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Limit Theory for $N$-Player $\alpha$-Potential Games

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Pith's one-line read Normalized N-player α-potential functions converge to a mean-field control problem whose objective is itself a potential for the limiting MFG when α_N vanishes.

desk verdict Solid limit theory that cleanly turns vanishing α_N into potential MFGs and gives PoC under common noise and non-separable costs. read the letter →

arxiv 2606.09815 v2 pith:FR3WRETZ submitted 2026-06-08 math.OC math.PR

Xin Guo , Meng Wang , Yufei Zhang This is my paper · ORCID
classification math.OCmath.PR MSC 60Fxx91A0691A1491A1591A16
keywords α-potentialgamepotentialmeanfieldcontrolmeasure-valuedPoincarélemmaWassersteinspacepropagationofchaos
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 shows that α-potential games, which turn approximate Nash search into minimization of one function, become potential mean-field games in the large-population limit. After dividing by N, both the optimal values and the approximate minimizers of the finite-player α_N-potential functions converge to those of a mean-field control problem that uses measure-valued controls. The condition that α_N itself tends to zero is equivalent to the classical closedness conditions that make an MFG potential; under that condition the limiting control objective serves as a potential function for the MFG. The technical backbone is a Poincaré lemma on Wasserstein space that reconstructs the potential by path integrals of the cost derivatives. The same limit also yields propagation of chaos for controlled diffusions that may include common noise and non-separable control interactions, giving a systematic route from finite-player games to potential MFGs.

What carries the argument

The Poincaré lemma on Wasserstein space (Theorems 4.1–4.2 and Proposition 5.1): closed differential forms built from the cost Hessians are exact, so the mean-field potential is recovered by path integrals of the cost derivatives along absolutely continuous curves of measures.

What would settle it

Construct an explicit family of N-player costs whose Hessians remain asymmetrically large so that α_N stays bounded away from zero, yet the normalized potentials still converge to an MFC objective that is a potential for the limiting MFG; any such example would break the claimed equivalence.

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Extended reading notes

Core claim

Both the optimal values and the minimizers of the normalized N-player α_N-potential functions converge to those of a mean-field control problem with measure-valued controls; moreover lim α_N = 0 is equivalent to the standard closedness conditions for potential MFGs, and the limiting MFC objective is itself a potential for the corresponding MFG.

Load-bearing premise

The idiosyncratic noise must be uniformly non-degenerate and the action set compact with bounded coefficients; without that non-degeneracy the lifted measure-valued controls may miss some accumulation points of the finite-player minimizers.

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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper studies the large-population limit of N-player α-potential stochastic differential games. It constructs an explicit α_N-potential via path integrals on the joint state-control space (Theorem 2.1), proves that the normalized potentials and their approximate minimizers converge to a mean-field control problem with measure-valued controls on a lifted canonical space (Theorem 3.1), and shows that lim α_N=0 is equivalent to the classical closedness/symmetry conditions for potential MFGs (Theorems 4.1–4.2). Under that condition the limiting MFC objective is itself a potential for an associated MFG with measure-valued controls (Theorem 4.3), yielding propagation of chaos from α_N-Nash equilibria to mean-field equilibria for controlled diffusions with common noise and non-separable control interactions (Corollary 4.1). A technical cornerstone is a Poincaré lemma / Green formula on Wasserstein space (Proposition 5.1).

Significance. If the results hold, the paper supplies a clean bridge from finite-player α-potential games to potential mean-field games, giving an explicit asymptotic construction of potential MFGs via vanishing α_N and a new route to propagation of chaos that covers common noise and non-separable state–control costs. The Poincaré lemma on Wasserstein space and the careful treatment of measure-valued controls (with concrete examples showing that lifting is necessary) are genuine technical contributions. The work therefore advances both the conceptual understanding of potential structures and the toolkit for large-population games.

