Prefer Accept Reject voting: Difference between revisions

# Each voter gives 1 point to each non-eliminateddisqualified candidate they prefer; and any voter who gave no such points (because their preferred candidates were all eliminateddisqualified) gives 1 point to each non-eliminateddisqualified candidate they accept. '''The winner is the candidate with the most points.'''

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

* It fails the [[later-no-help criterion]], but passes if there is at least one candidate above the eliminationqualification thresholds (which is always true, for instance, if there are some three candidates who get 3 different ratings on every ballot).

None are eliminateddisqualified, so C wins with 40 points (against 35, 25, 35 for A, B, and X). However, if 6 of the first group of voters strategically betrayed their true favorite A, the situation would be as follows:

Now, A is eliminateddisqualified with 51% rejection; so B (the CW) wins.

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 eliminateddisqualified without any betrayal.

Memphis is rejected by a majority, and is eliminateddisqualified. Chattanooga and Knoxville both get less than 25% preference, so they are also eliminateddisqualified. 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 eliminateddisqualified, 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.)

The 25%-preferred threshold in step 2 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 eliminateddisqualified. 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.