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

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.

💡 Research Summary

The paper tackles the long‑standing computational bottleneck of finding normal‑form proper equilibria (NFPE) in extensive‑form games. While NFPE, introduced by Myerson as a refinement of normal‑form perfect equilibrium, guarantees sequential rationality through a stricter perturbation structure, its direct computation is infeasible because the normal‑form representation grows exponentially with the number of parallel information sets.

To overcome this, the authors develop a compact sequence‑form proper equilibrium (SFPE). They first redefine expected payoffs over action sequences, turning the usual realization‑plan representation into a form where each sequence’s contribution to a player’s payoff is explicit. By proving strategic equivalence, they show that any SFPE corresponds exactly to an NFPE of the original game, thus preserving all equilibrium properties while reducing the representation size to linear in the game tree.

The second major contribution is an alternative formulation based on an ε‑permutahedron over sequences. This geometric object imposes a global perturbation on all realization probabilities, ensuring that the perturbed game’s Nash equilibria converge to an NFPE as ε → 0. Unlike earlier sequence‑form perturbations that act independently on each information set (yielding quasi‑perfect or quasi‑proper equilibria), the ε‑permutahedron respects the global strategic interdependence required by proper equilibrium.

Building on this perturbed‑game framework, the authors introduce two differentiable path‑following algorithms. Both construct artificial sequence‑form games whose expected payoff functions are augmented with a smooth regularization term: one uses a logarithmic barrier, the other an entropy term. An auxiliary scalar τ>0 controls the strength of the regularizer. For any positive τ, the artificial game possesses a unique interior equilibrium (all realization probabilities are strictly positive). As τ is continuously decreased toward zero, the equilibrium path remains smooth—thanks to the differentiability of the regularized payoff—and converges to an NFPE of the original game. The authors prove existence of this smooth path, uniqueness of the starting point (any positive realization plan), and convergence guarantees.

Algorithmically, the path is traced by solving a sequence of regularized optimization problems, typically via Newton‑type or quasi‑Newton methods, updating τ in a homotopy fashion. Because each subproblem is of linear size in the game tree, the overall computational burden is dramatically lower than enumerating the exponential normal‑form.

Experimental evaluation covers a suite of finite n‑player extensive‑form games with perfect recall, varying in depth, branching factor, and number of information sets. Results demonstrate that both methods dramatically reduce memory consumption compared with normal‑form approaches, achieve faster convergence, and reliably reach equilibria that match or improve upon those obtained by existing quasi‑perfect or proper‑equilibrium algorithms. The entropy‑regularized variant shows particularly robust performance in deeper games, while the logarithmic variant excels in games with many actions per information set.

In summary, the paper makes three substantive contributions: (1) a novel sequence‑form characterization of normal‑form proper equilibrium via redefined sequence payoffs; (2) an ε‑permutahedron‑based perturbation scheme that bridges sequence‑form games to proper equilibrium; and (3) two globally differentiable path‑following algorithms that compute NFPEs efficiently for multi‑player extensive‑form games. By collapsing the exponential normal‑form into a linear‑size sequence‑form while preserving the stringent refinement properties of proper equilibrium, the work opens the door to practical computation of highly refined equilibria in complex strategic settings.