{"id":"42c5a6d2-4ff5-44e3-be69-06ec60b8ba0e","arxiv_id":"2608.10293","paper_version":1,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.0,"correctness_risk":"low","formal_verification":"none","parameter_count":3,"one_line_summary":"A relative-entropy extragradient algorithm provably computes static mean field equilibria, with explicit trade-offs between grid fineness and iteration count.","lead":"This paper introduces a Kullback-Leibler version of the extragradient algorithm for computing equilibria of static mean field games directly on probability distributions. It proves convergence rates and explicit error bounds that jointly account for the number of iterations and the coarseness of the state-space grid.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Section 3's quantitative δ-MFE guarantee is sound; the load-bearing gap is Remark 5.1's claim that running KL Mirror-Prox on the regularized cost produces the entropy-selected equilibrium, a claim that lacks a proof of a uniform trajectory lower bound.","rationale":"The reader's weakest-assumption field focused on Assumption 3.3, the W1-Lipschitz continuity of J in the measure variable. That is indeed the condition on which the quantitative bounds in Section 3 rest, but it is an explicit hypothesis, and the proof of Theorem 3.2 does not contain a hidden reliance on it beyond what is stated; the paper is honest that the fixed-stepsize theory requires Lm. The more consequential unresolved item is the entropy-selection mechanism: Theorem 5.1 is a clean statement for exact regularized-VI solutions, but the bridge from the KL Mirror-Prox algorithm to those solutions is only sketched. The sketch assumes a uniform lower bound a_h on all iterates, which is essential because the entropic regularizer is not globally Lipschitz near the boundary of the simplex. This is not a numerical convenience but a logical requirement: without a_h, the local Lipschitz constant eL_h and the step-size λ_h cannot be fixed, and the residual bound s_{h,K}=O(K^{-1}) does not follow. The reader's rationale flagged exactly this gap and made the verdict conditional on a full proof or numerical verification; my stress-test agrees with that judgment. I do not find a flaw in the paper's main convergence results, so the verdict should remain CONDITIONAL rather than moving to ACCEPT or REJECT. The proposed computational check would settle whether the trajectory lower bound holds in practice and whether the selection claim lands.","tokens_in":33743,"tokens_out":24431,"duration_ms":245615,"concrete_test":"Implement the regularized KL Mirror-Prox update with F^{ε_h}_h(μ,·) = J(μ,·) + ε_h log(μ/ρ_h) for the arctangent example of Section 6.1 (Ω=[-1,1], M=8, ε_h=√h, uniform ρ_h and initial μ_{0,h}). Track the minimal coordinate of μ_{k,h} and ν_{k,h} for, say, 10^6 iterations. If the minimum coordinate decays to zero, or if any adaptive reduction of λ_h is required to keep iterates in the coordinate-wise region ≥ a_h/2, the uniform lower bound asserted in Remark 5.1 fails and the approximate-regularized-VI claim is not established. As a complement, verify numerically whether the ergodic average satisfies the regularized VI residual r_{h,K} ≤ ε_h τ_h with τ_h → 0.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central quantitative assertion of the paper, Theorem 3.2 combined with Propositions 3.2 and 3.5, is internally consistent: the mesh-projection argument, the Minty-residual bound, and the conversion to a δ-MFE all check out under Assumptions 3.2–3.4. The genuinely insecure step is the algorithmic realizability of the equilibrium-selection contribution in Section 5. Theorem 5.1 is proved for exact solutions of the regularized VI, but Remark 5.1 claims that the proposed KL Mirror-Prox algorithm itself can produce these solutions up to a vanishing residual. The argument requires applying the standard KL Mirror-Prox analysis to the regularized operator F^{ε_h}_h(μ,·) = J(μ,·) + ε_h log(μ/ρ_h). That operator is not globally W1-Lipschitz on P(X_h): the entropy gradient blows