Non-cooperative and semi-cooperative differential games

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In this paper we review some recent results on non-cooperative and semicooperative differential games. For the n-person non-cooperative games in one-space dimension, we consider the Nash equilibrium solutions. When the system of Hamilton-Jacobi equations for the value functions is strictly hyperbolic, we show that the weak solution of a corresponding system of hyperbolic conservation laws determines an n-tuple of feedback strategies. These yield a Nash equilibrium solution to the non-cooperative differential game. However, in the multi-dimensional cases, the system of Hamilton-Jacobi equations is generically elliptic, and therefore ill posed. In an effort to obtain meaningful stable solutions, we propose an alternative “semi-cooperative” pair of strategies for the two players, seeking a Pareto optimum instead of a Nash equilibrium. In this case, the corresponding Hamiltonian system for the value functions is always weakly hyperbolic.

Original languageEnglish (US)
Title of host publicationAnnals of the International Society of Dynamic Games
Number of pages20
StatePublished - 2009

Publication series

NameAnnals of the International Society of Dynamic Games
ISSN (Print)2474-0179
ISSN (Electronic)2474-0187

All Science Journal Classification (ASJC) codes

  • Statistics and Probability
  • Statistics, Probability and Uncertainty
  • Applied Mathematics


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