Geometry of Pareto equilibria and a maximum principle in N-person differential games

AbstractThis paper contains a discussion of a class of N-player differential games. The motion of the state x≜(x1,…,xn)∈E, where X is a domain in n-dimensional Euclidean space En, is determined by a given set of n ordinary differential equations. A play terminates when a state in given set of states θ, θ ⊂ ∂X, is reached.Given N integral payoffs, one for each of the players J1, …, Jn, optimality is expressed by the so called Pareto condition. An optimal strategy N-tuple is one satisfying the Pareto condition; the “value” of the game at state x is a N-tuple (V1∗(x),…, VN∗(x)) whose k-th component Vk∗(x), k = 1,…, N, is the corresponding value of the payoff for player Jk at state x.A Pareto surface in augmented state space EN + n is defined by gS(c) ≜ {z ≜ (xo, x): xo ≜ (xo1,…, xoN, x χ x ∪ θ, xok + Vk ∗(x) = Ck,k = 1,…,N} where C≜(C1,…,CN),and C1,…,CN are arbitrary constants. This definition generalizes the one of a game surface [3]. Some geometric properties of Pareto surfaces are exhibited. Necessary conditions that optimal strategy N-tuples must satisfy are deduced from these geometric properties