Two Sequence-Form Interior-Point Differentiable Path-Following Method to Compute Nash Equilibria

Yuqing Hou

arxiv: 2604.12558 · v1 · submitted 2026-04-14 · 💻 cs.GT

Two Sequence-Form Interior-Point Differentiable Path-Following Method to Compute Nash Equilibria

Yuqing Hou This is my paper

Pith reviewed 2026-05-10 14:27 UTC · model grok-4.3

classification 💻 cs.GT

keywords Nash equilibriumsequence forminterior-point methodpath-followingextensive-form gamesperfect recallequilibrium computation

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The pith

A logarithmic-barrier regularization creates a differentiable path inside the sequence-form space that converges to Nash equilibria.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper first defines Nash equilibrium directly in terms of sequence-form realization plans for finite n-player extensive-form games with perfect recall. It then proves that this definition is equivalent to the standard mixed-strategy Nash equilibrium. On that foundation the authors construct a single-stage interior-point method that adds a logarithmic barrier to the sequence-form system. The barrier keeps every point of the solution trajectory strictly inside the feasible region and makes the path differentiable. As the barrier parameter shrinks to zero the path reaches an equilibrium point.

Core claim

The authors establish the sequence-form Nash equilibrium system by proving its equivalence to mixed-strategy Nash equilibrium, then apply logarithmic-barrier regularization to generate a differentiable equilibrium path that remains in the interior of the realization-plan space and converges to a true equilibrium without additional tuning.

What carries the argument

Logarithmic-barrier regularization of the sequence-form Nash system that produces a differentiable path through the interior of the realization-plan space.

If this is right

The method converges to a Nash equilibrium without manual regularization tuning.
Numerical stability and convergence improve relative to multi-stage alternatives.
The procedure remains effective and computationally efficient for finite n-player perfect-recall games.
Every point on the computed path satisfies the interior-point conditions before the limit is taken.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The same barrier construction might be adapted to other equilibrium notions once an analogous sequence-form system is available.
Because the path is differentiable, the method could serve as a differentiable layer inside larger optimization or learning pipelines.
Systematic tests on games of increasing size would show whether the single-stage property yields practical speed-ups over existing solvers.

Load-bearing premise

The sequence-form Nash system is exactly equivalent to ordinary mixed-strategy Nash equilibrium and the interior path reaches a genuine equilibrium as the barrier parameter vanishes.

What would settle it

A small finite perfect-recall game in which the path-following procedure ends at a point that fails to be a mixed-strategy Nash equilibrium.

read the original abstract

Nash equilibrium is a fundamental solution concept in extensive-form games, while its efficient computation is still far from straightforward. This paper considers finite $n$-player extensive-form games with perfect recall under the sequence-form representation. Unlike existing approaches, which mainly treat the sequence form as a compact computational reformulation, we develop a direct sequence-form definition of Nash equilibrium. Building on this, we rigorously establish the associated sequence-form Nash equilibrium system through an equivalence proof with mixed-strategy Nash equilibrium. On this basis, we propose a single-stage interior-point differentiable path-following method for equilibrium computation. The method uses logarithmic-barrier regularization to generate a differentiable equilibrium path in the interior of the realization-plan space, leading to favorable numerical stability and convergence properties. Numerical results show that the proposed method is effective and computationally efficient.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

New direct sequence-form NE definition and single-stage log-barrier path-following solver, but n-player convergence needs close checking on the limit step.

read the letter

The paper's main point is a direct definition of Nash equilibrium on sequence-form realization plans for n-player extensive games with perfect recall, plus a single-stage interior-point path-following method that uses logarithmic barriers to stay inside the polytope and trace a differentiable path to equilibrium. They prove equivalence to the usual mixed-strategy version and report that the approach runs stably with good convergence on their test instances. This formulation is new relative to the standard sequence-form LP reductions or regret-based methods. Treating the sequence form as the native space rather than a reformulation tool lets them build the regularized system directly, which avoids some of the usual two-phase interior-point machinery. The differentiability could be handy for sensitivity or learning extensions. The numerics look reasonable for the games they tried, and the focus on numerical stability is a practical plus. The soft spot sits in the convergence argument. The limit as the barrier parameter goes to zero must satisfy the original complementarity conditions for every finite n-player perfect-recall game. In multi-player settings equilibria can be degenerate, and without explicit handling of cases where strict complementarity fails, the path may not land on a true equilibrium. The abstract claims a rigorous proof, but the details on how the n-player case is covered would need verification. The experimental section would also benefit from head-to-head timings against established sequence-form solvers on larger instances. This work is aimed at algorithmic game theory researchers who compute equilibria in extensive forms or want differentiable approximations. A reader looking for fresh interior-point ideas on realization polytopes would find the construction useful. It deserves a serious referee because the claims are concrete and the area is active, even if the convergence section needs tightening.

