On the stability of a triplet of scoring rules

Abstract

When choosing a voting rule in order to make subsequent decisions, the members of a committee may wish this rule to be self-selected when it is the object of a choice among a menu of different possible voting rules. Such concepts have recently been explored in Social Choice theory, and a menu of voting rule 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 the menu constituted by the three well-known scoring rules {Borda, Plurality, Antiplurality}. Under the Impartial Cultures Assumption (IC), which proposes an a priori model to estimate the likelihood of the profiles, we will derive a probability for the stability of this triplet of voting rules.

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

“…Hence, all the profiles have the same occurrence probability and the IC assumption specifies the scenario where the probability that a randomly selected voter has ith preference type is 1/6 for each 1 ≤ i ≤ 6. Recall that, the stability of the set {B, P , A} under the IC assumption was considered by Diss and Merlin (2010). The main conclusion of their work was rather negative: for a large population, where each voter has a probability of 1/6 to choose one of the six possible linear orders over the set E = {B, P , A} of voting rules, the set E is unstable for 15.51% of the profiles.…”

“…Next he assumes that voters have intrinsic preferences over the voting rules, regardless of the consequences on the ordinary level choice. This line of research has been subsequently developed by Diss and Merlin (2010), which highlights the notion of the stability of a set of voting rules by proposing an application of these new axioms to a set composed of three scoring voting rules. More precisely, they deal with the three most well-known rules in this category: the Borda rule (B), the Plurality rule (P ) and the Antiplurality rule (A).…”

“…First, we reconsider the result of Diss and Merlin (2010) by using another probability assumption, the so called Impartial Anonymous Culture. This model was proposed for the first time in the literature by Fishburn and Gehrlein (1976).…”

“…Section 2 formalizes the first ingredient of our paper dealing with preferences, scoring voting rules and the stability notion. It also presents the two major probability assumptions for the a priori assessment of voting paradoxes in social choice theory: the so-called Impartial Culture (IC) assumption (previously studied in Diss and Merlin, 2010), and the Impartial Anonymous Culture (IAC) assumption (used in this paper). In Section 3, we present Ehrhart's polynomial theory and use it to determine the probability of the stability for the {B, P , A} set using the nonconsequentialist assumption and the IAC model.…”

Strategic Manipulability of Self-­Selective Social Choice Rules

“…These questions were first considered by Koray [19], Koray and Unel [20], Barberà and Jackson [3], Barberà and Bevià [2] and Houy [14,15,16]. Considering some probabilistic models, these questions were also investigated by Diss and Merlin [5] and Diss et al [4].…”