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.

I think that simple PAR is close enough to FBC compliance to be an
acceptable proposal.

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).

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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