{"id":"bae1da86-647f-46e7-96c6-7d68acc656ef","arxiv_id":"2507.19123","paper_version":1,"verdict":"ACCEPT","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"Strong randomized mean-field equilibria exist for optimal-stopping games with countably generated common noise under continuity assumptions, and monotone comparative statics hold for strict equilibria.","lead":"This mathematics paper proves that a class of mean-field games of optimal stopping with common noise has an equilibrium when players may randomize the time they stop. The result matters because previous methods could only produce approximate equilibria in this setting, and the paper also shows how equilibria shift when rewards are ordered.","discovery_kind":"extension","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Assumption (2) of Theorem 3.4 does not imply the sup-norm continuity of the Bank-El Karoui input Y^m required by Theorem A.4(d); the martingale projection of the terminal reward is uncontrolled, so the fixed-point step invoked by Theorem 3.10 is not established.","rationale":"The reader accepted Theorem 3.10 at moderate confidence, identifying the countable-partition assumption and the X^m structural assumption as the weakest points. My stress-test finds a different, more internal gap: the transition from the epsilon-equilibria of Theorem 3.4 to the fixed-point theorem A.4 is not justified by the stated continuity assumption. In particular, the proof of Theorem 3.4 defines Y^m_t = e^{-ρt}g(t,m) - E[e^{-ρT}g(T,m)|F_t] and then claims that Assumption (2) implies condition (d) of Theorem A.4. That implication is false in general: the terminal martingale E[e^{-ρT}g(T,m)|F_t] can have a large maximal function even when the terminal rewards converge in L1, so the sup-norm expectation of Y^m differences need not vanish. The counterexample in the attack satisfies all standing assumptions with g supported only at the terminal time, so the gap is not merely cosmetic. Since Theorem 3.10's proof begins by invoking Theorem 3.4, the central existence claim is not proven as written. The result may be repairable: one can strengthen the continuity assumption to include the martingale maximal function, or prove that the optional/class-D/upper-semicontinuity hypotheses imply the required control. I would therefore make the verdict CONDITIONAL rather than ACCEPT or REJECT: the conclusion is plausible and the surrounding construction is coherent, but the main theorem requires an additional argument or hypothesis before it is fully supported.","tokens_in":19445,"tokens_out":56875,"duration_ms":598877,"concrete_test":"Test the implication claimed in Appendix B, Step b of Theorem 3.4 by instantiating it as follows. Set T=1, ρ=1, h=0, g(t,m)=0 for t<1, g(1,m)=e^ρ ξ_m with ξ_m≥0. Choose m_n→m_∞ in L0_G so that ξ_n-ξ_∞ = M_1/n, where M is a uniformly integrable martingale on [0,1] with E|M_1|<∞ and E[sup_{t∈[0,1]}|M_t|]=∞. Compute the two quantities: (i) E[sup_{t∈[0,1]} e^{-ρt}|g(t,m_n)-g(t,m_∞)|] = E|M_1|/n → 0, so Assumption (2) of Theorem 3.4 holds; (ii) E[sup_{t∈[0,1]}|Y^{m_n}_t-Y^{m_∞}_t|] = (1/n) E[sup_t |E[M_1|F_t]|] = ∞, which violates condition (d) of Theorem A.4. If (ii) is infinite, the proof of Theorem 3.4 needs an additional hypothesis controlling E[sup_t |E[e^{-ρT}(g(T,m_n)-g(T,m_∞))|F_t]|], and Theorem 3.10 should be stated with that condition or with a corrected proof.","verdict_should_be":"CONDITIONAL","load_bearing_attack":"The load-bearing gap is in the proof of Theorem 3.4, Appendix B, Step b, which Theorem 3.10 directly invokes. There the Bank-El Karoui input is Y^m_t = e^{-ρt}g(t,m) - E[e^{-ρT}g(T,m)|F_t]. Step b claims that Assumption (2) makes condition (d) of Theorem A.4 hold. But Assumption (2) controls only E[sup_t e^{-ρt}|g(t,m_n)-g(t,m_∞)|] and the integral of e^{-ρt}|h(t,m_n)-h(t,m_∞)|; it does not control the martingale correction M^m_t := E[e^{-ρT}g(T,m)|F_t]. L1 convergence of terminal values does not imply L1 convergence of the maximal function of their martingale: uniformly integrable martingales can have finite terminal mean and infinite supremum expectation, and this pathology survives scaling by 1/n. Concretely, take T=1, h=0, g(t,m)=0 for t<1, g(1,m)=e^ρ ξ_m with ξ_m≥0, and choose ξ_n-ξ_∞ = M_1/n, where M