major comments (2)
  1. Assumption 3.1(iii) (compact A, bounded coefficients, uniform non-degeneracy of idiosyncratic noise) is used for relative compactness of the empirical measures and for identifying all accumulation points of approximate minimizers of Φ_N. Remark 3.1 correctly notes that non-degeneracy can be dropped when the running cost depends only on the state law, but the main statements (Theorem 3.1, Corollary 4.1) are stated under the stronger hypothesis. A short additional remark clarifying which conclusions survive under mere Lipschitz coefficients (or citing the precise results of [18,22] that apply) would make the scope of the PoC claim more transparent without changing the theorems.
  2. In the adaptation of the propagation-of-chaos arguments of [17] (proof of Theorem 3.1), the N-dependent costs F_N, G_N differ from the limiting F_∞, G_∞ by O(1/N) terms. The paper asserts that these discrepancies vanish uniformly (display (5.2)), yet the uniform-integrability estimates that justify interchanging limits under the p>2 moment assumption are only sketched. A few additional lines verifying the uniform L^{p/2} bounds on the remainder (or an explicit reference to the corresponding estimates in [17]) would close a minor but load-bearing gap in the written proof.
minor comments (5)
  1. Notation for the lifted spaces (Ω̃, Ω̃′, Λ̃, Π̃, etc.) is dense; a short table or diagram summarizing the hierarchy of measures would help the reader.
  2. In Theorem 2.1 the upper bound on α_N is expressed with H^{2}-norms of state-control pairs; Corollary 2.1 then converts it into a more explicit constant. The dependence of C on the Lipschitz constants of (b,σ,γ) could be written out once for completeness.
  3. Examples 3.1–3.2 are illuminating but lengthy; the key message (that the barycentric projection loses the nonlinear cost) could be highlighted in a single sentence before the calculations.
  4. Typographical: “derivarive” (p. 2), “Poincar´e” inconsistently accented, and occasional missing spaces around “N-player”.
  5. References [3,5,16] to the authors’ earlier α-potential work are appropriate; a one-sentence comparison of the new path-integral construction with the sensitivity-process construction of [3] would orient readers familiar with that literature.

Circularity Check

1 steps flagged · score 1.0 of 10

No significant circularity: finite-N α-potential construction is cited as starting point, but limit theorems, Poincaré lemma on Wasserstein space, and identification of MFC objective as MFG potential are proved independently from first principles under stated assumptions.

  1. self citation load bearing [Section 2.2 / Theorem 2.1 and Introduction]
    "The framework of α-potential games has recently been introduced as a tool to analyze finite-player dynamic games... Recent work by [3] introduces the framework of α-potential game... Here we present a new construction approach. The α-potential function for the game (2.2) is derived by leveraging the decoupled structure..."

    The finite-N α-potential Φ is taken from the authors' prior paper [3] (with a new path-integral construction). This is the definitional starting point for the whole limit theory, but it is not used to force the asymptotic claims; Theorems 3.1 and 4.1–4.3 are proved independently. Minor and non-load-bearing for the paper's strongest claims.

full rationale

The paper's derivation chain begins from the authors' prior α-potential framework (cited as [3]) to construct Φ via path integrals of cost gradients (Theorem 2.1, eqs. (2.4)–(2.5)), then normalizes by N and passes to the limit under Assumption 3.1, obtaining convergence of values/minimizers to a lifted MFC problem (Theorem 3.1) by adapting external propagation-of-chaos arguments from [17] while controlling the explicit N-dependence of F^N, G^N. Equivalence of lim α_N=0 to the closedness conditions (4.1)–(4.2) is proved by direct computation of Hessians of empirical costs (eq. (5.4)) plus a self-contained Poincaré/Green lemma on Wasserstein space (Proposition 5.1, Theorems 4.1–4.2) that does not assume the conclusion. The potential property of the limiting objective (Theorem 4.3, eq. (4.6)) follows by differentiating the same path-integral functional along the interpolation κ_ε of Lemma 4.1; Remark 4.4 merely interprets this as the N→∞ limit of the finite-player α-condition (2.3). Self-citations supply the finite-N starting definition and are not load-bearing for the asymptotic statements, which rest on independent estimates, compactness, and the new differential-geometry arguments. No fitted parameters, no uniqueness imported to forbid alternatives, and no renaming of known results as predictions. Score 1 only for the ordinary (non-circular) reliance on the authors' earlier definition of α-potential games.

