Skip to main content
main-content
Top

Hint

Swipe to navigate through the articles of this issue

Published in: Dynamic Games and Applications 1/2021

17-02-2020

A Mean-Field Game Approach to Price Formation

Authors: Diogo A. Gomes, João Saúde

Published in: Dynamic Games and Applications | Issue 1/2021

Login to get access
share
SHARE

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.

To get access to this content you need the following product:

Springer Professional "Wirtschaft+Technik"

Online-Abonnement

Mit Springer Professional "Wirtschaft+Technik" erhalten Sie Zugriff auf:

  • über 69.000 Bücher
  • über 500 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Maschinenbau + Werkstoffe
  • Versicherung + Risiko

Testen Sie jetzt 15 Tage kostenlos.

Springer Professional "Technik"

Online-Abonnement

Mit Springer Professional "Technik" erhalten Sie Zugriff auf:

  • über 50.000 Bücher
  • über 380 Zeitschriften

aus folgenden Fachgebieten:

  • Automobil + Motoren
  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Elektrotechnik + Elektronik
  • Energie + Nachhaltigkeit
  • Maschinenbau + Werkstoffe




Testen Sie jetzt 15 Tage kostenlos.

Springer Professional "Wirtschaft"

Online-Abonnement

Mit Springer Professional "Wirtschaft" erhalten Sie Zugriff auf:

  • über 58.000 Bücher
  • über 300 Zeitschriften

aus folgenden Fachgebieten:

  • Bauwesen + Immobilien
  • Business IT + Informatik
  • Finance + Banking
  • Management + Führung
  • Marketing + Vertrieb
  • Versicherung + Risiko




Testen Sie jetzt 15 Tage kostenlos.

Literature
1.
go back to reference Almulla N, Ferreira R, Gomes D (2017) Two numerical approaches to stationary mean-field games. Dyn Games Appl 7(4):657–682 MathSciNetCrossRef Almulla N, Ferreira R, Gomes D (2017) Two numerical approaches to stationary mean-field games. Dyn Games Appl 7(4):657–682 MathSciNetCrossRef
2.
go back to reference 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 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.
go back to reference 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 MathSciNetCrossRef 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 MathSciNetCrossRef
4.
go back to reference 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 MathSciNetMATH 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 MathSciNetMATH
5.
go back to reference 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 MathSciNetCrossRef 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 MathSciNetCrossRef
6.
go back to reference 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 MathSciNetCrossRef 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 MathSciNetCrossRef
7.
go back to reference 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 MathSciNetCrossRef 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 MathSciNetCrossRef
8.
go back to reference Cagnetti F, Gomes D, Tran HV (2011) Aubry-Mather measures in the nonconvex setting. SIAM J Math Anal 43(6):2601–2629 MathSciNetCrossRef Cagnetti F, Gomes D, Tran HV (2011) Aubry-Mather measures in the nonconvex setting. SIAM J Math Anal 43(6):2601–2629 MathSciNetCrossRef
9.
go back to reference Cardaliaguet P (2011) Notes on mean-field games Cardaliaguet P (2011) Notes on mean-field games
10.
go back to reference Clemence A, Tahar Imen B, Anis M (2017) An extended mean field game for storage in smart grids. ArXiv e-prints Clemence A, Tahar Imen B, Anis M (2017) An extended mean field game for storage in smart grids. ArXiv e-prints
11.
go back to reference 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 CrossRef 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 CrossRef
12.
go back to reference 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 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
13.
go back to reference 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 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.
go back to reference Evans LC (1998) Partial differential equations. Graduate Studies in Mathematics. American Mathematical Society Evans LC (1998) Partial differential equations. Graduate Studies in Mathematics. American Mathematical Society
15.
go back to reference Evans LC (2010) Adjoint and compensated compactness methods for Hamilton–Jacobi PDE. Arch Ration Mech Anal 197(3):1053–1088 MathSciNetCrossRef Evans LC (2010) Adjoint and compensated compactness methods for Hamilton–Jacobi PDE. Arch Ration Mech Anal 197(3):1053–1088 MathSciNetCrossRef
16.
go back to reference Evans LC, Gomes D (2001) Effective Hamiltonians and averaging for Hamiltonian dynamics. I. Arch Ration Mech Anal 157(1):1–33 MathSciNetCrossRef Evans LC, Gomes D (2001) Effective Hamiltonians and averaging for Hamiltonian dynamics. I. Arch Ration Mech Anal 157(1):1–33 MathSciNetCrossRef
17.
go back to reference Evans LC, Gomes D (2002) Effective Hamiltonians and averaging for Hamiltonian dynamics. II. Arch Ration Mech Anal 161(4):271–305 MathSciNetCrossRef Evans LC, Gomes D (2002) Effective Hamiltonians and averaging for Hamiltonian dynamics. II. Arch Ration Mech Anal 161(4):271–305 MathSciNetCrossRef
18.
go back to reference Fleming WH, Soner HM (2006) Controlled Markov processes and viscosity solutions, vol 25. Stochastic Modelling and Applied Probability. Springer, New York MATH Fleming WH, Soner HM (2006) Controlled Markov processes and viscosity solutions, vol 25. Stochastic Modelling and Applied Probability. Springer, New York MATH
19.
go back to reference 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 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.
go back to reference 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] 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.
go back to reference 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 MathSciNetCrossRef 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 MathSciNetCrossRef
22.
go back to reference 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 MathSciNetCrossRef 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 MathSciNetCrossRef
23.
go back to reference Gomes D, Pimentel E, Voskanyan V (2016) Regularity theory for mean-field game systems. SpringerBriefs in Mathematics. Springer, Cham CrossRef Gomes D, Pimentel E, Voskanyan V (2016) Regularity theory for mean-field game systems. SpringerBriefs in Mathematics. Springer, Cham CrossRef
27.
go back to reference 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 MathSciNetCrossRef 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 MathSciNetCrossRef
28.
go back to reference 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 MathSciNetMATH 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 MathSciNetMATH
29.
go back to reference 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 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.
go back to reference 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 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.
go back to reference 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 CrossRef 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 CrossRef
32.
go back to reference 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 MathSciNetCrossRef 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 MathSciNetCrossRef
33.
34.
go back to reference 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 MathSciNetCrossRef 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 MathSciNetCrossRef
36.
go back to reference Lasry J-M, Lions P-L, Guéant O (2010) Mean field games and applications. Paris-Princeton lectures on Mathematical Finance Lasry J-M, Lions P-L, Guéant O (2010) Mean field games and applications. Paris-Princeton lectures on Mathematical Finance
37.
go back to reference Lions PL (2007–2011) Collège de France course on mean-field games Lions PL (2007–2011) Collège de France course on mean-field games
38.
go back to reference 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 MathSciNetCrossRef 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 MathSciNetCrossRef
39.
go back to reference 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 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.
go back to reference 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 MathSciNetCrossRef 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 MathSciNetCrossRef
Metadata
Title
A Mean-Field Game Approach to Price Formation
Authors
Diogo A. Gomes
João Saúde
Publication date
17-02-2020
Publisher
Springer US
Published in
Dynamic Games and Applications / Issue 1/2021
Print ISSN: 2153-0785
Electronic ISSN: 2153-0793
DOI
https://doi.org/10.1007/s13235-020-00348-x

Other articles of this Issue 1/2021

Dynamic Games and Applications 1/2021 Go to the issue

Premium Partner