up as any coordinate tends to zero. The sketch postulates a uniform lower bound a_h > 0 on every coordinate of every predictor and corrector along the entire trajectory, and then selects λ_h from the local Lipschitz constant on the set of vectors with coordinates at least a_h/2. No proof of this uniform lower bound is given; the assertion that boundedness of J and λ_h ε_h < 1 imply it is not demonstrated, and it is not obvious because the KL Mirror-Prox map can drive coordinates toward zero unless the entropy term exerts a sufficiently strong repulsion. Without this bound, the trajectory may leave the local Lipschitz regime, the step-size condition may be ill-defined, and the approximate-regularized-VI construction in Remark 5.1 is unjustified. Since equilibrium selection by KL Tikhonov penalty is stated as a contribution and no numerical experiment demonstrates the selection mechanism, this gap is the reason the paper should remain conditional rather than be accepted outright.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"This paper develops a KL-divergence version of Korpelevich's extragradient method for static mean-field equilibria, formulated as a variational inequality on the space of probability measures over a compact state space. On finite meshes X_h, each half-step reduces to an explicit exponential reweighting, and Proposition 3.1 gives a consistency result between discrete and continuous iterates. Under Lasry–Lions monotonicity and W1-Lipschitz continuity of the cost operator (Assumptions 3.2–3.4), the paper proves an O(1/K) ergodic Minty-residual bound (Lemma 3.3), a joint mesh/iteration bound (Theorem 3.2), and, via the approximate-MFE conversions in Propositions 3.2–3.5, explicit choices of h and K that certify a δ-MFE of the original compact-state problem. Under strong Lasry–Lions monotonicity (Assumption 4.1), it derives last-, best-, and averaged-iterate W1 rates (Theorems 4.1–4.2). Section 5 adds a KL-type Tikhonov regularization intended to select the entropy-minimizing equilibrium: Theorem 5.1 proves this selection for exact solutions of the regularized VI, and Remark 5.1 sketches an algorithmic construction via KL Mirror-Prox. The appendix contains the full proofs of the results in Sections 3 and 4.","tokens_in":34008,"tokens_out":18705,"duration_ms":180786,"significance":"If the central claims hold, this is a valuable contribution to computational static mean-field games. The quantitative analysis is genuinely joint in the mesh size h and the iteration count K, and it provides explicit, finite computation certificates without requiring PDE structure, potentiality, or convexity of the cost in the state variable. The framework covers non-Dirac equilibria and nonpotential operators. The proofs in Sections 3 and 4 are written out in full with explicit constants, the numerical examples satisfy the stated assumptions, and accompanying code is provided. I also note that the central quantitative claim, Theorem 3.2 together with Propositions 3.2 and 3.5, is internally consistent: the mesh-projection argument, the Minty-residual bound, and the conversion to a δ-MFE all check out under the stated assumptions. The main caveat is the equilibrium-selection part: Theorem 5.1 is nonconstructive, and the algorithmic claim in Remark 5.1 is only a sketch. Once that gap is resolved, the paper would be a strong addition to the literature.","major_comments":[{"comment":"Remark 5.1 claims that the KL Mirror-Prox algorithm applied to the regularized operator F^{ε_h}_h can produce a sequence satisfying the approximate discrete regularized VI with r_h/ε_h→0. The load-bearing step is the assertion that boundedness of J together with 0<λ_hε_h<1 implies a uniform positive lower bound a_h>0 on every coordinate of every predictor and corrector along the whole trajectory. No proof is given; the statement is not immediate because the corrector update contains the factor (ν_k/ρ_h)^{-λ_hε_h}, whose logarithm can in principle become unbounded if coordinates collapse. Furthermore, the proposed choice of