On non-uniqueness in mean field games

Erhan Bayraktar; Xin Zhang

REVIEW 3 major objections 5 minor 15 references

On non-uniqueness in mean field games

T0 review · 3 major / 5 minor · reviewed 2026-08-14 · deepseek-v4-flash

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Pith's one-line read This paper proves that a two-state mean field game with a follow-the-crowd running cost can admit many equilibria when the background jump rate is below one half, and that at zero jump rate the finite-player game selects the equilibrium…

desk verdict The multiplicity analysis is genuinely new and solid; the finite-player selection claim is the weak link, resting on an unverified citation. read the letter →

arxiv 1908.06207 v2 pith:DFERQ4ZZ submitted 2019-08-16 math.PR math.OCq-fin.MF

Erhan Bayraktar , Xin Zhang This is my paper · ORCID

classification math.PRmath.OCq-fin.MF MSC 60F9960J2760K3593E20

keywords meanfieldgamesentropysolutionmasterequationNashequilibriumnon-uniquenesstwo-statemodeljumpprocessanti-monotonerunningcost

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

The paper studies a two-state, follow-the-crowd mean field game in which each player's jump rate is their control plus a common background rate η. It establishes that the mean field game equation has a unique solution when η ≥ 1/2 but can have many solutions when η < 1/2, with the number of solutions growing without bound as the time horizon lengthens for a range of initial majority fractions. It then shows that at η = 0, the Nash equilibria of the N+1-player game converge to the mean field equilibrium induced by the entropy solution of the master equation, provided the initial fraction is not exactly one half. This resolves a conjecture from an earlier paper on the same model and leaves the regime 0 < η < 1/2 as an explicit open problem.

What carries the argument

The reduction of the two-state MFG system to a single second-order equation with absolute-value nonlinearities, d²y/dt² + y − (1/2)y³ − 3η|y|y − 4η²y = 0, solved by characteristics through the implicit relation dt/dy = ±(G(y) + v²)^(−1/2). The same characteristic ODE produces the entropy solution of the one-dimensional scalar conservation law obtained from the master equation; the entropy solution has a shock along x = 0 for times beyond T₁(0+), and the Rankine-Hugoniot and Lax conditions verify that this piecewise smooth function is the selected object.

What would settle it

Solve the finite N+1-player HJB system at η = 0 for a small N and an initial majority fraction θ̄ ≠ 1/2 and check whether the empirical fraction ever crosses 1/2 and whether the value functions track the entropy branch. Alternatively, count the initial velocities v solving x_v(T) = 2θ̄ − 1 for fixed η < 1/2: Proposition 3.2 predicts this number grows without bound as T increases, so a bounded count at large T would falsify the multiplicity claim.

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Extended reading notes

Core claim

The central claim is that non-uniqueness in mean field games can be massive even in the simplest two-state model: when the background jump rate is below one half, the forward-backward MFG system has a family of solutions whose size grows with the time horizon. The paper reduces the system, via x = 2θ − 1 and y = u(t,1) − u(t,0), to a single second-order ordinary differential equation with absolute-value nonlinearities, and shows that solutions correspond to initial velocities v. For small η the associated curves x_v(t) oscillate, and each new oscillation contributes additional solutions as T grows. At η = 0, the paper further claims that the only equilibrium supported by the N+1-player game is the one obtained from the entropy solution of the master equation, which it constructs explicitly by characteristics and verifies through the Rankine-Hugoniot and Lax conditions.

Load-bearing premise

The selection claim depends on a step that is asserted rather than proved: the paper borrows a convergence theorem from a similar two-state model in Remark 5.1 without verifying its hypotheses here.

Editorial extensions

If this is right

For η ≥ 1/2, the paper's Proposition 3.1 gives uniqueness of the forward-backward MFG system, so monotonicity is not necessary for uniqueness once the background jump rate is high enough.
For η < 1/2 and initial fractions satisfying |2θ̄ − 1| < 1 − η² − η√(η² + 2), the number of MFG solutions can be made arbitrarily large by choosing a long enough horizon T.
When η = 0 and θ̄ ≠ 1/2, only the entropy-solution-induced mean field equilibrium is charged by the finite-player Nash equilibrium; the other MFG solutions are not limits of the N-player game.
The finite-player empirical fraction stays on one side of one half when η = 0, which is the mechanism that lets the convergence argument run (Proposition 5.1).
The paper leaves 0 < η < 1/2 open because crossing the half line introduces jump terms that the current argument cannot control.

