JOURNAL OF OPTIMIZATIONTHEORY AND APPLICATIONS: Vol. 45, No. 2, FEBRUARY 1985 Tractable Classes of Nonzero-Sum Open-Loop Nash Differential Games: Theory and Examples I E. DOCKNER, 2 G. FEICHTINGER, 3 AND S. JORGENSEN 4 Communicated by G. Leitmann Abstract. This paper identifies some classes of N-person nonzero-sum differential games that are tractable, in the sense that open-loop Nash strategies can be determined, either explicitly or qualitatively in terms of a phase-diagram portrait. The classes are characterized by conditions imposed on the Hamiltonians. Also, the underlying game structures needed to satisfy these conditions are characterized. Key Words. Differential games, open-loop Nash equilibria, solvability, state separability, redundancy. 1. Introduction This paper deals with the following problem. Which are the structural assumptions that could be made to obtain solutions to an N-person, non- zero-sum, open-loop Nash differential game? By solutions we mean, loosely speaking, that the controls, the state variables, and the adjoint variables can be explicitly specified as functions of time. Also, we consider the possibilities of obtaining insights into the qualitative behavior of the solution trajectories, for example in terms of phase diagrams. We assume that the Hamiltonians are nonlinear in the control variables. Ctemhout and Wan write (Ref. 1, p. 419): "Much has been established both for the necessary conditions and the sulfcient conditions pertaining to the optimal play in N-person, general The authors wish to thank V. Kaitala, Helsinki University of Technology, and the referees. z Assistant Professor, Institute of Economic Theory and Policy, University of Economics, Vienna, Austria. 3 Professor, Institute for Econometrics and Operations Research, University of Technology, Vienna, Austria. 4 Associate professor, Institute of Theoretical Statistics, Copenhagen School of Economics and Business Administration, Copenhagen, Denmark. 179 0022-3239/85/0200-B179504,50/0 ~ 1985 PlenumPublishingCorporation