Assumptions & free parameters 0 free parameters · 3 assumptions · 2 invented entities

Pure-theory paper; the load-bearing ingredients are standard SDE well-posedness, Wasserstein calculus, and the authors’ prior α-potential framework. No free parameters are fitted. The only invented objects are the concrete path-integral α-potential and the triple hierarchy of measure-valued controls (the latter already present in Djete 2022).

assumptions (3)
  • domain assumption Lipschitz continuity of b,σ,γ and C^{0,2,2}/C^{2,2} regularity of the mean-field costs f,g (Assumptions 2.1, 2.2, 3.1)
    Standard for strong existence of controlled SDEs and for Lions derivatives to exist; invoked throughout Sections 2–4.
  • domain assumption Uniform non-degeneracy of idiosyncratic noise and compactness of the action set A (Assumption 3.1(iii))
    Used for relative compactness of empirical measures and continuity of the lifted objective; can be relaxed when costs depend only on state law (Remark 3.1).
  • standard math Existence of Lions derivatives and the fundamental theorem of calculus along absolutely continuous curves in P_2 (standard Wasserstein calculus)
    Taken from Carmona–Delarue and Gangbo et al.; extended in Proposition 5.1 to weaker regularity.
invented entities (2)
  • Path-integral α_N-potential Φ on the joint state-control space (Eqs. 2.4–2.5) independent evidence
    purpose: Provides an explicit finite-dimensional control problem whose minimizers are α_N-Nash equilibria without differentiating state processes w.r.t. controls.
    New construction relative to the sensitivity-process approach of Guo–Li–Zhang; independent evidence is the analytic upper bound on α_N in Theorem 2.1.
  • Triple hierarchy of measure-valued controls on the canonical space Ω̃ (Definition 3.1) independent evidence
    purpose: Captures all accumulation points of approximate minimizers when running costs depend nonlinearly on the control law.
    Adaptation of Djete’s lifted space; necessity shown by counter-examples 3.1–3.2 when the hierarchy is collapsed.

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Pith. "Pith review of Limit Theory for $N$-Player $\alpha$-Potential Games." pith.science (2026). https://pith.science/paper/FR3WRETZ

@misc{pith2026260609815,
  author       = {Pith},
  title        = {Pith review of: Limit Theory for $N$-Player $\alpha$-Potential Games},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/FR3WRETZ}},
  note         = {Machine review of arXiv:2606.09815}
}
abstract

The recently introduced framework of $\alpha$-potential games facilitates the analysis of finite-player dynamic games by reducing the search for approximate Nash equilibria to the minimization of a single $\alpha$-potential function. In this work, we investigate the large population limit of $\alpha$-potential games, and show that potential mean field games (MFGs) arise naturally. Specifically, we show that both the optimal values and the minimizers of normalized $N$-player $\alpha_N$-potential functions converge to those of a mean field control (MFC) problem with measure-valued controls. We further show that $\lim_{N\to\infty}\alpha_N= 0$ is equivalent to standard conditions for potential MFGs, and provide a unified construction of potential functions for MFGs. A key technical ingredient is the establishment of a Poincar\'e lemma for Wasserstein space. We also establish that the objective of the limiting MFC problem is a potential function for the corresponding MFGs. Together, our results not only yield new constructions of potential MFGs from finite-player games through the asymptotic condition $\lim_{N\to \infty}\alpha_N= 0$, but also establish propagation of chaos from $N$-player games to MFGs for general controlled diffusions with common noise and non-separable control interactions.

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Cited by 1 Pith paper

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Reference graph

Works this paper leans on

47 extracted references · 5 linked inside Pith · cited by 1 Pith paper

  1. [17]

    Extended mean field control problem: a propagation of chaos result

    Mao Fabrice Djete. Extended mean field control problem: a propagation of chaos result. Electronic Journal of Probability, 27:1 – 53, 2022

    2022

  2. [1]

    Potential games.Games and economic behavior, 14(1):124–143, 1996

    Dov Monderer and Lloyd S Shapley. Potential games.Games and economic behavior, 14(1):124–143, 1996

    1996

  3. [2]

    Towards an analytical framework for dynamic potential games

    Xin Guo and Yufei Zhang. Towards an analytical framework for dynamic potential games. SIAM Journal on Control and Optimization, 63(2):1213–1242, 2025

    2025

  4. [3]

    Anα-potential game framework forN-player dynamic games.SIAM Journal on Control and Optimization, 63(4):2964–3005, 2025

    Xin Guo, Xinyu Li, and Yufei Zhang. Anα-potential game framework forN-player dynamic games.SIAM Journal on Control and Optimization, 63(4):2964–3005, 2025

    2025

  5. [4]

    Xuan Di, Anran Hu, Zhexin Wang, and Yufei Zhang.α-potential games for decentralized control of connected and automated vehicles.arXiv preprint arXiv:2512.05712, 2025

    arXiv 2025

  6. [5]

    Distributed games with jumps: Anα-potential game approach.arXiv preprint arXiv:2508.01929, 2025

