Technology Category
Constraint Satisfaction and Optimization: Other Foundations of Constraint SatisfactionMultiagent Systems: Mechanism DesignGame Theory and Economic Paradigms: Fair Division
The (Exact) Price of Cardinality for Indivisible Goods: A Parametric Perspective
🤖 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
Application Category
📝 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
Research questions and friction points this paper is trying to address.
Fair Allocation
Indivisible Goods
Social Welfare Maximization
Innovation
Methods, ideas, or system contributions that make the work stand out.
Cardinality Price
Welfare Improvement
Category-based Allocation
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