Contradictory predictions with multiple agents

Let $X_1$, $X_2$, $\dots$, $X_n$ be a sequence of coherent random variables, i.e., satisfying the equalities $$X_j=\mathbb{P}(A\mathrm{\mid }{\mathcal{G}}_j),j=1,2,\dots ,n,$$ almost surely for some event $A$. The paper contains the proof of the estimate $$\mathbb{P}(\underset{1\le i<j\le n}{\max }\mathrm{\mid }{X}_i-X_j\mathrm{\mid }\ge \delta )\le \frac{n(1-\delta )}{2-\delta }\wedge 1,$$ where $\delta \in (\frac{1}{2},1]$ is a given parameter. The inequality is sharp: for any $\delta$, the constant on the right cannot be replaced by any smaller number. The argument rests on several novel combinatorial and symmetrization arguments, combined with dynamic programming. Our result generalizes the two-variate inequality of K. Burdzy and S. Pal and in particular provides its alternative derivation.