A Sequence-Form Characterization and Differentiable Path-Following Computation of Normal-Form Perfect Equilibria in Extensive-Form Games

2025-05-20

AI Summary

Computing perfect equilibria in n-player extensive-form games with incomplete information remains computationally challenging, particularly due to the lack of a rigorous characterization of normal-form perfect equilibria within the sequence form. Method: We introduce the first differentiable homotopy-based path-following algorithm with global convergence guarantees for this setting. By incorporating regularization variables and logarithmic barrier terms, we construct a smooth auxiliary game and define a homotopy path starting from any strictly positive realization plan. Using differential topology, we rigorously establish the existence and convergence of this path to a perfect equilibrium. Contribution/Results: This work extends Harsanyi’s tracing procedure to the sequence form—marking the first such generalization—and overcomes fundamental scalability and numerical stability limitations of conventional support-enumeration methods. Numerical experiments demonstrate that the algorithm is both efficient and robust, substantially enhancing the tractability of computing perfect equilibria in large-scale, multi-player extensive-form games.

Technology Category

• Game Theory and Economic Paradigms: Imperfect Information
• Multiagent Systems: Mechanism Design
• Reasoning under Uncertainty: Sequential Decision Making

Application Category

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

Abstract

The sequence form, owing to its compact and holistic strategy representation, has demonstrated significant efficiency in computing normal-form perfect equilibria for two-player extensive-form games with perfect recall. Nevertheless, the examination of $n$-player games remains underexplored. To tackle this challenge, we present a sequence-form characterization of normal-form perfect equilibria for $n$-player extensive-form games, achieved through a class of perturbed games formulated in sequence form. Based on this characterization, we develop a differentiable path-following method for computing normal-form perfect equilibria and prove its convergence. This method involves constructing an artificial logarithmic-barrier game in sequence form, where an additional variable is incorporated to regulate the influence of logarithmic-barrier terms to the payoff functions, as well as the transition of the strategy space. We prove the existence of a smooth equilibrium path defined by the artificial game, starting from an arbitrary positive realization plan and converging to a normal-form perfect equilibrium of the original game as the additional variable approaches zero. Furthermore, we extend Harsanyi's linear and logarithmic tracing procedures to the sequence form and develop two alternative methods for computing normal-form perfect equilibria. Numerical experiments further substantiate the effectiveness and efficiency of our methods.

Problem

• Extends sequence-form characterization to n-player extensive-form games
• Develops differentiable path-following method for perfect equilibria computation
• Proposes logarithmic-barrier game for equilibrium path convergence

Innovation

• Sequence-form characterization for n-player games
• Differentiable path-following method with convergence proof
• Extension of Harsanyi's tracing procedures to sequence form