Skip to main content
main-content
Top

Hint

Swipe to navigate through the articles of this issue

Published in: Dynamic Games and Applications 4/2020

08-01-2020

Stationary Equilibria of Mean Field Games with Finite State and Action Space

Author: Berenice Anne Neumann

Published in: Dynamic Games and Applications | Issue 4/2020

Login to get access
share
SHARE

Abstract

Mean field games formalize dynamic games with a continuum of players and explicit interaction where the players can have heterogeneous states. As they additionally yield approximate equilibria of corresponding N-player games, they are of great interest for socio-economic applications. However, most techniques used for mean field games rely on assumptions that imply that for each population distribution there is a unique optimizer of the Hamiltonian. For finite action spaces, this will only hold for trivial models. We propose a model with finite state and action space, where the dynamics are given by a time-inhomogeneous Markov chain that might depend on the current population distribution. We show existence of stationary mean field equilibria in mixed strategies under mild assumptions and propose techniques to compute all these equilibria. More precisely, our results allow—given that the generators are irreducible—to characterize the set of stationary mean field equilibria as the set of all fixed points of a map completely characterized by the transition rates and rewards for deterministic strategies. Additionally, we propose several partial results for the case of non-irreducible generators and we demonstrate the presented techniques on two examples.

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.

Appendix
Available only for authorised users
Literature
1.
go back to reference Asmussen S (2003) Applied probability and queues. Stochastic modelling and applied probability, vol 51, 2nd edn. Springer, New York MATH Asmussen S (2003) Applied probability and queues. Stochastic modelling and applied probability, vol 51, 2nd edn. Springer, New York MATH
3.
go back to reference Benazzoli C, Campi L, Persio LD (2018) Mean field games with controlled jump-diffusion dynamics: Existence results and an illiquid interbank market model. ArXiv preprint arXiv:​1703.​01919 Benazzoli C, Campi L, Persio LD (2018) Mean field games with controlled jump-diffusion dynamics: Existence results and an illiquid interbank market model. ArXiv preprint arXiv:​1703.​01919
4.
go back to reference Bensoussan A, Frehse J, Yam P (2013) Mean field games and mean field type control theory. Springerbriefs in mathematics. Springer, New York CrossRef Bensoussan A, Frehse J, Yam P (2013) Mean field games and mean field type control theory. Springerbriefs in mathematics. Springer, New York CrossRef
5.
go back to reference Berman A, Plemmons RJ (1979) Nonnegative matrices in the mathematical sciences. Computer science and applied mathematics. Academic Press Inc., New York MATH Berman A, Plemmons RJ (1979) Nonnegative matrices in the mathematical sciences. Computer science and applied mathematics. Academic Press Inc., New York MATH
7.
go back to reference Border KC (1985) Fixed point theorems with applications to economics and game theory. Cambridge University Press, Cambridge CrossRef Border KC (1985) Fixed point theorems with applications to economics and game theory. Cambridge University Press, Cambridge CrossRef
17.
go back to reference Durrett R (1999) Essentials of stochastic processes. Springer texts in statistics. Springer, New York MATH Durrett R (1999) Essentials of stochastic processes. Springer texts in statistics. Springer, New York MATH
24.
27.
go back to reference Guo X, Hernández-Lerma O (2009) Continuous-time Markov decision processes: theory and applications. Stochastic modelling and applied probability, vol 62. Springer, Berlin CrossRef Guo X, Hernández-Lerma O (2009) Continuous-time Markov decision processes: theory and applications. Stochastic modelling and applied probability, vol 62. Springer, Berlin CrossRef
30.
go back to reference Kelly FP (1979) Reversibility and stochastic networks. Wiley series in probability and mathematical statistics. Wiley, Chichester Kelly FP (1979) Reversibility and stochastic networks. Wiley series in probability and mathematical statistics. Wiley, Chichester
35.
go back to reference Norris JR (1997) Markov Chains. Cambridge series on statistical and probabilistic mathematics. Cambridge University Press, Cambridge Norris JR (1997) Markov Chains. Cambridge series on statistical and probabilistic mathematics. Cambridge University Press, Cambridge
36.
go back to reference Puterman ML (1994) Markov decision processes: discrete stochastic dynamic programming. Wiley series in probability and mathematical statistics. Wiley, New York CrossRef Puterman ML (1994) Markov decision processes: discrete stochastic dynamic programming. Wiley series in probability and mathematical statistics. Wiley, New York CrossRef
37.
go back to reference Resnick SI (1992) Adventures in stochastic processes. Birkhäuser, Boston MATH Resnick SI (1992) Adventures in stochastic processes. Birkhäuser, Boston MATH
38.
go back to reference Walker M, Wooders J (2008) Mixed strategy equilibrium. In: Durlauf SN, Blume LE (eds) The new Palgrave dictionary of economics, vol 5, 2nd edn. Macmillan Publishers Ltd., New York, pp 628–631 Walker M, Wooders J (2008) Mixed strategy equilibrium. In: Durlauf SN, Blume LE (eds) The new Palgrave dictionary of economics, vol 5, 2nd edn. Macmillan Publishers Ltd., New York, pp 628–631
39.
go back to reference Walter W (1998) Ordinary differential equations. Graduate texts in mathematics, vol 182. Springer, New York CrossRef Walter W (1998) Ordinary differential equations. Graduate texts in mathematics, vol 182. Springer, New York CrossRef
Metadata
Title
Stationary Equilibria of Mean Field Games with Finite State and Action Space
Author
Berenice Anne Neumann
Publication date
08-01-2020
Publisher
Springer US
Published in
Dynamic Games and Applications / Issue 4/2020
Print ISSN: 2153-0785
Electronic ISSN: 2153-0793
DOI
https://doi.org/10.1007/s13235-019-00345-9

Other articles of this Issue 4/2020

Dynamic Games and Applications 4/2020 Go to the issue

Premium Partner