Coherent distributions on the square $\unicode{x2013}$ extreme points and asymptotics

Abstract

Let $\mathcal{C}$ denote the family of all coherent distributions on the unit square $[0,1{]}^2$, i.e. all those probability measures $\mu$ for which there exists a random vector $(X,Y)\sim \mu$, a pair $(\mathcal{G},\mathcal{H})$ of $\sigma$-fields and an event $E$ such that $X=\mathbb{P}(E|\mathcal{G})$, $Y=\mathbb{P}(E|\mathcal{H})$ almost surely. In this paper we examine the set $\mathrm{ext}(\mathcal{C})$ of extreme points of $\mathcal{C}$ and provide its general characterisation. Moreover, we establish several structural properties of finitely-supported elements of $\mathrm{ext}(\mathcal{C})$. We apply these results to obtain the asymptotic sharp bound $$\underset{\alpha \to \infty }{\lim }\alpha \cdot (\underset{(X,Y)\in \mathcal{C}}{\sup }\mathbb{E}|X-Y{|}^{\alpha })=\frac{2}{e}.$$