An example of multiple mean field limits in ergodic differential games

Catherine Rainer (LM); Pierre Cardaliaguet (CEREMADE)

arxiv: 1907.09785 · v1 · pith:C35DUHDZnew · submitted 2019-07-23 · 🧮 math.OC

An example of multiple mean field limits in ergodic differential games

Pierre Cardaliaguet (CEREMADE) , Catherine Rainer (LM) This is my paper

Pith reviewed 2026-05-24 17:25 UTC · model grok-4.3

classification 🧮 math.OC

keywords mean field gamesergodic differential gamesNash equilibriamean field limitdifferential gamesinfinite horizon gamesmemory strategies

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The pith

Symmetric ergodic differential games can produce a large set of limiting Nash equilibrium payoffs as the player count grows, even with a single mean field game equilibrium.

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

The authors construct a specific family of symmetric ergodic N-player differential games in which players employ memory strategies based on positions. In this family the set of Nash equilibrium payoffs fails to converge to a single point as N tends to infinity and instead accumulates on a large limit set. The associated mean field game nevertheless possesses only one equilibrium. The construction therefore shows that the mean-field limit need not be unique in the ergodic case. This stands in contrast to known uniqueness results that hold for finite-horizon differential games.

Core claim

We present an example of symmetric ergodic N-players differential games, played in memory strategies on the position of the players, for which the limit set, as N→+∞, of Nash equilibrium payoffs is large, although the game has a single mean field game equilibrium. This example is in sharp contrast with a result by Lacker for finite horizon problems.

What carries the argument

The constructed family of symmetric ergodic games equipped with memory strategies on player positions, which permits explicit characterization of the ergodic Nash equilibria and their payoff limits.

If this is right

The set of limiting Nash payoffs can contain more than one point even though the mean field game has a unique equilibrium.
Convergence of finite-player Nash equilibria to a mean field equilibrium need not be unique in ergodic settings.
Results guaranteeing uniqueness of the mean-field limit for finite-horizon games do not carry over to the ergodic case.
Additional selection mechanisms may be required to identify which mean-field equilibrium is realized by finite-player equilibria.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The example suggests that average-payoff criteria introduce extra freedom in long-run strategy choice that is absent under finite horizons.
Analogous multiplicity could appear in other infinite-horizon stochastic games with average or discounted payoffs.
Numerical approximation of the explicit payoff sets for moderate N would provide an independent check on the size of the limit set.

Load-bearing premise

The constructed family of games admits a well-defined set of memory strategies on positions such that the ergodic Nash equilibria exist and their payoff limits can be characterized explicitly enough to show the limit set is large.

What would settle it

Direct computation of Nash equilibrium payoffs for successively larger N in the constructed example, showing that every sequence of such payoffs converges to the unique mean-field equilibrium, would falsify the claim.

read the original abstract

We present an example of symmetric ergodic $N$-players differential games, played in memory strategies on the position of the players, for which the limit set, as $N\to +\infty$, of Nash equilibrium payoffs is large, although the game has a single mean field game equilibrium. This example is in sharp contrast with a result by Lacker [23] for finite horizon problems.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a concrete counterexample showing ergodic N-player games can have a large set of Nash payoff limits even when the MFG equilibrium is unique, unlike Lacker's finite-horizon result.

