Cooperative solutions in a many-person positional differential game with continuous payment functions

Abstract

Continuing previous research /1–2/, we consider a non-zero-sum positional differential game which allows deviations of individual players and also groups of players from a coordinated solution∗∗. The proposed formalization of a non-zero-sum differential game is based on formalization and results of the theory of zero-sum positional differential games /3, 4/. We assume that in the course of the game each player may make continuous payments to other players. The payments are measured in units of the criterion of the player making the payments, and at any moment in the game they may not exceed a given fraction of the increment in that player's guaranteed outcome. Each player evaluates individually the payments received from other players. This approach covers, in particular, the case of transferable /5/ rewards. The proposed cooperative solution of the game is such that none of the admissible coalitions will be better off by deviating from this solution in the course of the game. The cooperative solutions provide for a certain penalty strategy /6/. An approach to the analysis of cooperative differential games with transferable rewards using the notation of dynamic stability of optimality principles was studied in /7/. The notion of a. strong equilibrium solution stable in relation to the deviations of coalitions was considered in /8/.