[EM] Quick Runoff (QR) LNHarm method and commentary

[Kevin Venzke]

Hello,

Here's a LNHarm method I came up with which has some interesting
properties. I actually came up with this before writing my latest method
generator, and hoped the generator would help find an even better method
in this vein, but no dice.

Here is the definition.
1. Voters rank the candidates. Equal ranking is not envisioned.
2. Label the top three candidates wrt first preferences as A, B, and C
in descending order. We don't care about other candidates.
3. Is there a >50% majority favoring B over A? If not, elect A.
For step 4 you have a choice.
4a. Is there a >50% majority favoring C over B? If not, elect B; else
elect C.
alternative:
4b. Eliminate A and transfer A votes to B or C as possible, and then
elect the one of those with the most votes.

The advantage of 4b is that it is more Condorcet-efficient: A voters
who want C to win don't have to be as numerous for it to happen. The
disadvantage is that 4b can violate Plurality if C gains enough votes
to defeat B but remains technically disqualified under Plurality due
to A's first preference count.

This is a LNHarm method: C bloc's second preference can only be used
to defeat A (i.e. when the second preference is for B; if it is for A
then it does nothing). B bloc's second preference isn't used. A bloc's
second preference is only used when A is eliminated.

This method has the nice property that I defined in my last post, that
(assuming there are just three candidates) if the entire A bloc has
the same strict last preference, that last preference will not be 
elected. This is the key difference from IRV, lessening the center squeeze
phenomenon: The A voters will get their second choice at worst, which
could well be the third place candidate.

In my last mail I cautioned that when A voters have such a guarantee,
it's important that they not be able to use it strategically to steal the
win. And in QR, that base is covered: Neither A nor B voters can use
C as a pawn to bury the other one of A/B.

The B voters can't benefit from creating a win for C over A, because the
A:C contest is never regarded at all. And the A voters can't benefit
from creating a C>B win because this preference will not be regarded
unless/until A has been eliminated.

Another pitfall with LNHarm methods is that they may technically provide
no incentive to truncate, while in practice there is still incentive
to make other voters believe that you may truncate. Unfortunately QR
does not completely escape from this.

Specifically, C voters who know which candidate is A (i.e. the first-
preference winner) can harmlessly vote for B as the second preference,
if they believe that A voters will give their second preference to C.
A voters can defend against this by making C voters believe that A won't
give any preferences to C. In that case, the C voters risk accidentally
giving the election to B by using their strategy.

A mitigating factor (which I've also brought up when discussing SPST)
is that this scenario has a less plausible thing about it. Usually when
we imagine three-candidate scenarios, we imagine that two of the 
candidates are closer to each other than to the third. There is overlap
in the voters that the two closer candidates compete for. Thus the
candidate with the most first-preferences will probably be the most
distant candidate. In that case, why would it happen that the distant
candidate would expect that they are the sincere second preference of
one of the smaller blocs? It seems more likely to me that the distant
candidate would simply expect that they will be voted against by the
other two blocs, in which case there is no point in withholding the C
preference (if that's who A voters like next best). A's truncation
deterrent is useless (and harms A voters) when C voters truly do like
B second best.

QR has a downside in that it's possible for B bloc to win by receiving
fewer votes. Here's the specific case:

40 A>B
35 B>C
25 C or C>A

A wins.

40 A>B
35 C or C>A
25 B>C

B wins.

I find these preferences a little odd personally. But few methods have a
problem like this one and I can imagine objections to QR over this.

My method comparer rates QR as much closer to IRV than to most FPP-like
methods. But I originally saw it as closer to SPST, and wanted to keep
"ST" ("single transfer") as part of the name to reflect that. Basically
they are the same method but with one difference: SPST eliminates A
when *both* B and C defeat A, while QR eliminates A when B defeats A,
and it doesn't matter whether C defeats A. So when there is truncation
QR is less likely to simply go with the FPP winner.

And even when there isn't truncation, QR still can be better:

40 A>B
35 B>A
25 C>B

DSC and SPST elect A, while QR elects B, the Condorcet(gross) winner.

Speaking of Condorcet efficiency: A lot can be lost when voters truncate,
since majority defeats are required at least for A:B (this is essential
for LNHarm). Of course we hope that nobody will truncate. In that case
a QR Condorcet violation looks like this:

40 A>C
35 B>C
25 C>A

Basically C bloc votes for the wrong second preference. A is allowed to 
autowin because B is so weak.

I hope you find this method interesting. Let me know if you have any
questions.

Kevin Venzke