# 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.

## Keywords

Computational social choice Resource allocation Fair division Indivisible goods Preferences## Notes

### 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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