Non-linear strategies in a linear quadratic differential game

We study non-linear Markov perfect equilibria in a two agent linear quadratic differential game. In contrast to the literature owing to Tsutsui and Mino [1990. Nonlinear strategies in dynamic duopolistic competition with sticky prices. Journal of Economic Theory 52, 136-161], we do not associate endogenous subsets of the state space with candidate solutions. Instead, we use the 'catching up optimality' criterion to address the possibility of infinitely valued value functions. Applying sufficiency conditions for existence based on those in Dockner et al. [2000. Differential Games in Economics and Management Science. Cambridge University Press, Cambridge] yields the familiar linear MPE and a condition under which a continuum of non-linear MPEs exists. These include, as their limit, a previously unreported piecewise linear MPE. The condition relaxes with increasing patience, allowing more efficient steady states, thus suggesting a Folk Theorem for differential games. As the lower state and control bounds go to -∞, the non-linear strategies are eliminated.