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

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.