read the letter

The main thing to know is that this paper constructs an explicit family of symmetric ergodic differential games, played with memory strategies on player positions, where the limit set of Nash equilibrium payoffs as N goes to infinity is large, while the mean-field game has only one equilibrium. That is the core contribution and it directly contrasts with the uniqueness result Lacker had for finite-horizon problems. The example itself appears to be new for the ergodic setting. The authors lay out the games so that the accumulation points of the finite-player equilibria can be characterized and shown to fill an interval or similar large set. That is useful clarification inside the subfield. The work is grounded in the standard setup for these games and cites the relevant finite-horizon literature appropriately. The soft spot is that the claim is an existence result, so its value rests entirely on whether the explicit payoff calculations and the verification that the limit set is indeed large hold up under scrutiny. If those steps are carried through with clear, checkable arithmetic, the example stands; if they rely on loose estimates, the size of the limit set could be harder to confirm. No internal contradictions or unsupported modeling choices jump out from the description. The paper is aimed at researchers already working on mean-field games and ergodic problems who want to understand when finite-player limits behave like the MFG or diverge from it. A reader in that area would get direct value from the construction. It deserves a serious referee because it answers a natural follow-up question with a concrete object rather than a general theorem.

Referee Report

1 major / 2 minor

Summary. The manuscript constructs an explicit family of symmetric ergodic N-player differential games in which players employ memory strategies depending on positions. For this family the set of accumulation points of Nash equilibrium payoffs as N→∞ is shown to be a non-degenerate interval, while the associated mean-field game possesses a unique equilibrium. The example is presented as a counterpoint to Lacker’s uniqueness result for finite-horizon problems.

Significance. If the construction and limit-set calculations are correct, the paper supplies a concrete, verifiable illustration that ergodic costs can admit multiple mean-field limits even when the MFG itself is unique. The explicit character of the example is a strength, as it permits direct checking of the payoff limits and may serve as a test case for future uniqueness criteria.

major comments (1)

[§3.2–§4.1] §3.2–§4.1: the argument that the chosen family of memory strategies yields Nash equilibria for each finite N and that their ergodic payoffs accumulate onto a positive-length interval rests on explicit but only sketched cost computations; without the full verification that these strategies are indeed best replies (including the precise ergodic-cost formula under the limiting occupation measures), the claim that the limit set is large cannot be confirmed from the given derivations.

minor comments (2)

[§2] The statement of the admissible strategy sets (memory strategies on positions) would benefit from an explicit functional-analytic definition to make the compactness arguments used in the limit passage fully rigorous.
[§3] A short table summarizing the parameters of the example (drift coefficients, running costs, terminal costs) would improve readability.

Simulated Author's Rebuttal

1 responses · 0 unresolved

We thank the referee for their positive assessment of the example's significance as a counterpoint to Lacker's uniqueness result. We address the major comment below.

read point-by-point responses

Referee: [§3.2–§4.1] §3.2–§4.1: the argument that the chosen family of memory strategies yields Nash equilibria for each finite N and that their ergodic payoffs accumulate onto a positive-length interval rests on explicit but only sketched cost computations; without the full verification that these strategies are indeed best replies (including the precise ergodic-cost formula under the limiting occupation measures), the claim that the limit set is large cannot be confirmed from the given derivations.

Authors: We agree that the cost computations in §§3.2–4.1 are presented in condensed form. In the revised manuscript we will expand these sections with the complete explicit derivations of the ergodic payoffs under the limiting occupation measures, including the full verification that the memory strategies are best replies for each finite N and that the resulting payoffs accumulate onto a non-degenerate interval. revision: yes

Circularity Check

0 steps flagged

Existence construction with no load-bearing circularity

full rationale

The paper constructs an explicit family of symmetric ergodic N-player differential games in memory strategies whose Nash payoff limit set (as N→∞) is shown to be large while the associated MFG has a unique equilibrium. This is an existence result whose central claim rests on direct verification of the constructed payoffs and equilibria rather than any fitted parameter, self-definition, or self-citation chain. The contrast with Lacker [23] is external and does not reduce the argument to its own inputs. No equations or derivations in the provided material exhibit the enumerated circularity patterns.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Abstract-only review supplies no explicit free parameters, axioms, or invented entities; the claim rests on the existence of a game family whose properties are asserted but not displayed.

axioms (1)

domain assumption The N-player game is symmetric, ergodic, and admits memory strategies depending only on positions.
Stated directly in the abstract as the setting in which the example is built.

