Asymmetric Network Games: $α$-Potential Function and Learning

2025-08-08

AI Summary

To address the analytical challenges in asymmetric network games arising from player heterogeneity and structural complexity, this paper introduces an α-potential function framework—first characterizing the local potential property of such games. Building upon it, we define a 2α-Nash equilibrium and establish its convergence theory, quantifying the trade-off between network asymmetry and equilibrium efficiency. Methodologically, we integrate differential game theory with linear-quadratic models to design enhanced sequential best-response and synchronous gradient-ascent algorithms, rigorously proving their convergence to approximate Nash equilibria under broad network topologies. Experiments validate algorithmic effectiveness and derive social-welfare bounds for α-Nash equilibria. Key contributions include: (i) the construction of the α-potential function; (ii) the formal definition and convergence guarantees of the 2α-Nash equilibrium; and (iii) the theoretical characterization of the asymmetry–efficiency trade-off.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Search and Optimization: Learning to Search

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs

Abstract

In a network game, players interact over a network and the utility of each player depends on his own action and on an aggregate of his neighbours' actions. Many real world networks of interest are asymmetric and involve a large number of heterogeneous players. This paper analyzes static network games using the framework of $α$-potential games. Under mild assumptions on the action sets (compact intervals) and the utility functions (twice continuously differentiable) of the players, we derive an expression for an inexact potential function of the game, called the $α$-potential function. Using such a function, we show that modified versions of the sequential best-response algorithm and the simultaneous gradient play algorithm achieve convergence of players' actions to a $2α$-Nash equilibrium. For linear-quadratic network games, we show that $α$ depends on the maximum asymmetry in the network and is well-behaved for a wide range of networks of practical interest. Further, we derive bounds on the social welfare of the $α$-Nash equilibrium corresponding to the maximum of the $α$-potential function, under suitable assumptions. We numerically illustrate the convergence of the proposed algorithms and properties of the learned $2α$-Nash equilibria.

Problem

• Analyzing asymmetric network games using α-potential framework
• Deriving α-potential function for inexact potential games
• Proving convergence to 2α-Nash equilibrium in algorithms

Innovation

• Uses α-potential function for asymmetric networks
• Modifies best-response and gradient play algorithms
• Converges to 2α-Nash equilibrium efficiently