λ_h from the 'trajectory-local step-size condition' depends on \\tilde L_h, which depends on a_h, which in turn depends on λ_h; the sketch does not demonstrate that a positive λ_h satisfying both conditions exists. Since Theorem 5.1 concerns exact solutions of the regularized VI, this remark is the only argument making the entropy-selection mechanism algorithmic. Please either provide a complete proof of the uniform lower bound and of the consistency of the step-size choice, or withdraw the algorithmic claim and present Theorem 5.1 as a nonconstructive selection result.","section":"Section 5, Remark 5.1 (residual estimates)"},{"comment":"The residual conversion in the same remark is also only sketched. Even assuming a uniform lower bound a_h, the proof must show that the local W1-to-L∞ Lipschitz constant \\tilde L_h on the set of probability vectors with coordinates at least a_h/2 is compatible with the chosen λ_h, that all iterates and the comparison mixtures ρ_{h,t} stay in the local Lipschitz regime for the full trajectory, and that the constants in s_{h,K} and r_{h,K} have the correct dependence on ε_h so that r_{h,K_h}/ε_h→0 is achievable. The remark asserts s_{h,K}=O(K^{-1}) and r_{h,K}≤2√(2\\tilde L_hD_Ωs_{h,K}) without tracking this dependence. A rigorous argument, or an equivalent lemma, is needed before the approximate-solution extension of Theorem 5.1 can be used.","section":"Section 5, Remark 5.1 (residual estimates)"}],"minor_comments":[{"comment":"The hypomonotone discussion is presented in two sentences and is explicitly conditional; please mark it as informal or provide a proof, since it currently reads as an unproved assertion about a modification of the main theorem.","section":"Section 5, Remark 5.2"},{"comment":"The symbol ∥μ−ν∥_1 is defined as ∫|dμ−dν|, which is twice total variation, but the paper later uses this norm together with W1 and Pinsker-type inequalities. Adding an explicit reminder in Lemma 3.2 or in the notation paragraph would prevent confusion with the standard L1 norm.","section":"Notation"},{"comment":"The sentence 'Applying Lemma 3.3 to P(X_h)' is formally unusual because Lemma 3.3 is stated for a general compact Ω; a one-line remark that X_h is compact and the same proof applies would remove ambiguity.","section":"Proof of Theorem 3.1"},{"comment":"The claimed equilibria μ*=δ_{−1/2} for Example (6.1) and μ*=1/2δ_{−1}+1/2δ_1 for Example (6.3) are stated without derivation; for a numerical section intended to illustrate the theory, adding the one-line verification would improve reproducibility.","section":"Section 6"},{"comment":"The paper does not discuss how L_m, L_x, and m_h are estimated in practice if they are not known a priori; since the step-size and mesh/iteration prescriptions depend on these constants, a short remark on this practical issue would be useful.","section":"Sections 3–4"},{"comment":"The asymptotic statement 'p_k ∼ 1/(λk)' is asserted from the recurrence p_{k+1}=p_k−λp_k^2+O(p_k^3); including a short asymptotic argument would make the impossibility statement self-contained.","section":"Remark 4.1"}],"recommendation":"major_revision","confidential_remarks":"The core contribution in Sections 3–4 is sound and well-written, and the Section 5 gap is local. If the authors can prove, or appropriately retract, the algorithmic claim in Remark 5.1, the paper could be accepted. I do not recommend rejection."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Bottom line: the main algorithmic content is solid, and the paper deserves a serious referee. The KL Mirror-Prox method on P(Omega) with explicit exponential-tilt updates, the O(1/K) Minty residual, and the joint mesh/iteration delta-MFE guarantee in Theorem 3.2 are rigorous and genuinely useful. The stress-test note is right: the load-bearing gap is Section 5, specifically Remark 5.1. What is new: the predictor-corrector extragradient idea transported to measure space using KL geometry, with both half-steps reducing to explicit reweightings. The ergodic theory works under only Lasry-Lions