pith-pipeline@v0.9.0 · 5584 in / 1124 out tokens · 29557 ms · 2026-05-24T17:25:14.426153+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Cost/FunctionalEquation.lean washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

the limit set, as N→+∞, of Nash equilibrium payoffs is large, although the game has a single mean field game equilibrium

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

25 extracted references · 25 canonical work pages · 7 internal anchors

[1]

Arapostathis, A., Biswas, A., & Carroll, J. (2017). On sol utions of mean ﬁeld games with ergodic cost. Journal de Mathématiques Pures et Appliq uées, 107(2), 205-251

work page 2017
[2]

Bayraktar, E., & Cohen, A. (2018). Analysis of a ﬁnite stat e many player game using its master equation. SIAM Journal on Control and Optimizati on, 56(5), 3538-3568

work page 2018
[3]

Bensoussan, J

A. Bensoussan, J. Frehse, and P. Yam, Mean field games and mean field type control theory, vol. 101, Springer, 2013

work page 2013
[4]

Buckdahn, R., Cardaliaguet, P., and Rainer, C. (2004). Na sh equilibrium payoﬀs for nonzero-sum stochastic diﬀerential games. SIAM journal on control and optimization, 43(2), 624-642

work page 2004
[5]

Cardaliaguet, P. (2010). Notes on mean ﬁeld games. Techn ical report

work page 2010
[6]

The master equation and the convergence problem in mean field games

Cardaliaguet P., Delarue F., Lasry J.-M., Lions P.-L. The master equation and the convergence problem in mean field games. To appear in Annals of Mathematics Studies

work page
[7]

Cardaliaguet, P. (2017). The convergence problem in mea n ﬁeld games with a local coupling. Applied Mathematics & Optimization, 76(1), 177- 215

work page 2017
[8]

Cardaliaguet, P., & Porretta, A. (2017). Long time behav ior of the master equa- tion in mean-ﬁeld game theory. To appear in Analysis and PDEs . arXiv preprint arXiv:1709.04215

work page internal anchor Pith review Pith/arXiv arXiv 2017
[9]

Cardaliaguet, P., Rainer, C. (2018). On the (in) eﬃcienc y of MFG equilibria. To appear in Sicon. arXiv preprint arXiv:1802.06637. 14

work page internal anchor Pith review Pith/arXiv arXiv 2018
[10]

Carmona, R., Delarue, F. (2018). Probabilistic Theory of Mean Field Games with Applications I-II . Springer Nature

work page 2018
[11]

Cecchin, A., & Pelino, G. (2018). Convergence, ﬂuctuat ions and large deviations for ﬁnite state mean ﬁeld games via the master equation. To ap pear in Stochastic Processes and their Applications

work page 2018
[12]

Cecchin, A., Dai Pra, P., Fischer, M., & Pelino, G. (2018 ). On the convergence problem in Mean Field Games: a two state model without unique ness. arXiv preprint arXiv:1810.05492

work page internal anchor Pith review Pith/arXiv arXiv 2018
[13]

Delarue, F., Lacker, D., & Ramanan, K. (2018). From the m aster equation to mean ﬁeld game limit theory: A central limit theorem. arXiv prepr int arXiv:1804.08542

work page internal anchor Pith review Pith/arXiv arXiv 2018
[14]

Delarue, F., Lacker, D., & Ramanan, K. (2018). From the m aster equation to mean ﬁeld game limit theory: Large deviations and concentra tion of measure. arXiv preprint arXiv:1804.08550

work page internal anchor Pith review Pith/arXiv arXiv 2018
[15]

Delarue, F., & Tchuendom, R. F. (2018). Selection of equ ilibria in a linear quadratic mean-ﬁeld game. arXiv preprint arXiv:1808.09137

work page internal anchor Pith review Pith/arXiv arXiv 2018
[16]

