The Condorcet Method The French mathematician Condorcet proposed the following method for determining a social preference order based on the strict preferences of the individuals in society. A voter i with a strict preference order Pi grants k Condorcet points to a strict preference relation P if there are exactly k pairs of alternatives a, b satisfying b ?P a and b ? Pi a. The number of Condorcet points amassed by the strict preference relation P is the total sum of the Condorcet points it receives from all the voters. The strict social preference order is the one that has amassed the greatest number of Condorcet points.
(a) Is the strict preference relation amassing the greatest number of Condorcet points unique? If yes, prove this claim. If no, present a counterexample.
(b) Show that if there exists a Condorcet winner, then every strict preference relation receiving the maximal number of Condorcet points ranks the Condorcet winner highest in its preference ordering.
(c) Find the number of Condorcet points that each strict preference relation receives in Examples 21.1 and 21.2, and determine the preference relation that the Condorcet Method chooses.