Pith. sign in

REVIEW 1 major objections 4 minor 2 cited by

Existence of Strong Randomized Equilibria in Mean-Field Games of Optimal Stopping with Common Noise

T0 review · 1 major / 4 minor · reviewed 2026-08-15 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗ Review JSON

Pith's one-line read This paper proves that mean-field games of optimal stopping with a countably generated common noise admit strong randomized mean-field equilibria, obtained as limits of approximate equilibria.

desk verdict The main existence theorem is not proven as written: the proof of Theorem 3.4 fails to control the martingale correction in the Bank-El Karoui input, so the epsilon-equilibria starting point for Theorem 3.10 is missing; the comparative statics section is independent and looks sound. read the letter →

arxiv 2507.19123 v1 pith:GRSKVQJM submitted 2025-07-25 math.OC math.PRq-fin.MF

Giorgio Ferrari , Anna Pajola This is my paper · ORCID
classification math.OCmath.PRq-fin.MF MSC 91A1660G4093E2091A55
keywords mean-fieldgamesoptimalstoppingrandomizedtimecommonnoiseBank-ElKarouirepresentationBaxter-Chaconconvergencestrongequilibriumcomparativestatics
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves that exact equilibria exist for a broad class of mean-field games of optimal stopping with common noise, provided the agents may randomize their stopping times. The common noise is required to be generated by a countable partition of the probability space, so Brownian common shocks are excluded, and the rewards must act on the mean-field term through a continuous-path auxiliary process. Starting from known approximate equilibria, the authors show that as the approximation error goes to zero a subsequence converges to a true equilibrium in which the interaction term is still adapted to the common noise and the stopping time is randomized. A separate monotone setting, valid for a general common noise, yields comparative statics: larger rewards shift the extremal equilibria in a defined order.

What carries the argument

The load-bearing object is the mean-field Bank-El Karoui representation: for each interaction $m$, an optional process $L^m$ represents the terminal reward $Y^m$ via $Y^m_\tau = \mathbb{E}[\int_\tau^T f^m(t,\sup_{v\in[\tau,t)} L^m_v)\,dt \mid \mathcal{F}_\tau]$, and the optimal stopping times are exactly the hitting times of the running supremum $\hat L^m$, with the largest and smallest optimal times corresponding to strict versus weak thresholds. Because those hitting times are not continuous in the L\'evy metric unless the limit process is strictly increasing, the paper works with hitting times of $\hat L^m+\delta_\varepsilon\,\mathrm{id}$ to obtain $\varepsilon$-equilibria. To pass to $\varepsilon=0$, it uses randomized stopping times on $\Omega\times[0,1]$ with the Baxter-Chacon topology, whose compactness replaces the lost continuity of ordinary stopping times, and Schauder's fixed point theorem on the space of $\mathcal{G}$-random probability measures, identified with a product of probability spaces when $\mathcal{G}$ is countably generated.

What would settle it

Solve the Bank-El Karoui representation explicitly for the two-atom case where the countable partition has two atoms and $\hat h,\hat g$ are chosen continuous, and check whether the fixed-point equation $m = \mathcal{L}(\tilde\tau(m)\mid\mathcal G\times\{\emptyset,[0,1]\})$ has a solution; a concrete instance with no fixed point would refute Theorem 3.10. A weaker but still decisive test is to take any $\varepsilon$-equilibrium sequence $(m_{\ast,\varepsilon},\tau_{\ast,\varepsilon})$ from the paper's Step a and compute whether every weak-in-probability cluster point $m^\ast$ satisfies $m^\ast = \mathcal{L}(\tilde\tau^\ast\mid\mathcal G\times\{\emptyset,[0,1]\})$ for some Baxter-Chacon limit $\tilde\tau^\ast$.

