Somewhere up this thread Blake Cretney brought up the idea that some Condorcet methods may satisfy a certain modified consistency criterion.
If the method gives a complete ranking as output, and if two subsets of ballots produce the same ranking, then the output based on the union of the two subsets should agree with the two partial results. The Kemeny Order is a Condorcet ranking that satisfies this modified consistency criterion. To see this, consider that in any metric space if H and K are finite sets of points and p is a point that minimizes the sum of distances from point x to points of H, as well as from x to points of K, then p minimizes the the sum of distances from x to the union of H and K. The Kemeny order is obtained by precisely this kind of minimization process. [The "points" are the permutations of candidates, and the "distances" are the required number of elementary swaps to go from one permutation to another.] Note that if the permutation is not a "beat path" then there is some pair of adjacent candidates in the permutation that would be preferred to be reversed by a majority of the voters. Performing that swap would decrease the distance by one each to that majority and increase the distance by one each to the minority that preferred it to remain unswapped. Therefore the total distance (hence average distance) is decreased by performing the swap. Therefore if the permutation is not a beat path, then it is not the Kemeny order. In other words, being a beat path is a necessary (though not sufficient) requirement for a permutation of candidates to be the Kemeny order. Therefore, the Kemeny order always picks a member of the Smith set as "winner." In summary, the Kemeny order is an example of a Condorcet order that is order consistent with subsets that unanimously agree on the order, which is what Blake was looking for, if I remember correctly. Forest ---- For more information about this list (subscribe, unsubscribe, FAQ, etc), please see http://www.eskimo.com/~robla/em
