Copeland's method: Difference between revisions

A beats B beats C beats A, so there is a [[Condorcet cycle]] between all candidates. Each candidate has one [[pairwise beat|pairwise victory]] and one defeat, so their Copeland scores are all 0, thus there is a tie. This example demonstrates why Copeland is almost never used for actual elections; it can guarantee someone in the [[Smith set]] will win, but says much less about who.

 

Copeland's method passes the [[Smith criterion]] because any candidate in the Smith set by definition beats everybody outside of the Smith set, but no candidate outside of it does so. For any candidate X in the Smith set and Y outside of it, Y is defeated by at least as many candidates as X, and X defeats at least one candidate that Y doesn't. Thus, every candidate in the Smith set must have a greater Copeland score than any candidate outside of it. Furthermore, the Copeland ranking of candidates (the ordering of candidates based on Copeland score) is a [[Smith set ranking]], since the above statement also holds with X being in the nth Smith set and Y in the (n+1)-th Smith set.

 

(Example showing Smith members having only 2 points more than non-Smith members: Suppose there are two candidates, one of whom is the Condorcet winner, and thus the only candidate in the Smith set. The CW has one victory and no defeats for a Copeland score of 1, while the other candidate has no victories and one defeat for a score of -1.)

 

Copeland's method also passes [[ISDA]]: since every candidate in the Smith set beats everybody outside it, eliminating a candidate outside of the Smith set will subtract one win from the score of every candidate in that Smith set. Thus eliminating a Smith-dominated candidate can never change the relative Copeland scores of candidates in the Smith set, and thus not change the winner either.

 

Further, Copeland always elects from the [[uncovered set]], and the Copeland ranking is an uncovered set ranking. This is because when one candidate covers another, the former candidate pairwise beats all candidates pairwise beaten by the latter candidate, and also either pairwise beats the latter candidate or beats someone who beats the latter candidate. Because of this, the covering candidate will have a minimum Copeland score of ((number of candidates beaten by latter candidate) + 1) - (number of candidates beating former candidate)), and the covered candidate will have a maximal Copeland score of ((number of candidates beaten by latter candidate) - ((number of candidates beating former candidate) + 1), resulting in the covering candidate having at least 2 more points than the covered candidate. This type of logic can be used to simplify the above Smith set-related proofs too.

 

==Generalizations==