Centralized Group Equitability and Individual Envy-Freeness in the Allocation of Indivisible Items

TL;DR

The paper addresses fair division of indivisible items under a centralized allocator who values group fairness via $CGEQ$/$CGEQ1$ and agent fairness via $EF1$. It introduces three algorithmic frameworks—$DM$ for identical valuations, $SPS$ for ordered valuations, and $CD^2P$ for binary allocator valuations—each guaranteeing an $EF1$ allocation together with $CGEQ1$, all in polynomial time. It also defines the allocator-centric $CGMMS$ objective, proving NP-hardness in general while showing tractability in the binary-allocator case using forward and reverse round-robin schemes. The work thus provides allocator-aware, group-sensitive fair division algorithms with practical implications for uneven resource distribution across departments, neighborhoods, or organizations, and outlines directions for extending to broader valuation models and approximate guarantees.

Abstract

We study the fair allocation of indivisible items for groups of agents from the perspectives of the agents and a centralized allocator. In our setting, the centralized allocator is interested in ensuring the allocation is fair among the groups and between agents. This setting applies to many real-world scenarios, including when a school administrator wants to allocate resources (e.g., office spaces and supplies) to staff members in departments and when a city council allocates limited housing units to various families in need across different communities. To ensure fair allocation between agents, we consider the classical envy-freeness (EF) notion. To ensure fairness among the groups, we define the notion of centralized group equitability (CGEQ) to capture the fairness for the groups from the allocator's perspective. Because an EF or CGEQ allocation does not always exist in general, we consider their corresponding natural relaxations of envy-freeness to one item (EF1) and centralized group equitability up to one item (CGEQ1). For different classes of valuation functions of the agents and the centralized allocator, we show that allocations satisfying both EF1 and CGEQ1 always exist and design efficient algorithms to compute these allocations. We also consider the centralized group maximin share (CGMMS) from the centralized allocator's perspective as a group-level fairness objective with EF1 for agents and present several results.

Figures (2)

• Figure 1: Example where $|G_p| = 2$ and $|G_q| = 5$, with $G_q$ receiving the first bundle during the reallocation process. We denote $B_{z}^{i}$ as the bundle received by $G_{z}$ in the $i$-th allocation. The red squares represent bundles received by $G_q$, and the blue squares represent bundles received by $G_p$.
• Figure 2: An illustration of the allocation process of the first twelve items. We use $\ell_i$ to denote the picking order of agent $i$ in her group. Assume that there are three groups $G_1$, $G_2$, and $G_3$, where each group has $3$, $4$, and $5$ agents respectively. The arrow means the sequence of the allocation of these twelve items. For example, in $G_1$, agent 1 with $\ell_1 = 1$ picks $o_1$, agent 6 with $\ell_6 = 2$ picks $o_6$, and agent 10 with $\ell_{10} = 3$ picks $o_{10}$. Then, in the following iterations, if $G_1$ receives some item, the algorithm will follow the order to select the target agent.

Theorems & Definitions (23)

• Definition 1: Envy-Freeness
• Definition 2: Envy-Freeness up to One Item
• Definition 3: Centralized Group Equitability
• Remark 1
• Definition 4: Centralized Group Equitability up to One Item
• Theorem 1
• Lemma 1
• proof
• Lemma 2
• proof