Abstract
We consider the fair division of indivisible items using the maximin shares measure. Recent work on the topic has focused on extending results beyond the class of additive valuation functions. In this spirit, we study the case where the items form an hereditary set system. We present a simple algorithm that allocates each agent a bundle of items whose value is at least 0.3667 times the maximin share of the agent. This improves upon the current best known guarantee of 0.2 due to Ghodsi et al. The analysis of the algorithm is almost tight; we present an instance where the algorithm provides a guarantee of at most 0.3738. We also show that the algorithm can be implemented in polynomial time given a valuation oracle for each agent.
Z. Li—The first author thanks McGill University for hosting them while conducting this research.
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Notes
- 1.
Observe that if the agents have sub-additive valuation functions then envy-freeness implies proportionality.
- 2.
For example, consider an hereditary set system \(H=(J, {\mathcal {F}})\) with three items \(J=\{a,b,c\}\) and let the maximal independent sets in \({\mathcal {F}}\) be \(\{a\}\) and \(\{b, c\}\). Suppose agent i has item values \(v_{i,a}=3, v_{i,b}=2\) and \(v_{i,c}=2\). Thus \(v_i(\{a,c\})=3\) and \(v_i(\{a,b,c\})=4\). Consequently, the marginal value of adding item c to the set \(\{a,b\}\) is larger than the marginal value of adding item c to the set \(\{a\}\). Thus the valuation function is not submodular.
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Acknowledgements
The authors thank Jugal Garg, Vasilis Gkatzelis and Richard Santiago for interesting discussions on fair division. We thank the anonymous reviewers for helpful suggestions.
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Li, Z., Vetta, A. (2018). The Fair Division of Hereditary Set Systems. In: Christodoulou, G., Harks, T. (eds) Web and Internet Economics. WINE 2018. Lecture Notes in Computer Science(), vol 11316. Springer, Cham. https://doi.org/10.1007/978-3-030-04612-5_20
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DOI: https://doi.org/10.1007/978-3-030-04612-5_20
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