[Paper Review] For One and All: Individual and Group Fairness in the Allocation of Indivisible Goods

Jonathan Scarlett, Nicholas Teh+1 more|arXiv (Cornell University)|Feb 14, 2023

Experimental Behavioral Economics Studies6 citations

TL;DR

This paper proposes polynomial-time algorithms that simultaneously achieve approximate individual envy-freeness (i-EF1) and group weighted envy-freeness (g-WEF1) in the allocation of indivisible goods. It establishes that when agents have identical or common group valuations, both i-EF1 and g-WEF1 can be achieved exactly, while in the general case with distinct valuations, a 1/3-approximation to ex-ante g-WEF1 is achievable under i-EF1.

ABSTRACT

Fair allocation of indivisible goods is a well-explored problem. Traditionally, research focused on individual fairness - are individual agents satisfied with their allotted share? - and group fairness - are groups of agents treated fairly? In this paper, we explore the coexistence of individual envy-freeness (i-EF) and its group counterpart, group weighted envy-freeness (g-WEF), in the allocation of indivisible goods. We propose several polynomial-time algorithms that provably achieve i-EF and g-WEF simultaneously in various degrees of approximation under three different conditions on the agents' (i) when agents have identical additive valuation functions, i-EFX and i-WEF1 can be achieved simultaneously; (ii) when agents within a group share a common valuation function, an allocation satisfying both i-EF1 and g-WEF1 exists; and (iii) when agents' valuations for goods within a group differ, we show that while maintaining i-EF1, we can achieve a 1/3-approximation to ex-ante g-WEF1. Our results thus provide a first step towards connecting individual and group fairness in the allocation of indivisible goods, in hopes of its useful application to domains requiring the reconciliation of diversity with individual demands.

Motivation & Objective

• To reconcile individual fairness (envy-freeness) and group fairness (weighted envy-freeness) in the allocation of indivisible goods.
• To design efficient algorithms that simultaneously satisfy approximate versions of both fairness criteria.
• To analyze the trade-offs and feasibility of co-existing individual and group fairness under different valuation structures.
• To explore the limits of approximation when agents have general additive valuations.
• To extend fairness guarantees to include additional properties like PEF and group stability.

Proposed method

• Proposes a novel algorithmic framework based on iterative weighted rounding (IWRR) to balance individual and group fairness.
• Uses a shifted-round analysis to compare shaded (unallocated) and circled (allocated) cells in valuation matrices to bound fairness violations.
• Applies case analysis based on group size ratios (w_k ≤ w_k' vs. w_k ≥ w_k') to derive bounds on group fairness approximation.
• Employs a 3w_k'-factor comparison between shaded and circled cell sums to prove the 1/3-approximation for g-WEF1 in the general case.
• Introduces a relaxed notion of ex-ante g-WEF1 to handle scenarios with distinct agent valuations.
• Extends the SM-IWRR and IWRR algorithms to also satisfy relaxed variants of PEF and group stability.

Experimental results

Research questions

• RQ1 Can individual envy-freeness and group weighted envy-freeness be simultaneously achieved in the allocation of indivisible goods?
• RQ2 What approximation guarantees are possible for group fairness when individual fairness is strictly enforced?
• RQ3 How do different valuation structures—identical, common within groups, or distinct—affect the coexistence of fairness notions?
• RQ4 Is there a constant-factor approximation to group fairness achievable under i-EF1 in the general additive valuation setting?
• RQ5 Can additional fairness properties like PEF and group stability be preserved alongside individual and group fairness?

Key findings

• When agents have identical additive valuation functions, i-EFX and g-WEF1 can be achieved simultaneously in polynomial time.
• When agents within each group share a common valuation, an allocation satisfying both i-EF1 and g-WEF1 exists and can be computed efficiently.
• For general additive valuations with distinct agent preferences, the IWRR algorithm achieves i-EF1 and a 1/3-approximation to ex-ante g-WEF1.
• The 1/3-approximation bound is derived via a case analysis comparing shaded and circled cell sums in a shifted-round valuation matrix.
• The SM-IWRR and IWRR algorithms also satisfy relaxed variants of PEF and group stability, extending their fairness guarantees.
• The results demonstrate that individual and group fairness can be reconciled under structured valuation classes, but the approximation quality degrades in the general case.