[EM] "Total Vote Runoff"
Forest Simmons
forest.simmons21 at gmail.com
Sun Jan 22 01:09:24 PST 2023
There is a very simple, transparrnt Condorcet method (and really only one
such method) along these lines that is both clone free and tournament
monotone: Worst Loser Elimination(WLE).
It's really the only RCV method worth serious consideration ... given the
current election reform environment ... which (among other considerstions)
only allows elimination methods.
For that reason I want to give a simple, transparent explanation of it in
sports tournament language.
A simple but thorough way to determine the championship winning basketball
team in a given league is by means of a tournament in which every team is
pitted in a head to head matchup with every other team.
If one team wins every one of its matches, then that team is the undisputed
universal champion.
Otherwise we eliminate one team at a time until there is such an undisputed
champion among the remaining teams.
Which team should be eliminated first?
If there is any team that lost every single one of its matchups (a
universal loser), then it qualifies as "worst loser" and is to be
eliminated first.
More generally, at any elimination stage a team that has lost every match
among the remaining teams should be the next team to be eliminated.
If at any stage there is no such Universal Loser, then we eliminate another
contestant for "worst loser".. the team closest to being "skunked" by the
other remaining teams ... the one closest to having zero points in one of
its matchups with another remaining team.
In this manner ... until there is an undisputed champ among the remaining
teams we eliminate (from among the remaining candidates) the worst loser
... defined as eithe the universal loser or (lacking that) the most nearly
skunked team.
The tournament finish order for the tournament is the reverse order of the
eliminations ... first eliminated finishes last ... last eliminated
finishes first. "The first shall be last, a d the last shall be first!"
There is one obvious question about our choice of which candidate to
eliminate when there is no universal loser: why not just eliminate the
candidate with the most lost matches?
There are two relevant reasons: 1. It is very likely that two teams will
be tied for the most losses among the remaining candidates at some stage
...as in a rock,paper,scissors cycle tie where each of three remaining team
loses to exactly one other team ... so then a tie breaking method must be
introduced ... complicating things. Remember, we place a high premium on
both decisiveness and simplicity in the public election context that we
have in the back of our minds. 2. This alternate elimination rule would not
do in the election methods context that we are contemplating, because itt
would create a clone independence violation of "teaming vulnerability" that
out of fairness considerations must be avoided at all costs in the election
methods context ... otherwise it would be like teams (think a set of
related candidates) colluding to prop up one of their buddies by carefully
selecting whom they lose to ... not trying very hard to win against their
agreed upon buddy. This distortion can happen accidentally too ...
independent of bad intentions.
So how do we convert this tournament championship method into a single
winner election method?
The number of points that candidate X gets in its matchup with candidate Y
is simply the number of ballots on which candidate X is ranked ahead of
candidate Y.
A rectangular table is created that has a row for each candidate listing
their points against the others. In the row for candidate X put a star next
to each candidate that loses a match to one or more of the other candidates.
If there is a candidate whose row has no stars, that candidate is the
undisputed champ.
Otherwise eliminate the candidate that has a star next to every entry in
its row ... unless there is no such candidate ... in which case eliminate
the candidate whose row contains the smallest starred number.
When you eliminate a candidate ... erase its entire row as well as all of
the points against it by other candidates ..
so that all remaining table entries refer only to the remaining candidates.
Then repeat the same process applied to the remaining table ... and since
the remaing stars refer only to the pairwise losses among the remaining
candidates, they are still valid ... no need to reestablish them at every
step.
Et Cetera!
Thanks!
Spread the word!
-Forest
On Fri, Jan 20, 2023, 11:40 PM Forest Simmons <forest.simmons21 at gmail.com>
wrote:
> A Condorcet Candidate will never have the lowest Borda Count ... not even
> below average ... so you could speed this up by eliminating at each step
> all candidates with below average "total votes."
>
> But the method is clone dependent ... unlike a similar but simpler
> Condorcet Compliant method that Benham calls "Gross Loser Elimination"
> (GLE) which (at each step) eliminates the candidate with the smallest
> single pairwise vote instead of eliminating the candidate with the smallest
> sum of pairwise votes.
