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A Mean-Field Game Approach to Price Formation

Abstract

Here, we introduce a price formation model where a large number of small players can store and trade a commodity such as electricity. Our model is a constrained mean-field game (MFG) where the price is a Lagrange multiplier for the supply versus demand balance condition. We establish the existence of a unique solution using a fixed-point argument. In particular, we show that the price is well defined, and it is a Lipschitz function of time. Then, we study linear-quadratic models that can be solved explicitly and compare our model with real data.

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References

  1. Almulla N, Ferreira R, Gomes D (2017) Two numerical approaches to stationary mean-field games. Dyn Games Appl 7(4):657–682

    MathSciNet  Article  Google Scholar 

  2. Bardi M, Capuzzo-Dolcetta I (1997) Optimal control and viscosity solutions of Hamilton–Jacobi–Bellman equations. Birkhäuser Boston Inc., Boston, MA. With appendices by Maurizio Falcone and Pierpaolo Soravia

  3. Burger M, Caffarelli LA, Markowich PA, Wolfram M-T (2014) On the asymptotic behavior of a Boltzmann-type price formation model. Commun Math Sci 12(7):1353–1361

    MathSciNet  Article  Google Scholar 

  4. Burger M, Caffarelli LA, Markowich PA, Wolfram M-T (2013) On a Boltzmann-type price formation model. Proc R Soc Lond Ser A Math Phys Eng Sci 469(2157):20130126, 20

    MathSciNet  MATH  Google Scholar 

  5. Caffarelli LA, Markowich PA, Pietschmann J-F (2011) On a price formation free boundary model by Lasry and Lions. C R Math Acad Sci Paris 349(11–12):621–624

    MathSciNet  Article  Google Scholar 

  6. Caffarelli LA, Markowich PA, Wolfram M-T (2011) On a price formation free boundary model by Lasry and Lions: the Neumann problem. C R Math Acad Sci Paris 349(15–16):841–844

    MathSciNet  Article  Google Scholar 

  7. Cagnetti F, Gomes D, Mitake H, Tran HV (2015) A new method for large time behavior of degenerate viscous Hamilton–Jacobi equations with convex Hamiltonians. Ann Inst H Poincaré Anal Non Linéaire 32(1):183–200

    MathSciNet  Article  Google Scholar 

  8. Cagnetti F, Gomes D, Tran HV (2011) Aubry-Mather measures in the nonconvex setting. SIAM J Math Anal 43(6):2601–2629

    MathSciNet  Article  Google Scholar 

  9. Cardaliaguet P (2011) Notes on mean-field games

  10. Clemence A, Tahar Imen B, Anis M (2017) An extended mean field game for storage in smart grids. ArXiv e-prints

  11. Couillet R, Perlaza SM, Tembine H, Debbah M (2012) Electrical vehicles in the smart grid: a mean field game analysis. IEEE J Sel Areas Commun 30(6):1086–1096 cited By 56

    Article  Google Scholar 

  12. De Paola A, Angeli D, Strbac G (2016) Distributed control of micro-storage devices with mean field games. IEEE Trans Smart Grid 7(2):1119–1127

    Google Scholar 

  13. De Paola A, Trovato V, Angeli D (2019) A mean field game approach for distributed control of thermostatic loads acting in simultaneous energy-frequency response markets. IEEE Trans Smart Grid

  14. Evans LC (1998) Partial differential equations. Graduate Studies in Mathematics. American Mathematical Society

  15. Evans LC (2010) Adjoint and compensated compactness methods for Hamilton–Jacobi PDE. Arch Ration Mech Anal 197(3):1053–1088

    MathSciNet  Article  Google Scholar 

  16. Evans LC, Gomes D (2001) Effective Hamiltonians and averaging for Hamiltonian dynamics. I. Arch Ration Mech Anal 157(1):1–33

    MathSciNet  Article  Google Scholar 

  17. Evans LC, Gomes D (2002) Effective Hamiltonians and averaging for Hamiltonian dynamics. II. Arch Ration Mech Anal 161(4):271–305

    MathSciNet  Article  Google Scholar 

  18. Fleming WH, Soner HM (2006) Controlled Markov processes and viscosity solutions, vol 25. Stochastic Modelling and Applied Probability. Springer, New York

    MATH  Google Scholar 

  19. Gomes D, Lafleche L, Nurbekyan L (2016) A mean-field game economic growth model. In: Proceedings of the American control conference, 2016-July, pp 4693–4698

  20. Gomes D, Nurbekyan L, Pimentel E (2015) Economic models and mean-field games theory. Publicações Matemáticas do IMPA. [IMPA Mathematical Publications]. Instituto Nacional de Matemática Pura e Aplicada (IMPA), Rio de Janeiro. 30\({^{{}}{\rm {o}}}\) Colóquio Brasileiro de Matemática. [30th Brazilian Mathematics Colloquium]