Watch

Extended reading notes

Core claim

The central claim, Theorem 3.10, is that under Assumption 2.5, a countable-partition common noise, and rewards depending on the interaction $m$ only through a continuous-path process $X^m$, there exists a strong randomized mean-field equilibrium $(m^\ast,\tilde\tau^\ast)$: the interaction satisfies $m^\ast = \mathcal{L}(\tilde\tau^\ast \mid \mathcal{G}\times\{\emptyset,[0,1]\})$ and the randomized stopping time maximizes the relaxed reward functional. The route is to take the strong $\varepsilon$-equilibria obtained by perturbing the running supremum process $\hat L^m+\delta_\varepsilon\,\mathrm{id}$ and let $\varepsilon\to 0$; randomization of stopping times restores compactness of the strategy space and turns the discontinuous hitting-time map into a continuous one on the enlarged space. The limit step also uses continuity of the rewards in $m$, obtained from the path-dependence structure on $X^m$. In the monotone part of the paper, for a general common noise, Tarski's fixed point theorem delivers strong equilibria with strict non-randomized stopping times, and comparative statics order the extremal equilibria under ordered reward functions.

Load-bearing premise

The construction requires the common noise $\mathcal G$ to be generated by a countable partition of $\Omega$; without that, the space of interaction terms has no compact convex structure to start the $\varepsilon$-to-zero limit.

Editorial extensions

If this is right

  • Exact strong randomized equilibria exist for the class covered by Theorem 3.10, upgrading the previously known $\varepsilon$-optimal equilibria to true equilibria.
  • The equilibria are strong: the mean-field interaction remains adapted to the common noise, and only the stopping time is mixed, so the randomization is a genuine mixed-strategy relaxation.
  • Randomization does not change the value of the single-agent problem, so the equilibrium outcome has the same payoff interpretation as the original stopping game.
  • In the monotone setting, strong equilibria with non-randomized stopping times exist even for general (for example Brownian) common noise, and the extremal equilibria move monotonically when the rewards are ordered.
  • The countable-partition assumption restricts the method: Brownian common shocks are excluded in the main existence result, but tail $\sigma$-algebras of recurrent Markov chains are admitted.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The proof suggests that the real obstruction to strong equilibria is the discontinuity of the hitting-time map rather than the mean-field interaction itself; if so, other devices that convexify stopping times, such as linear-programming relaxations, could yield exact equilibria in the Brownian common-noise case.
  • An immediate testable extension is to weaken structural assumption (a): rewards depending on $X^m$ through a discontinuous but c\`adl\`ag process might still be handled if the finite-dimensional convergence step can be carried out under weaker regularity.
  • Because the randomized equilibrium is a mixed-strategy equilibrium and the value is unchanged, this construction gives a candidate selection principle for large finite-player games: $N$-player approximate equilibria could be built by mixing over the $\varepsilon$-equilibria.
  • The comparative-statics result implies that equilibrium timing responds monotonically to reward shifts, which is directly testable in calibrated optimal-exercise or bank-run models.
Share X Bluesky LinkedIn Reddit HN

Signed reviews

No signed human review yet.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

1 major / 4 minor

Summary. The paper studies a mean-field game of optimal stopping with common noise, in which a representative agent chooses a stopping time to maximize a reward functional whose running and terminal rewards depend on a random probability measure m, and equilibrium requires m to coincide with the conditional law of the optimal stopping time given the common noise sigma-algebra G. Section 3 proves existence of strong randomized mean-field equilibria under continuity conditions, using the Bank-El Karoui representation to obtain epsilon-equilibria from the companion paper [19], then passing to the limit via Baxter-Chacon compactness and a path-dependent structural assumption on the rewards through an auxiliary continuous process X^m. Section 4 treats an ordered setting with monotone rewards and proves existence and comparative statics of strong equilibria via Tarski's fixed point theorem. The main result is Theorem 3.10; Theorem 3.4 supplies the epsilon-equilibrium bridge on which Theorem 3.10 relies.

