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

2025-01-03

AI Summary

This paper studies the worst-case impact of introducing cardinality constraints—limiting each agent to at most $k$ indivisible items—on social welfare in item allocation. It formalizes this degradation via the *cardinality cost*, a multiplicative measure quantifying welfare loss for both utilitarian and fairness-oriented objectives. Using parameterized analysis, extremal combinatorial optimization, and worst-case bounding techniques, the work provides the first exact characterization and tight bounds on the cardinality cost function. It refines and generalizes asymptotic bounds on the *balance cost*, previously studied in constrained allocation. Furthermore, the model is extended to multi-category hierarchical constraints, yielding exact or nearly tight bounds parameterized by $k$ and the number of agents. Collectively, these results establish a quantifiable theoretical framework for analyzing the fairness–efficiency trade-off under capacity constraints, supporting principled algorithmic design and policy decisions.

Technology Category

• Constraint Satisfaction and Optimization: Other Foundations of Constraint Satisfaction
• Multiagent Systems: Mechanism Design
• Game Theory and Economic Paradigms: Fair Division

Application Category

• Economics, Online Markets and Human Computation: Incentives in network design for Web infrastructures and ecosystems
• Graph Algorithms and Modeling for the Web: Algorithms and analysis for incomplete, noisy, or partially observed Web-related graphs
• Responsible Web: Human-perceived consequences of algorithmic deployment on the web

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.

Problem

• Fair Allocation
• Indivisible Goods
• Social Welfare Maximization

Innovation

• Cardinality Price
• Welfare Improvement
• Category-based Allocation