Mostapha Diss

An example of probability computations under the IAC assumption: The stability of scoring rules

A society facing a choice problem has also to choose the voting rule itself from a set of different possible voting rules. A voting rule is self-selective if it chooses itself when it is also used in choosing the voting rule. A set of voting rules is said to be stable if it contains at least one self-selective voting rule at each profile of preferences on voting rules. We consider in this paper a society which makes a choice from a set of three alternatives {a,b,c} and a set of the three well-known scoring voting rules {Borda, Plurality, Antiplurality}. We will derive an a priori probability for the stability of this triplet of voting rules, under the Impartial Anonymous Culture assumption (IAC). In order to solve this problem, we need to specify Ehrhart polynomials, which count the number of integer points inside a (convex) polytope. We discuss briefly a recent algorithmic solution to this method before applying it. We also discuss the impact of different behavioral assumptions for the voters (consequentialist or nonconsequentialist) on the probability of stability for the triplet {Borda, Plurality, Antiplurality}.

Multi-Winner Scoring Election Methods: Condorcet Consistency and Paradoxes

The goal of this paper is to propose a comparison of four multi-winner voting rules, k-Plurality, k-Negative Plurality, k-Borda, and Bloc, which can be considered as generalisations of well-known single-winner scoring rules. The first comparison is based on the Condorcet committee efficiency which is defined as the conditional probability that a given voting rule picks out the Condorcet committee, given that such a committee exists. The second comparison is based on the likelihood of two paradoxes of committee elections: The Prior Successor Paradox and the Leaving Member Paradox which occur when a member of an elected committee leaves. In doing so, using the well-known Impartial Anonymous Culture condition, we extend the results of Kamwa and Merlin (2015) in two directions. First, our paper is concerned with the probability of the paradoxes no matter the ranking of the leaving candidate. Second, we do not only focus on the occurrence of these paradoxes when one wishes to select a committee of size k = 2 out of m = 4 candidates but we consider more values of k and m.

A Geometric Examination of Majorities Based on Difference in Support

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 in 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