Fair Division with Allocator's Preference

Xiaolin Bu, Zihao Li, Shengxin Liu, Jiaxin Song, Biaoshuai Tao

Abstract: We consider the problem of fairly allocating indivisible resources to agents, which has been studied for years. Most previous work focuses on fairness and/or efficiency among agents given agents’ preferences. However, besides the agents, the allocator as the resource owner may also be involved in many real-world scenarios (e.g., government resource allocation, heritage division, company personnel assignment, etc.). The allocator has the inclination to obtain a fair or efficient allocation based on her own preference over the items and to whom each item is allocated. In this paper, we propose a new model and focus on the following two problems concerning the allocator’s fairness and efficiency: We consider the two fundamental fairness criteria: envy-freeness and proportionality. For the first problem, we study the existence of an allocation that is envy-free up to c goods (EF-c) or proportional up to c goods (PROP-c) from both the agents’ and the allocator’s perspectives, in which such an allocation is called doubly EF-c or doubly PROP-c respectively. When the allocator’s utility depends exclusively on the items (but not to whom an item is allocated), we prove that a doubly EF-1 allocation always exists. For the general setting where the allocator has a preference over the items and to whom each item is allocated, we prove that a doubly EF-1 allocation always exists for two agents, a doubly PROP-2 allocation always exists for binary valuations, and a doubly PROP- $$O(\log n)$$ allocation always exists in general. For the second problem, we provide various (in)approximability results in which the gaps between approximation and inapproximation ratios are asymptotically closed under most settings. Most of our results are based on some novel technical tools including the chromatic numbers of the Kneser graphs and linear programming-based analysis.