[EM] Fwd: Coombs. Elimination Examination.

Forest Simmons forest.simmons21 at gmail.com
Fri Sep 29 13:31:37 PDT 2023

Ironically, because Coombs is maximally vulnerable to burial, Coombs with
take-down is maximally punitive to buriers.

[If Coombs buriers had a 100 percent chance of success, then the take-down
version would be 100 percent burial proof ... which is clearly impossible
because of the double interpretation examples.]

In fact, the minute the buried candidate is eliminated, it takes down with
it all candidates defeated by it ... including the one that engineered the

You cannot beat that for burial resistance!

An equivalent formulation of Coombs is ...

Repeatedly eliminate the lowest implicit approval candidate, until only one

The max burial proof version is ...

Repeatedly eliminate the lowest implicit approval candidate along with all
candidates beaten by it, until only one remains.

[Stop here unless you want some advanced stuff]

If Implicit Approval is too messy for you, take advantage of the fact that
the lowest Implicit Approval candidate is for all practical purposes the
same as the candidate whose max pairwise support is the smallest ... which
is either the Condorcet Loser or the candidate whose best pairwise victory
(over another candidate) has the fewest winning votes.

All of this can be derived from the pairwise support matrix
m(j,k) = |j>k|
[Kristofer's notation] ...
thus obviating the need for consulting the ballots between elimination

Since Coombs with take down works by back-firing on the buriers, it does
not elect the buried candidate which is the one that would have won under
sincere votes.

But this buried candidate can easily be recovered: Simply elect the
head-head winner of a final two candidate runoff between the uneliminated
candidate a d the last pivot ... the last candidate to take down everybody
defeated by it.

It failed to eliminate the uneliminated candidate only because the
uneliminated candidate was the bus under which it was buried.  A sincere
final runoff corrects that mistake.


Every method has many equivalent formulations ... different procedures for
arriving at the same winner given the same ballot profile ... as well as
different ways of understanding  its most important advantages
andisadvantages when compared and contrasted with other methods

That said, ...I doubt that there is any better results practical method for
public proposal than some version of Coombs take-down ... though of course
the exact formulation details and language ... not to mention heuristic
explanation and presentation, have a long way to go!

The most important feature of an election method based on ordinal ballots
... and of the of the form ...
"Lacking an undefeated candidate ..." is to make it backfire on the faction
that subverted the sincere CW.

The next most important thing is to restore the Sincere CW by sincere

Sincere cycles are so rare in comparison with to sincere CW's that they
should always be treated as guilty until proven innocent by a sincere
runoff of the kind I mentioned above ... or similar.

This is why, for example, that simple Coombs with take-down (even without
the sincere runoff) is vastly superior to  Ranked Pairs as a public
proposal imho ;-)


On Wed, Sep 27, 2023, 6:52 AM Kristofer Munsterhjelm <km_elmet at t-online.de>

> On 9/25/23 20:41, Michael Ossipoff wrote:
> > Some academic authors have high praise for Coombs. One say that, with
> > sincere ranking, & fewer than 5 candidates, Coombs always elects the CW.
> That can't be right:
> 1: A>C>B>D
> 1: B>C>A>D
> 2: A>D>B>C
> 1: A>C>D>B
> 3: B>C>D>A
> 1: D>A>C>B
> A is the CW but has highest last preference count and so is eliminated
> first.
> However, it *is* true for three candidates.
> > But Coombs is obviously vulnerable to east burial strategy. In
> > particular, trust & betrayal perpetrated by the voters of a “
> lesser”-evil.
> Clearly then, knowing this fact, the voters who propose and enact Coombs
> must be tough voters who would never ever bury. Therefore Coombs' burial
> incentive is no problem wherever it would be proposed.
> I jest :-)
> My real point is "beware arguments that are too powerful; they might
> just lead to absurdity".
> Amusingly, according to James Green-Armytage, if we don't care about
> strategic susceptibility, then Coombs has a better VSE than IRV. So if
> we can simply discard strategy by the (too general) argument that
> strategy will never come into play, it's Coombs, not IRV, that ought to
> be enacted.
> -km
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