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Nonzero-sum differential games

  • A. W. Starr
  • Y. C. Ho
Contributed Papers

Abstract

The theory of differential games is extended to the situation where there areN players and where the game is nonzero-sum, i.e., the players wish to minimize different performance criteria. Dropping the usual zero-sum condition adds several interesting new features. It is no longer obvious what should be demanded of asolution, and three types of solutions are discussed:Nash equilibrium, minimax, andnoninferior set of strategies. For one special case, the linear-quadratic game, all three of these solutions can be obtained by solving sets of ordinary matrix differential equations. To illustrate the differences between zero-sum and nonzero-sum games, the results are applied to a nonzero-sum version of a simple pursuit-evasion problem first considered by Ho, Bryson, and Baron (Ref. 1).Negotiated solutions are found to exist which give better results forboth players than the usualsaddle-point solution. To illustrate that the theory may find interesting applications in economic analysis, a problem is outlined involving the dividend policies of firms operating in an imperfectly competitive market.

Keywords

Differential Equation Nash Equilibrium Nash Economic Analysis Performance Criterion 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Ho, Y. C., Bryson, A. E., andBaron, S.,Differential Games and Optimal Pursuit-Evasion Strategies, IEEE Transactions on Automatic Control, Vol. AC-10, No. 4, 1965.Google Scholar
  2. 2.
    Isaacs, R.,Differential Games, John Wiley and Sons, New York, 1965.Google Scholar
  3. 3.
    Case, J. H.,Equilibrium Points of N-Person Differential Games, University of Michigan, Department of Industrial Engineering, TR No. 1967-1, 1967.Google Scholar
  4. 4.
    Luce, R. D., andRaiffa, H.,Games and Decisions, John Wiley and Sons, New York, 1957.Google Scholar
  5. 5.
    Dacunha, N. O., andPolak, E.,Constrained Minimization Under Vector-Valued Criteria in Finite-Dimensional Spaces, University of California at Berkeley, Electronics Research Laboratory, Memorandum No. ERL-M188, 1966.Google Scholar
  6. 6.
    Zadeh, L. A.,Optimality and Non-Scalar-Valued Performance Criteria, IEEE Transactions on Automatic Control, Vol. AC-8, No. 1, 1963.Google Scholar
  7. 7.
    Klinger, A.,Vector-Valued Performance Criteria, IEEE Transactions on Automatic Control, Vol. AC-9, No. 1, 1964.Google Scholar

Copyright information

© Plenum Publishing Corporation 1969

Authors and Affiliations

  • A. W. Starr
    • 1
  • Y. C. Ho
    • 1
  1. 1.Division of Engineering and Applied PhysicsHarvard UniversityCambridge

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