Abstract
We consider the collective decision problem of a society choosing among three alternatives on a strict preference domain in which one preference ordering over alternatives is not admissible. We propose the family of Sequential Pareto Undominated Rules and characterize one of them as the unique full range, anonymous, tops-only, and strategy-proof voting rule.
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Notes
We thank Maurice Salles for having suggested this illustrative example.
See Sect. 2 for a formal definition.
To show the first observation, let \(P_{N}\) and \(P_{N}^{\prime }\) be such that they have the same set of top alternatives: a and c. In \(P_{N}\) some agents with top a prefer b to c but others c to b, while in \(P_{N}^{\prime }\) all agents with top a prefer c to b. Thus, \(U(P_{N})=\{a,b,c\}\) and \(U(P_{N}^{\prime })=\{a,c\}\). To show the second observation, note that \(b\in U(P_{N})\) but no agent consider b as the top alternative. To see that c can belong to the set of undominated alternatives of a profile in which no agent has c as top alternative, consider the profile \(P_{N}^{\prime \prime }\) with three agents such that \(aP^{\prime \prime }_{1}bP^{\prime \prime }_{1}c\), \(aP^{\prime \prime }_{2}cP^{\prime \prime }_{2}b\) and \(bP^{\prime \prime }_{3}cP^{\prime \prime }_{3}a\).
We thank an anonymous referee for suggesting this remark that simplifies the proof of the main result.
Note that one could require all agents to gain but allow for some of them not to change their preferences. That would turn out to be equivalent to the definition of manipulation we use.
Note that for any profile and any permutation of agents the permuted profile, say \(\left( P_{\rho (1)},P_{\rho (2)},\ldots ,P_{\rho (n)}\right) \in \mathcal {D}^{n}\) is in the domain.
This rule is the \(\succ\)-SPUR with associated linear order \(a\succ c\succ b\).
The proof is straightforward and left to the reader.
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Funding
We thank two anonymous referees for their detailed reading and suggestions. We also appreciate comments of M. Salles. D. Berga and B. Moreno thank the MOMA network and acknowledge the support from Programa operativo FEDER Andalucía PY18-2933. D. Berga and A. Nicolò thank the support from the Spanish Ministry of Economy, Industry and Competitiveness through Grants ECO2016-76255-P and PID2019-106642GB-I00. The usual disclaimer applies.
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Berga, D., Moreno, B. & Nicolò, A. Undominated rules with three alternatives in an almost unrestricted domain. Soc Choice Welf 60, 65–74 (2023). https://doi.org/10.1007/s00355-021-01330-1
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DOI: https://doi.org/10.1007/s00355-021-01330-1