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arxiv: 1907.10968 · v1 · pith:AIRCRWLHnew · submitted 2019-07-25 · 🧮 math.OC · math.PR

Submodular Mean Field Games: Existence and Approximation of Solutions

Jodi Dianetti , Giorgio Ferrari , Markus Fischer , Max Nendel This is my paper

Pith reviewed 2026-05-24 16:10 UTC · model grok-4.3

classification 🧮 math.OC math.PR
keywords mean field gamessubmodular costsTarski fixed point theorembest response iterationlattice of equilibriaexistence of solutionsrelaxed controlscommon noise
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The pith

Submodular costs in mean field games guarantee existence via Tarski's theorem and yield minimal and maximal solutions reachable by best-response iteration.

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

The authors consider mean field games driven by scalar Itô dynamics whose costs satisfy a submodularity condition with respect to a partial order on the state space and the space of probability measures. This single structural assumption lets them invoke Tarski's fixed-point theorem to obtain existence of equilibria even when the cost jumps discontinuously in the measure variable. The same assumption implies that the set of all equilibria forms a lattice, so a smallest and a largest solution exist, and that both extremes are recovered by iterating the best-response map. The argument is first given for ordinary controls, then extended to relaxed controls and to a class of games with common noise.

Core claim

When the running and terminal costs are submodular with respect to a suitable order relation on the product of the state space and the space of probability measures, the mean-field game admits at least one solution; moreover the collection of all solutions is a complete lattice, hence possesses a minimal and a maximal element, and these two distinguished solutions can be constructed by monotone iteration of the best-response operator.

What carries the argument

The submodularity assumption on the costs with respect to an order relation on the state-measure space, which makes the best-response map monotone and thereby permits direct application of Tarski's fixed-point theorem.

If this is right

  • Existence holds without any continuity requirement on the cost with respect to the measure variable.
  • The equilibrium set is a complete lattice under the natural pointwise order.
  • The minimal and maximal equilibria are obtained by iterating the best-response map from appropriate initial points.
  • The same lattice and iteration results apply to the relaxed-control formulation and to games with common noise.

Where Pith is reading between the lines

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

  • The monotone iteration may serve as a practical numerical scheme for locating extremal equilibria when the state space is moderate.
  • The lattice ordering could be used to compare welfare or cost properties across different equilibria in applied models.
  • Analogous monotonicity arguments might extend the existence and approximation results to certain non-mean-field stochastic games.

Load-bearing premise

The costs must be submodular with respect to the chosen order relation on states and measures.

What would settle it

A concrete mean-field game whose costs are submodular yet whose set of equilibria fails to contain both a minimal and a maximal element, or for which iteration of the best-response map does not converge to those extremes.

read the original abstract

We study mean field games with scalar It{\^o}-type dynamics and costs that are submodular with respect to a suitable order relation on the state and measure space. The submodularity assumption has a number of interesting consequences. Firstly, it allows us to prove existence of solutions via an application of Tarski's fixed point theorem, covering cases with discontinuous dependence on the measure variable. Secondly, it ensures that the set of solutions enjoys a lattice structure: in particular, there exist a minimal and a maximal solution. Thirdly, it guarantees that those two solutions can be obtained through a simple learning procedure based on the iterations of the best-response-map. The mean field game is first defined over ordinary stochastic controls, then extended to relaxed controls. Our approach allows also to treat a class of submodular mean field games with common noise in which the representative player at equilibrium interacts with the (conditional) mean of its state's distribution.

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.

Referee Report

2 major / 2 minor

Summary. The paper studies mean field games with Itô dynamics and costs submodular w.r.t. a suitable order on the state-measure space. It claims existence of equilibria via Tarski's fixed-point theorem applied to the best-response map (covering discontinuous measure dependence), shows the solution set forms a lattice with minimal and maximal elements, and proves these extremal solutions are obtained by iterating the best-response map from bottom/top. Results are first stated for ordinary controls, then extended to relaxed controls and to a class of common-noise games with conditional-mean interaction.

Significance. If the order construction and monotonicity hold, the result supplies an existence proof for MFGs with discontinuous interactions (where standard Schauder or Banach arguments may fail) together with a lattice structure and a constructive approximation procedure. The explicit use of Tarski plus best-response iteration is a methodological strength that could be useful in applications where submodularity is verifiable.

major comments (2)
  1. [§3] §3 (order construction): the claim that the chosen partial order turns the space of probability measures into a complete lattice (required for Tarski) is load-bearing; the verification that suprema/infima exist and are attained within the admissible set must be explicit, especially when the order is induced by the submodularity assumption.
  2. [§5] §5 (relaxed controls): the extension asserts that the best-response map remains monotone for relaxed controls, but the argument that submodularity is preserved under relaxation (and that the order on relaxed controls is still a complete lattice) is central to the extension claim and requires a self-contained check; without it the relaxed-control result does not follow automatically from the ordinary-control case.
minor comments (2)
  1. [§2] Notation for the order relation ≼ on measures (introduced in §2) could be accompanied by a short example illustrating how submodularity translates into monotonicity of the best-response map.
  2. [§6] The common-noise section would benefit from an explicit statement of the filtration and conditional-expectation operator used in the interaction term.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the positive evaluation and the recommendation of minor revision. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [§3] §3 (order construction): the claim that the chosen partial order turns the space of probability measures into a complete lattice (required for Tarski) is load-bearing; the verification that suprema/infima exist and are attained within the admissible set must be explicit, especially when the order is induced by the submodularity assumption.

