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On the Fairness of Additive Welfarist Rules

Karen Frilya Celine, Warut Suksompong, Sheung Man Yuen

TL;DR

This paper analyzes additive welfarist rules for fair division of indivisible goods, focusing on when they guarantee $EF1$ across varying instance classes. It proves that in the real-valued setting, the maximum Nash welfare (MNW) rule, corresponding to $f(x)=\alpha\log x+\beta$, uniquely yields $EF1$ for identical-good, two-value, and normalized-three-or-more-agent instances, connecting scale-invariance to fairness. In contrast, the integer-valued setting admits broader families of $EF1$-guaranteeing rules (e.g., modified logarithmic $\lambda_c$ and modified harmonic $h_c$) with precise necessary and sufficient conditions (Conditions 3–6) that vary by instance class (identical-good, binary, two-value, general). The work also frames these results via a network of implications and proposes directions for further study, including comparisons of fairness vs. efficiency trade-offs and extensions beyond additive welfarist rules. Overall, it deepens our understanding of how the choice of the welfare function $f$ shapes fairness guarantees in fair division problems, particularly distinguishing the rigid real-valued setting from the more flexible integer-valued regime.

Abstract

Allocating indivisible goods is a ubiquitous task in fair division. We study additive welfarist rules, an important class of rules which choose an allocation that maximizes the sum of some function of the agents' utilities. Prior work has shown that the maximum Nash welfare (MNW) rule is the unique additive welfarist rule that guarantees envy-freeness up to one good (EF1). We strengthen this result by showing that MNW remains the only additive welfarist rule that ensures EF1 for identical-good instances, two-value instances, as well as normalized instances with three or more agents. On the other hand, if the agents' utilities are integers, we demonstrate that several other rules offer the EF1 guarantee, and provide characterizations of these rules for various classes of instances.

On the Fairness of Additive Welfarist Rules

TL;DR

This paper analyzes additive welfarist rules for fair division of indivisible goods, focusing on when they guarantee across varying instance classes. It proves that in the real-valued setting, the maximum Nash welfare (MNW) rule, corresponding to , uniquely yields for identical-good, two-value, and normalized-three-or-more-agent instances, connecting scale-invariance to fairness. In contrast, the integer-valued setting admits broader families of -guaranteeing rules (e.g., modified logarithmic and modified harmonic ) with precise necessary and sufficient conditions (Conditions 3–6) that vary by instance class (identical-good, binary, two-value, general). The work also frames these results via a network of implications and proposes directions for further study, including comparisons of fairness vs. efficiency trade-offs and extensions beyond additive welfarist rules. Overall, it deepens our understanding of how the choice of the welfare function shapes fairness guarantees in fair division problems, particularly distinguishing the rigid real-valued setting from the more flexible integer-valued regime.

Abstract

Allocating indivisible goods is a ubiquitous task in fair division. We study additive welfarist rules, an important class of rules which choose an allocation that maximizes the sum of some function of the agents' utilities. Prior work has shown that the maximum Nash welfare (MNW) rule is the unique additive welfarist rule that guarantees envy-freeness up to one good (EF1). We strengthen this result by showing that MNW remains the only additive welfarist rule that ensures EF1 for identical-good instances, two-value instances, as well as normalized instances with three or more agents. On the other hand, if the agents' utilities are integers, we demonstrate that several other rules offer the EF1 guarantee, and provide characterizations of these rules for various classes of instances.

Paper Structure

This paper contains 22 sections, 53 theorems, 15 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Our Results
  3. Preliminaries
  4. Examples of Additive Welfarist Rules
  5. Real-Valued Instances
  6. Identical-Good Instances
  7. Two-Value Instances
  8. Normalized Instances
  9. Integer-Valued Instances
  10. Identical-Good Instances
  11. Binary Instances
  12. Two-Value Instances
  13. General Integer-Valued Instances
  14. Conclusion
  15. Omitted Proofs
  16. ...and 7 more sections

Key Result

theorem 1

Let $n \geq 2$ be given, and let $f: \mathbb{R}_{\geq 0} \to \mathbb{R} \cup \{-\infty\}$ be a strictly increasing function continuous on $\mathbb{R}_{>0}$. Then, the following statements are equivalent:

Figures (2)

  • Figure 1: Relationships between the different conditions (see \ref{['def:conditions']}).
  • Figure 2: Venn diagram showing the relationships between additive welfarist rules that guarantee EF1 for different classes of instances. Each region, labeled by $F$ with a subscript, represents the set of functions defining the additive welfarist rules that guarantee EF1 for the class of instances corresponding to the subscript. Some examples of these functions are given in the respective regions.

Theorems & Definitions (54)

  • definition 1
  • theorem 1
  • lemma 1
  • lemma 2
  • lemma 3
  • lemma 4
  • lemma 5
  • lemma 6
  • lemma 7
  • lemma 8
  • ...and 44 more