Reading between the lines

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

If this selection mechanism is robust, similar entropy-selection should appear in other finite-state anti-monotone mean field games whenever the master equation has a genuinely nonlinear conservation-law structure and shocks form.
The open interval 0 < η < 1/2 may be approachable by adding a small common noise and letting it vanish, in analogy with linear-quadratic models where common noise restores uniqueness; this would give a testable selection principle.
The unbounded multiplicity at small η means numerical value or policy iteration for these games can lock onto different mean field equilibria depending on initialization, even for short horizons; the paper's count formula supplies a benchmark for such solvers.

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Editorial analysis

A structured set of objections, weighed in public.

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

Desk Editor's Note

read the letter

What you should know: the multiplicity result is the real substance here, and it's good. The selection claim in the title and abstract is not fully proved; the paper delegates the key convergence step to a citation without verifying hypotheses.

The paper studies a two-state mean field game with follow-the-crowd running cost and a background jump rate η. When η < 1/2, the MFG equation can have arbitrarily many solutions as the horizon grows. That's genuinely new and extends [6]'s at-most-three result. The reduction to a planar ODE and the careful analysis of x_v(t) in Section 3 is solid; Lemma 3.3 gives a clean parametrization of non-intersecting characteristic curves. The entropy solution construction in Section 4 is also convincing: they write down a piecewise C1 function, check Rankine-Hugoniot and Lax conditions, and the shock at x=0 is handled cleanly.

Now the soft spot. The paper's punchline—that the finite-player game selects the entropy-solution equilibrium when η=0—depends on Section 5. Proposition 5.1 shows that rational players follow the majority, which is a nice monotonicity result. But Remark 5.1 then asserts convergence of the value functions by citing [6, Theorem 8] without stating the theorem or checking its hypotheses. The model here is a running-cost MFG, while [6] is a terminal-condition model, and the entropy solution is different. If the citation doesn't apply, the selection claim has no proof. The abstract's phrase 'resolve a conjecture' is stronger than what the body establishes, especially since the θ̄=1/2 case is left to a heuristic equal-probability claim and η∈(0,1/2) is left open. These are real gaps, but they are confined to the selection story. The multiplicity analysis stands on its own.

I'd send this to a serious referee. The main fix is either proving the convergence directly or carefully verifying that [6, Theorem 8] covers this setting. The math in Sections 3–4 is careful and the citation pattern looks honest; the only issue is an overreach in how the selection result is advertised.

This paper is for people working on non-unique mean field games and entropy selection. I would cite it for the multiplicity result, and I'd bring it to a reading group—the selection gap is a good discussion point.

Referee Report

3 major / 5 minor

Summary. The paper studies an (N+1)-player two-state mean field game with transition rate equal to the player's control plus a background jump rate eta, and with an anti-monotone running cost f(i,theta)=|1-theta-i|. It reduces the mean field game system to a planar ODE and proves that the forward-backward system has a unique solution for eta>=1/2, while for eta<1/2 it can have arbitrarily many solutions as the horizon grows. It constructs the entropy solution of the associated master equation by characteristics, verifies the Rankine-Hugoniot and Lax conditions, and then, for eta=0, argues that the finite-player Nash equilibrium stays on the initial side of theta=1/2 and that the finite-player value functions converge to the value function induced by the entropy solution. The abstract claims that this selects the entropy-solution equilibrium when eta=0, resolving the conjecture of Hajek and Livesay.