    Authors: We agree that an explicit verification strengthens the argument. In the revised manuscript we will expand the relevant paragraphs in §3 to construct the supremum and infimum of an arbitrary subset of probability measures under the given partial order, verify that these extrema remain admissible, and confirm that the resulting structure is a complete lattice. revision: yes

  2. Referee: [§5] §5 (relaxed controls): the extension asserts that the best-response map remains monotone for relaxed controls, but the argument that submodularity is preserved under relaxation (and that the order on relaxed controls is still a complete lattice) is central to the extension claim and requires a self-contained check; without it the relaxed-control result does not follow automatically from the ordinary-control case.

    Authors: We accept that a self-contained argument is needed. The revision will add a dedicated paragraph (or short appendix) that directly verifies preservation of submodularity under relaxation and shows that the space of relaxed controls, equipped with the induced order, remains a complete lattice; monotonicity of the best-response map will then be re-established from these properties. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper is a pure existence proof that invokes Tarski's fixed-point theorem on the best-response map once submodularity (an explicit modeling assumption) supplies monotonicity with respect to a suitably ordered complete lattice of measures. The derivation therefore rests on an external, independently verifiable theorem and on standard stochastic-control notions rather than any self-definition, fitted-parameter renaming, or load-bearing self-citation chain. No equation or step reduces the claimed result to its own inputs by construction.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claims rest on the domain assumption of submodularity and the invocation of Tarski's fixed point theorem; no free parameters or invented entities are introduced.

axioms (1)
  • standard math Tarski's fixed point theorem
    Invoked to obtain existence from the submodularity assumption on the best-response map.

pith-pipeline@v0.9.0 · 5691 in / 1326 out tokens · 27997 ms · 2026-05-24T16:10:57.276862+00:00 · methodology

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Reference graph

Works this paper leans on

33 extracted references · 33 canonical work pages

  1. [1]

    Adlakha, S., & Johari, R. (2013). Mean field equilibrium in dynamic games with strategic compl e- mentarities. Operations Research, 61(4), 971-989

    work page 2013

  2. [2]

    Amir, R. (1996). Cournot oligopoly and the theory of supermodular games. Gam es and Economics Behavior, 15(2), 132-148

    work page 1996

  3. [3]

    Amir, R. (2005). Supermodularity and complementarity in Economics: an elem entary survey. Southern Economic Journal, 71(3), 636-660

    work page 2005

  4. [4]

    Bardi, M., & Fischer, M. (2018) . On non-uniqueness and uniqueness of solutions in finite-ho rizon mean field games. ESAIM: Control, Optimisation and Calculus of Variations, pre-published on-line, https://doi.org/10.1051/cocv/2018026

    work page doi:10.1051/cocv/2018026 2018

  5. [5]

    Bensoussan, A., Sung, K.C.J., Yam, S.C.P., & Yung, S.P. (201 6). Linear-quadratic mean field games. Journal of Optimization Theory and Applications, 16 9(2), 496-529

    work page

  6. [6]

    Cardaliaguet, P. (2013). Notes on mean field games. Technical report, Universit´ e de P aris - Dauphine

    work page 2013

  7. [7]

    Cardaliaguet, P., & Hadikhanloo, S. (2017). Learning in mean field games: the fictitious play. ESAIM: Control, Optimisation and Calculus of Variations, 2 3(2), 569-591

    work page 2017

  8. [8]

    Carmona, R. (2016). Lectures on BSDEs, stochastic control, and stochastic diffe rential games with financial applications (Vol. 1). SIAM

    work page 2016

  9. [9]

    Carmona, R., & Delarue, F. (2018). Probabilistic theory of mean field games with applications. Volumes 83 and 84 of Probability Theory and Stochastic Model ing. Springer, Cham

    work page 2018

  10. [10]

    Carmona, R., Delarue, F., & Lacker, D. (2017). Mean field games of timing and models for bank runs. Applied Mathematics & Optimization, 76(1), 217-260

    work page 2017

  11. [11]

    Carmona, R., Delarue, F., & Lachapelle, A. (2013). Control of McKean–Vlasov dynamics versus mean field games. Mathematics and Financial Economics, 7(2) , 131-166

    work page 2013

  12. [12]

