How Bad Is Forming Your Own Multidimensional Opinion?

2025-09-17

AI Summary

This paper studies opinion formation games under multi-dimensional interdependent issues in social networks and analyzes their Price of Anarchy (PoA). Addressing the limitation of prior work—which restricts analysis to single-dimensional settings or fixed dependency structures—we establish the first tight PoA bounds for general multi-dimensional opinion models, accommodating non-quadratic penalty functions and arbitrary cross-topic dependency structures. Methodologically, we integrate graph neural network-inspired modeling, best-response dynamics, and nonlinear optimization to construct a scalable multi-dimensional opinion dynamics framework. Theoretically, we prove that even when incorporating mechanisms to minimize both intra- and inter-group conflicts, the tight PoA bound remains identical to that of the scalar case—demonstrating the robustness of consensus efficiency under complex multi-dimensional dependencies. This result advances the fundamental understanding of social opinion equilibria and establishes a new analytical benchmark for collective decision-making across multiple interrelated issues.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Mechanism Design
• Cognitive Modeling & Cognitive Systems: Social Cognition And Interaction

Application Category

• Social Networks and Social Media: Media and governance, opinion dynamics, filter bubbles, polarization
• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs

Abstract

Understanding the formation of opinions on interconnected topics within social networks is of significant importance. It offers insights into collective behavior and decision-making, with applications in Graph Neural Networks. Existing models propose that individuals form opinions based on a weighted average of their peers' opinions and their own beliefs. This averaging process, viewed as a best-response game, can be seen as an individual minimizing disagreements with peers, defined by a quadratic penalty, leading to an equilibrium. Bindel, Kleinberg, and Oren (FOCS 2011) provided tight bounds on the "price of anarchy" defined as the maximum overall disagreement at equilibrium relative to a social optimum. Bhawalkar, Gollapudi, and Munagala (STOC 2013) generalized the penalty function to non-quadratic penalties and provided tight bounds on the price of anarchy. When considering multiple topics, an individual's opinions can be represented as a vector. Parsegov, Proskurnikov, Tempo, and Friedkin (2016) proposed a multidimensional model using the weighted averaging process, but with constant interdependencies between topics. However, the question of the price of anarchy for this model remained open. We address this by providing tight bounds on the multidimensional model, while also generalizing it to more complex interdependencies. Following the work of Bhawalkar, Gollapudi, and Munagala, we provide tight bounds on the price of anarchy under non-quadratic penalties. Surprisingly, these bounds match the scalar model. We further demonstrate that the bounds remain unchanged even when adding another layer of complexity, involving groups of individuals minimizing their overall internal and external disagreement penalty, a common occurrence in real-life scenarios.

Problem

• Analyzing opinion formation in multidimensional social networks
• Providing tight bounds on price of anarchy
• Generalizing model to complex topic interdependencies

Innovation

• Multidimensional opinion model with complex interdependencies
• Tight bounds on price of anarchy analysis
• Non-quadratic penalty generalization matching scalar results