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.

Then the sets are 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.

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 and Smith criterion.

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

Example

Tennessee's four cities are spread throughout the state

Imagine that Tennessee is having an election on the location of its capital. The population of Tennessee is concentrated around its four major cities, which are spread throughout the state. For this example, suppose that the entire electorate lives in these four cities, and that everyone wants to live as near the capital as possible.

The candidates for the capital are:

• Memphis, the state's largest city, with 42% of the voters, but located far from the other cities
• Nashville, with 26% of the voters, near the center of Tennessee
• Knoxville, with 17% of the voters
• Chattanooga, with 15% of the voters

The preferences of the voters would be divided like this:

42% of voters (close to Memphis)	26% of voters (close to Nashville)	15% of voters (close to Chattanooga)	17% of voters (close to Knoxville)
Memphis Nashville Chattanooga Knoxville	Nashville Chattanooga Knoxville Memphis	Chattanooga Knoxville Nashville Memphis	Knoxville Chattanooga Nashville Memphis

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.