paper presents a differential game approach to the opinion formation problem in social networks to investigate the evolution of opinions as a result of a Nash equilibrium. The opinion of each individual is described by a differential equation, which is the continuous-time Hegselmann-Krause model for opinion dynamics with a time delay in input. The objective of each individual is to seek optimal strategies for her own opinion evolution by minimizing an individual cost function. Two differential game problems emerge, one for a population that is not stubborn and another for a population that is stubborn. The open-loop Nash equilibrium actions and their associated opinion trajectories are derived for both differential games using Pontryagin's principle. Additionally, the receding horizon control scheme is used to practice feedback strategies where the information flow is restricted by fixed and complete social graphs as well as the second neighborhood concept. The game strategies were executed on the well-known Zachary's Karate Club social network. The resulting opinion trajectories associated with the game strategies showed consensus, polarization, and disagreement in final opinions.

Differential Game Strategies for Social Networks with Self-Interested Individuals

TL;DR

This work develops a differential-game framework for continuous-time HK opinion dynamics with time-delayed input on graphs, deriving explicit open-loop Nash equilibria for both non-stubborn and stubborn agents. It shows how local neighbor information suffices to compute equilibrium actions and extends to receding-horizon (MPC) implementations to realize feedback-like strategies under fixed, complete, and second-neighborhood graph structures. Simulations on Zachary's Karate Club demonstrate how the interplay of delay, stubbornness, and graph topology yields consensus, polarization, or disagreement. The results provide a rigorous, distributed mechanism for strategic opinion formation with potential applications in networked social systems and control of collective behavior.

Abstract

A social network population engages in collective actions as a direct result of forming a particular opinion. The strategic interactions among the individuals acting independently and selfishly naturally portray a noncooperative game. Nash equilibrium allows for self-enforcing strategic interactions between selfish and self-interested individuals. This paper presents a differential game approach to the opinion formation problem in social networks to investigate the evolution of opinions as a result of a Nash equilibrium. The opinion of each individual is described by a differential equation, which is the continuous-time Hegselmann-Krause model for opinion dynamics with a time delay in input. The objective of each individual is to seek optimal strategies for her own opinion evolution by minimizing an individual cost function. Two differential game problems emerge, one for a population that is not stubborn and another for a population that is stubborn. The open-loop Nash equilibrium actions and their associated opinion trajectories are derived for both differential games using Pontryagin's principle. Additionally, the receding horizon control scheme is used to practice feedback strategies where the information flow is restricted by fixed and complete social graphs as well as the second neighborhood concept. The game strategies were executed on the well-known Zachary's Karate Club social network. The resulting opinion trajectories associated with the game strategies showed consensus, polarization, and disagreement in final opinions.
Paper Structure (11 sections, 2 theorems, 49 equations, 6 figures, 3 algorithms)

This paper contains 11 sections, 2 theorems, 49 equations, 6 figures, 3 algorithms.

Table of Contents

  1. Introduction
  2. Differential Game Formulation of Opinion Dynamics
  3. Open-loop Nash Equilibrium
  4. Non-stubborn social network
  5. Stubborn social network
  6. Receding Horizon Implementation of Game Strategies
  7. Fixed Social Graph
  8. Complete Social Graph
  9. Second Neighborhood in Social Graph
  10. Simulations
  11. Conclusions

Key Result

Theorem 1

The unique Nash equilibrium actions and the associated opinion trajectory with these equilibrium actions for opinion formation in a non-stubborn social network as the noncooperative differential game in (eq:nonstub-opt) are given by where

Figures (6)

  • Figure 1: Zachary's Karate Club network.
  • Figure 2: Opinion trajectories associated with the HK model for Zachary's network prior to optimization.
  • Figure 3: Opinion trajectories associated with the game strategies for the stubborn and totally stubborn Zachary's network.
  • Figure 4: Opinion trajectories associated with the game strategies for non-stubborn Zachary's network on a fixed graph.
  • Figure 5: Opinion trajectories associated with the game strategies for non-stubborn Zachary's network on a complete graph.
  • ...and 1 more figures

Theorems & Definitions (7)

  • Theorem 1
  • proof
  • Remark 1
  • Theorem 2
  • proof
  • Remark 2
  • Remark 3