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Generalized Differential Games

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Abstract

An important generalization of a Nash equilibrium is the case when the players must choose strategies which depend on the other players. The case in zero-sum differential games with players y and z when there is a constraint of the form \(g(y,z) \le 0\) is introduced. The Isaacs’ equations for the upper value and the lower value of a zero-sum differential game are derived and a condition guaranteeing existence of value is derived. It is also proved that the value functions are the limits of penalized games.

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References

  1. Bardi M, Capuzzo-Dolcetta I (1997) Optimal control and viscosity solutions of Hamilton-Jacobi-Bellman equations, with appendices by Maurizio Falcone and Pierpaolo Soravia, systems & control: foundations & applications. Birkhäuser Boston Inc, Boston, MA

    MATH  Google Scholar 

  2. Bressan A, Shen W (2003) Semi-cooperative strategies for differential games. Int J Game Theory 32:561–593

    MathSciNet  MATH  Google Scholar 

  3. Cardaliaguet P (2010) Introduction to differential games, https://www.ceremade.dauphine.fr/~cardalia/

  4. Evans LC, Souganidis P (1984) Differential games and representation formulas for solutions of Hamilton-Jacobi-Isaacs equations. Indiana Univ Math J 33:773–797

    Article  MathSciNet  MATH  Google Scholar 

  5. Elliott R, Kalton N (1972) The existence of value in differential games, Memoirs AMS, 126

  6. Engler H, Lenhart SM (1991) Viscosity solutions for weakly coupled systems of Hamilton-Jacobi equations. Proc London Math Soc 63:212–240

    Article  MathSciNet  MATH  Google Scholar 

  7. Facchinei F, Kanzow C (2010) Generalized Nash equilibrium problems. Ann Oper Res 175:177–211

    Article  MathSciNet  MATH  Google Scholar 

  8. Ishii H, Koike S (1991) Viscosity solutions for monotone systems of second-order parabolic PDE’s. Comm Partial Differ Equ 16:1095–1128

    Article  MATH  Google Scholar 

  9. Krasovskii NN, Subbotin AI (1988) Game-theoretical control problems. Springer, New York

    Book  Google Scholar 

  10. Lenhart SM (1988) Viscosity solutions for weakly coupled systems of first order PDEs. J Math Anal Appl 131:180–193

    Article  MathSciNet  MATH  Google Scholar 

  11. Subbotin AI (1995) Generalized solution of first-order PDEs. Birkhauser, Boston

    Book  Google Scholar 

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Loyola University Chicago, Chicago, IL, 60660, USA

    E. N. Barron

  2. Department of Mathematics, North Carolina State University, Raleigh, NC, 27695, USA

    K. T. Nguyen

Authors
  1. E. N. Barron
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  2. K. T. Nguyen
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Correspondence to E. N. Barron.

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K. T. Nguyen was partially supported by a Grant from the Simons Foundation/SFARI (521811, NTK)

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Barron, E.N., Nguyen, K.T. Generalized Differential Games. Dyn Games Appl 13, 705–720 (2023). https://doi.org/10.1007/s13235-022-00452-0

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  • DOI: https://doi.org/10.1007/s13235-022-00452-0

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