Extreme Points and Majorization

A key insight is that many, seemingly different, economic problems share a common mathematical structure: they all involve the maximization of a functional over sets of monotonic functions that are either majorized by, or majorize, a given function. We first present new, simpler proofs for the main characterization results of the extreme points of sets defined by monotonicity and majorization constraints obtained by Kleiner, Moldovanu, and Strack (2021). We then demonstrate how the characterization results can be fruitfully applied to a broad range of economic applications, from auction and information design to decision problems under risk such as optimal stopping. Finally, we conclude with an overview of recent, related work that extends these characterizations to settings with additional constraints, multidimensional state spaces, and alternative stochastic orders.