    Xin Guo, Xinyu Li, and Yufei Zhang. Distributed games with jumps: Anα-potential game approach.arXiv preprint arXiv:2508.01929, 2025

    arXiv 2025

  7. [6]

    Proceedings of Machine Learning Research vol, 283:1–18, 2025

    Dvij Kalaria, Chinmay Maheshwari, and Shankar Sastry.α-racer: Real-time algorithm for game-theoretic motion planning and control in autonomous racing usingα-potential function. Proceedings of Machine Learning Research vol, 283:1–18, 2025

    2025

  8. [7]

    Independent learning of nash equilibria in partially observable markov potential games with decoupled dynamics.arXiv preprint arXiv:2605.06377, 2026

    Philip Jordan and Maryam Kamgarpour. Independent learning of nash equilibria in partially observable markov potential games with decoupled dynamics.arXiv preprint arXiv:2605.06377, 2026

    arXiv 2026

Show all 47 references
  1. [8]

    Potential games on unimodular random graphs

    Eyal Neuman and Sturmius Tuschmann. Potential games on unimodular random graphs. arXiv preprint arXiv:2604.13836, 2026

    2026 arXiv

  2. [9]

    Springer Cham, 2018

    Ren´ e Carmona and Fran¸ cois Delarue.Probabilistic Theory of Mean Field Games with Appli- cations I. Springer Cham, 2018

    2018

  3. [10]

    Mean field games.Japanese Journal of Mathe- matics, 2(1):229–260, 2007

    Jean-Michel Lasry and Pierre-Louis Lions. Mean field games.Japanese Journal of Mathe- matics, 2(1):229–260, 2007

    2007

  4. [11]

    Learning in mean field games: The fictitious play.ESAIM: COCV, 23(2):569–591, 2017

    Cardaliaguet, Pierre and Hadikhanloo, Saeed. Learning in mean field games: The fictitious play.ESAIM: COCV, 23(2):569–591, 2017

    2017

  5. [12]

    Princeton University Press, 2019

    Pierre Cardaliaguet, Fran¸ cois Delarue, Jean-Michel Lasry, and Pierre-Louis Lions.The Mas- ter Equation and the Convergence Problem in Mean Field Games: (AMS-201), volume 2. Princeton University Press, 2019

    2019

  6. [13]

    Weak solutions to the master equation of potential mean field games.arXiv preprint arXiv:2204.04315, 2022

    Alekos Cecchin and Fran¸ cois Delarue. Weak solutions to the master equation of potential mean field games.arXiv preprint arXiv:2204.04315, 2022

    2022 arXiv

  7. [14]

    Jameson Graber

    P. Jameson Graber. Remarks on potential mean field games.Research in the Mathematical Sciences, 12(1):13, 2025. 35

    2025

  8. [15]

    Mete Soner

    Felix H¨ ofer and H. Mete Soner. Optimal control and potential games in the mean field. Stochastic Processes and their Applications, 199:104971, 2026

    2026

  9. [16]

    Bsde approach forα-potential stochastic differential games.arXiv preprint arXiv:2507.13256

    Xin Guo, Xun Li, and Liangquan Zhang. Bsde approach forα-potential stochastic differential games.arXiv preprint arXiv:2507.13256

    arXiv

  10. [18]

    Mckean–vlasov optimal control: Limit theory and equivalence between different formulations.Mathematics of Operations Research, 47(4):2891–2930, 2022

    Mao Fabrice Djete, Dylan Possama¨ ı, and Xiaolu Tan. Mckean–vlasov optimal control: Limit theory and equivalence between different formulations.Mathematics of Operations Research, 47(4):2891–2930, 2022

    2022

  11. [19]

    American Mathematical Society, 2011

    Wilfrid Gangbo, Hwa Kim, and Tommaso Pacini.Differential forms on Wasserstein space and infinite-dimensional Hamiltonian systems, volume 211. American Mathematical Society, 2011

    2011

  12. [20]

    Springer Cham, 2018

    Ren´ e Carmona and Fran¸ cois Delarue.Probabilistic Theory of Mean Field Games with Appli- cations II. Springer Cham, 2018

    2018

  13. [21]

    Springer New York, NY, 2017

    Jianfeng Zhang.Backward Stochastic Differential Equations. Springer New York, NY, 2017

    2017

  14. [22]

    Limit theory for controlled mckean–vlasov dynamics.SIAM Journal on Control and Optimization, 55(3):1641–1672, 2017

