Fair Division with Allocator's Preference

We study the fair allocation of indivisible resources among agents. Most prior work focuses on fairness and/or efficiency among agents. However, the allocator, as the resource owner, may also be involved in many scenarios (e.g., government resource allocation, heritage division, company personnel assignment, etc). The allocator inclines to obtain a fair or efficient allocation based on her preference over the items and to whom each item is allocated. We propose a model and study two problems: 1) Find an allocation fair to both agents and allocator; 2) Maximize allocator's efficiency under 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, called doubly EF-$c$ or doubly PROP-$c$. When the allocator's utility only depends on the items (not recipients), 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, a doubly EF-$1$ allocation always exists for two agents, a doubly PROP-$2$ allocation always exists for personalized bi-valued valuations, and a doubly PROP-$O(\log n)$ allocation always exists. For the second problem, we give (in)approximability results with asymptotically tight bounds in most settings. When agents' valuations are binary, maximizing the allocator's social welfare while ensuring agents' fairness criteria of PROP-$c$ (with a general number of agents) and EF-$c$ (with a constant number of agents) are both polynomial-time solvable for any integer $c$. Strong inapproximability holds for most of the other settings (general valuations, EF-$c$, etc).