is a uniformly integrable martingale with E|M_1|<∞ and E[sup_t |M_t|]=∞. Then Assumption (2) holds since E[sup_t e^{-ρt}|g(t,m_n)-g(t,m_∞)|] = E|M_1|/n → 0. However, E[sup_t |Y^{m_n}_t-Y^{m_∞}_t|] = (1/n) E[sup_t |E[M_1|F_t]|] = ∞ for every n, so condition (d) of Theorem A.4 fails. Thus the proof of Theorem 3.4 is incomplete, and Theorem 3.10 inherits the gap because its assumption (b) also fails to control the projection of the terminal reward difference onto the filtration.","agreement_with_reader":"disagree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":19884,"tokens_out":18239,"duration_ms":204509,"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":[{"comment":"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).","section":"Appendix B, proof of Theorem 3.4 Step b; Theorem A.4(d); Eq. (3.2)"}],"minor_comments":[{"comment":"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.","section":"Lemma 4.3(i)"},{"comment":"In the introduction, 'monography' should be 'monograph'.","section":"Introduction"},{"comment":"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.","section":"Proof of Theorem 3.10, Step c"},{"comment":"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.","section":"Definition 3.5"}],"recommendation":"major_revision","confidential_remarks":"The technical gap in Theorem 3.4 is real and load-bearing, but it appears repairable by adding a continuity condition on the conditional expectation of the terminal reward. I therefore recommend major revision rather than rejection. The authors should also check whether the strengthened condition propagates correctly through Theorem 3.10 and whether the statement should be modified accordingly."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Short version: this is a promising paper, but the main theorem is currently not proven. The idea is good: take the strong epsilon-equilibria of He-Tan-Zou [19], pass to the limit through Baxter-Chacon compactness for randomized stopping times, and get a strong randomized equilibrium. The compactness and approximation arguments in Theorem 3.10, Steps b-d, are coherent. But the paper leans on Theorem 3.4 for the existence of the epsilon-equilibria, and the proof of Theorem 3.4 has a load-bearing gap.\n\nThe problem is in Step b of the proof of Theorem 3.4. You invoke Theorem A.4(d), which requires E[sup_t |Y^{m_n}_t - Y^{m_∞}_t|] -> 0 for the Bank-El Karoui input Y^m_t = e^{-ρt}g(t,m) - E[e^{-ρT}g(T,m)|F_t]. Assumption (2) controls E[sup_t e^{-ρt}|g(t,m_n)-g(t,m_∞)|], but the second term, the martingale projection of the terminal reward, is not controlled. L1 convergence of the terminal values does not give L1 convergence of the maximal function of the martingale. The stress-test example is a valid counterexample within your assumptions: take T=1, h=0, g(t,m)=0 for t<1 and g(1,m)=e^ρ ξ_m, with ξ_n-ξ_∞ = M_1/n for a UI martingale M with E|M_1|<∞ and E[sup_t|M_t|]=∞. Assumption (2) holds; condition (d) fails. So Theorem 3.4 is not established, and Theorem 3.10 inherits the gap.\n\nThis is not a nitpick: the existence of the epsilon-equilibria is the starting point of the whole construction. You could likely fix it by strengthening the continuity assumption to control the martingale projection directly, or by assuming the payoff functions are bounded so Doob's inequality applies. But as written, the central claim needs a correction.\n\nCredit where it's due: Section 4 on comparative statics is independent, the monotonicity argument is clean, and the paper is honest about the countable-partition restriction. The exposition is generally clear, and the reliance on [19] is straightforward. The comparative statics result looks solid and could stand alone.