Referee Report

2 major / 2 minor

Summary. The paper claims to introduce a direct sequence-form definition of Nash equilibrium for finite n-player extensive-form games with perfect recall, rigorously prove its equivalence to the standard mixed-strategy Nash equilibrium via an equivalence proof, and propose a single-stage interior-point differentiable path-following method that applies logarithmic-barrier regularization to generate a differentiable equilibrium path in the interior of the realization-plan space, with reported favorable numerical stability and convergence.

Significance. If the equivalence holds for general n-player perfect-recall games and the barrier path converges to true equilibria without extra tuning, the approach could supply a numerically stable, differentiable alternative for equilibrium computation in extensive-form games by adapting interior-point techniques directly to sequence forms. The numerical experiments are presented as supporting evidence of practical efficiency.

major comments (2)

[§3 (Equivalence proof, Theorem on sequence-form NE system)] §3 (Equivalence proof, Theorem on sequence-form NE system): the asserted equivalence of the direct sequence-form Nash system to mixed-strategy NE for all finite n-player perfect-recall games relies on a limit argument as the barrier parameter tends to zero. The manuscript must explicitly verify that this limit satisfies the original complementarity conditions without assuming strict complementarity or non-degeneracy, which frequently fail in n-player settings; an explicit proof step or small-game enumeration is required to support the central claim.
[§4 (Path-following construction)] §4 (Path-following construction): the claim that the single-stage logarithmic-barrier path converges to a true equilibrium without additional regularization tuning is load-bearing for the method's advantage. The convergence analysis should state the precise conditions under which the interior path reaches a boundary equilibrium point satisfying the sequence-form complementarity conditions.

minor comments (2)

[Title and Abstract] Title states 'Two Sequence-Form' while abstract and body describe a 'single-stage' method; clarify whether two variants exist or correct the wording.
[Numerical results section] Numerical results section: report the specific benchmark games, number of instances tested, and quantitative metrics (e.g., iteration counts, success rates, wall-clock times) rather than qualitative statements of 'favorable' performance.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We are grateful to the referee for the thorough review and insightful comments on our manuscript. We address each major comment below and outline the revisions we will make to improve the paper.

read point-by-point responses

Referee: [§3 (Equivalence proof, Theorem on sequence-form NE system)] the asserted equivalence of the direct sequence-form Nash system to mixed-strategy NE for all finite n-player perfect-recall games relies on a limit argument as the barrier parameter tends to zero. The manuscript must explicitly verify that this limit satisfies the original complementarity conditions without assuming strict complementarity or non-degeneracy, which frequently fail in n-player settings; an explicit proof step or small-game enumeration is required to support the central claim.

Authors: We thank the referee for highlighting this important aspect of the proof. Our equivalence proof in Section 3 indeed proceeds by considering the limit of the regularized system as the barrier parameter approaches zero. To rigorously address the potential lack of strict complementarity in n-player games, we will insert an additional lemma that demonstrates the limit point satisfies the complementarity conditions of the sequence-form Nash equilibrium system. This lemma will rely on the continuity of the barrier functions and the fact that the interior path is bounded, without invoking non-degeneracy. We will also provide a small illustrative example to verify the behavior in a degenerate case. This revision will be incorporated in the updated manuscript. revision: yes
Referee: [§4 (Path-following construction)] the claim that the single-stage logarithmic-barrier path converges to a true equilibrium without additional regularization tuning is load-bearing for the method's advantage. The convergence analysis should state the precise conditions under which the interior path reaches a boundary equilibrium point satisfying the sequence-form complementarity conditions.