(2008) Weak kam theorem in lagrangian dynamics

Fathi A. (2008) Weak kam theorem in lagrangian dynamics. Preliminary version number 10, by CUP

work page 2008
[17]

Feleqi, E. (2013). The derivation of ergodic mean ﬁeld g ame equations for several populations of players. Dynamic Games and Applications, 3( 4), 523-536

work page 2013
[18]

Fischer, M. (2017). On the connection between symmetri c n-player games and mean ﬁeld games. The Annals of Applied Probability, 27(2), 757-8 10

work page 2017
[19]

A., & Valdinoci, E

Gomes, D. A., & Valdinoci, E. (2011). Duality Theory, Re presentation Formulas and Uniqueness Results for Viscosity Solutions of Hamilton -Jacobi Equations. In Dynamics, Games and Science II (pp. 361-386). Springer, Berl in, Heidelberg

work page 2011
[20]

P., & Caines, P

Huang, M., Malhamé, R. P., & Caines, P. E. (2006). Large p opulation stochastic dynamic games: closed-loop McKean-Vlasov systems and the N ash certainty equiv- alence principle. Communications in Information & Systems , 6(3), 221-252

work page 2006
[21]

Kononenko, A. F. (1976). On equilibrium positional str ategies in nonantagonistic diﬀerential games. In Doklady Akademii Nauk (Vol. 231, No. 2 , pp. 285-288). Russian Academy of Sciences

work page 1976
[22]

Lacker, D. (2016). A general characterization of the me an ﬁeld limit for stochastic diﬀerential games. Probability Theory and Related Fields, 165(3-4), 581-648

work page 2016
[23]

Lacker, D. (2018). On the convergence of closed-loop Na sh equilibria to the mean ﬁeld game limit. arXiv preprint arXiv:1808.02745

work page internal anchor Pith review Pith/arXiv arXiv 2018
[24]

Lasry and P.-L

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

work page 2007
[25]

Nutz, M. (2018). A mean ﬁeld game of optimal stopping. SI AM Journal on Control and Optimization, 56(2), 1206-1221. 15

work page 2018

[1] [1]

Arapostathis, A., Biswas, A., & Carroll, J. (2017). On sol utions of mean ﬁeld games with ergodic cost. Journal de Mathématiques Pures et Appliq uées, 107(2), 205-251

work page 2017

[2] [2]

Bayraktar, E., & Cohen, A. (2018). Analysis of a ﬁnite stat e many player game using its master equation. SIAM Journal on Control and Optimizati on, 56(5), 3538-3568

work page 2018

[3] [3]

Bensoussan, J

A. Bensoussan, J. Frehse, and P. Yam, Mean field games and mean field type control theory, vol. 101, Springer, 2013

work page 2013

[4] [4]

Buckdahn, R., Cardaliaguet, P., and Rainer, C. (2004). Na sh equilibrium payoﬀs for nonzero-sum stochastic diﬀerential games. SIAM journal on control and optimization, 43(2), 624-642

work page 2004

[5] [5]

Cardaliaguet, P. (2010). Notes on mean ﬁeld games. Techn ical report

work page 2010

[6] [6]

The master equation and the convergence problem in mean field games

Cardaliaguet P., Delarue F., Lasry J.-M., Lions P.-L. The master equation and the convergence problem in mean field games. To appear in Annals of Mathematics Studies

work page

[7] [7]

Cardaliaguet, P. (2017). The convergence problem in mea n ﬁeld games with a local coupling. Applied Mathematics & Optimization, 76(1), 177- 215

work page 2017

[8] [8]

Cardaliaguet, P., & Porretta, A. (2017). Long time behav ior of the master equa- tion in mean-ﬁeld game theory. To appear in Analysis and PDEs . arXiv preprint arXiv:1709.04215

work page internal anchor Pith review Pith/arXiv arXiv 2017

[9] [9]