Significance. If the central existence result is correct, the paper makes a useful contribution: it gives a concrete way to convert approximate strong equilibria into exact strong randomized equilibria in a common-noise environment, where compactness of the interaction terms and optimal stopping times is otherwise hard to obtain. The path-dependence through X^m is a natural structural condition, and the comparative-statics section is a clean application of order-theoretic fixed points. The proof of Theorem 3.10 is mostly coherent once the epsilon-equilibria are available. However, the key bridge, Theorem 3.4, contains a genuine gap in verifying the compactness/fixed-point condition of the auxiliary Bank-El Karoui representation, and this gap propagates to Theorem 3.10. The issue is local and repairable by strengthening the continuity assumptions, so the paper merits major revision rather than rejection.

major comments (1)
  1. [Appendix B, proof of Theorem 3.4 Step b; Theorem A.4(d); Eq. (3.2)] In Step b of the proof of Theorem 3.4, the authors assert that condition (d) of Theorem A.4 'clearly holds' from assumption (3.2). This is not established. The process defined in Step a is Y^m_t = e^{-ρt}g(t,m) - E[e^{-ρT}g(T,m)|F_t], so |Y^{m_n}_t - Y^{m_∞}_t| contains the term |E[e^{-ρT}(g(T,m_n)-g(T,m_∞))|F_t]|. Assumption (3.2) controls sup_t e^{-ρt}|g(t,m_n)-g(t,m_∞)| and the integral of e^{-ρt}|h(t,m_n)-h(t,m_∞)|, but it gives no control of the maximal function of the terminal martingale. L^1 convergence of terminal values does not imply L^1 convergence of the supremum of their martingale projections: a uniformly integrable martingale can have E|M_1|<∞ and E[sup_{t<1}|M_t|]=∞. Concretely, take T=1, h=0, m_n=δ_{1/n}, m_∞=δ_0, and g(1,m_n)-g(1,m_∞)=e^ρ M_1/n with g(t,·)=0 for t<1, where M is such a martingale. Then (3.2) holds with value E|M_1|/n→0, while for t<1 we have Y^{m_n}_t-Y^{m_∞}_t = -E[M_1|F_t]/n, so E[sup_{t<1}|Y^{m_n}_t-Y^{m_∞}_t|] = (1/n)E[sup_{t<1}|M_t|] = ∞ for every n. This example is compatible with the other hypotheses of Theorem 3.4, and it shows that condition (d) of Theorem A.4 fails. Since Theorem 3.10 invokes Theorem 3.4 directly, its proof inherits the gap; hypothesis (b) of Theorem 3.10 similarly controls g through X^m but does not control the conditional expectation of the terminal reward difference. A repair could add an explicit continuity assumption, e.g. E[sup_t |E[e^{-ρT}(g(T,m_n)-g(T,m_∞))|F_t]|]→0, or prove that the stopping-time structure of K yields such control; the authors must then re-verify every subsequent use of Theorem A.4(d).
minor comments (4)
  1. [Lemma 4.3(i)] In the proof of Lemma 4.3(i), the displayed inequality contains the term h1(t,m)-h1(t,m) where h1(t,m)-h2(t,m) is clearly intended; please correct this typo.
  2. [Introduction] In the introduction, 'monography' should be 'monograph'.
  3. [Proof of Theorem 3.10, Step c] The sentence 'both the sequences X^k and τ_{*,εk} converge weakly and thus are tight' is imprecise for X^k, which converges in probability; please rephrase to distinguish weak convergence of the stopping times from convergence in probability of the processes.
  4. [Definition 3.5] The regularity condition on v ↦ τ̃(ω,v) should explicitly state that it holds for P-almost every ω, not pointwise for all ω, to match the measure-theoretic setting.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the main existence theorem is derived from external results and verified directly, not assumed by construction.

full rationale

No circularity found. Theorem 3.10 is proved by taking a subsequence of strong epsilon-mean-field equilibria supplied by Theorem 3.4, which is explicitly credited to Proposition 2.15 of [19] by He, Tan, and Zou (not the present authors), then extracting a Baxter-Chacon limit of randomized stopping times via Theorem 3.7 (Baxter and Chacon), and finally verifying the two defining conditions of a strong randomized mean-field equilibrium directly. Step b of the proof derives the consistency condition m* = L(tau_tilde* | G x {empty, [0,1]}) from the convergence m_{*,epsilon_k} -> m* together with the definition of Baxter-Chacon convergence, rather than postulating it. Step d derives optimality from the epsilon-equilibrium inequalities and the continuity assumptions (a)-(b); the limiting inequality J_tilde(tau_tilde*, m*) >= sup_tau J(tau, m*) is obtained by passing to the limit in the epsilon-inequality, not by renaming an optimality condition as an equilibrium. The structural assumptions (countable partition for G, dependence of rewards on m only through X^m, and the L1 continuity of the reward functionals) are genuine hypotheses imposed before the proof and are not fitted to the target conclusion. The citations to [14] and [15], which include one of the present authors as coauthor, are background references and carry no load-bearing weight in the existence proof. A skeptical reviewer's concern that Assumption (2) of Theorem 3.4 may not imply condition (d) of Theorem A.4 concerns the correctness or completeness of an imported theorem's verification; even if valid, it is a proof gap about the uncontrolled martingale projection, not a circularity, because the desired equilibrium is never assumed as an input. Accordingly, the appropriate finding is no significant circularity, score 0.

