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Opinion Dynamics for Utility Maximizing Agents: Exploring the Impact of the Resource Penalty

Prashil Wankhede, Nirabhra Mandal, Sonia Martínez, Pavankumar Tallapragada

TL;DR

It is shown that for any arbitrary social influence network, opinions are ultimately bounded and for networks with weak antagonistic relations, there exists a globally exponentially stable equilibrium using contraction theory.

Abstract

We propose a continuous-time nonlinear model of opinion dynamics with utility-maximizing agents connected via a social influence network. A distinguishing feature of the proposed model is the inclusion of an opinion-dependent resource-penalty term in the utilities, which limits the agents from holding opinions of large magnitude. This model is applicable in scenarios where the opinions pertain to the usage of resources, such as money, time, computational resources etc. Each agent myopically seeks to maximize its utility by revising its opinion in the gradient ascent direction of its utility function, thus leading to the proposed opinion dynamics. We show that, for any arbitrary social influence network, opinions are ultimately bounded. For networks with weak antagonistic relations, we show that there exists a globally exponentially stable equilibrium using contraction theory. We establish conditions for the existence of consensus equilibrium and analyze the relative dominance of the agents at consensus. We also conduct a game-theoretic analysis of the underlying opinion formation game, including on Nash equilibria and on prices of anarchy in terms of satisfaction ratios. Additionally, we also investigate the oscillatory behavior of opinions in a two-agent scenario. Finally, simulations illustrate our findings.

Opinion Dynamics for Utility Maximizing Agents: Exploring the Impact of the Resource Penalty

TL;DR

It is shown that for any arbitrary social influence network, opinions are ultimately bounded and for networks with weak antagonistic relations, there exists a globally exponentially stable equilibrium using contraction theory.

Abstract

We propose a continuous-time nonlinear model of opinion dynamics with utility-maximizing agents connected via a social influence network. A distinguishing feature of the proposed model is the inclusion of an opinion-dependent resource-penalty term in the utilities, which limits the agents from holding opinions of large magnitude. This model is applicable in scenarios where the opinions pertain to the usage of resources, such as money, time, computational resources etc. Each agent myopically seeks to maximize its utility by revising its opinion in the gradient ascent direction of its utility function, thus leading to the proposed opinion dynamics. We show that, for any arbitrary social influence network, opinions are ultimately bounded. For networks with weak antagonistic relations, we show that there exists a globally exponentially stable equilibrium using contraction theory. We establish conditions for the existence of consensus equilibrium and analyze the relative dominance of the agents at consensus. We also conduct a game-theoretic analysis of the underlying opinion formation game, including on Nash equilibria and on prices of anarchy in terms of satisfaction ratios. Additionally, we also investigate the oscillatory behavior of opinions in a two-agent scenario. Finally, simulations illustrate our findings.
Paper Structure (18 sections, 17 theorems, 38 equations, 3 figures)

This paper contains 18 sections, 17 theorems, 38 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Modeling and Problem Setup
  4. Asymptotic Behavior of Opinions
  5. Consensus and Nash Equilibria
  6. Consensus Equilibria
  7. Nash Equilibria
  8. Price of Anarchy
  9. Oscillatory Behavior of Opinions
  10. Simulations
  11. Conclusions
  12. Proof of Theorem \ref{['thm:exis_uniq_of_eqm']}
  13. Proof of Proposition \ref{['prop:ub_c_pos']}
  14. Proof of Lemma \ref{['lem:convexcost']}
  15. Proof of Theorem \ref{['thm:poa_bound']}
  16. ...and 3 more sections

Key Result

Theorem II.3

(Equilibrium of a strongly contracting system2023_FB_CTDS_book) Suppose $\mathcal{C} \subset \mathbb{R}^{n}$ is convex, closed and positive $\mathbf{f}$-invariant. If $\mathbf{f}:\mathcal{C}\to\mathbb{R}^{n}$ is strongly infinitesimally contracting on $\mathcal{C}$ with rate $\alpha>0$ then, $\mathb

Figures (3)

  • Figure 1: A social network consisting of 6 agents. The direction of any link denotes the direction of influence and the number near arrowhead of any directed link $(k,i)$ represents the corresponding link weight $a_{ik}$.
  • Figure 2: Convergence of opinions. (a) Consensus equilibrium. (b) Disagreement equilibrium within $\mathcal{M}^{n}$.
  • Figure 3: Oscillatory behavior of opinions. (a) Opinion trajectory. (b) Intersection of corresponding trajectory with the ellipse.

Theorems & Definitions (31)

  • Definition II.1
  • Definition II.2
  • Theorem II.3
  • Definition III.1
  • Example III.2
  • Remark III.3
  • Theorem IV.1
  • Lemma IV.2
  • Theorem IV.3
  • Remark IV.4
  • ...and 21 more