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

Yuqing Hou, Yiyin Cao+1 more|arXiv (Cornell University)|Feb 11, 2026

Game Theory and Applications0 citations

TL;DR

The paper develops a compact sequence-form framework for normal-form proper equilibria in extensive-form games, introduces a perturbed ε-permutahedron representation, and presents two differentiable path-following methods to compute these equilibria with convergence guarantees.

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.

Motivation & Objective

• Motivate the study of normal-form proper equilibria in finite extensive-form games with perfect recall.
• Provide a compact sequence-form characterization that is equivalent to normal-form proper equilibria via strategic equivalence.
• Introduce a perturbed sequence-form formulation using an ε-permutahedron to characterize proper equilibria.
• Develop two differentiable path-following algorithms to compute equilibria by leveraging logarithmic/entropy regularization.
• Demonstrate effectiveness and efficiency through experiments.

Proposed method

• Redefine expected payoffs over sequences to obtain a compact sequence-form proper equilibrium and prove its equivalence to the normal-form proper equilibrium.
• Introduce an ε-permutahedron over sequences to formulate a perturbed game capturing proper-equilibrium structure.
• Construct artificial sequence-form games with payoff functions augmented by logarithmic or entropy regularization via an auxiliary variable.
• Derive two differentiable path-following methods that generate a smooth equilibrium path from any positive realization plan to a normal-form proper equilibrium as the auxiliary variable → 0.
• Establish existence of a smooth path for each artificial game and convergence to the original game’s normal-form proper equilibrium.

Experimental results

Research questions

• RQ1 Can a compact sequence-form representation characterize normal-form proper equilibria in extensive-form games?
• RQ2 Does the sequence-form proper equilibrium coincide with the normal-form proper equilibrium via strategic equivalence?
• RQ3 Can perturbed sequence-form games (ε-permutahedron) capture the landscape of proper equilibria?
• RQ4 Do differentiable path-following methods exist to compute normal-form proper equilibria in this framework, with convergence guarantees?
• RQ5 Do the proposed methods converge to a normal-form proper equilibrium as the auxiliary variable tends to zero, and are they efficient in practice?

Key findings

• A compact sequence-form proper equilibrium is developed and proven to be strategically equivalent to the normal-form proper equilibrium.
• A new ε-permutahedron based perturbed-game formulation for normal-form proper equilibria is proposed for the first time in this context.
• Two differentiable path-following methods are introduced using logarithmic/entropy regularization to compute equilibria.
• Each method yields a smooth equilibrium path starting from a positive realization plan and converges to a normal-form proper equilibrium as the auxiliary variable approaches zero.
• Experimental results validate the effectiveness and efficiency of the proposed approaches.