Assumptions & free parameters 0 free parameters · 9 assumptions · 0 invented entities

The central claim carries no fitted numerical parameters. The quantity delta_epsilon = epsilon / (3 times the integral of t e^{-rho t} dt) is a proof constant chosen in Appendix B to control the epsilon-optimality gap, not a parameter fit to data. The model inputs rho, T, and the reward functions are external data of the problem. The auxiliary process X^m is an assumption on the reward structure, not a new physical entity, and it carries no independent falsifiable prediction outside the model.

assumptions (9)
  • standard math Bank-El Karoui representation theorem (Theorem 3 in [3]) provides the optional process L^m satisfying (A.1) for each fixed m.
    Used in Section 3 and Appendix A to characterize optimal stopping times as hitting times of the running supremum of L^m.
  • standard math Schauder fixed point theorem applies to the locally convex space of G-random probability measures when G is generated by a countable partition.
    Used in Theorem 3.4 and Theorem A.4 to obtain fixed points of m -> L(tau^{m,epsilon}|G); requires a compact convex set of interaction terms.
  • standard math Tarski fixed point theorem applies to increasing maps on complete lattices of conditional laws under first-order stochastic dominance.
    Used in Section 4 and Theorem A.5 to prove existence of strong equilibria under monotonicity conditions.
  • standard math Baxter-Chacon compactness: the space of randomized stopping times is sequentially compact when the underlying sigma-algebra is countably generated (Theorem 1.5 in [5]).
    Essential for extracting a limit randomized stopping time in Theorem 3.10, Step a.
  • standard math Prokhorov theorem and tightness identify compact subsets of L^0_G(P([0,T])) under the countable-partition assumption.
    Used to extract a convergent subsequence of interaction terms m_{*,epsilon} in the proof of Theorem 3.10.
  • domain assumption G is generated by a countable partition of Omega (Theorem 3.4 condition (1)).
    Makes the space of G-random probability measures a locally convex product space and enables Schauder fixed point arguments; it excludes Brownian common noise.
  • ad hoc to paper Rewards depend on m only through a continuous-path adapted process X^m, with continuity of m -> X^m under weak convergence in probability (Theorem 3.10 assumptions (a) and (b)).
    Imposed to get continuity of the reward functional with respect to the pair (randomized stopping time, interaction term) when passing to the epsilon-to-zero limit.
  • domain assumption Assumption 2.5: e^{-rho t} g is optional, of class (D), and upper semicontinuous in expectation; e^{-rho t} h is progressively measurable and integrable.
    Standard regularity conditions ensuring the optimal stopping problem and Bank-El Karoui representation are well posed.
  • domain assumption The common noise is represented by a single sigma-algebra G rather than a filtration, relying on an immersion or H-property (Remark 2.4).
    Needed to identify conditional laws given G_t with those given G_T; standard under immersion, but restricts the class of common-noise models.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Existence of Strong Randomized Equilibria in Mean-Field Games of Optimal Stopping with Common Noise." pith.science (2026). https://pith.science/paper/GRSKVQJM

@misc{pith2026250719123,
  author       = {Pith},
  title        = {Pith review of: Existence of Strong Randomized Equilibria in Mean-Field Games of Optimal Stopping with Common Noise},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GRSKVQJM}},
  note         = {Machine review of arXiv:2507.19123}
}
read the original abstract

We study a mean-field game of optimal stopping and investigate the existence of strong solutions via a connection with the Bank-El Karoui's representation problem. Under certain continuity assumptions, where the common noise is generated by a countable partition, we show that a strong randomized mean-field equilibrium exists, in which the mean-field interaction term is adapted to the common noise and the stopping time is randomized. Furthermore, under suitable monotonicity assumptions and for a general common noise, we provide a comparative statics analysis of the set of strong mean-field equilibria with strict equilibrium stopping times.