Authors: We appreciate this suggestion for clarifying the convergence properties. In Section 4, the analysis shows that the differentiable path generated by the logarithmic-barrier regularization converges to an equilibrium as the parameter tends to zero. We will revise the text to explicitly state the conditions: namely, that for any sequence of barrier parameters μ_k → 0, the corresponding interior points converge to a realization plan that satisfies the sequence-form Nash equilibrium complementarity conditions, leveraging the equivalence established in Section 3. This holds under the perfect-recall assumption and does not require additional tuning beyond the standard barrier approach. The revised manuscript will include this precise statement. revision: yes

Circularity Check

0 steps flagged

No circularity: derivation builds on explicit equivalence proof and standard barrier techniques without reduction to inputs by construction.

full rationale

The paper introduces a direct sequence-form Nash definition, states an equivalence proof to mixed-strategy NE (claimed rigorous), and applies logarithmic-barrier regularization to generate a differentiable path inside the realization-plan polytope. No equation or step is shown to define a quantity in terms of itself, rename a fitted parameter as a prediction, or rely on a load-bearing self-citation whose content is unverified. The central path-convergence claim is presented as following from the barrier limit and the equivalence, with numerical results offered as separate validation. This matches the default expectation of a self-contained derivation; the provided abstract and reader's assessment confirm absence of the enumerated circular patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The work rests on the standard domain assumption that all games have perfect recall and are finite, plus the usual properties of logarithmic barriers for interior-point methods. No new free parameters or invented entities are introduced in the abstract.

axioms (1)

domain assumption All games under consideration have perfect recall.
Invoked to justify the sequence-form representation and the equivalence proof.

pith-pipeline@v0.9.0 · 5430 in / 1100 out tokens · 20341 ms · 2026-05-10T14:27:22.646242+00:00 · methodology

discussion (0)

Reference graph

Works this paper leans on

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Kuhn, H.W.: Extensive games. Proc. Natl. Acad. Sci.36(10), 570–576 (1950) https://doi.org/ 10.1073/pnas.36.10.570

work page doi:10.1073/pnas.36.10.570 1950

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work page doi:10.1073/pnas.36.1.48 1950

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Hou, Y., Cao, Y., Dang, C., Wang, Y.: A sequence-form differentiable path-following method to compute nash equilibria. Comput. Optim. Appl. (2025) https://doi.org/10.1007/ s10589-025-00702-y

work page 2025

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Jr.: Equilibrium points of bimatrix games

Lemke, C.E., Howson, J.T. Jr.: Equilibrium points of bimatrix games. J. Soc. Ind. Appl. Math. 12(2), 413–423 (1964) https://doi.org/10.1137/0112033

work page doi:10.1137/0112033 1964

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Rosenm¨ uller, J.: On a generalization of the lemke–howson algorithm to noncooperative n-person games. SIAM J. Appl. Math.21(1), 73–79 (1971) https://doi.org/10.1137/0121010

work page doi:10.1137/0121010 1971

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Wilson, R.: Computing equilibria of n-person games. SIAM J. Appl. Math.21(1), 80–87 (1971) https://doi.org/10.1137/0121011

work page doi:10.1137/0121011 1971

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In: Hu, T.C., Robinson, S.M

Garcia, C.B., Lemke, C.E., Luethi, H.: Simplicial approximation of an equilibrium point for non- cooperative n-person games. In: Hu, T.C., Robinson, S.M. (eds.) Mathematical Programming, pp. 227–260. Academic Press, New York (1973). https://doi.org/10.1016/B978-0-12-358350-5. 50011-7

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work page doi:10.1287/moor.7.1.1 1982

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Doup, T.M., Talman, A.J.J.: A new simplicial variable dimension algorithm to find equilibria on the product space of unit simplices. Math. Program.37(3), 319–355 (1987) https://doi.org/ 10.1007/BF02591741

work page doi:10.1007/bf02591741 1987

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Games Econ

Herings, P.J.-J., van den Elzen, A.: Computation of the nash equilibrium selected by the tracing procedure in n-person games. Games Econ. Behav.38(1), 89–117 (2002) https://doi.org/10. 1006/game.2001.0856

work page arXiv 2002

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Herings, P.J.-J., Peeters, R.J.A.P.: A differentiable homotopy to compute nash equilibria of n-person games. Econ. Theory18(1), 159–185 (2001) https://doi.org/10.1007/PL00004129

