Abstract
Reciprocal preferences have been introduced in the literature of social choice theory in order to deal with preference intensities. They allow individuals to show preference intensities in the unit interval among each pair of options. In this framework, majority based on difference in support can be used as a method of aggregation of individual preferences into a collective preference: option \(a\) is preferred to option \(b\) if the sum of the intensities for \(a\) exceeds the aggregated intensity of \(b\) by a threshold given by a real number located between 0 and the total number of voters. Based on a three dimensional geometric approach, we provide a geometric analysis of the non-transitivity of the collective preference relations obtained by majority rule based on difference in support. This aspect is studied by assuming that each individual reciprocal preference satisfies a \(g\)-stochastic transitivity property, which is stronger than the usual notion of transitivity.
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Notes
In this article, we assume that individuals vote sincerely, so that the possibility of strategic voting is not considered.
By symmetry, we also consider the case \(bP_k a\), \(cP_k b\) and \(\lnot (cP_k a)\).
Recall that a trirectangular tetrahedron is a tetrahedron where all three face angles at one vertex are right angles.
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Acknowledgments
For helpful comments received, the authors want to thank an associate editor, two anonymous referees, W.V. Gehrlein, Y. Koriyama, V. Merlin, and the participants of the 13th SAET conference. Financial support by the National Agency for Research (ANR)—research program “Dynamic Matching and Interactions: Theory and Experiments” (DynaMITE) ANR. BLANC—and the “Mathématiques de la décision pour l’ingénierie physique et sociale” (MODMAD) project is gratefully acknowledged.
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Baron, R., Diss, M., Rémila, E. et al. A geometric examination of majorities based on difference in support. Soc Choice Welf 45, 123–153 (2015). https://doi.org/10.1007/s00355-015-0870-y
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DOI: https://doi.org/10.1007/s00355-015-0870-y