[EM] Instant pairwise bracket

[Kristofer Munsterhjelm]

On 2018-12-02 08:15, Rob Lanphier wrote:
> Hi folks,
> 
> Check out this editorial from Harold Meyerson at the LA Times:
> "A Democratic victory in 2020 demands a new form of primary"
> <https://www.latimes.com/opinion/op-ed/la-oe-meyerson-primary-ranking-20181129-story.html>
> 
> Meyerson describes what a mess the 2020 Democratic primary for U.S.
> President promises to be, with potentially up to 30 viable candidates
> all competing for the Democratic Party nomination, and how even in a
> field of 10-12 candidates, it's not hard to win with only 15% of the
> vote. He openly muses about a tournament that would seem reasonably
> certain to select the Condorcet winner in a crowded field (or at least
> one of the candidates from the Smith set):
> 
>> I suppose the Democrats could adopt NCAA-like brackets, with the winner
>> of the billionaires’ primary (which might feature Bloomberg, Tom
>> Steyer and Starbucks’ Howard Schultz) facing off against the winner
>> of the left primary (Sanders, Warren and maybe Brown), with the winner
>> of that going up against the victor of the center-left senator bracket
>> (Kirsten Gillibrand, Amy Klobuchar and Cory Booker), who...
> 
> ...oooh, this is really appealing!  Is he about to describe a pairwise system?
> 
>> ... well, this system has problems, too.
> 
> Hrm....nope.  What is he about to propose?
> 
>> The one way to ensure that the nominee actually is favored by a majority
>> of Democratic voters is for the party to adopt a form of ranked-choice
>> voting.
> 
> D'oh!  He seems to be referring to Instant Runoff.  And indeed:

We may be angry at FairVote for appropriating the term "ranked-choice
voting"; but that just there is how it works. Get people to think ranked
ballots = IRV.

> Certainly, Approval is a really good choice.  Still, given the
> direction Meyerson started to go, I'd like to riff on the NCAA runoff
> idea.  A lonnng time ago (in 2006), I proposed a tournament-style
> version (more about that below).  The NCAA-tournament idea and using
> that as a framework for a method could allow us to create a really
> good Condorcet-flavored proposal.  Here's a set of rules I cobbled
> together tonight:
> 
> 1.  Create a playoff bracket with room for all candidates, using rules
> similar to the ones the NCAA uses for March Madness:
> <https://en.wikipedia.org/wiki/March_Madness>
> 
> ....or maybe on the ones Wimbledon uses:
> <https://en.wikipedia.org/wiki/Wimbledon_championship>
> 
> In short, use a single-elimination tournament:
> <https://en.wikipedia.org/wiki/Single-elimination_tournament>
> 
> 2.  Seed the candidates in the tournament such that all members of the
> Smith set are guaranteed to advance to the final rounds.  I'm guessing
> that the Copeland score could be used:
> <https://en.wikipedia.org/wiki/Copeland_method>
> 
> 3. Calculate the winner of each contest using the standard ways of
> inferring pairwise matchup results based on ranked/rated ballots
> 
> These steps may seem like a lot of theatrical extras (especially in
> contests where there is a single Condorcet winner) but I think this
> framework could provide a useful mental model for people whose eyes
> glaze over when try to describe some of the mathematical
> vulnerabilities of systems like Instant Runoff.

I have a hunch that some criteria (independence of clones, mainly) in
combination with Condorcet/Smith depend upon doing at least n^2 pairwise
comparisons. If so, then the magic has to happen in the seeding, not in
the tournament itself, since playoff formats generally have log(n)
stages and do no more than n comparisons in each stage.

Of course, if we're talking practical methods, maybe eating the
imperfection bullet and accepting some clone dependence as long as it's
not too bad is the way to go. Or I could be wrong; it's only a hunch.

The problem with doing n^2 comparisons by clever seeding is that the
complexity that (it's suspected, at least) makes Condorcet uninteresting
would just be moved into the seeding phase.