On eligibility by the de Borda voting rules (Q933759)

Let \(A = \{a_1, a_2, \ldots, a_n\}\) be a set of candidates, \(s = (s_0, s_1, \ldots, s_{n-1})\) be a score vector with \(s_0 = 0 < s_1 < \ldots < s_{n-1}\) and \(\beta_s\) be the corresponding scoring rule. Given a profile \(p\), a candidate \(a\) is a \textit{winner} if the \(s\)-score \(B_{s}(a)\) of \(a\) is maximal. Given a profile \(p\), an alternative \(a\) is \textit{eligible} if there exists a score vector \(s\) such that \(a\) is a winner (given profile \(p\)). A \textit{cumulative rank vector} for \(a\) is the vector \(r(a) \in \mathbb{R}^{n-1}\) such that its first component \(r_{1}(a)\) is the number of voters who regard \(a\) as best, the second component \(r_{2}(a)\) is the number of voters for whom \(a\) is either first or second, etc. The \textit{Pareto boundary} of a set \(R \subset \mathbb{R}^{m}\) consists of all \(r' \in R\) such that there are no points \(r \in R\) with all coordinates not less than the respective coordinates of \(r'\) and \(r_{j} > r'_{j}\) for some \(j\). \textit{H. Moulin} [Axioms of cooperative decision making, Cambridge University Press, Cambridge (1988; Zbl 0699.90001)] showed that if a candidate \(a\) is eligible, then its cumulative rank vector \(r(a)\) belongs to the Pareto boundary of the set \(R = \{r(x) : x \in A \}\). The author of this paper gives an example illustrating that the converse of Moulin's statement does not hold and strengthens Moulin's result as follows: \(a\) is eligible if and only if \(r(a)\) belongs to the Pareto boundary of the convex hull of \(R = \{ r(x) : x \in A \}\).