## Abstract

We study strategy-proofness in a model of resource allocation due to Brams and King (Ration Soc 17:387–421, 2005) and Brams et al. (Theory Decis 55:147–180, 2003), further developed by Baumeister et al. (J Auton Agents Multi Agent Syst 31(3):628–655, 2017). We assume resources to be indivisible and nonshareable and that agents have responsive preferences over the power set of the resources, but only submit ordinal preferences over single resources to the social planner. Using scoring vectors, these ordinal preferences induce additive utility functions. We then focus on allocation correspondences that maximize utilitarian social welfare, and we use extension principles (from social choice theory, such as the Kelly and the Gärdenfors extension) for preferences to study manipulation of allocation correspondences. We characterize strategy-proofness of the utilitarian allocation correspondence: It is Gärdenfors/Kelly-strategy-proof if and only if the number of different values in the scoring vector is at most two or the number of occurrences of the greatest value in the scoring vector is larger than half the number of resources.

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## Notes

- 1.
The common definition of Borda scoring in voting is based on the vector \((m-1,m-2,\ldots ,1,0)\). However, we follow Brams et al. (2003) by setting the score of the bottom-rank resource to the value 1. Note that scoring vectors in voting can be shifted or scaled without changing the winner set (Hemaspaandra and Hemaspaandra 2007); but for scoring allocation correspondences such an operation would have an impact in general (Baumeister et al. (2014, 2017)).

- 2.
We don’t explicitly consider the extension principle proposed by Fishburn (1972), which for a weak order \(\ge \) over \(2^R\) and any two sets \(A, B \subseteq 2^R\) of bundles of resources says that

*A*be weakly preferred to*B*if and only if for all \(x \in A{\smallsetminus } B\), for all \(y \in A \cap B\), and for all \(z \in B{\smallsetminus } A\), we have \(x \ge y \ge z\). Comparing this extension principle with those of Kelly’s and Gärdenfors’s in Definition 2, it is clear that it is intermediate between them. Therefore, our results apply to it as well. - 3.
In fact, this example even shows that manipulability using the Fishburn extension is more demanding than Gärdenfors-manipulability.

- 4.
In social choice theory, a

*k*-approval scoring vector for elections with \(m \ge k\) candidates is defined by \( (\underbrace{1, \ldots , 1}_{k}, 0, \ldots , 0)\). By*k*-*approval-like scoring vector*we mean a vector of the form \((\underbrace{s_1, \ldots , s_1}_{k}, s_2, \ldots , s_2)\) for \(s_1> s_2 > 0\), since we have excluded zero as a scoring value.

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## Acknowledgements

We thank the anonymous Social Choice and Welfare, IJCAI-2015, and COMSOC-2016 reviewers (of the earlier versions) for helpful comments and Jérôme Lang for interesting discussions and feedback on an earlier draft of this paper. This work was supported in part by the NRW Ministry for Innovation, Science, and Research and DFG Grants RO 1202/14-1, RO 1202/14-2, and RO 1202/15-1.

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Nguyen, N., Baumeister, D. & Rothe, J. Strategy-proofness of scoring allocation correspondences for indivisible goods.
*Soc Choice Welf* **50, **101–122 (2018). https://doi.org/10.1007/s00355-017-1075-3

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