CPAR voting

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

1. Voters can Prefer, Accept, or Reject each candidate. Default is Accept.
2. Candidates with a majority of Reject, or with under 25% Prefer, are eliminated, unless that would eliminate all candidates.
3. Tally "prefer" ratings for all non-eliminated candidates.
4. Find the leader in this tally, and add in "accept" ratings on ballots that don't prefer the leader (if they haven't already been counted).
5. Repeat step 4 until the leader doesn't change. The winner is the final leader.

So, preferring a candidate will always add 1 to their tally if possible; rejecting a candidate never will; and accepting them will add to their tally unless a candidate you prefer wins (or at least was the leader at some point).

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 favorite betrayal criterion, 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 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 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

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

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.)