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

[Jameson Quinn]

Whoops. When I stated that PAR meets FBC on a restricted domain, I forgot
to stipulate that one of the restrictions is that there should be no honest
Condorcet cycles. I also didn't state that the voters should all have one
ideological stripe for which they reject none of the candidates.

2016-11-11 6:44 GMT-05:00 Jameson Quinn <jameson.quinn at gmail.com>:

> 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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