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Prefer Accept Reject (PAR) voting works as follows:
# '''Voters can Prefer, Accept, or Reject each candidate.''' BlanksOn countballots aswhich don't explicitly use "Reject", ifor nofor rivalcandidates iswith explicitlyless rejectedthan 25% "Prefer", blanks count as "Reject"; otherwise, blankblanks iscount as "Accept".
# '''CandidatesTally with1 atpoint leastfor 25%each "Prefer, and no more than 50% reject, are "viable"'''. Thefor most-preferred viableeach candidate (if any) is the leader.
# EachOut "prefer"of isthe worthcandidates 1(if point.any) with no more than Each50% "acceptReject", forfind athe viableone candidatewith onthe most points. '''For aevery ballot which doesn't prefer"Prefer" the leader isthis alsofrontrunner, worthadd 1 point. '''Mostfor pointseach wins"Accept".'''
# 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.
Note that the procedure above will always elect a viable candidate if any exist. It will usually, but not always, elect the leader. Each candidate's score at the end can be seen as an approval total.
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.
== 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 O(N²) [[summability]]. (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 [[Independence of irrelevant alternatives]], but passes [[Local independence of irrelevant alternatives]].
* It fails the [[Condorcet criterion]], but forif anythere setis ofa voters such that an honestvoted majority Condorcet winner existsX, therethen alwaysany existsfaction a(defined strongas equilibriumthe set of strictlyvoters semi-honestwho ballotsprefer thatX>Y,Z, electsor thatalternately CW.as (Notethe thatset thiswho isprefer not trueY>X>Z, for anysome strictlyY and Z) can ensure that X beats Z, using semi-rankedhonest Condorcetballots system!(X>>Z or YX>>Z, respectively).
* It fails the [[participation criterion]] but passes the [[semi-honest participation criterion]].
* It fails O(N)the [[summabilitylater-no-help criterion]], but canpasses getif thatthere summabilityis withat two-passleast tallyingone (firstcandidate determinerejected who's disqualified,by thenunder retally)50%.
* It may passfails the majority[[consistency criterion]], the [[Condorcet loser criterion]], (?)[[reversibility]], the [[Strategy-free criterion]], and the [[later-no-harm criterion]].
* It fails the [[later-no-help criterion]], but passes if there is at least one candidate above the qualification thresholds (which is always true, for instance, if there are some three candidates who get 3 different ratings on every ballot).
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.
=== Favorite betrayal? ===
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)
* 105: BAX>AC
* 10: B>AC
* 5: B>C
* 40: C>B
None are disqualifiedmajority-rejected, soand C winsis withthe 40frontrunner. pointsPoints (againstare: 35A, 2560; B, 3555; for AC, B55; X, and35. X)A wins. However, if 611 of the firstlast group of voters strategically betrayed their true favorite AC, 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)
* 29: AX>B
* 65: XAX>BC
* 1015: B>A
* 10: B>AC
* 529: B>C>B
* 4011: C>B
Now, AC is disqualifiednot viable with 51% rejection; so B (is the CW)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 "compromisestand aside" option to the ballot, as described in [[FBPPAR]].
For another, wethe B>AC voters could restrictsimply reject C, the domainstrongest torival votingof scenariostheir whichfavorite, meetand theB would win with no need for followingfavorite restrictions:betrayal.
And finally, note 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. For instance, in first scenario above, if 1116 of the AXC>B voters switch to >AXBCB, then AB is disqualifiedthe leader and wins without anythem having to rate C below their true betrayalfeelings. ▼# Each candidate either comes from one of no more than 3 "ideological categories", or is "nonviable".
# No "nonviable" candidate is preferred by more than 25%.
# Each voter rejects at least one of the 3 "ideological categories" (that is, rejects all candidates in that category) and does not reject at least one of them (rejects none of the candidates in that category).
# There are no honest Condorcet cycles.
If the above restrictions hold, then PAR voting would meet FBC. It is arguably likely that real-world voting scenarios will meet the above restrictions, except for a negligible fraction of "ideologically atypical" voters. For instance, in the first scenario above, the categories appear to be {XA}, {B}, and {C}, so the B>AC voters would probably actually vote either B>A or B>C.
▲And finally, note 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. For instance, in first scenario above, if 11 of the AX>B voters switch to >AXB, then A is disqualified without any betrayal.
== An example ==
{{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">
!City
!P
!(A)
!A
!(R)
!R
!tally
!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
!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 disqualifiednonviable. Chattanooga and Knoxville both get less than 25% preference, so they are also disqualifiednonviable. 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 disqualified 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.)
=== Logic for 25%-preferred threshold (step 2) ===
The 25%-preferred threshold in step 21 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.
[[Category:Graded Bucklin systemsmethods]]
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