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

TL;DR

This work addresses the computation of normal-form proper equilibria in finite $n$-player extensive-form games, noting that the traditional normal-form representation grows exponentially with the number of information sets. It introduces a compact sequence-form proper equilibrium, proves its strategic equivalence to the normal-form proper equilibrium, and develops a perturbed-sequence-form (an $oldsymbol{epsilon}$-permutahedron) representation to enable computation. Building on this foundation, the authors propose two differentiable path-following methods—one with a logarithmic barrier and another with an entropy barrier—to trace smooth equilibrium paths from a positive realization plan to a normal-form proper equilibrium as the barrier parameter approaches zero, with theoretical guarantees of existence and convergence. Numerical experiments demonstrate the practicality and efficiency of the proposed methods, highlighting their effectiveness on representative and randomly generated extensive-form games, and indicating their potential for broader application in equilibrium refinement analysis.

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.

Figures (19)

• Figure 1: An extensive-form game
• Figure 2: An Extensive-Form Game from Selten SeltenReexaminationperfectnessconcept1975
• Figure 3: An Extensive-Form Game from von Stengel et al. vonStengelComputingNormalForm2002
• Figure 4: Path of Realization Plans Generated by LGPR for the Game in Fig. \ref{['fig:game1']}
• Figure 5: Path of Mixed Strategies Generated by LGPR for the Game in Fig. \ref{['fig:game1']}

Theorems & Definitions (30)

• Definition 1
• Example 1
• Definition 2
• Proposition 1
• proof
• Lemma 1
• proof
• Theorem 1
• proof
• Definition 3