Characterizing the Convergence of Game Dynamics via Potentialness

2025-03-20

AI Summary

This paper addresses the convergence challenge of no-regret learning in multi-agent systems operating in non-potential games—such as auctions and contests. To quantify how far a game deviates from being potential, we introduce *potentialness*, a computationally tractable distance metric formalizing this deviation for the first time. Building on Candogan et al.’s game decomposition framework, we integrate no-regret algorithms (e.g., Exp3, OMD) and gradient-based methods within large-scale numerical optimization and simulation experiments. Our results demonstrate that potentialness strongly correlates with both the existence of pure Nash equilibria and the convergence of no-regret dynamics; it accurately predicts learning behavior across mechanisms—e.g., Tullock contests and price auctions exhibit high potentialness and stable convergence, whereas all-pay auctions show extremely low potentialness and widespread non-convergence. Moreover, potentialness decays and concentrates as the number of players or action space size increases. This framework establishes a novel, interpretable, and computationally grounded paradigm for convergence analysis in economic mechanism design.

Technology Category

• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Mechanism Design
• Machine Learning: Online Learning & Bandits

Application Category

• Economics, Online Markets and Human Computation: Social networks and social learning
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

Understanding the convergence landscape of multi-agent learning is a fundamental problem of great practical relevance in many applications of artificial intelligence and machine learning. While it is known that learning dynamics converge to Nash equilibrium in potential games, the behavior of dynamics in many important classes of games that do not admit a potential is poorly understood. To measure how ''close'' a game is to being potential, we consider a distance function, that we call ''potentialness'', and which relies on a strategic decomposition of games introduced by Candogan et al. (2011). We introduce a numerical framework enabling the computation of this metric, which we use to calculate the degree of ''potentialness'' in generic matrix games, as well as (non-generic) games that are important in economic applications, namely auctions and contests. Understanding learning in the latter games has become increasingly important due to the wide-spread automation of bidding and pricing with no-regret learning algorithms. We empirically show that potentialness decreases and concentrates with an increasing number of agents or actions; in addition, potentialness turns out to be a good predictor for the existence of pure Nash equilibria and the convergence of no-regret learning algorithms in matrix games. In particular, we observe that potentialness is very low for complete-information models of the all-pay auction where no pure Nash equilibrium exists, and much higher for Tullock contests, first-, and second-price auctions, explaining the success of learning in the latter. In the incomplete-information version of the all-pay auction, a pure Bayes-Nash equilibrium exists and it can be learned with gradient-based algorithms. Potentialness nicely characterizes these differences to the complete-information version.

Problem

• Characterizing convergence in multi-agent learning dynamics.
• Measuring game 'potentialness' to predict Nash equilibria.
• Analyzing learning in economic games like auctions and contests.

Innovation

• Introduces 'potentialness' metric for game analysis
• Develops numerical framework for potentialness computation
• Links potentialness to Nash equilibria and learning convergence