\n\nMy recommendation: send it to a good referee, but the referee should be told that the proof of Theorem 3.4 needs a fix. The paper is not ready in current form; a major revision is required.","headline":"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.","tokens_in":20392,"tokens_out":5212,"would_cite":false,"duration_ms":51446,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["91A16","60G40","93E20","91A55"],"pacs":[],"model":"deepseek-v4-flash","headline":"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.","keywords":["mean-field games","optimal stopping","randomized stopping time","common noise","Bank-El Karoui representation","Baxter-Chacon convergence","strong equilibrium","comparative statics"],"falsifier":"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 $\\tilde\\tau^\\ast$.","tokens_in":19208,"feed_emoji":"🎲","tokens_out":10356,"duration_ms":105699,"temperature":0.7,"pith_summary":"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.","feed_headline":"Randomized stopping times yield exact mean-field equilibria","feed_subtitle":"Limits of ε-optimal stopping games become exact equilibria, with the mean-field term still adapted to the common noise.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Supplies the strong $\\varepsilon$-mean-field equilibria and the mean-field Bank-El Karoui representation framework that the paper passes to the limit.","marker":"[19]"},{"why":"Introduces randomized stopping times and proves the compactness of their space in the Baxter-Chacon topology, the key strategy-space compactness.","marker":"[5]"},{"why":"Provides the stochastic representation theorem by which optimal stopping times are hitting times of the running supremum of $L^m$.","marker":"[3]"},{"why":"Connects the representation to optimal stopping and underpins Proposition A.7's identification of the largest and smallest optimal stopping times.","marker":"[4]"},{"why":"Supplies the convergence-argument template used to prove continuity of the limit reward functional in Step c.","marker":"[13]"},{"why":"Gives the closest prior strong-equilibrium notion for mean-field timing games with common noise, whose definition the present equilibria extend.","marker":"[10]"}],"fun_headline_variants":["Randomized stopping times make mean-field equilibria exact","ε-optimal limits become strong equilibria via randomization","Countable common noise yields strong randomized equilibria","Monotone rewards give strict equilibria in stopping games","Relaxed stopping restores compactness for mean-field equilibria"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Randomized stopping times make mean-field equilibria exact","ε-optimal limits become strong equilibria via randomization","Countable common noise yields strong randomized equilibria","Monotone rewards give strict equilibria in stopping games","Relaxed stopping restores compactness for mean-field equilibria"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000883,"raw_usage":{"total_tokens":3784,"prompt_tokens":888,"completion_tokens":2896,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":504,"completion_tokens_details":{"reasoning_tokens":2817}},"tokens_in":504,"tokens_out":2896,"duration_ms":23551,"temperature":1.0,"reasoning_tokens":2817,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-15T18:01:11.112778+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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 $\\tilde\\tau^\\ast$.","supporting_citations":[{"cited_title":"A mean-field version of Bank-El Karoui's representation of stochastic processes","cited_arxiv_id":"2302.03300","evidence_quote":"Supplies the strong $\\varepsilon$-mean-field equilibria and the mean-field Bank-El Karoui representation framework that the paper passes to the limit."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Introduces randomized stopping times and proves the compactness of their space in the Baxter-Chacon topology, the key strategy-space compactness."},{"cited_title":"Bank and N","cited_arxiv_id":null,"evidence_quote":"Provides the stochastic representation theorem by which optimal stopping times are hitting times of the running supremum of $L^m$."},{"cited_title":"Bank and H","cited_arxiv_id":null,"evidence_quote":"Connects the representation to optimal stopping and underpins Proposition A.7's identification of the largest and smallest optimal stopping times."},{"cited_title":"Coquet and S","cited_arxiv_id":null,"evidence_quote":"Supplies the convergence-argument template used to prove continuity of the limit reward functional in Step c."},{"cited_title":"Carmona, F","cited_arxiv_id":null,"evidence_quote":"Gives the closest prior strong-equilibrium notion for mean-field timing games with common noise, whose definition the present equilibria extend."}],"review_version":2}