[EM] The general form of Quick Runoff

[Kevin Venzke]

Hello,

I realized that QR can be generalized for any number of candidates and
still retain LNHarm, Plurality, and resistance to the usual type of
burial strategy. To me this makes the method surprisingly good.

The philosophy is to elect the candidate with the fewest first-preferences
(think center-squeeze here) who has a very specific majority beatpath
to the first-preference winner.

Here is the new definition:

1. Rank the candidates. Truncation is allowed. Equal ranking is not 
planned for (but we could come up with something).
2. Label the candidates A, B, C, ... Z in descending order of first
preference count.
3. Let the current leader be A.
4. While the current leader has a majority pairwise loss to the very
next candidate, set the current leader to the latter candidate. (In
other words step 4 must be repeated until there is no loss or no other
candidates.)
5. Elect the current leader.

Proof of LNHarm satisfaction: Let's say you were voting B>Y (retaining
the meaning of the alphabetical ordering) and you consider changing
your ballot to B>Y>M. The sole effect this may have is to create a
majority for M>L, causing L to lose. You didn't rank L, so you didn't
harm any higher preferences. (And if you had ranked L, then adding the 
M preference could not have created a majority M>L. Also note that
adding preferences cannot reverse or remove any majorities.)

Who wins instead? Let's talk about burial. Typically the concern is that
voters for a strong candidate will rank a weak candidate insincerely
high in an effort to make a strong competitor lose. For example, you
would vote A>C to confuse the method into defeating B and electing A.
In QR your added C preference can only help elect a candidate who was
even weaker (in first preferences) than C. This makes burial a useless
strategy for the largest factions.

Proof of Plurality satisfaction (a second advantage over MMPO): If X has
more first preferences than Y has votes total, then Y can't have a
majority win over anybody and can never be the current leader.

Monotonicity: We still have an unusual monotonicity problem in that a
candidate who lacks a majority over the candidate previous to him in
first-preference order, may wish he had received fewer first preferences
in order to sit behind a candidate that he did defeat (and who can
still provide the necessary majority beatpath to the top). He may also
wish he received *more* first preferences. Is it a wash?

In any case, getting additional second or third (etc) preferences can't
hurt a candidate.

QR doesn't satisfy Condorcet(gross) (i.e. a candidate with a majority
over every other candidate is not guaranteed to win unless he is one
of the top two candidates in first-preference order) but it does satisfy
Condorcet(gross) Loser.

It doesn't satisfy minimal defense in general. A candidate barred
according to minimal defense can only win if he places first (since he
will be unable to take the win from any other candidate) and he does
not lose by a majority to second-place. (If the latter candidate is the
majority's common candidate under minimal defense, then the barred
candidate will lose.)

It doesn't satisfy SFC generally (because a majority win is only enforced
against one other candidate) but it does work when the involved candidates
place first and second in some order. (If the suspected sincere CW is
A, then A has a majority over B and wins immediately; if the suspected
sincere CW is B, then B takes the win from A and B cannot lose it to
anybody.)

Fairly obviously it satisfies Majority Favorite and Majority Last
Preference. It doesn't satisfy Majority for Solid Coalitions due to the
possibility that the majority's first preferences are so fragmented that
none of their candidates place first or second, and the necessary 
beatpath is not created (B>A).

Due to this it doesn't satisfy Clone-Winner. It may not satisfy 
Clone-Loser either, since cloning a candidate could adjust the first-
preference order to the benefit of the clones as well as to their
detriment.

(It's conceivable that another way of ordering the candidates could
preserve all the properties plus clone independence, but I'm not very
optimistic at the moment.)

It should be clear that the method doesn't satisfy Later-no-Help. If you
change D to D>C you could change the winner from B to D.

What remains is the criterion I defined in my method generator to 
differentiate IRV from QR, which says that the largest faction's last
choice is never elected. I'm not sure how to reformulate it for more
than three candidates.

When we are dealing with scenarios in issue space, the general behavior
of IRV is to remove tiny center candidate(s) (as well as other 
miscellaneous losers) until all that's left are two enormous flanks, and
then we pick the lesser evil.

With QR we hope to see one flank knock out its counterpart and "reveal"
additional preferences. We'd like to see our finalists near the median,
not the flanks. This can fail (imagine that the subsequent candidate
is further from the median). But a criterion would be based on the goals
of this approach.

The MMPO (Minmax(pairwise opposition)) approach is basically similar,
and more successful theoretically, but the lack of structure creates
unacceptable oddities and also strategic vulnerabilities.

The DSC (Descending Solid Coalitions) approach is also kind of similar,
though its focus on solid coalitions makes it less sensitive to majority
opinions. (A pairwise preference of dissimilar factions is not likely to
be counted.) It does satisfy nice criteria though (monotonicity,
participation, clone independence).

Thanks for any comments. Hopefully I haven't made any errors.

Kevin Venzke