About some properties of truthful voting rules for area of preferences that unimodal on a tree
Let U be a set of all single-picked proﬁles u on a tree G and ϕ : U → A be a voting rule. We consider a set of 'middle points' of G by the proﬁle u and investigate the structure of the set when n is even (in the case of odd n there is a unique middle point corresponding the Condorcet winner by u). So we completely describe the set of all weak Condorcet winners in case of even n.