On eligibility by the de Borda voting rules

We show that a criterion for eligibility of a candidate by the set of de Borda’s voting rules in [H. Moulin (1988) Axioms of cooperative decision making] is erroneous and we obtain the correct version of this criterion. Let r(ai) be the score vector of a candidate ai, R be the set of all vectors r(ai), and let R ′ be the Pareto boundary of the convex hull convR. Then there is a scoring s such that a candidate a wins with respect to the de Borda voting rule βs if and only if r(a) ∈ R . Introduction. Suppose the set A of candidates and the profile u of the voters’ preferences are fixed. Let s be a system of scores and βs be the de Borda rule assigned to s. Further, let βs(u) ⊂ A be the set of winners w.r.t. this rule. A candidate a ∈ A is eligible w.r.t. the set of de Borda’s rules βs if there is a scoring s such that a ∈ βs(u). The book [1] suggests the following criterion for eligibility of a given candidate a w.r.t. the set of de Borda’s rules: a winner a has the score vector r(a) that belongs to the Pareto boundary of the set R = {r(aj), aj ∈ A}; see pages 2 – 3 for rigorous definitions. Unfortunately, the criterion in this formulation is incorrect. On page 5 we give a counter-example using the profile u defined in Eq. (1). In Theorems 2 and 3 below we prove the correct version of this eligibility criterion. The difference between the correct statement and the erroneous one contained in [1] is that the Pareto boundary of the convex hull convR must be used instead of the Pareto boundary of the set R itself, here R is the set of all score vectors. 1. Definitions 1.1. Profiles. Let P1, . . . , Pn be electors (or voters) and A = {a1; . . . ; ap} be the set of candidates in some elections. Suppose that every elector Pi has an opinion about each candidate such that the candidates are arranged by the strict order >i : the first candidate in this rearrangement is the most favourable for Pi, etc. This strict linear order >i on A is called the preference of the elector Pi and is denoted by ui. The order ui is given by the sequence aj1 >i aj2 >i . . . >i ajp, where J = (j1; j2; . . . ; jp) is a rearrangement of (1; 2; . . . ; p); generally, J depends on the elector Pi. In what follows, we write down the elements of the preferences ui in columns and thus we compose the matrix u = ∣