[EM] PAR is awesome part 1/2: FBC?

[Jameson Quinn]

Here's the definition of PAR
<http://wiki.electorama.com/wiki/Prefer_Accept_Reject_voting> again:


   1. *Voters can Prefer, Accept, or Reject each candidate.* Default is
   "Reject" for voters who do not explicitly reject any candidates, and
   "Accept" otherwise.
   2. *Candidates with a majority of Reject, or with under 25% Prefer, are
   disqualified*, unless that would disqualify all candidates.
   3. Each voter gives 1 point to each non-eliminated candidate they
   prefer; and any voter who gave no such points (because their preferred
   candidates were all eliminated) gives 1 point to each non-eliminated
   candidate they accept. *The winner is the candidate with the most
   points.*


Note that since originally proposing this method, the only substantive
change to the process above has been a slight adjustment in the default
rule: the part where default is "Reject" for voters who do not explicitly
reject any candidates.

As previously discussed, this method does not meet FBC. For instance,
consider the following "non-disqualifying center-squeeze" scenario:

35: AX>B
10: B>A
10: B>AC
5: B>C
40: C>B

None are eliminated, 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:

29: AX>B
6: X>B
10: B>A
10: B>AC
5: B>C
40: C>B

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

Is this violation of FBC a serious defect in the system? I would argue it
isn't. In the above scenario pair, candidates A, B, and C are the clear
frontrunners, with X being merely a distraction. In that context, the 10
B>AC voters are clearly not using their full voting power. If they voted
their true preferences, whether those are B>A>C or B>C>A, then either A or
C would have to be eliminated, and B would win.

More generally, one can "rescue" FBC-like behavior for this system by
restricting the domain to voting scenarios which meet the following three
restrictions:

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).

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 three categories
would be {AX}, {B}, and {C}, and the B>AC voters, who violate the third
restriction, would probably actually vote either B>A or B>C, which wouldn't
violate that restriction.

Also, note that in any scenario where PAR fails FBC 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 eliminated without any betrayal.

If you're really concerned about FBC failure, then you can always use FBPPAR
<http://wiki.electorama.com/wiki/FBPPAR> instead:


   1. Voters can Prefer, Accept, or Reject each candidate. Default is
   "Accept"; except that for voters who do not explicitly reject any
   candidates, default is "Reject". Voters can also mark a global option that
   says: "I believe that voters like me should be the first to compromise."
   2. Candidates with a majority of Reject, or with under 25% Prefer, are
   eliminated, unless that would eliminate all candidates. If a candidate
   would have been eliminatable considering all the "prefer" votes they got on
   "compromise" ballots as "rejects", then they are considered "eager to
   compromise"
   3. The winner is the non-eliminated candidate with the highest score.
   Voters give 1 point to each candidate whom they prefer; and, if all the
   candidates they gave points to are "eager to compromise", they also give 1
   point to each candidate whom they accept.


However, I think that FBPPAR is just a theoretical curiosity. The
"compromise" option adds significant extra complexity, and would almost
never be used. I think that simple PAR is close enough to FBC compliance to
be an acceptable proposal.

Other than FBC, PAR has some pretty excellent properties. It elects the CW
in most realistic chicken dilemma scenarios, giving a strong Nash
equilibrium with naive/honest/strategyless ballots, as shown in the
Tennessee example. It elects the "correct" winner in a chicken dilemma
scenario, naive/honest/strategyless ballots, without a "slippery slope"
(though of course, this is no longer a strong Nash equilibrium).

PAR voting passes the majority criterion, the mutual majority 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, and resolvability.

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 for any set of voters such that
   an honest majority Condorcet winner exists, there always exists a strong
   equilibrium set of strictly semi-honest ballots that elects that CW. (Note
   that though this is in some sense a "weaker" criterion, it is actually not
   met by most strictly-ranked Condorcet systems!)
   - It fails the participation criterion but passes the semi-honest
   participation criterion.
   - It fails O(N) summability, but can get that summability with two-pass
   tallying (first determine who's eliminated, then retally).
   - It may pass the majority Condorcet loser criterion (?). If not, it
   certainly passes some weakened version.
   - It fails the later-no-help criterion, but passes if there is at least
   one candidate above the elimination 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, reversibility, the majority loser
criterion, the Strategy-free criterion, and later-no-harm.

All-in-all, I think it's a great method: reasonably simple and intuitive,
passes FBC on a restricted but essentially-realistic domain, handles
center-squeeze and CD with naive ballots, and cloneproof.
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