Writing study: Nash Equilibria in Games with Playerwise Concave Coupling Constraints: Existence and Computation

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← ICML 2026 oral writing corpus

Philip Jordan, Maryam Kamgarpour · Official ICML 2026 oral record

Observed writing choices, not acceptance rules. This study examines the inspected manuscript version. It does not establish why the paper was accepted or that its presentation choices should be copied unchanged.

Figure, table and section ledger

Editorial contribution type: Theory & formal analysis (primary).

Layout extraction, not a required paper template. Counts are distinct caption labels; a multi-panel figure counts once. Section spans are estimated from heading positions, including floats and whitespace. Method and limitations.

Canonical PDF

25 physical pages

Before references / appendix

≈ 8.29 page-equivalents

Main figures / tables

6 / 0

Attached appendix figures / tables

0 / 1

Attached appendix span

≈ 13.91 page-equivalents

Before references / appendix: ≈ 33.16%References / intervening material: ≈ 11.2%Attached appendix: ≈ 55.64%Estimated layout envelopes, not text/artwork area. Unresolved appendix boundaries remain within the undivided post-body material.

Section-by-section lengths and source pages

The title/abstract/front matter precede the first numbered heading. A span ends at the next detected top-level heading, not at the last sentence; end-page whitespace is included. References and statements are listed separately. These are not measurements of text-only or artwork area.

Top-level heading	Starts on PDF page	Estimated span	Share of entire PDF
1. Introduction	1	≈ 1.86 pp.	7.44%
2. Preliminaries	3	≈ 0.41 pp.	1.64%
3. Existence of Constrained Nash Equilibria	3	≈ 1.8 pp.	7.2%
4. Learning Constrained Nash Equilibria	5	≈ 2.28 pp.	9.12%
5. Simulations	7	≈ 1.38 pp.	5.52%
6. Conclusion	9	≈ 0.15 pp.	0.6%
Acknowledgements	9	≈ 0.04 pp.	0.16%
Impact Statement	9	≈ 0.05 pp.	0.2%
References	9	≈ 2.8 pp.	11.2%
A. Overview of Notation	12	≈ 1 pp.	4%
B. Discussion on Constraint Structure	13	≈ 1 pp.	4%
C. Background for Section 3	14	≈ 1.22 pp.	4.88%
D. Proof of Existence Result	15	≈ 2.19 pp.	8.76%
E. Background for Section 4	17	≈ 2.59 pp.	10.36%
F. Proofs for Section 4	20	≈ 5.91 pp.	23.64%

All detected figure and table locations

Main text

• Figure 1 · p. 4
• Figure 2 · p. 4
• Figure 3 · p. 8
• Figure 4 · p. 8
• Figure 5 · p. 8
• Figure 6 · p. 8

Attached appendix

• Table 1 · p. 12

Argument and organization

1. 1 Introduction; 2 Preliminaries; 3 Existence

   Introduces shared constraints, boundary regularity and a contractibility-based existence argument.

2. 4 Learning Constrained Nash Equilibria

   Narrows to potential games and specifies feedback, feasible initialization, adaptive steps and the rate.

3. 5 Simulations; 6 Conclusion

   Illustrates feasibility and Nash-gap behavior, then names stochastic-feedback and broader-game extensions.

Problem, claim and contribution

Existence and computation are separated: playerwise concavity and a boundary qualification support existence, while learning is studied in the narrower potential-game class.

Sections 3-4, pp.3-6 · Main paper

Argument and section order

Counterexamples motivate the constraint qualification before a geometric proof overview, with the formal convergence assumptions collected beside the theorem.

Examples 3.2-3.3 and Figures 1-2, pp.3-4; Theorem 4.7, p.6 · Main paper

Figures, captions and visual explanation

Feasible-set sketches and colored contraction arrows illustrate the topological argument; simulations pair trajectories or network paths with gap and constraint traces.

Figures 1-2, p.4; Figures 3-6, p.8 · Main paper

Tables, comparisons and evidence

The rate is best-iterate under exact gradient information and strictly feasible initialization, not a last-iterate guarantee under noisy decentralized sensing.

Sections 4.2-4.4, pp.5-7 · Main paper

Appendix and reproducibility

A notation table precedes constraint counterexamples, background and proof blocks that mirror the two principal theorem sections.

Appendices A-B, pp.12-13; E-F, pp.18,25 · Main paper

Clarity and writing conventions

The future-work paragraph keeps stochastic feedback outside the established claim; a caption-direction inconsistency remains visible in Figure 4, whose trace and setup use a lower-bound constraint despite the word below.

Figure 4, p.8; Conclusion, p.9 · Main paper

What to reuse—and what not to infer

Transferable lesson: Give existence and algorithmic claims separate assumptions, then use conceptual geometry and feasibility traces to explain what each guarantee actually covers.

Boundary: The formal rate is a best-iterate result for the stated potential-game and feedback assumptions. Figure 4 has a below/above wording inconsistency, so its caption should not be copied as a correct generic template.

Apply this lesson using the argument and evidence worksheet.

Inspection record and sources

Checked

2026-10-07

Reading status

Writing-focused manuscript scan + selected visuals inspected

Main text

Writing-oriented scan of the main argument through physical p.9, with focused physical-page text inspection on 1, 3, 4, 5, 6, 7, 8, 9. Framing, section progression, mechanism/proof explanation, displays, comparison conditions and conclusions were inspected. This is not exhaustive technical reading, proof verification or reproduction.

Appendix / supplement

Appendix organization and main-text pointers were checked; focused text inspection of physical pages 12, 13, 18, 25. Remaining appendix proofs and details were not exhaustively read. No separately linked supplement inspected.

Visually inspected PDF pages

4, 8

This is a writing-focused scan of the manuscript and the stated supplementary scope: framing, organization, visual explanation, evidence presentation and wording. It is not a line-by-line proof audit, reproduction of results or exhaustive scientific review.

• Published proceedings PDF, PMLR volume 306, retrieved 2026-10-07 · 25 pages