The (Exact) Price of Cardinality for Indivisible Goods: A Parametric Perspective
Alexander Lam, Bo Li, Ankang Sun
TL;DR
This paper introduces the price of cardinality, the worst-case multiplicative loss in social welfare when enforcing a cardinality constraint on allocations of indivisible goods, for both utilitarian and egalitarian metrics. It adopts a parametric approach, deriving tight or almost-tight, exact bounds as functions of instance parameters $n$, $m$, $k$, and per-category sizes $m_j$, $k_j$, in both single- and multi-category settings, and it shows how welfare improves as the constraint loosens ($k$ increases). Notably, it provides a closed-form characterization in the single-category case for utilitarian welfare $\frac{1}{2}\left(1+\sqrt{1+\frac{m-1}{k}}\right)$ and egalitarian welfare $\max\left\{\frac{m-n+1}{k},1\right\}$, and extends to multi-category scenarios with exact bounds such as $\frac{2}{\frac{k_1}{m_1}+\frac{k_2}{m_2}}$ for two agents (utilitarian) and $\frac{m_1}{k_1}$ (general $n$), as well as a tight egalitarian bound across categories. These results yield a practical framework for decision-makers to select an appropriate cardinality level by quantifying the trade-off between fairness and welfare and extend prior asymptotic bounds on the price of balancedness. The work also outlines linear-time pathways to compute cardinal allocations that guarantee welfare matching the derived bounds via maximum weight bipartite matchings per category, enhancing applicability in real-world fair division tasks.
Abstract
We adopt a parametric approach to analyze the worst-case degradation in social welfare when the allocation of indivisible goods is constrained to be fair. Specifically, we are concerned with cardinality-constrained allocations, which require that each agent has at most $k$ items in their allocated bundle. We propose the notion of the price of cardinality, which captures the worst-case multiplicative loss of utilitarian or egalitarian social welfare resulting from imposing the cardinality constraint. We then characterize tight or almost-tight bounds on the price of cardinality as exact functions of the instance parameters, demonstrating how the social welfare improves as $k$ is increased. In particular, one of our main results refines and generalizes the existing asymptotic bound on the price of balancedness, as studied by Bei et al. [BLMS21]. We also further extend our analysis to the problem where the items are partitioned into disjoint categories, and each category has its own cardinality constraint. Through a parametric study of the price of cardinality, we provide a framework which aids decision makers in choosing an ideal level of cardinality-based fairness, using their knowledge of the potential loss of utilitarian and egalitarian social welfare.