Dynamic Games and Applications

, Volume 8, Issue 2, pp 254–279 | Cite as

Potential Differential Games

  • Alejandra Fonseca-Morales
  • Onésimo Hernández-Lerma


This paper introduces the notion of a potential differential game (PDG), which roughly put is a noncooperative differential game to which we can associate an optimal control problem (OCP) whose solutions are Nash equilibria for the original game. If this is the case, there are two immediate consequences. Firstly, finding Nash equilibria for the game is greatly simplified, because it is a lot easier to deal with an OCP than with the original game itself. Secondly, the Nash equilibria obtained from the associated OCP are automatically “pure” (or deterministic) rather than “mixed” (or randomized). We restrict ourselves to open-loop differential games. We propose two different approaches to identify a PDG and to construct a corresponding OCP. As an application, we consider a PDG with a certain turnpike property that is obtained from results for the associated OCP. We illustrate our results with a variety of examples.


Differential games Nash equilibria Potential games Optimal control Maximum principle 

Mathematics Subject Classification

91A23 91A10 49N70 49N90 34H05 



Funding was provided by CONACyT (Grant No. 221291).


  1. 1.
    Amir R, Nannerup N (2006) Information structure and the tragedy of the commons in resource extraction. J Bioecon 8:147–165CrossRefGoogle Scholar
  2. 2.
    Charalambous CD (2016) Decentralized optimality conditions of stochastic differential decision problems via Girsanov’s measure transformation. Math Control Signals Syst 28:19.  doi:10.1007/s00498-016-0168-3
  3. 3.
    Clarke F (2013) Functional analysis, calculus of variations and optimal control. Springer, BerlinCrossRefMATHGoogle Scholar
  4. 4.
    Dockner E, Feischtinger G, Jørgensen S (1985) Tractable classes of nonzero-sum open-loop Nash differential games: theory and examples. J Optim Theory Appl 45:179–197MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Dockner EJ, Jørgensen S, Long NV, Sorger G (2000) Differential games in economics and management science. Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar
  6. 6.
    Dragone D, Lambertini L, Leitmann G, Palestini A (2009) Hamiltonian potential functions for differential games. IFAC Proc 42:1–8CrossRefMATHGoogle Scholar
  7. 7.
    Dragone D, Lambertini L, Palestini A (2012) Static and dynamic best-response potential functions for the non-linear Cournot game. Optimization 61:1283–1293MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Dragone D, Lambertini L, Leitmann G, Palestini A (2015) Hamiltonian potential functions for differential games. Automatica 62:134–138MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Fonseca-Morales A, Hernández-Lerma O. A note on differential games with Pareto-optimal Nash equilibria: deterministic and stochastic models. J Dyn Games (to appear)Google Scholar
  10. 10.
    Friedman A (2013) Differential games. Dover Publications, Inc., Mineola, New YorkGoogle Scholar
  11. 11.
    González-Sánchez D, Hernández-Lerma O (2013) Discrete-time stochastic control and dynamic potential games: the Euler-equation approach. Springer, BerlinCrossRefMATHGoogle Scholar
  12. 12.
    González-Sánchez D, Hernández-Lerma O (2013) An inverse optimal problem in discrete-time stochastic control. J Differ Equ Appl 19:39–53MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    González-Sánchez D, Hernández-Lerma O (2014) Dynamic potential games: the discrete-time stochastic case. Dyn Games Appl 4:309–328MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    González-Sánchez D, Hernández-Lerma O (2016) A survey of static and dynamic potential games. Sci China Math 59:2075–2102MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Gopalakrishnan R, Marden JR, Wierman A (2014) Potential games are necessary to ensure pure Nash equilibria in cost sharing games. Math Oper Res 39:1252–1296MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Jørgensen S, Zaccour G (2012) Differential games in marketing, vol 15. Springer, BerlinGoogle Scholar
  17. 17.
    La QD, Chew YH, Soong BH (2016) Potential game theory: applications in radio resource allocation. Springer, BerlinMATHGoogle Scholar
  18. 18.
    Long NV (2011) Dynamic games in the economics of natural resources: a survey. Dyn Games Appl 1:115–148MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Mangasarian OL (1969) Nonlinear programming. McGraw-Hill, New YorkMATHGoogle Scholar
  20. 20.
    Monderer D, Shapley LS (1996) Potential games. Game Econ Behav 14:124–143MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Mou L, Yong J (2007) A variational formula for stochastic controls and some applications. Pure Appl Math Q 3:539–567MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Potters JAM, Raghavan TES, Tijs SH (2009) Pure equilibrium strategies for stochastic games via potential functions. In: Advances in dynamic games and their applications. Birkhauser, Boston, pp 433–444Google Scholar
  23. 23.
    Rosenthal RW (1973) A class of games possessing pure-strategy Nash equilibria. Int J Game Theory 2:65–67MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Slade EM (1994) What does an oligopoly maximize? J Ind Econ 42:45–61CrossRefGoogle Scholar
  25. 25.
    Sundaram RK (1996) A first course in optimization theory. Cambridge University Press, CambridgeCrossRefMATHGoogle Scholar
  26. 26.
    Tauchnitz N (2015) The Pontryagin maximum principle for nonlinear optimal control problems with infinite horizon. J Optim Theory Appl 167:27–48MathSciNetCrossRefMATHGoogle Scholar
  27. 27.
    Trélat E, Zuazua E (2015) The turnpike property in finite-dimensional nonlinear optimal control. J Differ Equ 258:81–114MathSciNetCrossRefMATHGoogle Scholar
  28. 28.
    Yong J, Zhou XY (1999) Stochastic controls: Hamiltonian systems and HJB equations, vol 43. Springer, BerlinCrossRefMATHGoogle Scholar
  29. 29.
    Zazo S, Zazo J, Sánchez-Fernández M (2014) A control theoretic approach to solve a constrained uplink power dynamic game. In: 22nd European Signal processing conference on IEEE (EUSIPCO), pp 401–405Google Scholar
  30. 30.
    Zazo S, Valcarcel S, Sánchez-Fernández M, Zazo J (2015) A new framework for solving dynamic scheduling games. In: IEEE international conference on acoustics, speech and signal processing (ICASSP), pp 2071–2075Google Scholar

Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  • Alejandra Fonseca-Morales
    • 1
  • Onésimo Hernández-Lerma
    • 1
  1. 1.Mathematics DepartmentCINVESTAV-IPNMexico CityMexico

Personalised recommendations