Review history (2 revisions) →

Discussion (0). Continue with ORCID to comment.

Forward citations

Cited by 2 Pith papers

Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. Fast and slow mean-field games

    math.OC 2026-08 conditional novelty 7.0 of 10

    An averaged, common-noise-free mean-field game equilibrium induces an approximate Nash equilibrium for a two-scale common-noise mean-field game, with error O(delta^(1/6)) or O(delta^(1/3)).

  2. A new probabilistic approach for mean field games of optimal stopping

    math.PR 2026-07 conditional novelty 7.0 of 10

    Randomized mean-field equilibria of optimal-stopping games are characterized by a coupled reflected McKean–Vlasov forward-backward SDE system whose survival process L is an endogenous part of the solution.

Reference graph

Works this paper leans on

32 extracted references · 30 canonical work pages · cited by 2 Pith papers

  1. [19]

    X. He, X. Tan, and J. Zou , A mean-field version of Bank-El Karoui’s representation of stochastic processes, arXiv preprint arXiv:2302.03300, (2023)

    work page Pith review arXiv 2023

  2. [1]

    Aumann, Mixed and behavior strategies in infinite extensive games, Annals of Mathematics Studies, 52 (1961), p

    R. Aumann, Mixed and behavior strategies in infinite extensive games, Annals of Mathematics Studies, 52 (1961), p. 1

    work page 1961

  3. [2]

    R. Aïd, R. Dumitrescu, and P. Tankov , The entry and exit game in the electricity markets: A mean-field game approach, Journal of Dynamics and Games, 8 (2021), pp. 331–358

    work page 2021

  4. [3]

    Bank and N

    P. Bank and N. El Karoui , A stochastic representation theorem with applications to optimization and obstacle problems, The Annals of Probability, 32 (2004)

    work page 2004

  5. [4]

    Bank and H

    P. Bank and H. Föllmer , American Options, Multi–armed Bandits, and Optimal Consumption Plans: A Unifying View, Springer Berlin Heidelberg, Berlin, Heidelberg, 2003, pp. 1–42

    work page 2003

  6. [5]

    J. R. Baxter and R. V. Chacon , Compactness of stopping times, Zeitschrift für Wahrscheinlichkeits- theorie und verwandte Gebiete, 40 (1977), pp. 169–181

    work page 1977

  7. [6]

    Bertucci, Optimal stopping in mean field games, an obstacle problem approach, Journal de Mathé- matiques Pures et Appliquées, 120 (2018), pp

    C. Bertucci, Optimal stopping in mean field games, an obstacle problem approach, Journal de Mathé- matiques Pures et Appliquées, 120 (2018), pp. 165–194

    work page 2018

  8. [7]

    Bouveret, R

    G. Bouveret, R. Dumitrescu, and P. Tankov , Mean-field games of optimal stopping: a relaxed solution approach, SIAM Journal on Control and Optimization, 58 (2020), pp. 1795–1821

    work page 2020

Show all 32 references
  1. [8]

    Carmona and F

    R. Carmona and F. Delarue , Probabilistic Theory of Mean Field Games with Applications I-II, vol. 3, Springer, 2018

    2018

  2. [9]

    Carmona, F

    R. Carmona, F. Delarue, and D. Lacker , Mean field games with common noise, The Annals of Probability, 44 (2014)

    2014

  3. [10]

    Carmona, F

    R. Carmona, F. Delarue, and D. Lacker , Mean field games of timing and models for bank runs, Applied Mathematics & Optimization, 76 (2017), pp. 217–260

    2017

  4. [11]

    Y. S. Chow, H. Robbins, and D. Siegmund , The theory of optimal stopping, Dover, 1991

    1991

  5. [12]

    J. B. Conw ay, A course in functional analysis, vol. 96, Springer Science & Business Media, 1994

    1994

  6. [13]

