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 (Appl Math Optim 6:241–255, 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 (Econ Theory 13: 345–363, 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 limiting 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.
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Notes
We give a formal definition of a voting situation in Sect. 2.
Assume \((X_1, X_2,\cdots , X_n)\) a vector of n random variables with a nonsingular multivariate normal distribution. Plackett (1954) evaluated the probability \(P(X_1>x_1, X_2>x_2,\cdots , X_n >x_n)\); he ended with a reduction formula of this probability based on the numerical quadrature for \(n=3, 4\).
A limiting probability is computed when the size of the electorate tends to infinity.
We provide a supplement document in which the reader can see how our formulas look like.
The methodology developed in this paper was recently used and applied by Kamwa and Merlin (2015) to perform probability computations for some voting events under some popular voting rules.
A linear order is a binary relation that is transitive, complete and antisymmetric. The binary relation R on A is transitive if for \(a,b,c\in A\), if aRb and bRc then aRc. R is antisymmetric if for all for \(a\ne b\), \(aRb\Rightarrow \lnot bRa\); if we have aRb and bRa, then \(a=b\). R is complete if and only if for all \(a,b\in A\), we have aRb or bRa.
We were not able to find a reduced form of this formula. This was also the case for all the other general formulas.
Given a collective ranking over a set of four candidates, they determine under the impartial culture condition, the probability of each of the six possible rankings when one candidate is dropped.
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Acknowledgements
The authors are grateful to two anonymous reviewers for their helpful comments. Thanks to two anonymous members of the Maple-group for their assistance. Eric Kamwa acknowledges the support of the Hitotsubashi Institute of Economic Research during his stay under the “JSPS Program for Advancing Strategic International Networks to Accelerate the Circulation of Talented Researchers”. Vincent Merlin acknowledges support from the project ANR-14-CE24-0007-01 CoCoRICo-CoDec.
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Appendix
Appendix
For space constraints, here are the links to:
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An external sheet where the reader can see how our formulas look.
https://www.dropbox.com/s/mkdbxuwfrsevmk3/general%20formula.pdf?dl=0
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The MAPLE-sheet we used for all our computations.
https://www.dropbox.com/s/byyvtoex90h4je5/MapleSheet_generic.pdf?dl=0
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Kamwa, E., Merlin, V. 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. Comput Econ 53, 1377–1395 (2019). https://doi.org/10.1007/s10614-018-9816-7
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DOI: https://doi.org/10.1007/s10614-018-9816-7