This paper revisits the problem of how to select an equilibrium in a differential game in the case of multiplicity of Nash equilibria. Most of the previous applied dynamic games literature has considered pre-play negotiations between players, implicitly or explicitly, with the aim of reaching an agreement on the selection of the pair of strategies. The main objective of this paper is to determine what would be the equilibrium to be played without pre-play communications. We study the linear and nonlinear Markov perfect Nash equilibria for a class of well-known models in the literature if pre-play communications are eliminated. We analyze both symmetric and nonsymmetric strategies. We show that the nonlinear strategies are not always the optimal strategies implemented when pre-play communications are removed. We conclude that in the presence of multiple equilibria and without pre-play communications the equilibria actually implemented are symmetric piecewise linear Markov perfect Nash equilibria at least for a range of initial values of the state variable.
