[EM] PAR theory

[Jameson Quinn]

I want to discuss the reasoning behind PAR, and to prove some lemmas about
it.

(Note: I've included the latest version of the PAR rules at bottom. I
apologize that these rules have now gone through many iterations, and even
changed how they deal with some edge cases; but I think the basic ideas of
PAR, and the results in most simple realistic scenarios, have remained
consistent.)

First off: there are two basic ideas in PAR: the default rule, and the
tally rule. These ideas are more-or-less independent conceptually, but they
both work together to give the method a good tradeoff between
non-slippery-slope performance in the chicken dilemma without damaging its
performance in a center squeeze situation.

The default rule is intended to "nudge" reasonably-lazy voters to cooperate
in a chicken dilemma. A voter in a chicken dilemma faction has at least one
candidate they support, and one candidate they strongly and saliently
oppose. If they explicitly vote these two "prefer" and "reject", and leave
all others blank, how should that be interpreted? In PAR, that is
interpreted as "accept" for candidates with significant support (defined as
at least 25% top-ranks), but "reject" for relatively-unknown candidates or
clear chicken-dilemma losers (under 25% top-ranks).

If highly-engaged strategic voters know that a large portion of their
faction will vote in the "lazy" style above, it becomes unlikely that
strategy will benefit them. I believe that many voters will be aggressively
strategic if and only if they expect most of the "other side" to be so. If
that's true, nudging the lazy voters away from aggressive strategy will
serve to nudge such "copycat" voters in the same way.

The tally rule in the latest version of PAR is: tally all "prefer" ratings,
and all "accept" ratings except on ballots which prefer the frontrunner,
where the frontrunner is the candidate X (if any) who meets the following
three criteria:
- X has the highest tally using the rules above when considering X as
frontrunner
- X is not rejected by a majority
- X has more top-ranks than any candidate Y not rejected by a majority

Thus, if no candidate meets these criteria, the PAR winner is simply the
least-rejected candidate.

This method will always elect a voted majority Condorcet winner C, because
such a candidate will always either meet the three criteria, or win by the
fallback rule.
- For any Z, C's tally when considering C as a frontrunner will include
(more than) all ballots with C>Z, by assumption a majority; but Z's tally
will include only ballots with Z>C, by assumption a minority.
- C will not be rejected by a majority.
- If Z has more top-ranks than C, and Z is not rejected by a majority, then
Z will not have the highest tally when considering Z as a frontrunner, and
C will have fewer rejections than Z.

(I realize that the above "proof" is not actually solid. I have not shown
it's impossible for Z to have more top-ranks than C, so that the fallback
tally ends up being used, and then for some other candidate Y to have fewer
rejections than C. But it's pretty hard to do that without creating a
Condorcet cycle.)

However, PAR does not always elect the CW if it is not a majority CW.
Consider the following election:

33: A>B
22: B>A
12: C>B
33: C

A is the most-preferred non-majority-rejected, so start the tally with A as
the frontrunner. A tallies 55, B tallies 34, C tallies 45; A is still in
the lead, so A wins. But B would beat A pairwise by (a nonmajority tally
of) 34 to 33.

In a sense, this is a failure of center squeeze. But if all the C voters
voted C>B, B would win; it is only in cases where it's a minority of them
who do so that A can still win. And this behavior makes it less necessary
for the A voters to "defensively" truncate B, which I think will lead to
more honestly cooperative voting overall.

...

Here are the rules


   1. *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".
   2. *Tally 1 point for each "Prefer"* for each candidate.
   3. 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".*
   4. If the frontrunner still has the most points, they win. Otherwise,
   the winner is the candidate with fewest "Reject" ratings.
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.electorama.com/pipermail/election-methods-electorama.com/attachments/20161115/ea3be1c7/attachment.htm>