[EM] The general form of Quick Runoff

[Juho]

On May 23, 2010, at 5:14 AM, Kevin Venzke wrote:

> 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.

How about this example and LNH.

6: A>C
5: B>A
2: C>B
2: C

Candidate names indicate the order in first preferences. B beats A. C  
beats B. C wins.

6: A>C
5: B>A
2: C>B
2: C>A

Two "C" voters have changed their vote to "C>A". B does not beat A. A  
wins. The "C" voters were harmed when they included their later  
preferences.

Juho



>
> 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
>
>
>
>
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