Contradictory predictions with multiple agents

TL;DR

This paper proves a sharp probabilistic inequality for the maximum difference among a sequence of coherent random variables, extending previous two-variable results using novel combinatorial and symmetrization techniques.

Contribution

It introduces a new sharp inequality for the maximum difference of multiple coherent random variables, generalizing earlier two-variable bounds with innovative combinatorial methods.

Findings

Proves a sharp upper bound for the probability of large deviations among multiple coherent variables.

Extends the two-variable inequality of Burdzy and Pal to multiple variables.

Uses novel combinatorial and symmetrization techniques in the proof.

Abstract

Let $X_1$, $X_2$, $\ldots$, $X_n$ be a sequence of coherent random variables, i.e., satisfying the equalities $$ X_j=\mathbb{P}(A|\mathcal{G}_j),\qquad j=1,\,2,\,\ldots,\,n,$$ almost surely for some event $A$. The paper contains the proof of the estimate $$\mathbb{P}\Big(\max_{1\le i < j\le n}|X_i-X_j|\ge \delta\Big) \leq \frac{n(1-\delta)}{2-\delta} \wedge 1,$$ 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.