Descending Acquiescing Coalitions

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Descending Acquiescing Coalitions (or DAC) is a voting system devised by Douglas Woodall for use with ranked ballots. It is a variation of Descending Solid Coalitions (DSC), another voting system devised by Woodall.

Procedure

Every possible set of candidates is given a score equal to the number of voters who acquiesce to the candidates in that set. A voter "acquiesces" to a set of candidates if he or she does not rank any candidate outside of the set strictly above any candidate within the set.

Then 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.

Properties

DAC satisfies the Plurality criterion, the Majority criterion, Mono-raise, Mono-add-top, the Participation criterion, the Later-no-help criterion and Clone Independence.

DAC fails the Condorcet criterion, the Smith criterion and the Later-no-harm criterion.

DAC can be considered a First-Preference Plurality variant that satisfies Clone Independence. It is (along with DSC) 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 on Wikipedia, the state's largest city, with 42% of the voters, but located far from the other cities
  • Nashville on Wikipedia, with 26% of the voters, near the center of Tennessee
  • Knoxville on Wikipedia, with 17% of the voters
  • Chattanooga on Wikipedia, 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)
  1. Memphis
  2. Nashville
  3. Chattanooga
  4. Knoxville
  1. Nashville
  2. Chattanooga
  3. Knoxville
  4. Memphis
  1. Chattanooga
  2. Knoxville
  3. Nashville
  4. Memphis
  1. Knoxville
  2. Chattanooga
  3. Nashville
  4. 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.

Since DAC fails the Later-no-harm criterion, a voter can hurt the chances of a candidate already ranked by ranking additional candidates below that candidate, and can thus get a better result in some cases by witholding lower preferences. Since DAC satisfies the Later-no-help criterion, however, a voter cannot increase the probability of election of a candidate already ranked by ranking additional candidates below that candidate, and cannot hurt the chances of a candidate already ranked by withholding or equalizing lower preferences.