Closed-loop Nash equilibrium for a partial differential game with application to competitive personalized advertising

Abstract

This paper is devoted to an N-person partial differential game whose dynamics of the state variable is described by a hyperbolic differential equation with certain boundary and initial conditions while the objective of each player is given by a finite horizon accumulated payoff functional with discounting. We extend the concept of a closed-loop Nash equilibrium for a partial differential game with the dynamics of the states described by a hyperbolic differential equation (a transport equation). We propose the definition of a dual closed-loop Nash equilibrium for which we give sufficient conditions. Moreover, we present the relationship between the Nash equilibria with the dual closed-loop and the classical closed-loop information structure. We apply the new results to the goodwill dynamics model in which the goodwill is influenced by personalized advertising and consumers’ recommendations for which we construct a dual closed-loop Nash equilibrium and we examine its economic properties.