EF1 Allocations for Identical Trilean and Separable Single-Peaked Valuations

TL;DR

This paper extends the existence results of EF1 allocations to new classes of valuations, specifically identical trilean and separable single-peaked valuations, while also showing EFX allocations do not exist for these classes.

Contribution

It introduces and proves EF1 existence for two new valuation classes, expanding the understanding of fair division under these valuation types.

Findings

EF1 allocations exist for identical trilean valuations with any number of agents.

EF1 allocations exist for three agents with separable single-peaked valuations.

EFX allocations do not exist for these valuation classes.

Abstract

In the fair division of items among interested agents, envy-freeness is possibly the most favoured and widely studied formalisation of fairness. For indivisible items, envy-free allocations may not exist in trivial cases, and hence research and practice focus on relaxations, particularly envy-freeness up to one item (EF1). A significant reason for the popularity of EF1 allocations is its simple fact of existence. It is known that EF1 allocations exist for two agents with arbitrary valuations; agents with doubly-monotone valuations; agents with Boolean valuations; and identical agents with negative Boolean valuations. We consider two new but natural classes of valuations, and partly extend results on the existence of EF1 allocations to these valuations. Firstly, we consider trilean valuations - an extension of Boolean valuations - when the value of any subset is 0, $a$, or $b$ for any integers $a$ and $b$. Secondly, we define separable single-peaked valuations, when the set of items is partitioned into types. For each type, an agent's value is a single-peaked function of the number of items of the type. The value for a set of items is the sum of values for the different types. We prove EF1 existence for identical trilean valuations for any number of agents, and for separable single-peaked valuations for three agents. For both classes of valuations, we also show that EFX allocations do not exist.