Prefer Accept Reject voting

Prefer Accept Reject (PAR) voting works as described below:

1. Voters can Prefer, Accept, or Reject each candidate. Default is "Reject" for voters who do not explicitly reject any candidates, and "Accept" otherwise.
2. Candidates with a majority of Reject, or with under 25% Prefer, are eliminated, unless that would eliminate all candidates.
3. The winner is the non-eliminated candidate with the highest score, counting 1 point for each voter who prefers a candidate, and 1 point for each voter who accepts a candidate while preferring only eliminated candidates.

Note that the 25%-preferred threshold in step 2 is exactly enough so that, in a 3-way election where all voters preferred at least 1 candidate and rejected at least 1 candidate, there will always be at least 1 candidate who passes the thresholds to not be eliminated. This does not hold for an election with 4 or more candidates; but hopefully, even in those cases, the top 3 candidates combined will usually get enough preferences to ensure that at least one of them is above the thresholds.

Relationship to NOTA

If all the candidates in the first round got a majority of reject, then the voters have sent a message that none of the candidates are good, akin to a result of "none of the above" (NOTA). PAR still gives a winner, but it might be good to have a rule that such a winner could only serve one term, or perhaps a softer rule that if they run for the same office again, the information of what percent of voters rejected should be next to their name on the ballot.

Criteria compliance

PAR voting passes the majority criterion, the mutual majority criterion, Local independence of irrelevant alternatives (under the assumption of fixed "honest" ratings for each voter for each candidate), Independence of clone alternatives, Monotonicity, polytime, resolvability, and the later-no-help criterion.

There are a few criteria for which it does not pass as such, but where it passes related but weaker criteria. These include:

• It fails the favorite betrayal criterion, but in any scenario where it fails that for some small group, there is a rational strategy for some superset of that group which does not involve betrayal. (Also, the cases of such failure would arguably be quite rare in practice.) Also, in a 3-way election where all voters preferred at least 1 candidate and rejected at least 1 candidate, there is never a favorite-betrayal incentive unless there's a Condorcet cycle. (This holds even if you add weak also-ran candidates to such an election, because of the following property.)

• It fails Independence of irrelevant alternatives, but passes Local independence of irrelevant alternatives.

• It fails the Condorcet criterion, but for any set of voters such that an honest majority Condorcet winner exists, there always exists a strong equilibrium set of strictly semi-honest ballots that elects that CW.

• It fails the participation criterion but passes the semi-honest participation criterion.

• It fails but, as shown in the center squeeze scenario below, in a 3-candidate scenario it does at least offer viable strategies to each of the subgroups of the majority that prefers X>Y, such that either of the potentially-strategic subgroups has a strategy to ensure Y loses, even if the other potentially-strategic subgroup does not maximally cooperate. ("Subgroup" in this sense is characterized by whether they prefer Z over or under both. The assumption is that the "honest" vote is Support, Accept, Reject in some order for the three candidates, or only Support and Reject in case of indifference between two of them. This guarantees that any X>Z>Y voters will maximally cooperate under honesty, so this subgroup is not potentially-strategic.)

• It fails O(N) summability, but can get that summability with two-pass tallying (first determine who's eliminated, then retally).

• It may pass the majority Condorcet loser criterion (?).

It fails the consistency criterion, the Condorcet loser criterion, reversibility, the majority loser criterion, the Strategy-free criterion, and the later-no-harm and later-no-help criteria.

An 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

Assume voters in each city preferred their own city; rejected any city that is over 200 miles away or is the farthest city; and accepted the rest.

City	P	A	R	tally
Memphis	42	0	58	NA
Nashville	26	74	0	100
Chattanooga	15	43	42	NA
Knoxville	17	41	42	NA

Memphis is rejected by a majority, and is eliminated. Chattanooga and Knoxville both get less than 25% preference, so they are also eliminated. Nashville wins with a tally of 100%. This is a strong equilibrium; no rational strategy from any faction or combination thereof would change the winner. Knoxville and/or Chattanooga could each prevent the other from being eliminated, but Nashville would still win with a tally of at least 68 (the ballots of Nashville and Memphis).

(If Memphis voters rejected Nashville, then Chattanooga or Knoxville could win by conspiring to reject Nashville and accept Memphis. However, Nashville could stop this by rejecting them. Thus this strategy would not work without extreme foolishness from both Memphis and Nashville voters, and extreme amounts of strategy from the others.)