    Daniel Lacker. Limit theory for controlled mckean–vlasov dynamics.SIAM Journal on Control and Optimization, 55(3):1641–1672, 2017

    2017

  15. [23]

    Mean field games via controlled martingale problems: Existence of markovian equilibria.Stochastic Processes and their Applications, 125(7):2856–2894, 2015

    Daniel Lacker. Mean field games via controlled martingale problems: Existence of markovian equilibria.Stochastic Processes and their Applications, 125(7):2856–2894, 2015

    2015

  16. [24]

    A probabilistic weak formulation of mean field games and applications.The Annals of Applied Probability, 25(3):1189 – 1231, 2015

    Ren´ e Carmona and Daniel Lacker. A probabilistic weak formulation of mean field games and applications.The Annals of Applied Probability, 25(3):1189 – 1231, 2015

    2015

  17. [25]

    Mean field game of controls and an appli- cation to trade crowding.Mathematics and Financial Economics, 12(3):335–363, 2018

    Pierre Cardaliaguet and Charles-Albert Lehalle. Mean field game of controls and an appli- cation to trade crowding.Mathematics and Financial Economics, 12(3):335–363, 2018

    2018

  18. [26]

    Convergence of large population games to mean field games with interaction through the controls.SIAM Journal on Mathematical Analysis, 54(3):3535–3574, 2022

    Mathieu Lauri` ere and Ludovic Tangpi. Convergence of large population games to mean field games with interaction through the controls.SIAM Journal on Mathematical Analysis, 54(3):3535–3574, 2022

    2022

  19. [27]

    Mean field games of controls: On the convergence of Nash equilibria

    Mao Fabrice Djete. Mean field games of controls: On the convergence of Nash equilibria. The Annals of Applied Probability, 33(4):2824 – 2862, 2023

    2023

  20. [28]

    Non-asymptotic convergence rates for mean-field games: weak formulation and mckean–vlasov bsdes.Applied Mathematics & Optimization, 91(3):58, 2025

    Dylan Possama¨ ı and Ludovic Tangpi. Non-asymptotic convergence rates for mean-field games: weak formulation and mckean–vlasov bsdes.Applied Mathematics & Optimization, 91(3):58, 2025

    2025

  21. [29]

    Convergence for linear quadratic potential mean field games.arXiv preprint arXiv:2602.14842, 2026

    Alekos Cecchin and Jodi Dianetti. Convergence for linear quadratic potential mean field games.arXiv preprint arXiv:2602.14842, 2026

    2026

  22. [30]

    On approximate nash equilibria in mean field games

    Mao Fabrice Djete and Nizar Touzi. On approximate nash equilibria in mean field games. arXiv preprint arXiv:2601.20910, 2026

    2026

  23. [31]

    Selection of equilibria in a linear quadratic mean-field game.Stochastic Processes and their Applications, 130(2):1000–1040, 2020

    Fran¸ cois Delarue and Rinel Foguen Tchuendom. Selection of equilibria in a linear quadratic mean-field game.Stochastic Processes and their Applications, 130(2):1000–1040, 2020. 36

    2020

  24. [32]

    Selection by vanishing common noise for potential finite state mean field games.Communications in Partial Differential Equations, 47(1):89– 168, 2022

    Alekos Cecchin and Fran¸ cois Delarue. Selection by vanishing common noise for potential finite state mean field games.Communications in Partial Differential Equations, 47(1):89– 168, 2022

    2022

  25. [33]

    Synchronization in a kuramoto mean field game.Communications in Partial Differential Equations, 48(9):1214–1244, 2023

    Rene Carmona, Quentin Cormier, and H Mete Soner. Synchronization in a kuramoto mean field game.Communications in Partial Differential Equations, 48(9):1214–1244, 2023

    2023

  26. [34]

    A cucker–smale inspired deterministic mean field game with velocity interactions.SIAM Journal on Control and Optimization, 59(6):4155– 4187, 2021

    Filippo Santambrogio and Woojoo Shim. A cucker–smale inspired deterministic mean field game with velocity interactions.SIAM Journal on Control and Optimization, 59(6):4155– 4187, 2021

    2021

  27. [35]

    Mean-field type modeling of nonlocal crowd aversion in pedestrian crowd dynamics.SIAM Journal on Control and Optimization, 56(1):434–455, 2018

