Expected Natural Density of Countable Sets after Infinitely Iterated de Finetti Lotteries, Computed via Matrix Decomposition

TL;DR

This paper introduces a novel matrix decomposition method to compute the expected natural density of countable sets after infinitely iterated de Finetti lotteries, generalizing previous specific case results.

Contribution

The work develops a new framework using infinite matrices to calculate expected densities in iterated lotteries, extending prior findings to more general scenarios.

Findings

Provides a general formula for expected densities after infinite iterations

Extends previous specific case results to broader settings

Introduces a matrix decomposition approach for analysis

Abstract

Consider a fair lottery over the natural numbers in which the selected number is removed. This lottery is iterated countably infinite times, with a known ratio of iterations to natural numbers. Removed numbers are not replaced. The natural numbers are partitioned into two sets with a given ratio of elements, which is tracked along each iteration of the lottery. Hess and Polisetty considered and investigated such a process and reported the expected values of the densities for some particular cases. In this work, we provide a novel framework for computing these expected densities using infinite matrices. The results presented in this work generalize previous results.