monotonicity plus W1-Lipschitz continuity, without potential structure or individual-state convexity. Theorem 3.2 together with Propositions 3.2 and 3.5 gives an explicit finite recipe: choose h and K as stated, and the ergodic average is a delta-MFE. The appendix proofs check out; assumptions are stated before each result, and the step-size conditions are consistent throughout. The strong-monotonicity section is also clean, including the two-point counterexample showing that without an active-set gap you only get O(1/k) KL decay, not geometric convergence. The numerical examples satisfy the assumptions, confirm the rates, and the code is available. No constant fitting to make the theory look good. The soft spots, in proportion. Main one: the equilibrium-selection contribution in Section 5 is incomplete. Theorem 5.1 is fine for exact solutions of the regularized VI, but Remark 5.1 claims the KL Mirror-Prox algorithm itself produces those solutions up to vanishing residual. The argument requires a uniform lower bound a_h on every coordinate of every predictor and corrector along the whole trajectory. That bound is asserted, not proved. The entropy gradient is not globally Lipschitz, and boundedness of J plus lambda_h epsilon_h < 1 does not obviously keep coordinates away from zero. This is a genuine gap: the selection mechanism is stated as a contribution, and no numerical experiment demonstrates it. The paper should either supply the proof or clearly demote this to a conjecture or experimental claim. Minor: the last-iterate entry threshold in Theorem 4.1 is very conservative on fine meshes; the paper acknowledges this, and the best/averaged-iterate bounds are the more practical ones. Also, the stepsize requires L_m and D_Omega, so an unknown Lipschitz constant would break the guarantees, but that is standard for constant-stepsize mirror-prox methods. Who this is for: anyone working on computational MFGs, VI-based learning in anonymous games, or entropic proximal methods. I would send it to a referee without hesitation. The core ergodic results are strong enough to publish regardless of whether Section 5 gets fixed; but if the selection claim stays as is, it should be flagged as unproven.","headline":"A rigorous and useful KL Mirror-Prox method for static MFGs; the core ergodic and mesh guarantees hold up, but Section 5's algorithmic selection step is asserted rather than proved.","tokens_in":870,"tokens_out":879,"would_cite":true,"duration_ms":40543,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["91A16","49J40","65K15"],"pacs":[],"model":"deepseek-v4-flash","headline":"The paper proves that two explicit exponential reweightings per step, run on a fine enough mesh for enough iterations, compute static mean-field equilibria to any prescribed accuracy; a KL penalty selects among multiple equilibria.","keywords":["mean field games","variational inequalities","extragradient method","relative entropy","Lasry-Lions monotonicity","mirror-prox","equilibrium selection","mesh refinement"],"falsifier":"On the arctangent example (6.1) with $M=1001$ mesh points, compute $\\sup_\\eta \\langle J(\\eta,\\cdot),\\bar\\nu_{K,h}-\\eta\\rangle$ and check that it stays below $\\log M/(\\lambda K)+5/(M-1)$: a violation at the predicted $K$ would falsify the Theorem 3.2 bound. A second test: run the fixed-stepsize method on a monotone static game whose cost is continuous but only Hölder in the measure, e.g., $J(\\mu,x)=\\int |x-y|^{1/2}\\,\\mu(dy)$ on $\\Omega=[-1,1]$; this operator has no finite $L_m$, so the paper's step-size rule offers no guarantee, and observing divergence or the need for a line search would confirm that the Lipschitz assumption is truly load-bearing.","tokens_in":33474,"feed_emoji":"⚖️","tokens_out":20552,"duration_ms":166432,"temperature":0.7,"pith_summary":"This paper establishes that static mean field equilibria—population distributions in which every agent's chosen state is a best response to that same distribution—can be computed by a two-step reweighting scheme with fully quantitative