On the Non-Uniqueness of Linear Markov Perfect Equilibria in Linear-Quadratic Differential Games : A Geometric Approach

Although the possibility of multiple nonlinear equilibria inlinear-quadratic differential games is extensively discussed, the literatureon models with multiple linear Markov perfect equilibria (LMPEs) isscarce. And indeed, almost all papers confined to a single state (a verylarge majority of the application of differential games to economicproblems) find a unique LMPE. This paper explains this finding and derivesconditions for multiplicity based on the analysis of the phase plane in thestate and the derivative of the value function. The resulting condition isapplied to derive additional examples using pathways different from the(two) known ones. However, all these arithmetical examples, more precisely,their underlying pathways, contradict usual assumptions in economic models.However, by extending the state space, we provide an economic setting (learning by doing) thatleads to multiple LMPEs naturally without arithmetical tricks