A Sequence-Form Characterization and Differentiable Path-Following Method for Computing Normal-Form Perfect Equilibria in Extensive-Form Games

TL;DR

This work develops a sequence-form characterization of normal-form perfect equilibria (NFPE) for finite $n$-player extensive-form games with perfect recall and introduces globally differentiable path-following methods to compute NFPEs. The approach constructs perturbed sequence-form games and, via an artificial logarithmic-barrier game, yields smooth equilibrium paths that converge to NFPE as barrier parameters vanish; it also extends Harsanyi's linear and logarithmic tracing procedures to the sequence form. The authors prove convergence of the smooth paths using fixed-point theorems and implicit-function arguments and validate the methods with extensive numerical experiments, showing the logarithmic-barrier path (LOGB) approach offers superior stability and efficiency. Overall, the paper provides a principled, scalable framework for computing refined equilibria in multi-player extensive-form games with improved computational performance.

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 formulates an artificial logarithmic-barrier game in sequence form, introducing an additional variable to regulate the impact of logarithmic-barrier terms on 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 computational efficiency of our methods.

Figures (18)

• Figure 1: An Extensive-Form Game from van Damme vandammeRelationPerfectEquilibria1984
• Figure 2: An Extensive-Form Game
• Figure 3: Path of Mixed Strategies Generated by LOGB for the Game in Fig. \ref{['nfpe-fig-gam1']}
• Figure 4: Path of Mixed Strategies Generated by HLTP for the Game in Fig. \ref{['nfpe-fig-gam1']}
• Figure 5: Path of Mixed Strategies Generated by HLOG for the Game in Fig. \ref{['nfpe-fig-gam1']}

Theorems & Definitions (29)

• Definition 1
• Lemma 1
• proof
• Lemma 2
• proof
• Definition 2
• Lemma 3
• proof
• Lemma 4
• proof