guarantees. The method, KL Mirror-Prox, replaces the Euclidean projections of the classical extragradient algorithm with relative-entropy proximal steps, so each half-iteration is an explicit exponential reweighting of the current measure: no projection, line search, PDE solve, or best-response computation is needed. Under Lasry–Lions monotonicity, a $W_1$-Lipschitz condition on the measure variable, and spatial Lipschitz continuity of the cost, the averaged iterate satisfies a Minty residual bound of order $\\log(1/m_h)/(\\lambda K) + (2L_m+L_x)h$, which the paper converts into a certificate: for any accuracy $\\delta>0$, explicit choices of mesh $h$ and iterations $K$ produce a $\\delta$-approximate mean field equilibrium. Under strong monotonicity the same machinery yields $W_1$-metric convergence rates for the last, best, and averaged iterates, and a KL-type Tikhonov penalty is shown to select, in the vanishing-penalty limit, the equilibrium of minimal relative entropy with respect to a chosen reference measure.","feed_headline":"KL Mirror-Prox computes mean-field equilibria to any accuracy","feed_subtitle":"Two explicit reweightings per step, with the mesh size and iteration count that certify accuracy.","key_machinery":"The machinery is the KL three-point identity for the relative-entropy Bregman geometry: for a cost $g$ and its exponential tilt $\\zeta(dx)\\propto e^{-\\lambda g(x)}\\rho(dx)$, the identity $\\lambda\\langle g,\\zeta-\\eta\\rangle = D_{\\rm KL}(\\eta\\|\\rho)-D_{\\rm KL}(\\eta\\|\\zeta)-D_{\\rm KL}(\\zeta\\|\\rho)$ holds exactly for every competing measure $\\eta$. Applied to the predictor and corrector half-steps, the two identities add up so that intermediate KL terms cancel and the only leftover is the predictor–corrector mismatch $\\langle J(\\nu_k,\\cdot)-J(\\mu_k,\\cdot),\\nu_k-\\mu_{k+1}\\rangle$, bounded by $\\lambda L_m D_\\Omega$ times the KL dissipation; summing over iterations telescopes the relative-entropy terms into the ergodic Minty residual. A measurable mesh projection $Q_h$ with $\\sup_x|Q_h(x)-x|\\le h$ carries these finite-mesh estimates to a general compact state space at the cost of the $(2L_m+L_x)h$ term, and the paper's approximate MFE/VI/MVI implications (with the square-root conversion $\\varepsilon\\mapsto 2\\sqrt{2L_m D_\\Omega\\varepsilon}$) complete the chain from residual to certified $\\delta$-approximate equilibrium.","core_discovery":"The paper's central claim is that a static mean field game, written as the variational inequality $\\langle J(\\mu^*,\\cdot),\\eta-\\mu^*\\rangle\\ge 0$ for all $\\eta\\in\\mathcal P(\\Omega)$, is solved by iterating two explicit exponential tilts—the predictor $\\nu_k\\propto \\mu_k e^{-\\lambda J(\\mu_k,\\cdot)}$ and the corrector $\\mu_{k+1}\\propto \\mu_k e^{-\\lambda J(\\nu_k,\\cdot)}$—and that the error of the iterates is explicitly quantifiable from the algorithm's constants. Under the standard Lasry–Lions monotonicity condition $\\langle J(\\mu,\\cdot)-J(\\nu,\\cdot),\\mu-\\nu\\rangle\\ge 0$ and a stepsize $\\lambda<1/(L_m D_\\Omega)$, the KL three-point identity telescopes over iterations into an $O(1/K)$ Minty residual bound on any finite mesh; projecting an arbitrary comparison measure onto the mesh and using the measure- and spatial-Lipschitz constants $L_m,L_x$ upgrades this to a uniform bound $\\varepsilon_{K,h}=\\log(1/m_h)/(\\lambda K)+(2L_m+L_x)h$ on the full compact state space. The approximate-equivalence results then convert this residual into an equilibrium statement: an $\\varepsilon$-Minty solution is a $\\delta$-VI solution with $\\delta=2\\sqrt{2L_m D_\\Omega\\,\\varepsilon}$, so for every $\\delta>0$ the choices $h\\le \\varepsilon^*_\\delta/(2(2L_m+L_x))$ and $K\\ge 2\\log(1/m_h)/(\\lambda\\varepsilon^*_\\delta)$ certify that the averaged iterate is a $\\delta$-MFE of the original problem. Under strong Lasry–Lions monotonicity the same one-step estimate yields sublinear and then geometric last-iterate $W_1$-convergence on a fixed mesh, and a separate discretization bound $W_1(\\mu^*_h,\\mu^*)\\le \\sqrt{L_x/\\alpha}\\,h$ splits the total metric error into optimization plus approximation terms. Finally, adding $\\varepsilon\\,D_{\\rm KL}(\\cdot\\|\\rho)$ as a Tikhonov penalty and letting $\\varepsilon\\downarrow 0$ selects the equilibrium of minimal relative entropy with respect to the reference measure $\\rho$, whenever that minimizer is unique and finite.","pith_inferences":["Because the updates are explicit tilts, the algorithm is an oracle method: I would expect the same iteration bounds to survive if $J$ were evaluated by Monte Carlo or particle samples, with the sampling error folded into the $W_1$-Lipschitz mismatch term; the paper does not analyze this stochastic variant, so this is a conjecture to test.","The uniform Minty bound acts as a run-time certificate: on a finite mesh one can evaluate the gap $\\langle J(\\eta,\\cdot),\\bar\\nu_{K,h}-\\eta\\rangle$ at adversarial test measures such as point masses and verify the $\\delta$-MFE claim during the run, a cheap a posteriori check the paper leaves implicit.","The explicit dependence on $\\log(1/m_h)$ makes the initialization a computational lever: for a quasi-uniform mesh with uniform initialization ($D_{0,h}=O(d|\\log h|)$), the minimal atom $m_h$ rather than the mesh resolution can dominate the iteration count, so designing exploration measures with controlled small masses may matter as much as refining the grid.","For non-unique equilibria the reference measure $\\rho$ is a selection dial; taking $\\rho$ uniform selects the maximum-entropy equilibrium (minimal relative entropy to uniform), which is plausibly the equilibrium reached by vanishing-noise or learning dynamics—an identification that would connect this paper's selection mechanism to the common-noise selection literature."],"forward_implications":["Any prescribed accuracy is reachable: for each $\\delta>0$ the paper specifies the mesh size and the iteration count explicitly, so a user can compute a $\\delta$-approximate mean field equilibrium without tuning, line search, or a posteriori selection.","The method is projection-free and needs only cost evaluations: each half-step is one evaluation of $J$ plus an exponential reweighting, so it applies to potential and nonpotential costs, to costs that are neither convex nor differentiable in the state variable, and to problems with no PDE structure.","Genuinely non-Dirac equilibria are in scope: because the analysis never imposes displacement monotonicity or spatial convexity, equilibria supported on several states (such as the $\\frac12\\delta_{-1}+\\frac12\\delta_1$ double-well example) are computed directly rather than excluded by assumption.","Under strong monotonicity the metric error separates into an optimization term and a mesh term, yielding, under the paper's scaling assumptions, explicit iteration counts of order $d^2\\varepsilon^{-2}|\\log\\varepsilon|$ for $\\varepsilon$-accuracy in $W_1$, while storage on a quasi-uniform grid still scales like $\\varepsilon^{-d}$.","The KL Tikhonov penalty turns equilibrium selection into a modeling choice: the vanishing-penalty limit is the equilibrium minimizing $D_{\\rm KL}(\\cdot\\|\\rho)$, so the reference measure $\\rho$ explicitly controls which equilibrium is returned when several exist."],"supporting_citations":[{"why":"Supplies the extragradient predictor–corrector template that KL Mirror-Prox transports from Euclidean space to the space of probability measures.","marker":"[21]"},{"why":"Supplies the mirror-prox construction and the $O(1/t)$ Minty-residual analysis for monotone variational inequalities that the ergodic bound is built on.","marker":"[31]"},{"why":"Supplies the mean field equilibrium concept and the Lasry–Lions monotonicity condition under which the VI and Minty VI solution sets coincide.","marker":"[23]"},{"why":"Supplies the approximate VI/MVI/equilibrium equivalence framework that Propositions 3.2–3.5 adapt, converting Minty residuals into approximate-MFE certificates.","marker":"[6]"},{"why":"Supplies the dual-averaging/FTRL