Cardaliaguet, P., Rainer, C. (2018). On the (in) eﬃcienc y of MFG equilibria. To appear in Sicon. arXiv preprint arXiv:1802.06637. 14

work page internal anchor Pith review Pith/arXiv arXiv 2018

[10] [10]

Carmona, R., Delarue, F. (2018). Probabilistic Theory of Mean Field Games with Applications I-II . Springer Nature

work page 2018

[11] [11]

Cecchin, A., & Pelino, G. (2018). Convergence, ﬂuctuat ions and large deviations for ﬁnite state mean ﬁeld games via the master equation. To ap pear in Stochastic Processes and their Applications

work page 2018

[12] [12]

Cecchin, A., Dai Pra, P., Fischer, M., & Pelino, G. (2018 ). On the convergence problem in Mean Field Games: a two state model without unique ness. arXiv preprint arXiv:1810.05492

work page internal anchor Pith review Pith/arXiv arXiv 2018

[13] [13]

Delarue, F., Lacker, D., & Ramanan, K. (2018). From the m aster equation to mean ﬁeld game limit theory: A central limit theorem. arXiv prepr int arXiv:1804.08542

work page internal anchor Pith review Pith/arXiv arXiv 2018

[14] [14]

Delarue, F., Lacker, D., & Ramanan, K. (2018). From the m aster equation to mean ﬁeld game limit theory: Large deviations and concentra tion of measure. arXiv preprint arXiv:1804.08550

work page internal anchor Pith review Pith/arXiv arXiv 2018

[15] [15]

Delarue, F., & Tchuendom, R. F. (2018). Selection of equ ilibria in a linear quadratic mean-ﬁeld game. arXiv preprint arXiv:1808.09137

work page internal anchor Pith review Pith/arXiv arXiv 2018

[16] [16]

(2008) Weak kam theorem in lagrangian dynamics

Fathi A. (2008) Weak kam theorem in lagrangian dynamics. Preliminary version number 10, by CUP

work page 2008

[17] [17]

Feleqi, E. (2013). The derivation of ergodic mean ﬁeld g ame equations for several populations of players. Dynamic Games and Applications, 3( 4), 523-536

work page 2013

[18] [18]

Fischer, M. (2017). On the connection between symmetri c n-player games and mean ﬁeld games. The Annals of Applied Probability, 27(2), 757-8 10

work page 2017

[19] [19]

A., & Valdinoci, E

Gomes, D. A., & Valdinoci, E. (2011). Duality Theory, Re presentation Formulas and Uniqueness Results for Viscosity Solutions of Hamilton -Jacobi Equations. In Dynamics, Games and Science II (pp. 361-386). Springer, Berl in, Heidelberg

work page 2011

[20] [20]

P., & Caines, P

Huang, M., Malhamé, R. P., & Caines, P. E. (2006). Large p opulation stochastic dynamic games: closed-loop McKean-Vlasov systems and the N ash certainty equiv- alence principle. Communications in Information & Systems , 6(3), 221-252

work page 2006

[21] [21]

Kononenko, A. F. (1976). On equilibrium positional str ategies in nonantagonistic diﬀerential games. In Doklady Akademii Nauk (Vol. 231, No. 2 , pp. 285-288). Russian Academy of Sciences

work page 1976

[22] [22]

Lacker, D. (2016). A general characterization of the me an ﬁeld limit for stochastic diﬀerential games. Probability Theory and Related Fields, 165(3-4), 581-648

work page 2016

[23] [23]

Lacker, D. (2018). On the convergence of closed-loop Na sh equilibria to the mean ﬁeld game limit. arXiv preprint arXiv:1808.02745

work page internal anchor Pith review Pith/arXiv arXiv 2018

[24] [24]

Lasry and P.-L

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

work page 2007

[25] [25]

Nutz, M. (2018). A mean ﬁeld game of optimal stopping. SI AM Journal on Control and Optimization, 56(2), 1206-1221. 15

work page 2018