Potential Differential Games
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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.
KeywordsDifferential games Nash equilibria Potential games Optimal control Maximum principle
Mathematics Subject Classification91A23 91A10 49N70 49N90 34H05
Funding was provided by CONACyT (Grant No. 221291).
- 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
- 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.Friedman A (2013) Differential games. Dover Publications, Inc., Mineola, New YorkGoogle Scholar
- 16.Jørgensen S, Zaccour G (2012) Differential games in marketing, vol 15. Springer, BerlinGoogle Scholar
- 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
- 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.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