Randomization in Optimal Execution Games

Abstract.

We study optimal execution in markets with transient price impact in a competitive setting with [Formula: see text] traders. Motivated by prior negative results on the existence of pure Nash equilibria, we consider randomized strategies for the traders and whether allowing such strategies can restore the existence of equilibria. We show that given a randomized strategy, there is a nonrandomized strategy with strictly lower expected execution cost, and moreover this derandomization can be achieved by a simple averaging procedure. As a consequence, Nash equilibria cannot contain randomized strategies, and nonexistence of pure equilibria implies nonexistence of randomized equilibria. Separately, we also establish uniqueness of equilibria. Both results hold in a general transaction cost model given by a strictly positive definite impact decay kernel and a convex trading cost.