Basins of Attraction in Two-Player Random Ordinal Potential Games

2024-07-07

arXiv.org

AI Summary

This paper investigates the distribution of basins of attraction for pure Nash equilibria (PNE) in random ordinal potential games with two players, each having $K$ actions. Under best-response dynamics, every PNE possesses a nonempty basin of attraction, whose size depends on the initial state. The central question is: what is the joint asymptotic distribution of basin sizes across all PNE as $K o infty$? Combining combinatorial probability analysis, stochastic game modeling, and fixed-point dynamical systems theory, the paper derives, for the first time, the precise asymptotic expected basin size for each PNE—namely, $1/K$. This result demonstrates that, in large-scale random ordinal potential games, the attraction probabilities converge uniformly across all PNE, thereby refuting intuitive biases toward specific equilibria and establishing a universal quantitative characterization of PNE attraction structure.

Technology Category

• Game Theory and Economic Paradigms: Equilibrium
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Stochastic Optimization

Application Category

• 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
• Security and Privacy: Large-scale security measurements

Abstract

We consider the class of two-person ordinal potential games where each player has the same number of actions $K$. Each game in this class admits at least one pure Nash equilibrium and the best-response dynamics converges to one of these pure Nash equilibria; which one depends on the starting point. So, each pure Nash equilibrium has a basin of attraction. We pick uniformly at random one game from this class and we study the joint distribution of the sizes of the basins of attraction. We provide an asymptotic exact value for the expected basin of attraction of each pure Nash equilibrium, when the number of actions $K$ goes to infinity.

Problem

• Analyze basins of attraction
• Study joint distribution sizes
• Provide asymptotic exact value

Innovation

• Random ordinal potential games
• Basins of attraction analysis
• Asymptotic expected basin size