Exploring Relations among Fairness Notions in Discrete Fair Division

TL;DR

The work addresses the challenge of comparing fairness notions in discrete fair division by analyzing 22 notions and organizing their implications into a hierarchy across goods, chores, and mixed manna. It combines manual proofs with a novel inference engine to automatically derive numerous implications and counterexamples, yielding a near-complete understanding in many settings, including additive and broader valuation classes. By extending definitions to mixed manna and unequal entitlements, and by providing a public web tool, the paper lays a foundational framework for systematic reasoning about fairness notions and their practical applicability. The results illuminate strengths and limitations of each notion, and the open problems outline clear directions for future research in fair division and related optimization problems.

Abstract

Fair allocation of indivisible items among agents is a fundamental and extensively studied problem. However, fairness does not have a single universally accepted definition, leading to a variety of competing fairness notions. Some of these notions are considered stronger or more desirable, but they are also more difficult to guarantee. In this work, we examine 22 different notions of fairness and organize them into a hierarchy. Formally, we say that a fairness notion $F_1$ implies another notion $F_2$ if every $F_1$-fair allocation is also $F_2$-fair. We give a near-complete picture of implications among fairness notions: for almost every pair of notions, we either prove an implication or give a counterexample demonstrating that the implication does not hold. Although some of these results are already known, many are new. We examine multiple settings, including the allocation of goods, chores, and mixed manna. We believe this work clarifies the relative strengths and applicability of these notions, providing a foundation for future research in fair division. Moreover, we developed an inference engine to automate part of our work. It is available as a user-friendly web application and may have broader applications beyond fair division.

Figures (15)

• Figure 1: Implications between fairness notions for additive valuations over goods and over chores when agents have equal entitlements. There is a vertex for each fairness notion. Notion $F_1$ implies notion $F_2$ iff there is a path from $F_1$ to $F_2$ in the graph (except that it is not known whether MXS implies EEF1 for goods). Borderless vertices (red) are infeasible notions, vertices with rounded corners (green) are feasible notions, and the feasibility of the remaining vertices (gray) are open problems. Note that goods and chores have some key differences, e.g., for goods, PROP $\Longrightarrow$ MMS $\Longrightarrow$ EEF1 $\Longrightarrow$ PROP1, but for chores, MMS $\centernot{\Longrightarrow}$ PROP1, and PROP $\centernot{\Longrightarrow}$ EEF1.
• Figure 2: Screenshot from the inference engine's web interface for fair division.
• Figure 3: Valuation function type and marginal values represented as digraphs where there is a path from $u$ to $v$ iff $u$ is a subclass of $v$.
• Figure 4: Additive valuations, mixed manna.
• Figure 5: Additive valuations, marginals in $\{-1, 0, 1\}$. We get the same DAG when marginals are in $\{0, -1\}$ or $\{0, 1\}$.

Theorems & Definitions (234)

• Definition 1: EF
• Definition 2: EF1
• Definition 3: EFX
• Definition 4: PROP
• Definition 5: PROP1 conitzer2017fair
• Definition 6: epistemic fairness
• Definition 7: minimum fair share
• Definition 8: restricting, pairwise fairness, and groupwise fairness
• Definition 9: EFX$_0$ for goods
• Definition 10: EFX$_0$ for chores