  21. Gomes D, Pimentel E, Sánchez-Morgado H (2015) Time-dependent mean-field games in the subquadratic case. Commun Partial Differ Equ 40(1):40–76

    MathSciNet  Article  Google Scholar 

  22. Gomes D, Pimentel E, Sánchez-Morgado H (2016) Time-dependent mean-field games in the superquadratic case. ESAIM Control Optim Calc Var 22(2):562–580

    MathSciNet  Article  Google Scholar 

  23. Gomes D, Pimentel E, Voskanyan V (2016) Regularity theory for mean-field game systems. SpringerBriefs in Mathematics. Springer, Cham

    Book  Google Scholar 

  24. Gomes D, Saúde J (2014) Mean field games models—a brief survey. Dyn Games Appl 4(2):110–154

    MathSciNet  Article  Google Scholar 

  25. Gomes D, Saude J (2017) Monotone numerical methods for finite-state mean-field games. arXiv preprint. arXiv:1705.00174

  26. Graber J, Mouzouni C (Jul 2017) Variational mean field games for market competition. arXiv e-prints, page. arXiv:1707.07853

  27. Huang M, Caines PE, Malhamé RP (2007) Large-population cost-coupled LQG problems with nonuniform agents: individual-mass behavior and decentralized \(\epsilon \)-Nash equilibria. IEEE Trans Automat Control 52(9):1560–1571

    MathSciNet  Article  Google Scholar 

  28. Huang M, Malhamé RP, Caines PE (2006) Large population stochastic dynamic games: closed-loop McKean–Vlasov systems and the Nash certainty equivalence principle. Commun Inf Syst 6(3):221–251

    MathSciNet  MATH  Google Scholar 

  29. Kizilkale AC, Malhame RP (2014) A class of collective target tracking problems in energy systems: cooperative versus non-cooperative mean field control solutions. In: Proceedings of the IEEE conference on decision and control, 2015-February (February), pp 3493–3498

  30. Kizilkale AC, Malhame RP (2014) Collective target tracking mean field control for electric space heaters. In: 2014 22nd Mediterranean conference on control and automation, MED 2014, pp 829–834

  31. Kizilkale AC, Malhame RP (2014) Collective target tracking mean field control for markovian jump-driven models of electric water heating loads. IFAC Proc Vol (IFAC-PapersOnline) 19:1867–1872

    Article  Google Scholar 

  32. Lachapelle A, Lasry J-M, Lehalle C-A, Lions P-L (2016) Efficiency of the price formation process in presence of high frequency participants: a mean field game analysis. Math Financ Econ 10(3):223–262

    MathSciNet  Article  Google Scholar 

  33. Lasry J-M, Lions P-L (2006) Jeux à champ moyen. I. Le cas stationnaire. C R Math Acad Sci Paris 343(9):619–625

    MathSciNet  Article  Google Scholar 

  34. Lasry J-M, Lions P-L (2006) Jeux à champ moyen. II. Horizon fini et contrôle optimal. C R Math Acad Sci Paris 343(10):679–684

    MathSciNet  Article  Google Scholar 

  35. Lasry J-M, Lions P-L (2007) Mean field games. Jpn J Math 2(1):229–260

    MathSciNet  Article  Google Scholar 

  36. Lasry J-M, Lions P-L, Guéant O (2010) Mean field games and applications. Paris-Princeton lectures on Mathematical Finance

  37. Lions PL (2007–2011) Collège de France course on mean-field games

  38. Malhamé R, Chong C-Y (1988) On the statistical properties of a cyclic diffusion process arising in the modeling of thermostat-controlled electric power system loads. SIAM J Appl Math 48(2):465–480

    MathSciNet  Article  Google Scholar 

  39. Malhamé R, Kamoun S, Dochain D (1989) On-line identification of electric load models for load management. In: Advances in computing and control (Baton Rouge, LA, 1988), vol 130 of Lect. Notes Control Inf. Sci. Springer, Berlin, pp 290–304

  40. Markowich PA, Matevosyan N, Pietschmann J-F, Wolfram M-T (2009) On a parabolic free boundary equation modeling price formation. Math Models Methods Appl Sci 19(10):1929–1957

    MathSciNet  Article  Google Scholar 

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Correspondence to Diogo A. Gomes.

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Diogo A. Gomes was partially supported by KAUST baseline funds and KAUST OSR-CRG2017-3452. João Saúde was partially supported by FCT/Portugal through the CMU-Portugal Program.

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Gomes, D.A., Saúde, J. A Mean-Field Game Approach to Price Formation. Dyn Games Appl 11, 29–53 (2021). https://doi.org/10.1007/s13235-020-00348-x

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  • DOI: https://doi.org/10.1007/s13235-020-00348-x

Keywords

  • Mean-field games
  • Price formation
  • Monotonicity methods

Mathematics Subject Classification

  • 91A13
  • 91A10
  • 49M30