Simultaneously Fair Allocation of Indivisible Items Across Multiple Dimensions

TL;DR

This paper introduces new fairness notions for allocating indivisible items across multiple criteria, providing bounds, algorithms, and complexity results for these multidimensional fairness concepts.

Contribution

It proposes two relaxed envy-freeness notions for multidimensional settings, analyzes their existence bounds, and studies the computational complexity of verifying these allocations.

Findings

Bounds on the relaxation parameter c for existence of sEFc allocations.

Algorithms for checking the existence of weak and strong sEFc allocations.

NP-hardness results for verifying sEF1 allocations.

Abstract

This paper explores the fair allocation of indivisible items in a multidimensional setting, motivated by the need to address fairness in complex environments where agents assess bundles according to multiple criteria. Such multidimensional settings are not merely of theoretical interest but are central to many real-world applications. For example, cloud computing resources are evaluated based on multiple criteria such as CPU cores, memory, and network bandwidth. In such cases, traditional one dimensional fairness notions fail to capture fairness across multiple attributes. To address these challenges, we study two relaxed variants of envy-freeness: weak simultaneously envy-free up to c goods (weak sEFc) and strong simultaneously envy-free up to c goods (strong sEFc), which accommodate the multidimensionality of agents' preferences. Under the weak notion, for every pair of agents and for each dimension, any perceived envy can be eliminated by removing, if necessary, a different set of goods from the envied agent's allocation. In contrast, the strong version requires selecting a single set of goods whose removal from the envied bundle simultaneously eliminates envy in every dimension. We provide upper and lower bounds on the relaxation parameter c that guarantee the existence of weak or strong sEFc allocations, where these bounds are independent of the total number of items. In addition, we present algorithms for checking whether a weak or strong sEFc allocation exists. Moreover, we establish NP-hardness results for checking the existence of weak sEF1 and strong sEF1 allocations.