🤖 AI Summary
This paper investigates the joint optimization of fairness and efficiency in indivisible resource allocation, addressing fairness for both agents and the allocator. It introduces the novel concept of “double fairness” (doubly EF-c/PROP-c), explicitly modeling the allocator’s preferences as agent-dependent functions and requiring allocations to satisfy fairness constraints for both parties. Methodologically, the work integrates Kneser graph chromatic number analysis, combinatorial existence proofs, linear programming formulations, and approximation algorithm design. Theoretically, it establishes that doubly EF-1 allocations always exist—even under single-item preferences or two-agent settings—while doubly PROP-2 allocations exist for binary valuations, and doubly PROP-O(log n) allocations exist generally. Moreover, it closes approximation gaps in most settings by providing tight (non-)approximability bounds for double fairness.
📝 Abstract
We consider the fair allocation problem of indivisible items. 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., heritage division. 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: 1) Is it possible to find an allocation that is fair for both the agents and the allocator? 2) What is the complexity of maximizing the allocator's social welfare while satisfying the agents' fairness? 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 results are based on novel technical tools including the chromatic numbers of the Kneser graphs and linear programming-based analysis.
