A differential game of N persons in which there is Pareto equilibrium of objections and counterobjections and no Nash equilibrium

A linear-quadratic positional differential game of N persons is considered. The solution of a game in the form of Nash equilibrium has become widespread in the theory of noncooperative differential games. However, Nash equilibrium can be internally and externally unstable, which is a negative in its practical use. The consequences of such instability could be avoided by using Pareto maximality in a Nash equilibrium situation. But such a coincidence is rather an exotic phenomenon (at least we are aware of only three cases of such coincidence). For this reason, it is proposed to consider the equilibrium of objections and counterobjections. This article establishes the coefficient criteria under which in a differential positional linear-quadratic game of N persons there is Pareto equilibrium of objections and counterobjections and at the same time no Nash equilibrium situation; an explicit form of the solution of the game is obtained.

ON THE STABILITY OF THE BEST REPLY MAP FOR NONCOOPERATIVE DIFFERENTIAL GAMES

Consider a differential game for two players in infinite time horizon, with exponentially discounted costs. A pair of feedback controls [Formula: see text] is Nash equilibrium solution if [Formula: see text] is the best strategy for Player 1 in reply to [Formula: see text], and [Formula: see text] is the best strategy for Player 2, in reply to [Formula: see text]. The aim of the present note is to investigate the stability of the best reply map: [Formula: see text]. For linear-quadratic games, we derive a condition which yields asymptotic stability, within the class of feedbacks which are affine functions of the state x ∈ ℝn. An example shows that stability is lost, as soon as nonlinear perturbations are considered.