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Prefer Accept Reject voting: Difference between revisions

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Prefer Accept Reject voting (view source)

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Prefer Accept Reject (PAR) voting works as described belowfollows:
 
# '''Voters can Prefer, Accept, or Reject each candidate.''' On ballots which don't explicitly use "Reject", or for candidates with less than 25% "Prefer", blanks count as "Reject"; otherwise, blanks Defaultcount isas "Accept".
# '''Tally 1 point for each "Prefer"''' for each candidate.
# Candidates with a majority of Reject, or with under 25% Prefer, are eliminated, unless that would eliminate all candidates.
# Out of the candidates (if any) with no more than 50% "Reject", find the one with the most points. '''For every ballot which doesn't "Prefer" this frontrunner, add 1 point for each "Accept".'''
# The winner is the non-eliminated candidate rated equal-or-above any other non-eliminated candidate on the most ballots.
# If the frontrunner still has the most points, they win. Otherwise, the winner is the candidate with fewest "Reject" ratings.
 
To express it in a single sentence: if the most-preferred non-majority-rejected candidate, X, has more non-reject votes than any other candidate has non-reject votes that aren't below X, then X wins; otherwise, the least-rejected candidate wins.
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 if and only if all candidates you preferred are eliminated.
 
Note that the procedure above will always elect a candidate with no more than 50% "Reject", if any exist. This is because, if any exist, one of them will be the frontrunner, and they will thus score points equal to at least 50% of the voters.
 
Each candidate's score at the end can be seen as an approval total, and is thus suitable for combining with approval totals from other jurisdictions in a system like the National Popular Vote Interstate Compact.
 
A related system which passes [[FBC]] is [[FBPPAR]]. This has the same steps, except that voters can choose to mark any of their "preferred" candidates as "stand aside". "Stand aside" preferences are counted as rejections when finding the leader, but as preference when assigning points.
 
== Relationship to NOTA ==
Line 13 ⟶ 20:
== Criteria compliance ==
 
PAR voting passes the [[majority criterion]], the [[mutual majority criterion]], the [[majority loser 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 theO(N²) [[later-no-help criterionsummability]]. (It is also possible to run it in no more than 3 counting rounds, each of which is O(N) summable.)
 
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.)▼
 
* It fails [[Independence of irrelevant alternatives]], but passes [[Local independence of irrelevant alternatives]].
 
* It fails the [[Condorcet criterion]], but forif there is a voted majority Condorcet winner X, then any faction (defined as the set of voters suchwho thatprefer anX>Y,Z, honestor majorityalternately Condorcetas winnerthe set who prefer existsY>X>Z, therefor alwayssome existsY aand strongZ) equilibriumcan setensure ofthat strictlyX beats Z, using semi-honest ballots that(X>>Z electsor thatYX>>Z, CWrespectively).
 
* It fails the [[participation criterion]] but passes the [[semi-honest participation criterion]].
 
* It fails the [[later-no-help criterion]], but passes if there is at least one candidate rejected by under 50%.
* 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 the [[consistency criterion]], the [[Condorcet loser criterion]], [[reversibility]], the [[majority loser criterion]], the [[Strategy-free criterion]], and the [[later-no-harm criterion|later-no-harm]] and [[later-no-help criterion|later-no-help]] criteria.▼
* It fails O(N) [[summability]], but can get that summability with two-pass tallying (first determine who's eliminated, then retally).
 