Significance. If the selection step is supplied, the paper resolves the two-state anti-monotone mean field game conjecture by exhibiting unbounded non-uniqueness below the jump-rate threshold and identifying the selected equilibrium in the vanishing-eta limit. The ODE reduction in Section 3 and the entropy construction in Section 4 are detailed, internally consistent, and parameter-free: the threshold eta=1/2, the critical initial fraction 1-eta^2-eta*sqrt(eta^2+2), and the counting formula in Proposition 3.2 are derived rather than fitted, and the entropy solution is checked against the Rankine-Hugoniot and Lax conditions. These parts provide concrete, falsifiable predictions about the number of MFG solutions. The main weakness is that the finite-player selection claim in Section 5 is not actually proved; it is delegated to an unstated citation theorem whose hypotheses are not verified.

major comments (3)

[Section 5, Remark 5.1] The central selection claim is not established as written. The only bridge from the finite-player game to the entropy-solution master equation is the sentence "it can be easily seen that V^{N+1}(t,1,theta) converges to U(t,1,theta) if theta≠1/2 (see e.g. [6, Theorem 8])". The cited theorem is neither stated nor are its hypotheses checked for the present model, which has a running cost f(i,theta)=|1-theta-i| and a state-dependent flux in (4.1), whereas the paper's own description of [6] concerns an anti-monotone terminal condition. Moreover, the mode of convergence is left unspecified (pointwise in theta, uniform in t, along which grid of theta?). Since this is the only argument connecting the finite-player Nash equilibria of Section 5 with the entropy solution of Section 4, the claim that the entropy-solution-induced MFG equilibrium is selected in the N-player limit is unproved. The propagation-of-chaos assertion in the same remark is likewise asserted without proof.
[Abstract and Section 6] The abstract states that only the entropy-solution equilibrium is charged "when eta=0", but the argument in Section 5 assumes the initial fraction satisfies theta-bar≠1/2, and Section 6 explicitly says that at theta-bar=1/2 the two solutions are charged with equal probability. The abstract should include the noncritical initial-condition caveat; as written it overstates the result. Section 6 also attributes the resolved conjecture to [7] while Section 1 attributes it to [10]; the two statements should be reconciled.
[Section 5, after Proposition 5.1] Even apart from the citation issue, Proposition 5.1 only establishes the sign of Y^{N+1}(t,theta) at fixed grid points theta, and the proof is written only for even N with the remaining cases dismissed as "similarly". The step from this sign condition to the statement that the random empirical fraction theta^{N+1}(t) stays on one side of 1/2 pathwise should be spelled out, as this pathwise monotonicity is used to avoid the shock region in the convergence argument.

minor comments (5)

[Eq. (3.5)] In the definition of x_v(t), the term "2 eta y_v(T)" should read "2 eta y_v(t)"; as printed the formula mixes t and the final horizon T.
[Lemma 3.2] The displayed definition H(v):=∫_{y(v)}^{v} dz/sqrt(G(z)+v^2) has the integration limits reversed: since y(v)≥v, this integral is negative, while H(v) is later used as a positive time increment. The intended definition appears to be H(v)=∫_{v}^{y(v)} dz/sqrt(G(z)+v^2), and the proof's change of variables rewrites it with the opposite orientation.
[Section 4, Proposition 4.1] The paragraph before Proposition 4.1 says the shock curve is taken to be gamma(t)=0 for all t in R_+, while the proposition states the shock exists for t>T_1(0+). Since the proof shows Y is continuous across x=0 for t≤T_1(0+), the proposition's statement is the correct one and the earlier sentence should be adjusted.
[Section 5] The sentence "the system can be uniquely solved with terminal condition V^{N+1}(T,0,theta)=0" is inaccurate: the terminal condition in (HJB) is V(T,i,theta)=0 for both i=0,1, and the displayed system in (5.1) is for V(t,1,theta). The phrase should read V^{N+1}(T,1,theta)=0 (with the symmetric terminal data understood).
[General] There are several presentation slips: in the abstract "We also prove that that although" has a doubled "that"; Section 2 has "It is can be easily seen"; and Section 6's "conjecture of [7]" conflicts with the conjecture attribution in Section 1. The reference inconsistency should be fixed before publication.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the non-uniqueness, entropy-solution, and selection arguments are either derived in-paper or rest on an external, non-self cited theorem.