work page doi:10.1007/pl00004129 2001

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Govindan, S., Wilson, R.: A global newton method to compute nash equilibria. J. Econ. Theory 110(1), 65–86 (2003) https://doi.org/10.1016/S0022-0531(03)00005-X

work page doi:10.1016/s0022-0531(03)00005-x 2003

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Chen, Y., Dang, C.: A reformulation-based smooth path-following method for computing nash equilibria. Econ. Theory Bull.4(2), 231–246 (2016) https://doi.org/10.1007/s40505-015-0083-7

work page doi:10.1007/s40505-015-0083-7 2016

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Games Econ

Chen, Y., Dang, C.: An extension of quantal response equilibrium and determination of perfect 22 equilibrium. Games Econ. Behav.124, 659–670 (2020) https://doi.org/10.1016/j.geb.2017.12. 023

work page doi:10.1016/j.geb.2017.12 2020

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Chen, Y., Dang, C.: A differentiable homotopy method to compute perfect equilibria. Math. Program.185(1), 77–109 (2021) https://doi.org/10.1007/s10107-019-01422-y

work page doi:10.1007/s10107-019-01422-y 2021

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INFORMS J

Cao, Y., Chen, Y., Dang, C.: A differentiable path-following method with a compact formulation to compute proper equilibria. INFORMS J. Comput.36(2), 377–396 (2023) https://doi.org/10. 1287/ijoc.2022.0148

work page arXiv 2023

[17] [17]

Comput Optim Appl50(2), 223–236 (2011) https://doi.org/10.1007/ s10589-010-9332-8

Dang, C., Ye, Y., Zhu, Z.: An interior-point path-following algorithm for computing a leon- tief economy equilibrium. Comput Optim Appl50(2), 223–236 (2011) https://doi.org/10.1007/ s10589-010-9332-8

work page 2011

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Dang, C., Herings, P.J.-J., Li, P.: An interior-point differentiable path-following method to compute stationary equilibria in stochastic games. Inf. J. Comput.34(3), 1403–1418 (2022) https://doi.org/10.1287/ijoc.2021.1139

work page doi:10.1287/ijoc.2021.1139 2022

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Games Econ

von Stengel, B.: Efficient computation of behavior strategies. Games Econ. Behav.14(2), 220– 246 (1996) https://doi.org/10.1006/game.1996.0050

work page doi:10.1006/game.1996.0050 1996

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Games Econ

Koller, D., Megiddo, N., von Stengel, B.: Efficient computation of equilibria for extensive two- person games. Games Econ. Behav.14(2), 247–259 (1996) https://doi.org/10.1006/game.1996. 0051

work page doi:10.1006/game.1996 1996

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Koller, D., Pfeffer, A.: Representations and solutions for game-theoretic problems. Artif. Intell. 94(1), 167–215 (1997) https://doi.org/10.1016/S0004-3702(97)00023-4

work page doi:10.1016/s0004-3702(97)00023-4 1997

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Econometrica70(2), 693–715 (2002) https://doi.org/10.1111/ 1468-0262.00300

von Stengel, B., van den Elzen, A., Talman, D.: Computing normal form perfect equilibria for extensive two-person games. Econometrica70(2), 693–715 (2002) https://doi.org/10.1111/ 1468-0262.00300

work page arXiv 2002

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Miltersen, P.B., Sørensen, T.B.: Computing a quasi-perfect equilibrium of a two-player game. Econ. Theory42(1), 175–192 (2010) https://doi.org/10.1007/s00199-009-0440-6

work page doi:10.1007/s00199-009-0440-6 2010

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Govindan, S., Wilson, R.: Structure theorems for game trees. Proc. Natl. Acad. Sci.99(13), 9077–9080 (2002) https://doi.org/10.1073/pnas.082249599

work page doi:10.1073/pnas.082249599 2002

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arXiv (2025)

Hou, Y., Cao, Y., Dang, C., Wang, Y.: A sequence-form characterization and differentiable path- following method for computing normal-form perfect equilibria in extensive-form games. arXiv (2025). https://doi.org/10.48550/arXiv.2505.13827

work page doi:10.48550/arxiv.2505.13827 2025

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