[EM] I'm liking PAR

[Jameson Quinn]

I said earlier that I couldn't think of a realistic scenario where PAR
fails to choose the CW. I've now thought of one:

22: A>B
4: A>C
25: B>A
49: C>AB (or C>anything including at least 3% which accept C, and at least
as many C>A as C>B)

This can be thought of as a variety of center squeeze, with A as the
center. (Sorry, I know that the convention is to use B as the center, but I
don't want to rewrite this whole email.)

The B>A voters did not account for the second preference of the A>C voters,
so, unexpectedly to them, neither B nor C is eliminated. If C had been
eliminated, A would not have needed the B>A ballots to win; and if B had
been, the B>A ballots would have rolled over to A. But since neither of
these things happened, C wins, instead of the CW A.

In order to ensure A can beat C, the B voters would have to
almost-unanimously top-rate A. But that would mean that B couldn't win if
the B faction happened to outweigh the A factions.

I find this scenario plausible, but still not very likely. I think that in
most cases, either B would get less than 25% preferences and be eliminated;
or, if A and B are both comfortably over 25% preferred, C would get over
50% rejection and be eliminated.

I guess that a slightly more-plausible version of the scenario is:


22: A>B
4: A>C
5: BA
20: B>A
49: C>AB (or C>anything including at least 3% which accept B, and at least
as many C>A as C>B)

In this case, C wins, but the 5 BA voters could elect A by voting A>B.

Whoops! That's an FBC violation! Note that it's not a violation if all the
voters who honestly prefer B>A can strategize as a bloc, but it is a
violation for any individuals in that group if they know that the rest of
the group will be using sub-optimal strategy.

So in general, PAR violates FBC in a center-squeeze scenario in situations
where the Condorcet loser is not majority-rejected. In a situation where
the honest preferences are roughly as in the second scenario above, there
are several ways that the CW could still win:

   1. The A voters largely reject B (defensive truncation)
   2. The B voters almost unanimously prefer A (defensive compromise)
   3. A few B or AB voters say A>B (defensive betrayal or truncation)
   4. The C voters almost unanimously reject B (strategically suboptimal
   over-truncation)
   5. A few B voters say ∅>AB (defensive, um, I don't know what that is
   called. "Denormalization"? "Weakening"? Technically, this could be seen as
   restoring FBC compliance, but that's a stretch. I'd call that "semi-FBC" at
   best.)
   6. Combination of 1 and 4 above
   7. Combination of 3 and 5 above (although since either one requires
   relatively few voters, it's unlikely that both would be needed.)

In general, I still think that PAR does exceptionally well with naive
ballots, because I think that cases where the problem arises, but none of
the above solutions happens naturally, would be rare. But hmmm... failing
FBC... I recognize that that looks bad.

Is there a way to fix this? I guess you could run PAR's step 3 as IRV-style
successive eliminations. In that case, B would be eliminated first, and the
votes would go to A, so C would lose. But.... I suspect it's still
possible, although massively unlikely in practice, to make a pathology
where the elimination would go in the wrong order, so that the resulting
system fails FBC.

Oh... and I guess that solution 5 above doesn't actually give even
"semi-FBC", because in theory the C voters could counter it by voting
CB>... in exactly the right proportions (offensive turkey-raising). I don't
think this would ever work in practice because it requires knowing exactly
how many of another faction will strategize AND sophisticated
within-faction vote-management, but it still blows the criterion.

Anybody have any thoughts on any of this?


2016-11-02 9:03 GMT-04:00 Jameson Quinn <jameson.quinn at gmail.com>:

> I made the electowiki page for PAR voting
> <http://wiki.electorama.com/wiki/Prefer_Accept_Reject_voting>, and I'm
> liking the system more and more. In the Tennessee voting example, Nashville
> wins in a strong equilibrium; as far as I can recall, this is the only
> non-Condorcet system where that's true.
>
> Compared to MAS, it loses later-no-help, IIA (though it still has LIIA),
> and summability (though it has two-pass O(N) summability). It may gain
> majority voted Condorcet loser, though I don't yet have a proof for that.
>
> In practice, I expect that it would be rare to have elections with other
> than exactly 1 or 2 non-eliminated candidates. In such elections, this
> system elects the voted pairwise winner of the non-eliminated candidates.
> In fact, I can't think of a single realistic scenario where the most
> obvious naive votes don't elect an existing CW, including nonmajority CWs.
> In particular, it handles both CD and center squeeze using naive votes.
>
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