=== Favorite betrayal? ===
* 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 criterion|later-no-harm]] and [[later-no-help criterion|later-no-help]] criteria.
PAR voting fails the [[favorite betrayal criterion]] (FBC). For instance, consider the following "non-disqualifying center-squeeze" scenario: (
 
* 30: AX>B (That is, on 35 ballots, A and X are preferred, B is accepted, and C is rejected)
* 5: AX>C
* 15: B>A
* 10: B>AC
* 40: C>B
 
None are majority-rejected, and C is the frontrunner. Points are: A, 60; B, 55; C, 55; X, 35. A wins. However, if 11 of the last group of voters strategically betrayed their true favorite C, the situation would be as follows:
 
* 30: AX>B (That is, on 35 ballots, A and X are preferred, B is accepted, and C is rejected)
* 5: AX>C
* 15: B>A
* 10: B>AC
* 29: C>B
* 11: B
 
Now, C is not viable with 51% rejection; so B is the leader. Since C is no longer the leader, B gets the 34 points from C voters, and wins. The strategy succeeded; the strategic voters are better off.
 
However, there are several ways to "rescue" FBC-like behavior for this system.
 
For one, we could add a "stand aside" option to the ballot, as described in [[FBPPAR]].
 
For another, the B>AC voters could simply reject C, the strongest rival of their favorite, and B would win with no need for favorite betrayal.
 
▲*And It fails the [[favorite betrayal criterion]]finally, butnote that 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. (AlsoFor instance, thein casesfirst scenario above, if 16 of suchthe failureC>B wouldvoters arguablyswitch beto quiteCB, rarethen inB is the leader and wins without them having to rate C below their true practicefeelings.)
 
== An example ==
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{{Tenn_voting_example}}
 
Assume voters in each city preferred their own city; rejected any city that is over 200 miles away or is the farthest city; andexplicitly accepted the closest other city; and left the rest blank. In the following table, the quantities in parentheses are explicit votes, while the implicit totals (including blank votes) are unparenthesized.
 
<div class="floatright">
Line 43 ⟶ 74:
!City
!P
!(A)
!A
!(R)
!R
!tally
Line 49 ⟶ 82:
!bgcolor="#fff"|Memphis
|bgcolor="#fff"|42
|bgcolor="#fff"|(0)
|bgcolor="#fff"|0
|bgcolor="#fcc"|(58)
|bgcolor="#fcc"|58
|bgcolor="#fcc"|NA42
|-
!bgcolor="#fff"|Nashville
|bgcolor="#fff"|26
|bgcolor="#fff"|(42)
|bgcolor="#fff"|74
|bgcolor="#fff"|(0)
|bgcolor="#fff"|0
|bgcolor="#cfc"|100
Line 61 ⟶ 98:
!bgcolor="#fff"|Chattanooga
|bgcolor="#fcc"|15
|bgcolor="#fff"|(43)
|bgcolor="#fff"|43
|bgcolor="#fff"|(0)
|bgcolor="#fff"|42
|bgcolor="#fccfff"|NA58
|-
!bgcolor="#fff"|Knoxville
|bgcolor="#fcc"|17
|bgcolor="#fff"|41(15)
|bgcolor="#fff"|4215
|bgcolor="#fccfff"|NA(0)
|bgcolor="#fcc"|68
|bgcolor="#fcc"|32
|}
</div>
 
Memphis is rejected by a majority, and is eliminatednonviable. Chattanooga and Knoxville both get less than 25% preference, so they are also eliminatednonviable. Nashville is the leader, and 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 eliminateddisqualified and even get the other to temporarily be the frontrunner, 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.)▼
 
== Discussion ==
 
=== Logic for 25%-preferred threshold (step 2) ===
 
The 25%-preferred threshold in step 1 is not purely arbitrary; it is exactly enough so that, in a 3-candidate election where all voters give all three grades, there will always be at least 1 candidate who passes the thresholds to not be disqualified. In other words: if a minority supports a rejected candidate, while a majority divides preferences between two candidates while accepting the other, then at least one of those two will not be disqualified. This does not hold for an election with 4 or more candidates, because the majority could split its preferences more than two ways; but even in those cases, it is usually reasonable to hope that the top 3 candidates combined will get enough preferences to ensure that at least one of them is above the 25% threshold.
 
▲(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.)
[[Category:Graded Bucklin systemsmethods]]
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