3-2-1 voting

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

• Find 3 Semifinalists: the candidates with the most “good” ratings.
• Find 2 Finalists: the semifinalists with the fewest "bad" ratings.
• Find 1 winner: the finalist who is rated above the other on more ballots.

There are two extra qualifications for semifinalists: their "good" ratings should be more than anyone else in their party (that is, only one semifinalist per party), and at least 15% of the electorate. Usually there will be three eligible semifinalists will easily pass these qualifications naturally, but if only two of them do, you can just treat them as finalists and skip step 2. In the unlikely event that fewer than two candidates get 15% "good" ratings, and re-running the election with new candidates is not an option, then the three highest become semifinalists and the election proceeds to step 2 normally.

Blank ratings

There are two ways to handle blank ratings: "Delegated", which makes voting easier for voters by letting them choose to give some of their voting power to their favorite candidate; and "Undelegated", which does its best to infer voter intentions directly. "Delegated" is suggested unless there are reasons against it.

Delegated 3-2-1

In this method, each candidate can pre-rate other candidates "OK", "conditionally OK", or "bad". If they do not explicitly pre-rate, they are considered to rate all others "conditionally OK". Once all ratings have been submitted, all "conditionally OK" ratings are turned to "Bad" if the rating coming the other way is "Bad", and to "OK" otherwise. Candidate ratings are public information.

When a voter leaves a candidate X blank/unrated, X receives the lowest rating that they got from any candidate that voter rated "Good". So if the voter had rated candidates A and B "good", and both A and B rated X as "OK", then X would get an "OK" from that voter; while if either A and/or B had rated X as "Bad", then X would get a "Bad" from that voter.

Undelegated 3-2-1

For voters who do not explicitly use the "Bad" rating, blank ratings count as "bad". For those who do use "bad", blank ratings count as "OK", except that in step 3 they count as lower than an explicit "OK".

Tiebreaker

In all cases, ties are broken by score, with each "Good" counting as 2 points and each "OK" counting as 1. If two candidates are tied in score as well (highly unlikely), the tie is broken randomly.

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.

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, 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

This leads to the following outcome:

Candidate	"Good" ratings	"OK" 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	"OK" candidates	"Bad" candidates
2	Knights	Bulldogs	Lions, Tigers
38	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	"OK" ratings	"Bad" ratings	2-way score
Lions	25	35	40	25
Tigers	35	25	40	35
Knights	2	38	60
Bulldogs	38	2	60

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

This shows a "chicken dilemma" between the felines (Lions and Tigers); together, they can beat Bulldogs, but separately they can't. In 3-2-1, as in almost any voting methods which successfully elect Nashville in the example above, it is possible for the Lions voters to win by strategically rating Tigers as "bad". However, it would take at least 20 of the 25 Lions voters to accomplish this; any fewer, and Tigers would still win. Thus, unlike many voting methods, as long as each Lions voter expects even a third of other Lions voters to vote honestly, there is no incentive for them to "defect" individually.

Properties

This method 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. Thus, the only strategic reason not to add any "OK" 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.

This method fails the favorite betrayal criterion, in that in steps 1 or 2 it could in theory be necessary to rate your favorite below "Good" in order to leave room for a more-viable compromise candidate to be a semifinalist or finalist. However, in order for that to be a worthwhile strategy, the compromise would have to do better in a pairwise race against the other finalist, but have a worse chance of becoming a semifinalist or finalist under your honest vote. This combination of strength in one context and weakness in another is akin to a Condorcet cycle, and like such cycles, it may be rare in real-world elections, and even rarer that it is predictable enough a priori to make a favorite-betrayal strategy feasible.

In terms of summability, this can be done in one of two ways. They method can work with one count that is O(N²) summable, or with two consecutive tallies that are each O(N) summable (one for "3-2" and the second for "1"). The latter could make this feasible to run even on older ballot machines; though full counts of the second step might involve some configuration and a couple of passes over the ballots, in many cases the "3-2" tallies would make it obvious who wins the "1" step, so voters would not have to be kept in suspense as the second step proceeded.

For US presidential elections

In order to be usable for US presidential elections, a voting method should be able to work as an interstate compact alongside other methods. Such an interstate compact would have at most the following steps:

1. Voters in each state vote using the state's particular voting method.
2. Each state publishes raw totals in some format.
3. Possibly looking at the raw totals of other states, each state publishes its final totals.
4. Final totals for each state are added and the national winner is found
5. If they constitute a majority of the electoral college, signatory states are bound by compact to give all their electors to the national winner.

In order for a voting method to work with this, it must have a feasible way to work with steps 2, 3, and 4.

Step 2: "raw totals in some format": many voting methods exist, and many of them require different information from the ballots for summability. One reasonable lowest common denominator would be that all states must publish the rating or ranking levels available, and the raw tallies — the number of times each candidate is rated or ranked at each level. This is far less information than would be required to find a winner under IRV or Condorcet, but it is enough for 3-2-1, when combined with the following steps. It is also information that naturally would always be available from states using simpler methods such as plurality or approval.

Step 4: In order to add to provide national totals, each state's final totals should be in the form of a point method - that is, approval, score, or borda ballots, normalized so that each vote is in the range 0-1. This is not an endorsement of approval, score, or borda as voting methods; it's simply because these point methods are the only methods natively compatible with ballots from states still using plurality.

Step 3: So a state using 3-2-1 must be able to look at the raw tallies from other states, and provide final local tallies, such that the following properties are satisfied:

• Each individual local ballot contributes between 0 and 1 points to each candidate's final local tally.
• A ballot will always contribute 1 point to its most-preferred candidate and 0 points to its least-preferred candidate.
• A ballot will never contribute more points to a less-preferred candidate than to a more-preferred one.
• If all states used the same final local tally procedure, the winner would be the 3-2-1 winner.

It's easy to give tallies that satisfy the properties above. First, you find the semifinalists — the 3 candidates with the most top-ratings nationwide — and the finalists — the two semifinalists with the fewest bottom-ratings nationwide. Then, tally 1 point each time a candidate is rated "good"; 0 points each time they're rated "bad"; and for "OK" ratings tally 1 point if that ballot didn't rate either of the finalists "good", and 0 points otherwise.

This procedure works fine in combination with other states using approval voting, plurality voting, or various other methods. It makes it easier for the voters in 3-2-1 states to cast a strategically-optimal vote, but does not give any greater voting power to a 3-2-1 voter over a strategically-optimal plurality or approval voter. In other words, it is still a matter of "one person one vote"; states would have an incentive to adopt 3-2-1 voting, but voters would not be artificially disenfranchised for not passing it, any more than they are already disenfranchised by inferior voting methods like plurality.