Egalitarianism in the rank aggregation problem: a new dimension for democracy

Abstract: Winner selection by majority, in an election between two candidates, is the only rule compatible with democratic principles. Instead, when the candidates are three or more and the voters rank candidates in order of preference, there are no univocal criteria for the selection of the winning (consensus) ranking and the outcome is known to depend sensibly on the adopted rule. Building upon XVIII century Condorcet theory, whose idea was to maximize total voter satisfaction, we propose here the addition of a new basic principle (dimension) to guide the selection: satisfaction should be distributed among voters as equally as possible. With this new criterion we identify an optimal set of rankings. They range from the Condorcet solution to the one which is the most egalitarian with respect to the voters. We show that highly egalitarian rankings have the important property to be more stable with respect to fluctuations and that classical consensus rankings (Copeland, Tideman, Schulze) often turn out to be non optimal. The new dimension we have introduced provides, when used together with that of Condorcet, a clear classification of all the possible rankings. By increasing awareness in selecting a consensus ranking our method may lead to social choices which are more egalitarian compared to those achieved by presently available voting systems.