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Positional scoring-based allocation of indivisible goods

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Abstract

We define a family of rules for dividing m indivisible goods among agents, parameterized by a scoring vector and a social welfare aggregation function. We assume that agents’ preferences over sets of goods are additive, but that the input is ordinal: each agent reports her preferences simply by ranking single goods. Similarly to positional scoring rules in voting, a scoring vector \(s = (s_1, \ldots , s_m)\) consists of m nonincreasing, nonnegative weights, where \(s_i\) is the score of a good assigned to an agent who ranks it in position i. The global score of an allocation for an agent is the sum of the scores of the goods assigned to her. The social welfare of an allocation is the aggregation of the scores of all agents, for some aggregation function \(\star \) such as, typically, \(+\) or \(\min \). The rule associated with s and \(\star \) maps a profile to (one of) the allocation(s) maximizing social welfare. After defining this family of rules, and focusing on some key examples, we investigate some of the social-choice-theoretic properties of this family of rules, such as various kinds of monotonicity, and separability. Finally, we focus on the computation of winning allocations, and on their approximation: we show that for commonly used scoring vectors and aggregation functions this problem is NP-hard and we exhibit some tractable particular cases.

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Notes

  1. Note that the Borda scoring vector in voting is usually defined as \((m-1,m-2,\dots ,1,0)\). Here, together with Brams et al. [14], we define the Borda scoring vector by fixing the score of the bottom-rank object to 1, meaning that getting it is better than getting nothing. For scoring voting rules, a translation of the scoring vector has obviously no impact on winner determination (see Observation 2.2 in the work of Hemaspaandra and Hemaspaandra [27]); for scoring allocation rules, however, it does.

  2. This choice comes with a loss of generality, as there are tie-breaking mechanisms that are not defined this way (we thank a reviewer for this remark). Also, we rule out the possibility of randomly breaking ties.

  3. This is the case for all properties expressing that an agent prefers a set of allocations to another set of allocations (and applies, e.g., to object monotonicity); for these properties there is not a unique way of generalizing the property, unlike in voting where this is well-known, e.g., for strategy-proofness. For a study of strategy-proofness for scoring allocation correspondences, we refer to the work of Nguyen et al. [34].

  4. If the scoring vector s is part of the input then the problem \(F_{s, \star }\)-FOA, \(\star \in \{\min , {{\mathrm{leximin}}}\}\), is \(\mathrm {NP}\)-hard (though not strongly \(\mathrm {NP}\)-hard in the sense of Garey and Johnson [23, 24]), even for two agents having identical preferences, by a direct reduction from Partition.

  5. Here and later, we slightly abuse notation, as X and \(C_i\) will refer both to the initial sets and their corresponding sets of goods.

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Acknowledgments

We are grateful to the anonymous ECAI’14 and COMSOC’14 reviewers for their helpful comments. In particular, we thank the reviewer who pointed out a proof sketch of Theorem 6 for her or his consent to include the result and its proof. This work was supported in part by Deutsche Forschungsgemeinschaft under grants RO 1202/14-1, RO 1202/14-2, and RO 1202/15-1, by a project of the DAAD-PPP / PHC PROCOPE program entitled “Fair Division of Indivisible Goods: Incomplete Preferences, Communication Protocols and Computational Resistance to Strategic Behavior,” by COST Action IC1205 on Computational Social Choice, by ANR project CoCoRICo-CoDec, by a grant for gender-sensitive universities funded by the NRW Ministry for Innovation, Science, and Research, and by Vietnam’s National Foundation for Science and Technology Development (NAFOSTED Project No. 102.01-2015.33).

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Authors and Affiliations

  1. Heinrich-Heine Universität Düsseldorf, Universitätsstr. 1, 40225, Düsseldorf, Germany

    Dorothea Baumeister, Nhan-Tam Nguyen & Jörg Rothe

  2. Univ. Grenoble Alpes, CNRS, LIG, 38000, Grenoble, France

    Sylvain Bouveret

  3. LAMSADE, Université Paris-Dauphine, Place du Maréchal de Lattre de Tassigny, 75775, Paris Cedex 16, France

    Jérôme Lang

  4. Hai Phong University, Hai Phong, Vietnam

    Trung Thanh Nguyen

  5. University of New South Wales, Sydney, NSW, 2052, Australia

    Abdallah Saffidine

Authors
  1. Dorothea Baumeister
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  2. Sylvain Bouveret
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  4. Nhan-Tam Nguyen
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  6. Jörg Rothe
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Correspondence to Nhan-Tam Nguyen.

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A preliminary version of this paper appeared in the proceedings of the 21st European Conference on Artificial Intelligence (ECAI’14) [6] and has also been presented at the 5th International Workshop on Computational Social Choice (COMSOC’14) [5].

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Baumeister, D., Bouveret, S., Lang, J. et al. Positional scoring-based allocation of indivisible goods. Auton Agent Multi-Agent Syst 31, 628–655 (2017). https://doi.org/10.1007/s10458-016-9340-x

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