3-2-1 voting

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In 3-2-1 voting, voters may rate each candidate “Good”, “Acceptable”, or “Rejected”. It has three steps:

  • Find 3 Semifinalists: the candidates with the most “good” ratings. (If this is a partisan election, no two semifinalists may come from the same party).
  • Find 2 Finalists: the semifinalists with the fewest rejections.
  • Find 1 winner: the finalist who is rated above the other on more ballots.

Motivation for each step

Step 1: A winner should have strong support; at least some voters who have paid attention and are enthusiastic. But if you keep fewer than 3 at this stage, you'd risk prematurely eliminating a centrist and leaving only the two extremes.

Step 2: This allows a majority of the electorate to have a veto on any candidate. Also, candidates that are eliminated here would usually have little chance in step 3 anyway.

Step 3: This is like a runoff between the two strongest candidates. If you know which two candidates will be finalists, you have no incentive not to rank them honestly, and everybody who made a distinction between them gets equal voting power.

Properties

This system satisfies the Majority criterion; the Condorcet Loser criterion; monotonicity; and local independence of irrelevant alternatives.

It satisfies the mutual majority criterion as long as any member of the mutual majority set of candidates is among the 3 semifinalists. In practice, this is almost guaranteed to be the case.

Steps 1 and 3 satisfy the later no-harm criterion, so that the only strategic reason not to add any "acceptable" ratings would be if your favorite was one of the two most-rejected semifinalists but also was able to beat the least-rejected semifinalist in step 3. This combination of weak and strong is unlikely to happen in real life, and even less likely to be predictable enough a priori to be a basis for strategy.

Examples

Tennessee capital (center squeeze)

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

This leads to the following outcome:

Candidate "Good" ratings "Acceptable" ratings "Bad" ratings 2-way score
Memphis 42 0 58
Nashville 26 74 0 68
Chattanooga 15 85 0
Knoxville 17 41 42 32

The three most-endorsed are Memphis (42), Nashville (26), and Knoxville (17). Of those three, the two least-rejected are Nashville (0 rejections) and Knoxville (42 rejections). Of those two, Nashville is preferred by 68 to 32.

High school mascot (chicken dilemma)

Imagine an election for a high school mascot, in which the options are “Bulldogs”, “Lions”, “Tigers”, or “Knights”, with the following votes:

Faction size "Good" candidates "Acceptable" candidates "Bad" candidates
20 Bulldogs, Knights Lions, Tigers
20 Bulldogs Knights Lions, Tigers
35 Tigers Lions Bulldogs, Knights
25 Lions Tigers Bulldogs, Knights

The votes above lead to the following outcome:

Candidate "Good" ratings "Acceptable" ratings "Bad" ratings 2-way score
Lions 25 35 40 25
Tigers 35 25 40 35
Knights 20 20 60
Bulldogs 40 0 60

The semifinalists are Lions, Tigers, and Bulldogs. The finalists are Lions and Tigers. The winner is Tigers.