    Cecchin, A., Dai Pra, P., Fischer, M., & Pelino, G. (2019). On the convergence problem in mean field games: a two state model without uniqueness. SIAM J ournal on Control Optimization, 57(4), 2443–2466

    work page 2019

  13. [13]

    Chandra, T.K. (2015). De La Vall´ ee Poussin’s theorem, uniform integrability, ti ghtness and moments. Statistics & Probability Letters, 107, 136-141

    work page 2015

  14. [14]

    Delarue, F., & Tchuendom, R.F. (2019). Selection of equilibria in a linear quadratic mean-field gam e. Stochastic Processes and their Applications, in press

    work page 2019

  15. [15]

    El Karoui, N., Du Huu, N., & Jeanblanc-Picqu ´e, M. (1987). Compactification methods in the control of degenerate diffusions: existence of an optimal co ntrol. Stochastics, 20(3), 169-219

    work page 1987

  16. [16]

    Fuhrman, M., & Tessitore, G. (2004) . Existence of optimal stochastic controls and global solut ions of forward-backward stochastic differential equations. SI AM Journal on Control and Optimization, 43(3), 813-830

    work page 2004

  17. [17]

    Haussmann, U.G., & Lepeltier, J.P. (1990). On the existence of optimal controls. SIAM Journal on Control and Optimization, 28(4), 851-902

    work page 1990

  18. [18]

    Hofbauer, J., & Sandholm, W.H. (2002). On the global convergence of stochastic fictitious play. Econometrica, 70(6), 2265-2294. 22 DIANETTI, FERRARI, FISCHER, AND NENDEL

    work page 2002

  19. [19]

    Huang, M., Malham ´e, R.P., & Caines, P.E. (2006). Large population stochastic dynamic games: Closed-loop McKean-Vlasov systems and the Nash certainty e quivalence principle. Communications in Information and Systems, 6(3), 221-252

    work page 2006

  20. [20]

    Lacker, D. (2015). Mean field games via controlled martingale problems: Existe nce of Markovian equi- libria. Stochastic Processes and Their Applications, 125( 7), 2856-2894

    work page 2015

  21. [21]

    Lasry, J.-M., & Lions, P.-L. (2007). Mean field games. Japanese Journal of Mathematics, 2(1), 229 -260

    work page 2007

  22. [22]

    Leskel¨a, L., & Vihola, M. (2013). Stochastic order characterization of uniform integrabili ty and tight- ness. Statistics & Probability Letters, 83(1), 382-389

    work page 2013

  23. [23]

    Milgrom, P., & Roberts, J. (1990). Rationalizability, learning, and equilibrium in games wit h strategic complementarities. Econometrica, 58(6), 1255-1277

    work page 1990

  24. [24]

    Nendel, M. (2019). A note on stochastic dominance and compactness. In preparat ion

    work page 2019

  25. [25]

    Shaked, M., & Shanthikumar, J.G. (2007). Stochastic orders. Springer Science & Business Media

    work page 2007

  26. [26]

    Tarski, A. (1955). A lattice-theoretical fixpoint theorem and its application s. Pacific Journal of Mathe- matics, 5(2), 285-309

    work page 1955

  27. [27]

    Tchuendom, R.F. (2018). Uniqueness for linear-quadratic mean field games with commo n noise. Dy- namic Games and Applications, 8(1), 199-210

    work page 2018

  28. [28]

    Topkis, D.M. (1979). Equilibrium points in nonzero-sum n-person submodular gam es. SIAM Journal on Control and Optimization, 17(6), 773-787

    work page 1979

  29. [29]

    Topkis, D.M. (2011). Supermodularity and complementarity. Princeton Universi ty Press

    work page 2011

  30. [30]

    Vives, X. (1990). Nash equilibrium with strategic complementarities. Journ al of Mathematical Econom- ics, 19(3), 305-321

    work page 1990

  31. [31]

    Vives, X. (2001). Oligopoly pricing: old ideas and new tools. MIT press

    work page 2001

  32. [32]

    (2017).Total reward semi-Markov mean field games with complementar ity properties

    Wie ¸cek, P. (2017).Total reward semi-Markov mean field games with complementar ity properties. Dy- namic Games and Applications, 7, 507-529

    work page 2017

  33. [33]

    Yong, J., & Zhou, X.Y. (1999). Stochastic controls: Hamiltonian systems and HJB equation s (Vol. 43). Springer Science & Business Media. J. Dianetti: Center for Mathematical Economics (IMW), Biel efeld University, Universit ¨atsstrasse 25, 33615, Bielefeld, Germany E-mail address : jodi.dianetti@uni-bielefeld.de G. Ferrari: Center for Mathematical Economi...

    work page 1999