How to cut a discrete cake fairly

TL;DR

This paper proves that a fair division of indivisible items arranged on a path, satisfying a discrete envy-freeness condition, always exists for any number of agents with monotone valuations, extending previous results and addressing open questions.

Contribution

It establishes the existence of connected EF1 divisions for any number of agents, including secretive and extra-agent scenarios, solving an open problem in fair division.

Findings

Connected EF1 division always exists for any number of agents.

The result extends to secretive and extra-agent versions.

The proof settles an open question from prior research.

Abstract

Cake-cutting is a fundamental model of dividing a heterogeneous resource, such as land, broadcast time, and advertisement space. In this study, we consider the problem of dividing a discrete cake fairly in which the indivisible goods are aligned on a path and agents are interested in receiving a connected subset of items. We prove that a connected division of indivisible items satisfying a discrete counterpart of envy-freeness, called envy-freeness up to one good (EF1), always exists for any number of agents n with monotone valuations. Our result settles an open question raised by Bil\`o et al. (2019), who proved that an EF1 connected division always exists for the number of agents at most 4. Moreover, the proof can be extended to show the following (1) secretive and (2) extra versions: (1) for n agents with monotone valuations, the path can be divided into n connected bundles such that an EF1 assignment of the remaining bundles can be made to the other agents for any selection made by the secretive agent; (2) for n+1 agents with monotone valuations, the path can be divided into n connected bundles such that when any extra agent leaves, an EF1 assignment of the bundles can be made to the remaining agents.