EF1 Allocations for Identical Trilean and Separable Single-Peaked Valuations

TL;DR

The paper advances fair division by proving EF1 existence for two new valuation families: identical trilean valuations (values in $\{0, a, b\}$) for any $n$, and separable single-peaked valuations (SSP) for three agents. It delivers constructive algorithms, including NegBooleanEF1 for Boolean valuations and TrileanNegEF1/TrileanPosEF1 for trilean valuations, with phase-based analyses and FixEF1Violations subroutines to ensure EF1. It also shows that EFX allocations do not exist for these valuation classes, and provides SSP-specific results: EF1 exists under common per-type thresholds and also for three agents with differing thresholds. The work further clarifies the limitations of EFX variants (e.g., $\textrm{EFX}^{+}_{-}$, $\textrm{EFX}^{0}_{-}$) via counterexamples, highlighting important directions for future research on EF1 in nonmonotone and SSP settings and the remaining open cases (notably three agents with nonidentical valuations).

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.

Figures (2)

• Figure 1: The edges depict possible EF1 violations between different sets of agents.
• Figure 2: The edges show possible EF1 violations between different sets of agents

Theorems & Definitions (32)

• Proposition 0
• Definition 1
• Theorem 2
• Theorem 3
• Lemma 3
• Proposition 3
• Proposition 4
• Claim 5
• Claim 6
• Claim 6