Consistent collective decisions under majorities based on difference of votes

Abstract

The main criticism to the aggregation of individual preferences under majority rules refers to the possibility of reaching inconsistent collective decisions from the election process. In these cases, the collective preference includes cycles and even could prevent the election of any alternative as the collective choice. The likelihood of consistent outcomes under a class of majority rules constitutes the aim of this paper. Specifically, we focus on majority rules that require certain consensus in individual preferences to declare an alternative as the winner. Under majorities based on difference of votes, the requirement asks to the winner alternative to obtain a difference in votes with respect to the loser alternative taking into account that individuals are endowed with weak preference orderings. Same requirement is asked to the restriction of these rules to individual linear preferences.

Probabilities of Consistent Election Outcomes with Majorities Based on Difference in Support

“…In the present paper, we extend the study in Diss and Pérez-Asurmendi (2015) to the majorities based on difference in support M k . However, as long as the intensities of preference between each pair of alternatives can take any value in the continuous interval [0, 1], the IAC model can not be applied.…”

“…If r p ij = 0.5, individual p is indifferent between both alternatives. Notice that in the first case, these majorities are equivalent to supermajorities (see for a formal proof of that Diss and Pérez-Asurmendi (2015)); in the second case, these voting rules are majorities based on difference of votes.…”

“…Assuming weak or linear individual orderings and using the well-known Impartial Anonymous Culture (IAC) condition (Gehrlein and Fishburn 1976), calculations of the probabilities of consistent outcomes (transitivity and tripleacyclicity) under majorities based on difference in votes have been conducted in Diss and Pérez-Asurmendi (2015). The IAC condition assumes that all voting situations are drawn independently and uniformly given that each voting situation indicates a specific combination of the number of voters associated with each order of the alternatives.…”