# The Likelihood of the Consistency of Collective Rankings under Preferences Aggregation with Four Alternatives using Scoring Rules: A General Formula and the Optimal Decision Rule

*Eric Kamwa* and
*Vincent Merlin* ()

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**Abstract:**
In most of the social choice literature dealing with the computation of the exact probability of voting events under the impartial culture assumption, authors deal with no more than four constraints to describe voting events. With more than four constraints, most of the authors rely on Monte-Carlo simulations. It is usually more tricky to estimate the probability of events described by five constraints. Gehrlein and Fishburn (1980) have tried, but their conclusions are based on conjectures. In this paper, we circumvent this conjecture by having recourse to the technique suggested by Saari and Tataru (1999) in order to compute the limit probability of the consistency of collective rankings when there are four competing alternatives given that the decision rule is a scoring rule. We provide a general formula for the limit probability of the consistency and we determine the optimal decision rules among the scoring rules that provide the best guarantee of consistency. Given the collective ranking on a set A, we have consistency if the collective ranking on B a proper subset of A is not altered after some alternatives are removed from A.

**Date:** 2019

**New Economics Papers:** this item is included in nep-cdm

**Note:** View the original document on HAL open archive server: https://hal.univ-antilles.fr/hal-01757742

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**Citations:** View citations in EconPapers (1)

**Published** in Computational Economics, 2019, 53 (4), pp.1377-1395. ⟨10.1007/s10614-018-9816-7⟩

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Journal Article: The Likelihood of the Consistency of Collective Rankings Under Preferences Aggregation with Four Alternatives Using Scoring Rules: A General Formula and the Optimal Decision Rule (2019)

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**Persistent link:** https://EconPapers.repec.org/RePEc:hal:journl:hal-01757742

**DOI:** 10.1007/s10614-018-9816-7

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