    Coquet and S

    F. Coquet and S. Toldo ,Convergence of values in optimal stopping and convergence of optimal stopping times, Electronic Journal of Probability, 12 (2005)

    2005

  7. [14]

    Dianetti , Strong solutions to submodular mean field games with common noise and related mck- ean–vlasov fbsdes, The Annals of Applied Probability, 35 (2025)

    J. Dianetti , Strong solutions to submodular mean field games with common noise and related mck- ean–vlasov fbsdes, The Annals of Applied Probability, 35 (2025)

    2025

  8. [15]

    Dianetti, G

    J. Dianetti, G. Ferrari, M. Fischer, and M. Nendel , A unifying framework for submodular mean field games, Mathematics of Operations Research, 48 (2023), pp. 1679–1710

    2023

  9. [16]

    Dumitrescu, M

    R. Dumitrescu, M. Leutscher, and P. Tankov , Control and optimal stopping mean field games: a linear programming approach, Electronic Journal of Probability, 26 (2021), pp. 1–49

    2021

  10. [17]

    Durrett, Probability: Theory and Examples, Cambridge University Press, USA, 4th ed., 2019

    R. Durrett, Probability: Theory and Examples, Cambridge University Press, USA, 4th ed., 2019

    2019

  11. [18]

    E. B. Dynkin , The optimum choice of the instant for stopping a Markov process, Soviet Mathematics, 4 (1963), pp. 627–629

    1963

  12. [20]

    Huang, R

    M. Huang, R. P. Malhamé, and P. E. Caines , Large population stochastic dynamic games: closed-loop McKean-Vlasov systems and the Nash certainty equivalence principle, Communications in Information& Systems, 6 (2006), pp. 221 – 252

    2006

  13. [21]

    Kallenberg, Foundations of Modern Probability, Springer, 1997

    O. Kallenberg, Foundations of Modern Probability, Springer, 1997

    1997

  14. [22]

    77, Springer Cham, 2017

    , Random Measures, Theory and Applications, vol. 77, Springer Cham, 2017. 20 G. FERRARI AND A. PAJOLA

    2017

  15. [23]

    Kamae and U

    T. Kamae and U. Krengel , Stochastic Partial Ordering, The Annals of Probability, (1978), pp. 1044– 1049

    1978

  16. [24]

    N. E. Karoui and I. Karatzas , A new approach to the Skorohod problem, and its applications, Stochas- tics and Stochastic Reports, 34 (1991), pp. 57–82

    1991

  17. [25]

    Laraki and E

    R. Laraki and E. Solan , The value of zero-sum stopping games in continuous time, SIAM Journal on Control and Optimization, 43 (2005), pp. 1913–1922

    2005

  18. [26]

    Lasry and P.-L

    J.-M. Lasry and P.-L. Lions , Mean field games, Japanese Journal of Mathematics, 2 (2007), pp. 229– 260

    2007

  19. [27]

    Nutz, A mean field game of optimal stopping, SIAM Journal on Control and Optimization, 56 (2018), pp

    M. Nutz, A mean field game of optimal stopping, SIAM Journal on Control and Optimization, 56 (2018), pp. 1206–1221

    2018

  20. [28]

    Peskir and A

    G. Peskir and A. Shiryaev , Optimal stopping and free-boundary problems, Springer, 2006

    2006

  21. [29]

    Possamaï and M

    D. Possamaï and M. Talbi , Mean-field games of optimal stopping: master equation and weak equilibria, arXiv preprint arXiv:2307.09278, (2023)

    2023 arXiv

  22. [30]

    A. N. Shiryaev , Optimal stopping rules, in International Encyclopedia of Statistical Science, Springer, 2011, pp. 1032–1034

    2011

  23. [31]

    Touzi and N

    N. Touzi and N. Vieille , Continuous-time Dynkin games with mixed strategies, SIAM Journal on Control and Optimization, 41 (2002), pp. 1073–1088

    2002

  24. [32]

    X. Yu, J. Zhang, K. Zhang, and Z. Zhou , Major-minor mean field game of stopping: An entropy regularization approach, arXiv preprint arXiv:2501.08770, (2025)

    2025

Pith tools

Reviewed August 15, 2026 · model on record in the stance chip above.