Nonzero-sum differential games

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.