Correlated relaxed equilibria in nonzero-sum linear differential games

Abstract

Intrinsic to the theory of nonzero-sum games is an underlying desire among players to form some cooperation, which in no-sense is a binding contract. This notion of cooperation without compulsion requires either initiation by a third party or some form of preplay communication. In a seminal paper [l], Aumann introduced one such scheme for normal form (matrix) games, called correlation in randomized strategies. Correlated equilibria are, in some sense, extensions of Nash equilibrium solutions. The advantages of using correlated equilibria in matrix games are many. They form a compact convex set for any matrix game and can be computed via linear programming methods. Very often some correlated equilibria yield higher expected payoffs for all players compared to Nash equilibrium payoffs (see [I, 11, 121 for details). Some interesting variations and extensions of Aumann’s approach to correlation in nonzero-sum games are included in [7, 111 and some applications to economic theory can be found, for example, in [9]. In this paper we define correlated strategies for differential games and introduce a solution concept, which is inspired by Moulin’s approach to correlated equilibria [9, 11, 121. We consider linear differential games of prescribed duration. In this section we assume that the players receive no information about the state variables during the play, except for their initial values which are common knowledge from the start. Hence, open-loop strategies are considered. Our correlation scheme is based on relaxed control theory. In Section 3, we also discuss correlated equilibria in closed-loop strategies. Let .N= { 1, 2, . . . . m} be the set of players. Consider an m-person linear differential nonzero-sum game of prescribed duration with 104 0022-247X/92 $3.00