On eligibility by the de Borda voting rules

We show that a necessary condition for eligibility of a candidate by the set of de Borda's voting rules in [H. Moulin (1988), Axioms of cooperative decision making] is not sufficient and we obtain a version of the criterion. Let $r(a_i)$ be the score vector of a candidate $a_i$, $R$ be the set of all vectors $r(a_i)$, and let $R'$ be the Pareto boundary of the convex hull $\conv R$. Then there is a scoring $s$ such that a candidate $a$ wins with respect to the de Borda voting rule $\beta_s$ if and only if $r(a)\in R'$.