The (Exact) Price of Cardinality for Indivisible Goods: A Parametric Perspective

TL;DR

This paper introduces the concept of the price of cardinality to quantify the worst-case social welfare loss due to fairness constraints in allocating indivisible goods, providing bounds and insights for decision-making.

Contribution

It defines and characterizes the price of cardinality, extending existing bounds and analyzing multi-category item constraints for fair and efficient allocations.

Findings

Derived tight bounds on the price of cardinality as functions of instance parameters.

Generalized the asymptotic bound on the price of balancedness from prior work.

Extended analysis to multi-category item partitions with individual constraints.

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.