# The solution set of the N-player scalar feedback Nash algebraic Riccati equations.

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fbnsn3.dvi The solution set of the N -player scalar feedback Nash algebraic Riccati equations J.C. Engwerda Tilburg University Department of Econometrics P.O. Box 90153 5000 LE Tilburg The Netherlands phone: 31-134662174 fax: 31-134663280 email: [email protected] Abstract In this paper we analyze the set of scalar algebraic Riccati equations (ARE) that play an important role in finding feedback Nash equilibria of the scalar N -player linear-quadratic differential game. We show that in general there exist at most 2N − 1 solutions of the (ARE) that give rise to a Nash equi- librium. In particular we analyze the number of equilibria as a function of the autonomous growth parameter and present both necessary and sufficient conditions for the existence of a unique solution of the ARE. Keywords: Differential games, Linear-quadratic control, Feedback Nash equilibrium, Algebraic Riccati equations 1 Introduction During the last decade there has been an increasing interest in studying sev- eral problems in economics using a dynamic game theoretical setting. In particular in the area of environmental economics and macro-economic pol- icy coordination this is a very natural framework for modeling problems (see e.g. Engwerda et al. [2] for references). In, e.g., policy coordination prob- lems usually two basic questions arise: first, are policies coordinated and, second, which information do the participating parties have. Usually both these points are rather unclear and, therefore, strategies for different possible scenarios are calculated and compared with each other. One of these scenar- ios is the so-called feedback Nash scenario (see Bas¸ar and Olsder [1] for a precise definition and survey of the relevant literature). Since according to this scenario the participating parties can react to each other’s policies, its relevance is in economics usually larger than that of the open-loop Nash scenario. In particular the feedback Nash scenario is ve

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