Optimal Transport in Mechanism Design, Differential Games, and Ensemble Control

Abstract

In this thesis, we study optimal transport problems arising in game and control-theoretic settings. The thesis has three parts; in part one we study a variation of the matching problem over a continuum of consumers and producers, where the producers are uncertain about their cost of production, in part two we study optimal transport problem in game-theoretic settings, and in part three we study optimal transport in ensemble control-theoretic settings. In part one, using optimal transport theory, we provide conditions under which matching equilibrium exists. In part two and three, we provide sufficient condition for the existence and uniqueness of an optimal transport map and prove that it can be characterized as the gradient of some convex function.