dynamics whose $O(K^{-1/2})$ rate and vanishing learning rate motivate the constant-stepsize optimistic correction.","marker":"[16]"},{"why":"Supplies the finite-alphabet reverse Pinsker inequality used in Lemma E.1, which yields the eventual geometric last-iterate rates under strong monotonicity.","marker":"[11]"}],"fun_headline_variants":["Two exponential tilts solve mean-field games","KL Mirror-Prox: explicit tilts compute mean-field equilibria","Entropic extragradient with proven convergence to mean-field equilibrium","Double tilt algorithm: mean-field equilibrium with error bounds","Provable mean-field equilibria via KL reweighting"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The load-bearing premise is that the cost operator changes at a bounded, known rate as the population measure varies—$W_1$-Lipschitz continuity with a known constant $L_m$—together with the fixed stepsize restriction $\\lambda<1/(L_m D_\\Omega)$; if the operator is only continuous in the measure, or if $L_m$ is unknown, every quantitative guarantee in the paper collapses and stepsize selection would need a different mechanism.","fun_headline_variants_meta":{"raw":{"variants":["Two exponential tilts solve mean-field games","KL Mirror-Prox: explicit tilts compute mean-field equilibria","Entropic extragradient with proven convergence to mean-field equilibrium","Double tilt algorithm: mean-field equilibrium with error bounds","Provable mean-field equilibria via KL reweighting"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000643,"raw_usage":{"total_tokens":3104,"prompt_tokens":1242,"completion_tokens":1862,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":858,"completion_tokens_details":{"reasoning_tokens":1782}},"tokens_in":858,"tokens_out":1862,"duration_ms":12063,"temperature":1.0,"reasoning_tokens":1782,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-14T04:10:54.907236+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"On the arctangent example (6.1) with $M=1001$ mesh points, compute $\\sup_\\eta \\langle J(\\eta,\\cdot),\\bar\\nu_{K,h}-\\eta\\rangle$ and check that it stays below $\\log M/(\\lambda K)+5/(M-1)$: a violation at the predicted $K$ would falsify the Theorem 3.2 bound. A second test: run the fixed-stepsize method on a monotone static game whose cost is continuous but only Hölder in the measure, e.g., $J(\\mu,x)=\\int |x-y|^{1/2}\\,\\mu(dy)$ on $\\Omega=[-1,1]$; this operator has no finite $L_m$, so the paper's step-size rule offers no guarantee, and observing divergence or the need for a line search would confirm that the Lipschitz assumption is truly load-bearing.","supporting_citations":[{"cited_title":"Prox-method with rate of convergenceO(1/t) for variational inequalities with Lipschitz continuous monotone operators and smooth convex-concave saddle point problems","cited_arxiv_id":null,"evidence_quote":"Supplies the mirror-prox construction and the $O(1/t)$ Minty-residual analysis for monotone variational inequalities that the ergodic bound is built on."},{"cited_title":"Approxi- mate variational inequalities and equilibria.Computational Management Science, 20(1):43, 2023","cited_arxiv_id":null,"evidence_quote":"Supplies the approximate VI/MVI/equilibrium equivalence framework that Propositions 3.2–3.5 adapt, converting Minty residuals into approximate-MFE certificates."},{"cited_title":"Learning in nonatomic games, part i: Finite action spaces and population games.Journal of Dynamics and Games, 9(4):433–460, 2022","cited_arxiv_id":null,"evidence_quote":"Supplies the dual-averaging/FTRL dynamics whose $O(K^{-1/2})$ rate and vanishing learning rate motivate the constant-stepsize optimistic correction."},{"cited_title":"Context tree estimation for not necessarily finite memory processes, via BIC and MDL.IEEE Transactions on Information Theory, 52(3):1007–1016, 2006","cited_arxiv_id":null,"evidence_quote":"Supplies the finite-alphabet reverse Pinsker inequality used in Lemma E.1, which yields the eventual geometric last-iterate rates under strong monotonicity."}],"review_version":1}