Cutting a cake for infinitely many guests

TL;DR

This paper introduces new, simpler algorithms for fair division with irrational entitlements and proves that fair division is always possible even with infinitely many players, extending classical methods.

Contribution

The paper presents novel algorithms based on the Last diminisher technique for fair division with irrational entitlements and establishes the existence of fair division among infinitely many players.

Findings

New algorithms for irrational entitlements

Existence of fair division with infinitely many players

Simpler methods than previous approaches

Abstract

Fair division with unequal shares is an intensively studied recourse allocation problem. For $ i\in [n] $, let $ \mu_i $ be an atomless probability measure on the measurable space $(C,\mathcal{S}) $ and let $ t_i $ be positive numbers (entitlements) with $ \sum_{i=1}^{n}t_i=1 $. A fair division is a partition of $ C $ into sets $ S_i\in \mathcal{S} $ with $ \mu_i(S_i)\geq t_i$ for every $ i\in [n] $. We introduce new algorithms to solve the fair division problem with irrational entitlements. They are based on the classical Last diminisher technique and we believe that they are simpler than the known methods. Then we show that a fair division always exists even for infinitely many players.