full rationale

The MFG non-uniqueness result (Section 3) is a self-contained ODE analysis: the auxiliary quantities x, y, G, v0, T(v), and H(v) are defined from the model, no parameter is fitted to a target answer, and Propositions 3.1 and 3.2 derive uniqueness for eta>=1/2 and multiplicity for eta<1/2 from explicit monotonicity of x_v(T). The entropy solution (Section 4) is not defined as the N-player limit: the authors construct Y(x,t) by characteristics and verify the Rankine-Hugoniot and Lax conditions explicitly, so the entropic status is checked against an independent criterion. Remark 4.1's 'easily seen' correspondence to an ME solution is a routine identification, not an assumption equivalent to the conclusion. The selection step (Remark 5.1) cites [6, Theorem 8] for convergence of V^{N+1} to U; [6] is authored by Cecchin, Pra, Fischer, and Pelino, none of whom are the present authors, so this is external support rather than a self-citation chain. The fact that the hypotheses of [6, Theorem 8] are not verified for this specific running cost is a legitimate correctness/rigor concern, but it is not circularity: the paper does not define U to match V^{N+1}, and the cited theorem is not used as an unexamined self-referential premise. The only self-citation ([1], used for background on the master equation and convergence under monotonicity) is not load-bearing for the novel claims. Honest limitations are stated for eta in (0,1/2) and theta_bar=1/2, which further supports that the paper does not smuggle its conclusions into its assumptions.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

No free parameters are fitted; the jump rate is an exogenous model parameter and all thresholds such as v0 and the critical initial fraction are derived. The main unproved inputs are standard conservation-law and ODE theory, plus the imported convergence theorem [6, Theorem 8]. No invented entities are introduced.

assumptions (5)

domain assumption The finite-N HJB system (5.1) has a unique solution and gives a unique Nash equilibrium.
Invoked in Section 5; relies on standard finite-state MFG theory from [1], [5], [9] rather than being proved in this paper.
domain assumption The convergence theorem [6, Theorem 8] transfers to this model and yields convergence of the finite-player value function to the entropy solution outside the shock when the initial fraction is not one half.
Section 5, Remark 5.1 makes this the load-bearing selection step with the words it can be easily seen; the hypotheses are not verified.
domain assumption The entropy solution of the scalar conservation law (4.1) is the correct selection criterion for the N-player limit.
The paper verifies Rankine-Hugoniot and Lax conditions but does not derive entropy admissibility from game-theoretic first principles; it is imported from [6].
standard math Standard existence and uniqueness theory for scalar conservation laws applies to (4.1).
Used implicitly in Proposition 4.1 and Remark 4.1 to talk about the unique entropy solution.
standard math The ODE (3.4) has unique C1 solutions depending continuously on initial velocity v.
Used in Lemma 3.1 and Lemma 3.3; standard but not stated as a theorem in the paper.

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Cite this review

Pith. "Pith review of On non-uniqueness in mean field games." pith.science (2026). https://pith.science/paper/DFERQ4ZZ

@misc{pith2026190806207,
  author       = {Pith},
  title        = {Pith review of: On non-uniqueness in mean field games},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/DFERQ4ZZ}},
  note         = {Machine review of arXiv:1908.06207}
}

abstract

We analyze an $N+1$-player game and the corresponding mean field game with state space $\{0,1\}$. The transition rate of $j$-th player is the sum of his control $\alpha^j$ plus a minimum jumping rate $\eta$. Instead of working under monotonicity conditions, here we consider an anti-monotone running cost. We show that the mean field game equation may have multiple solutions if $\eta < \frac{1}{2}$. We also prove that that although multiple solutions exist, only the one coming from the entropy solution is charged (when $\eta=0$), and therefore resolve a conjecture of ArXiv: 1903.05788.

Figures

Figures reproduced from arXiv: 1908.06207 by the authors.

**Figure 1.** Characteristic curves, η = 0.1, T = 3 on the left; η = 0.6, T = 1 on the right. Proof. It is clear that the function Y (x, t) is C 1 outside the shock curve, and we only need to check the Rankine-Hugoniot condition and the Lax condition (see [6, Proposition 3]). Define Y+(t) := lim x↓0 Y (x, t), Y− := lim x↑0 Y (x, t). If t > T1(0+), there exists a v > 0 such that t = T1(v) since v 7→ T1(v) is increasing to +∞ as v … view at source ↗

Discussion (0). Continue with ORCID to comment.

