Characterization and Computation of Normal-Form Proper Equilibria in Extensive-Form Games via the Sequence-Form Representation

2026-02-11

AI Summary

Computing normal-form proper equilibrium in extensive-form games is challenging due to its representation size growing exponentially with the number of information sets. This work establishes, for the first time, the strategic equivalence between normal-form and sequence-form proper equilibria, enabling a compact sequence-form formulation. By constructing perturbed games via ε-perturbed polyhedra, the authors develop a differentiable path-following algorithm based on logarithmic or entropy regularization. They theoretically prove the existence of a smooth equilibrium path and demonstrate that the original problem can be solved efficiently through this approach. Experimental results confirm the method’s significant advantages in both solution accuracy and computational efficiency.

Technology Category

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

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• 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

Abstract

Normal-form proper equilibrium, introduced by Myerson as a refinement of normal-form perfect equilibrium, occupies a distinctive position in the equilibrium analysis of extensive-form games because its more stringent perturbation structure entails the sequential rationality. However, the size of the normal-form representation grows exponentially with the number of parallel information sets, making the direct determination of normal-form proper equilibria intractable. To address this challenge, we develop a compact sequence-form proper equilibrium by redefining the expected payoffs over sequences, and we prove that it coincides with the normal-form proper equilibrium via strategic equivalence. To facilitate computation, we further introduce an alternative representation by defining a class of perturbed games based on an $\varepsilon$-permutahedron over sequences. Building on this representation, we introduce two differentiable path-following methods for computing normal-form proper equilibria. These methods rely on artificial sequence-form games whose expected payoff functions incorporate logarithmic or entropy regularization through an auxiliary variable. We prove the existence of a smooth equilibrium path induced by each artificial game, starting from an arbitrary positive realization plan and converging to a normal-form proper equilibrium of the original game as the auxiliary variable approaches zero. Finally, our experimental results demonstrate the effectiveness and efficiency of the proposed methods.

Problem

• extensive-form games
• normal-form proper equilibrium
• sequence-form representation
• equilibrium computation
• exponential complexity

Innovation

• sequence-form representation
• proper equilibrium
• path-following method
• extensive-form games
• regularization