Descending Solid Coalitions

Descending Solid Coalitions (or DSC) is a voting system devised by Douglas Woodall for use with ranked ballots.

Procedure
Every possible set of candidates is given a score equal to the number of voters who are solidly committed to the candidates in that set. A voter is solidly committed to a set of candidates if he ranks every candidate in this set strictly above every candidate not in the set.

The sets are then considered in turn, from those with the greatest score to those with the least. When a set is considered, every candidate not in the set becomes ineligible to win, unless this would cause all candidates to be ineligible, in which case that set is ignored.

When only one candidate is still eligible to win, that candidate is elected.

A variation of this method is Descending Acquiescing Coalitions (DAC).

Properties
DSC satisfies the Plurality criterion, the Majority criterion, Mono-raise, Mono-add-top, the Participation criterion, Later-no-harm and Clone Independence.

DSC fails the Condorcet criterion, the Smith criterion and the Later-no-help criterion.

DSC can be considered a First-Preference Plurality variant that satisfies Clone Independence. It is (along with DAC) the most complicated method satisfying the Participation criterion.

Example
The sets have the following strengths:


 * 100 {M,N,C,K}
 * 58 {N,C,K}
 * 42 {M,N,C}
 * 42 {M,N}
 * 42 {M}
 * 32 {C,K}
 * 26 {N,C}
 * 26 {N}
 * 17 {K}
 * 15 {C}

{N,C,K} is the strongest set that excludes a candidate. Memphis becomes ineligible.

No matter in which order we consider the sets with 42% of the voters solidly committed to them, we will arrive at the same result, which is that Nashville will be the only candidate remaining. So Nashville is the winner.

Notice that more than half of the votes held Memphis to be the worst alternative, yet the Memphis supporters' votes were still useful in securing their second choice, Nashville. If the Memphis voters had not listed any preferences after Memphis, the winner would have been Chattanooga.

Since DSC satisfies Later-no-harm, it's not possible for a voter to get a better result by withholding lower preferences, or to hurt the chances of a candidate already ranked by ranking additional candidates below that candidate. Since DSC fails the Later-no-help criterion, however, it is possible in some cases for a voter to get a better result for a candidate already ranked by ranking additional candidates below that candidate, or by changing the ranking of candidates ranked below that candidate such that at least one candidate is ranked above another candidate instead of being ranked the same as or below that candidate. The corollary to this statement is that it is possible in some cases for a voter to get a worse result for a candidate by withholding preferences for candidates ranked below that candidate, or by changing the ranking of candidates ranked below that candidate such that at least one candidate is ranked the same as or below another candidate instead of being ranked above that candidate. If the Nashville voters had not listed any preferences after either Nashville or Chattanooga, or had ranked Memphis the same as or above either Knoxville or Chattanooga or both, the winner would have been Memphis, as the number of candidates strictly committed to the set {N,C,K} would have been only 32, which is less than the number of candidates strictly committed to at least the sets {M} and {M,N}. Regardless of the order in which those two sets (along with the set {M,N,C} if Nashville's voters had still all ranked Chattanooga ahead of Knoxville) were considered, Memphis would have been the only candidate remaining after those two or three sets were considered and so would have been the winner.