On the non-uniqueness of linear Markov perfect equilibria in linear-quadratic differential games: a geometric approach

Abstract

Although the possibility of multiple nonlinear equilibria in linear-quadratic differential games is extensively discussed, the literature on models with multiple linear Markov perfect equilibria (LMPEs) is scarce. Indeed, almost all papers confined to a single state (the vast majority of the application of differential games to economic problems) find a unique LMPE. This paper explains this finding and derives conditions for multiplicity based on the analysis of the phase plane in the state and the derivative of the value function. The resulting condition is applied to derive additional pathways different from the (two) known ones. All these examples, more precisely, their underlying pathways or the resulting outcomes, contradict usual assumptions in economic models. However, by extending the state space, we provide an economic setting (learning by doing) that gives rise to multiple LMPEs.