The Borda rule, Condorcet consistency and Condorcet stability

Summary.

The Borda rule is known to be the least vulnerable scoring rule to Condorcet inconsistency, Saari (2000). Such inconsistency occurs when the Condorcet winner (the alternative which is preferred to any other alternative by a simple majority) is not selected by the Borda rule. This note exposes the relationship between the Borda rule and the Condorcet q-majority principle as well as the Condorcet q-majority voting rule. The main result establishes that the Borda rule is Condorcet q-majority consistent when $q \ge (k - 1) / k$ where k is the number of alternatives. The second result establishes that $(k - 1) / k$ is the minimal degree of majority decisiveness corresponding to the Borda rule under sincere voting. The same majority is required to ensure decisiveness under the Borda rule and to ensure that a q-rule (the generalized q-majority Condorcet rule) is a voting rule.