Prefer Accept Reject voting: Difference between revisions
Category:Graded Bucklin systems → Category:Graded Bucklin methods
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Prefer Accept Reject (PAR) voting works as follows:
# '''Voters 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 count as "Accept".
# '''Tally 1 point for each "Prefer"''' for each candidate.
#
# 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 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.
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== 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:
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* It fails [[Independence of irrelevant alternatives]], but passes [[Local independence of irrelevant alternatives]].
* It fails the [[Condorcet criterion]], but
* It fails the [[participation criterion]] but passes the [[semi-honest participation criterion]].
* It fails
=== Favorite betrayal? ===
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* 40: C>B
None are
* 30: AX>B (That is, on 35 ballots, A and X are preferred, B is accepted, and C is rejected)
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* 15: B>A
* 10: B>AC
*
*
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.
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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 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 16 of the C>B voters switch to CB, then B is the leader and wins without them having to rate C below their true feelings.
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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;
<div class="floatright">
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!City
!P
!(A)
!A
!(R)
!R
!tally
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!bgcolor="#fff"|Memphis
|bgcolor="#fff"|42
|bgcolor="#fff"|(0)
|bgcolor="#fff"|0
|bgcolor="#fcc"|(58)
|bgcolor="#fcc"|58
|bgcolor="#fcc"|42
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!bgcolor="#fff"|Nashville
|bgcolor="#fff"|26
|bgcolor="#fff"|(42)
|bgcolor="#fff"|74
|bgcolor="#fff"|(0)
|bgcolor="#fff"|0
|bgcolor="#cfc"|100
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!bgcolor="#fff"|Chattanooga
|bgcolor="#fcc"|15
|bgcolor="#fff"|(43)
|bgcolor="#fff"|43
|bgcolor="#fff"|(0)
|bgcolor="#fff"|42
|bgcolor="#
|-
!bgcolor="#fff"|Knoxville
|bgcolor="#fcc"|17
|bgcolor="#fff"|
|bgcolor="#fff"|
|bgcolor="#
|bgcolor="#fcc"|68
|bgcolor="#fcc"|32
|}
</div>
Memphis is rejected by a majority, and is nonviable. Chattanooga and Knoxville both get less than 25% preference, so they are also nonviable. 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.)
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=== Logic for 25%-preferred threshold (step 2) ===
The 25%-preferred threshold in step
[[Category:Graded Bucklin
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