>
> In my opinion GLE is the simplest elimination method worth supporting.
>
> Operationally you start by constructing the same precinct summable
> pairwise matrix that you would use for Borda Elimination.
>
> But instead of eliminating (at each step) the remaining candidate with the
> smallest row sum, you eliminate (at each step) the remaining candidate with
> the smallest row min ... the Gross Loser.
>
> The margins versions of these two related methods are slightly more
> transparent, as well as precisely equivalent in the case of complete
> rankings.. The pairwise margins matrix is obtained from the pairwise vote
> matrix by subtracting is transpose from it.
>
> In a margins matrix every negative entry in the row of a candidate
> represents a pairwise loss for that candidate.
>
> The row of a Condorcet candidate will have no negative entry, but every
> other row will. In particular the margins row of the Gross Loser must have
> a negative entry ... so the Gross Loser cannot be a Condorcet Candidate ...
> therefore eliminating the Gross Loser (or any other candidate with a
> negative entry) cannot eliminate a Condorcet candidate.
>
> The average of all of the rows in the margins matrix is a row of zeros ...
> and therefore has a zero row summ. But the row of a Condorcet candidate
> has a positive row sum ... therefore above average.
>
> So you no longer have to take on blind faith the veracity of these
> assertions about the Condorcet efficiency of Borda Elimination and GLE....
> you can see for yourself the complete rationale!
>
> Next time you see Maskin ... alert him to the existence of GLE ... so he
> can break out of his rut of supporting clone depend methods ... like
> Copeland, Borda, Black, Nanson, Baldwin, etc.
>
> At least he hasn't yet recommended Kemeny-Young! ... (as far as I am aware)
>
> I have to believe that if he knew better, he would do better!
>
> -Forest
>
>
> On Fri, Jan 20, 2023, 7:54 PM robert bristow-johnson <
> rbj at audioimagination.com> wrote:
>
>>
>>
>> > On 01/20/2023 10:31 PM EST Bob Richard (lists) <
>> lists001 at robertjrichard.com> wrote:
>> >
>> >
>> > If Robert (or anyone else) can demonstrate that this is not Condorcet,
>> > they should let Foley know pronto. He is a Condorcet enthusiast.
>> >
>>
>> I am that. But I, also, am not trained in social choice theory or
>> psephology. I am trained in the mathematics surrounding digital signal
>> processing and my experience is DSP of audio and music signals (stuff like
>> guitar effects and music synthesis).
>>
>> I have trouble reading some of the math in other papers coming from the
>> likes of Maskin or Tideman or Schulze. Sometimes it's just the way it's
>> setup, but I am not as good at discrete mathematics as I am into the kinda
>> math that engineers and meatball physicists use (that comes outa calculus
>> and diff eq and linear system theory)
>>
>> > The method appears to be to eliminate the current Borda loser in each
>> > round, until one remaining candidate has a majority. Or something very
>> > close to that. Foley is a lawyer, not a social choice theorist, and his
>> > explanations are a little hard to follow.
>>
>> He got Maskin on his side. I almost got Maskin and Tideman to appear
>> (via Zoom) before the Vermont Senate Government Operations committee last
>> April but the chair of the committee *snubbed* them. Unbelievable.
>>
>> I am reading (or skimming) Foley's paper now and will probably send him
>> an email. His email address appears to be foley.33 at osu.edu . I would
>> hope that folks here better than me might pipe in. It appears that Foley
>> is unaware of Burlington 2009 and of Minneapolis 2021 (which had a cycle).
>> And since Maskin was certainly aware of Burlington 2009 (because of my
>> conversations with him), I am surprized that there is all this talk in the
>> paper of Alaska 2022 but none of Burlington 2009.
>>
>>
>> --
>>
>> r b-j . _ . _ . _ . _ rbj at audioimagination.com
>>
>> "Imagination is more important than knowledge."
>>
>> .
>> .
>> .
>> ----
>> Election-Methods mailing list - see https://electorama.com/em for list
>> info
>>
>
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