Reference graph

Works this paper leans on

15 extracted references · 15 canonical work pages

[6]

Cecchin, P

A. Cecchin, P. D. Pra, M. Fischer, and G. Pelino , On the Convergence Problem in Mean Field Games: A Two State Model without Uniqueness , SIAM J. Control Optim., 57 (2019), pp. 2443–2466

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Delarue and R

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[10]

On non-unique solutions in mean field games

B. Hajek and M. Livesay , On non-unique solutions in mean ﬁeld games , arXiv e-prints, (2019), p. arXiv:1903.05788

work page Pith review arXiv 2019
[1]

Bayraktar and A

E. Bayraktar and A. Cohen , Analysis of a ﬁnite state many player game using its master eq uation, SIAM J. Control Optim., 56 (2018), pp. 3538–3568

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Cardaliaguet, F

P. Cardaliaguet, F. Delarue, J.-M. Lasry, and P.-L. Lions , The master equation and the convergence problem in mean ﬁeld games , vol. 201 of Annals of Mathematics Studies, Princeton Unive rsity Press, Princeton, NJ, 2019

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Carmona and F

R. Carmona and F. Delarue , Probabilistic theory of mean ﬁeld games with applications. I, vol. 83 of Probability Theory and Stochastic Modelling, Springer, Cham, 2018. Mea n ﬁeld FBSDEs, control, and games

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Cecchin and G

A. Cecchin and G. Pelino , Convergence, ﬂuctuations and large deviations for ﬁnite st ate mean ﬁeld games via the master equation , Stochastic Process. Appl., 129 (2019), pp. 4510–4555

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J.-M. Lasry and P.-L. Lions , Mean ﬁeld games , Jpn. J. Math., 2 (2007), pp. 229–260. Department of Mathematics, University of Michigan E-mail address : erhan@umich.edu Department of Mathematics, University of Michigan E-mail address : zxmars@umich.edu

2007

Pith tools

Reviewed August 14, 2026 · model on record in the stance chip above.

[1] [6]

Cecchin, P

A. Cecchin, P. D. Pra, M. Fischer, and G. Pelino , On the Convergence Problem in Mean Field Games: A Two State Model without Uniqueness , SIAM J. Control Optim., 57 (2019), pp. 2443–2466

work page 2019

[2] [7]

Delarue and R

F. Delarue and R. Foguen Tchuendom , Selection of equilibria in a linear quadratic mean-ﬁeld gam e, Sto- chastic Process. Appl., 130 (2020), pp. 1000–1040

work page 2020

[3] [10]

On non-unique solutions in mean field games

B. Hajek and M. Livesay , On non-unique solutions in mean ﬁeld games , arXiv e-prints, (2019), p. arXiv:1903.05788

work page Pith review arXiv 2019

[4] [1]

Bayraktar and A

E. Bayraktar and A. Cohen , Analysis of a ﬁnite state many player game using its master eq uation, SIAM J. Control Optim., 56 (2018), pp. 3538–3568

work page 2018

[5] [2]

Cardaliaguet, F

P. Cardaliaguet, F. Delarue, J.-M. Lasry, and P.-L. Lions , The master equation and the convergence problem in mean ﬁeld games , vol. 201 of Annals of Mathematics Studies, Princeton Unive rsity Press, Princeton, NJ, 2019

work page 2019

[6] [3]

Carmona and F

R. Carmona and F. Delarue , Probabilistic theory of mean ﬁeld games with applications. I, vol. 83 of Probability Theory and Stochastic Modelling, Springer, Cham, 2018. Mea n ﬁeld FBSDEs, control, and games

work page 2018

[7] [4]

, Probabilistic theory of mean ﬁeld games with applications. II, vol. 84 of Probability Theory and Stochastic Modelling, Springer, Cham, 2018. Mean ﬁeld games with commo n noise and master equations

work page 2018

[8] [5]

Cecchin and G

A. Cecchin and G. Pelino , Convergence, ﬂuctuations and large deviations for ﬁnite st ate mean ﬁeld games via the master equation , Stochastic Process. Appl., 129 (2019), pp. 4510–4555

work page 2019