Nash Equilibria over Unbounded Measure Spaces

Nash Equilibria over Unbounded Measure Spaces

Definitions and mathematical statement

Shared notation.

Standard number systems are denoted by $\mathbb N,\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$; index ranges, computational models and specialized notation follow the cited problem source and the definitions below.

A strategy here is a probability measure on the designated action space. A Nash equilibrium is a strategy profile where no player improves its payoff by unilateral change. Gradient descent-ascent minimizes one variable and maximizes another in the stated zero-sum or saddle-point formulation.

For a differentiable loss $L$, gradient flow solves $\dot\theta=-\nabla L(\theta)$, while gradient descent updates $\theta_{t+1}=\theta_t-\eta_t\nabla L(\theta_t)$. Global optimality refers to the minimum over the whole admissible parameter space.

Statement recorded in the supplied source.

Establish general conditions for existence of Nash equilibria for games whose strategies are probability measures on unbounded sets. Under those conditions, determine whether gradient descent–ascent converges to an equilibrium, beyond the known displacement-convex–concave cases. [S2]

Scope qualification.

The payoff regularity, moment restrictions, topology on measures, and exact gradient geometry must be supplied by the source; existence and convergence are separate targets.

Source status.

The supplied catalogue labels this record partially solved. Its source-status metadata was checked against the supplied snapshot on 2026-10-11. This import does not independently establish current open status or mathematical correctness. The catalogue literature assessment is dated 2026-08-22.

Short English statement

Find equilibrium-existence conditions on unbounded measure spaces and prove convergence of the proposed dynamics beyond the known convexity regime.

Sources

• S1 Ulam AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 30006091 (OWR-14298800-003). Catalogue curation and metadata: CC BY 4.0. The supplied source question is retained; definitions and the English summary are editorial additions. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 Mini-Workshop: High-Dimensional Control Problems and Mean-Field Equations with Applications in Machine Learning $2025$, Oberwolfach Reports, DOI 10.4171/owr/2024/56, pp. 3211--3254 https://doi.org/10.4171/owr/2024/56.
• S3 Conger et al., Coupled Wasserstein gradient flows https://arxiv.org/abs/2411.07403. Underlying Oberwolfach Reports material retains its source-specific licence, including CC BY-SA 4.0 where applicable; see THIRD_PARTY_NOTICES.md.

Preparation and review.

Original question attributed to the cited source and UnsolvedMath contributors. Codex (GPT-6; exact serving revision not exposed) assisted with definitions, English summaries, source inspection and formatting. No new human mathematical authorship or independent proof verification is asserted. Canonical ID 30006091 and source code OWR-14298800-003 checked against the complete supplied registry and existing repository statements; known flagged repetitions and obvious identical conjectures excluded. Full mathematical-equivalence/literature review has not been performed. Statement preparation and final review followed contribution prompts C and D; automated checks verify structure and identity, not open status.

Research attempt

Literature, source context and scope

The full original catalogue body and source qualifications are preserved above. The relevant primary workshop passages were read in the cached publisher report. Historical claims and subsequent attributed results are distinguished from the mathematical work below. A restricted example, standard lemma or known theorem is not counted as new progress on the entire original target. Missing assumptions and repeated or rhetorical extracts are not silently turned into different solved problems.

The primary report already proves existence and exponential convergence under suitable displacement convexity and concavity. It explicitly discusses unbounded Wasserstein spaces and the importance of lower curvature bounds. This attempt examines which elementary assertions survive when those hypotheses are weakened; it does not claim to extend the reported theorem.

An exactly solvable unbounded measure game

Let both strategies lie in $\mathcal P_2(\mathbb R)$, with $\rho$ maximizing and $\mu$ minimizing \[ F(\rho,\mu)=b m_\rho m_\mu-\frac a2\int z^2d\rho(z)+\frac c2\int x^2d\mu(x), \quad m_\rho=\int z\,d\rho, \] where $a,c>0$ and $b\in\mathbb R$. There is no entropy term in this restricted example.

Failed relaxations

Removing the positive quadratic terms leaves the bilinear game $F=m_\rho m_\mu$. For Dirac strategies its flow is $z'=x,x'=-z$, whence $z(t)^2+x(t)^2$ is constant. Starting at $(1,0)$ gives $(\cos t,-\sin t)$, which does not converge despite the saddle $(0,0)$. Non-strict convex-concave structure alone cannot justify the previous contraction argument.

Unbounded strategy spaces alone do not even ensure existence: the maximizing payoff $\int\arctan z\,d\rho$ has supremum $\pi/2$ but no maximizer. Every probability measure has strictly smaller expectation because $\pi/2-\arctan z$ is everywhere positive. Dirac masses at increasing positive integers approach the supremum and all belong to $\mathcal P_2$. An irrelevant second player turns this into a zero-sum example.

Verification and remaining gaps

Checks verify the moment dissipation identity and the explicit rotation. The examples identify missing coercivity and curvature steps, but do not classify general games, relax the source's regularity assumptions, or supply algorithms beyond this linear-quadratic family.

References and source attribution

Primary workshop report: https://doi.org/10.4171/owr/2024/56. Relevant passages read: OWR 2024/56, printed pages 3247--3249, Hoffmann on two-player Wasserstein games.

All retrievals were checked on 11 October 2026. Original question attribution remains in the preserved Sources section. External theorems are attributed and not claimed as new discoveries or fully verified merely by citation.

Integration review.

Codex (GPT-6; exact serving revision not exposed) reviewed the written arguments and expanded layout and source-reference wrappers. Repository metadata and the production web renderer were checked using Python 3.13 and JavaScript with MathJax 3.2.2. This model-assisted review is not independent human proof verification; no PDF compilation or formal proof checking is asserted.

Completion Estimate

COMPLETION: 0% No substantiated new progress toward the entire original target is claimed. The worked results and failed route are delimited above; a known result, illustrative model, or auxiliary obstruction is not counted as original-target completion.

Final status and attribution

Attempt status: unresolved. This describes this research attempt, not a new ruling on the source catalogue's status or on external literature claims. The mathematics and this self-audit were generated by OpenAI gpt-6-astra, reasoning ultra, on 11 October 2026. Codex (GPT-6; exact serving revision not exposed) orchestrated the batch, read and reviewed the written mathematical arguments, independently reran the finite checks, and contributed formatting/repository review. No human mathematical authorship, independent human proof checking, or formal verification is claimed. The historical preparation paragraphs in the reproduced catalogue body concern that body only.

Contribution prompt A was applied to the complete definition and supplied source record, followed by the adversarial checks of prompt B. Local reproduction files are batch026/checks.py, computations.json, and the per-ID audit JSON. They were not public links at preparation time. Numerical checks are finite and do not prove unrestricted results. No LaTeX engine was found locally; compilation was not performed by this research agent.