    Alexander Aurell and Boualem Djehiche. Mean-field type modeling of nonlocal crowd aversion in pedestrian crowd dynamics.SIAM Journal on Control and Optimization, 56(1):434–455, 2018

    2018

  28. [36]

    On a mean field game approach modeling congestion and aversion in pedestrian crowds.Transportation research part B: methodological, 45(10):1572–1589, 2011

    Aim´ e Lachapelle and Marie-Therese Wolfram. On a mean field game approach modeling congestion and aversion in pedestrian crowds.Transportation research part B: methodological, 45(10):1572–1589, 2011

    2011

  29. [37]

    Springer International Publishing, Cham, 2020

    Filippo Santambrogio.Lecture Notes on Variational Mean Field Games, pages 159–201. Springer International Publishing, Cham, 2020

    2020

  30. [38]

    Springer Cham, 2021

    Olav Kallenberg.Foundations of Modern Probability. Springer Cham, 2021

    2021

  31. [39]

    Bertsekas and Steven E

    Dimitri P. Bertsekas and Steven E. Shreve.Stochastic Optimal Control: The Discrete-Time Case. Athena Scientific, 2007

    2007

  32. [40]

    Cambridge Series in Statistical and Prob- abilistic Mathematics

    Rick Durrett.Probability: Theory and Examples. Cambridge Series in Statistical and Prob- abilistic Mathematics. Cambridge University Press, 5 edition, 2019

    2019

  33. [41]

    R. M. Dudley.Real Analysis and Probability. Cambridge Studies in Advanced Mathematics. Cambridge University Press, 2 edition, 2002. A Proofs of Examples 3.1 and 3.2 Proof of Example 3.1.LetX N i :=X uN i i for eachi∈[N]. First, define (Y i)i∈N∗ ⊂C([0,1];R) and Y∈C([0,1];R) as:...

    2002

  34. [42]

    Moreover, ΦN(uN) =E  − 1 N 1 8 + 1 81B 1 2 ≥0 + 1 41B 1 2 <0 +   1 N X i∈[N] W1,i   2 − 1 4 + 1 N 2 X i∈[N] 1 2 +W 1,i 2   = 31 16N − 1 4 , which implies thatV N ≤Φ N(uN) = 31 16N − 1

  35. [43]

    Therefore, Φ N(uN)≤V N + 47 16N and limN→∞ V N = − 1 4. For the MFC problem, a direct computation shows thatF ∞ andG ∞ in (3.7) are given by F ∞(t, x, a, ν) = Z A (a′)2ν(R2, da′)− Z A a′ν(R2, da′) 2 , G∞(x, µ) = Z R2 1 −1 ⊤ x′µ(dx′) ! Z R2 1 −1 ⊤ x′µ(dx′)−1 ! . SinceF ∞ ≥0 and...

  36. [44]

    Proof of Example 3.2.LetX N i :=X uN i i for eachi∈[N]

    This along with ˜Φ(P∞) =− 1 4 implies that VV =V ∞ = ˜Φ(P∞) =− 1 4. Proof of Example 3.2.LetX N i :=X uN i i for eachi∈[N]. First, define (Y i)i∈N∗ ⊂C([0,1];R) andY∈C([0,1];R) as: for eachi∈N ∗ andt∈[0,1], letY t,i := 1 W 1 2 ,1≥0 (2t−1) + ∧ 1 2 + 1W 1 2 ,1<0 t− 1 2 + +W t,i, ...

  37. [45]

    Moreover, ΦN(uN) =E  − 1 N +   1 N X i∈[N] W1,i   2 − 1 4 + 1 N 2 X i∈[N] 1 2 +W 1,i 2   = 5 4N − 1 4 , whenV N ≤Φ N(uN) = 5 4N − 1

  38. [46]

    Therefore, lim N→∞ V N =− 1 4, andu N is anε N-optimal control for someε N ≤ 9 4N . For the MFC problem, a direct computation shows thatF ∞ andG ∞ in (3.7) are given by F ∞(t, x, a, ν) = Z A (a′)2ν(R2, da′)− Z A a′ν(R2, da′) 2 , G ∞(x, µ) = Z R x′µ(dx′) Z R x′µ(dx′)−1 . 40 Sin...

  39. [47]

    The proof of (3.15) is analogous to that of (3.13)

    This along ˜Φ(P∞) =− 1 4 yieldsV V =V ∞ = ˜Φ(P∞) =− 1 4. The proof of (3.15) is analogous to that of (3.13). This finishes the proof. 41

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