# Mixtures of Mean-Preserving Contractions

## TL;DR

This paper proves that any mean-preserving contraction of a discrete probability measure can be expressed as a mixture of simpler contractions, with implications for economic modeling.

## Contribution

It introduces a novel decomposition of mean-preserving contractions into mixtures of simpler contractions, enhancing understanding of their structure.

## Key findings

- Any mean-preserving contraction with support on more points can be decomposed into mixtures of contractions with support on fewer points.
- The decomposition has practical applications in economic theory.
- Theoretical proof of the mixture representation for atomic probability measures.

## Abstract

Given a purely atomic probability measure with support on n points, P, any mean-preserving contraction (mpc) of P, Q, with support on m > n points is a mixture of mpcs of P, each with support on most n points. We illustrate an application of this result in economics.