The (Exact) Price of Cardinality for Indivisible Goods: A Parametric Perspective
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866912176138092544 |
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| author | Lam, Alexander Li, Bo Sun, Ankang |
| author_facet | Lam, Alexander Li, Bo Sun, Ankang |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_01660 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The (Exact) Price of Cardinality for Indivisible Goods: A Parametric Perspective Lam, Alexander Li, Bo Sun, Ankang Computer Science and Game Theory 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. |
| title | The (Exact) Price of Cardinality for Indivisible Goods: A Parametric Perspective |
| topic | Computer Science and Game Theory |
| url | https://arxiv.org/abs/2501.01660 |