Extreme Points and Majorization: Economic Applications

Abstract

We characterize the set of extreme points of monotonic functions that are either majorized by a given function f or themselves majorize f and show that these extreme points play a crucial role in many economic design problems. Our main results show that each extreme point is uniquely characterized by a countable collection of intervals. Outside these intervals the extreme point equals the original function f and inside the function is constant. Further consistency conditions need to be satisfied pinning down the value of an extreme point in each interval where it is constant. We apply these insights to a varied set of economic problems: equivalence and optimality of mechanisms for auctions and (matching) contests, Bayesian persuasion, optimal delegation, and decision making under uncertainty.

Optimal measure preserving derivatives revisited

“…The results in Post and Longarela (2021) and Kleiner et al (2021) discussed in this article establish that risk aversion is indeed the central issue relevant to understanding pricing kernel nonmonotonicity. Propositions 2.1 and 2.2 and Examples 2.3 and 2.4 in this article establish that it is the technical condition of adequacy imposed in Dybvig (1988) which broadens the set of stochastic arbitrage opportunities available under pricing kernel nonmonotonicity to include replicating portfolios.…”

“…The equivalence of conditions (ii) and (iii) in Proposition 2.1 is implied by Theorem 1 in Post and Longarela (2021) under the additional requirement that 𝜇 has finite support. The equivalence of conditions (i) and (iii) in Proposition 2.1 follows from the discussion in Section S.2.2 of the supplementary material to Kleiner et al (2021) under the additional requirements that 𝜇 is atomless, that 𝜋 is 𝜇-essentially bounded, and that 𝑋 is integrable.…”

“…This result is contained in Proposition 2.2 given here, which relaxes atomlessness to adequacy and drops the essential boundedness and integrability conditions. The proof given by Kleiner et al (2021) relies upon the Hardy-Littlewood rearrangement inequality as in Carlier and Dana (2005) and in Beare (2011), and also upon an inequality of Fan and Lorentz (1954), which they apply fruitfully to a range of problems beyond the one discussed here. The role played by the Fan-Lorentz inequality is replaced in the proofs of Propositions 2.1 and 2.2 given here by an older inequality due to Hardy.…”

“…The second recent article motivating this one is Kleiner et al. (2021). The most relevant part is Section S.2.2 in the supplementary material.…”

The comparative statics of persuasion

“…A great deal of effort has been devoted to characterising optimal signals (e.g. Kolotilin, 2018;Gentzkow & Kamenica, 2016;Dworczak & Martini, 2019;Kleiner, Moldovanu & Strack, 2021;Arieli, Babichenko, Smorodinsky & Yamashita, 2022), yielding sharp descriptions of information provision in a few special cases as well as some high-level general insights. Comparative statics has been an open question.…”