Fair Division with Subjective Divisibility
Xiaohui Bei, Shengxin Liu, Xinhang Lu
TL;DR
This work introduces fair division with subjective divisibility, where agents differ on whether a given good is divisible or indivisible. It adapts maximin share (MMS) and envy-freeness relaxations to this model and provides tight or near-tight guarantees: a worst-case MMS approximation of $\frac{2}{3}$ for $n\ge 2$ with exactness for $n=2,3$, and a $\frac{1}{2}$-MMS allocation for any $n$; it also analyzes envy-freeness relaxations (EF, EF1M, EFM, EFXM) and non-wastefulness, proving incompatibility between EFM and non-wastefulness but showing that EFXM with at most one discarded good exists for two agents, while EF1M is compatible with non-wastefulness for any $n$. The paper develops algorithmic tools, such as High-Valued-Alloc and reducible-bundle concepts, and discusses computation via PTAS to approximate MMS, tying fairness with economic efficiency (PO). It also extends results to heterogeneous divisible goods (cakes) and outlines future directions for broader subjective divisibility models and constraints. Overall, the work broadens fair division theory to settings where divisibility is agent-relative, revealing fundamental limits and constructive possibilities for fair and efficient allocations.
Abstract
The classic fair division problems assume the resources to be allocated are either divisible or indivisible, or contain a mixture of both, but the agents always have a predetermined and uncontroversial agreement on the (in)divisibility of the resources. In this paper, we propose and study a new model for fair division in which agents have their own subjective divisibility over the goods to be allocated. That is, some agents may find a good to be indivisible and get utilities only if they receive the whole good, while others may consider the same good to be divisible and thus can extract utilities according to the fraction of the good they receive. We investigate fairness properties that can be achieved when agents have subjective divisibility. First, we consider the maximin share (MMS) guarantee and show that the worst-case MMS approximation guarantee is at most $2/3$ for $n \geq 2$ agents and this ratio is tight in the two- and three-agent cases. This is in contrast to the classic fair division settings involving two or three agents. We also give an algorithm that produces a $1/2$-MMS allocation for an arbitrary number of agents. Second, we study a hierarchy of envy-freeness relaxations, including EF1M, EFM and EFXM, ordered by increasing strength. While EF1M is compatible with non-wastefulness (an economic efficiency notion), this is not the case for EFM, even for two agents. Nevertheless, an EFXM and non-wasteful allocation always exists for two agents if at most one good is discarded.