[EM] DMTBR again: Voice of reason

Kristofer Munsterhjelm km_elmet at t-online.de
Thu Jun 30 09:44:22 PDT 2022

Monroe's voice of reason method is a method that Monroe proposed for its
ability to never elect a universally despised candidate in some
strategic equilibrium. I.e. it is immune to DH3-type outcomes.

The voice of reason method finds the ballot that agrees most often with
the pairwise relations (e.g. A is ranked over B on the ballot and A
beats B pairwise). It then returns this ballot as its social ordering,
so the winner must be someone's first preference, hence avoiding DH3.

(It's also possible to devise a Kemeny variant where a ballot's ranking
has a score equal to the sum of the number of voters on the winning side
for all pairwise relations that the ballot agrees with, e.g. in an
election with three candidates in an A>B>C>A cycle, the ballot A>B>C
scores the number of voters preferring A>B plus the number of voters
preferring B>C. Then the ballot with the highest score is selected as
the method's social ordering.)

So I thought its strategy resistance might imply DMT or DMT candidate
burial resistance that I could perhaps transplant into a Condorcet
method (the voice of reason doesn't pass Condorcet).

But to my surprise, no!

3: A>B>C>D
3: B>A>C>D
1: C>D>A>B

A is the Condorcet winner and DMT candidate. A beats everybody, B beats
everybody but A, C beats D, and D is the Condorcet loser, so the
Condorcet ordering is A>B>C>D.

The A>B>C>D ballot ranked by three voters is in perfect agreement with
the Condorcet order, so it is the voice of reason's social order and wins.

But now let the B-first faction bury A:

3: A>B>C>D
3: B>C>D>A	<-- burial
1: C>D>A>B

Then the pairwise victories are:
	A>B (4)
	B>C (6), B>D (6)
	C>A (4) C>D (7)
	D>A (4)

The ballot A>B>C>D agrees with A>B, B>C, B>D, and C>D: 4 of them.
The ballot B>C>D>A agrees with B>C, B>D, C>D, C>A, and D>A: 5 of them.
and the ballot C>D>A>B agrees with C>D, C>A, D>A, and A>B: 4 of them.

So B wins, hence the burial worked, hence there's a DMTCBR failure.

(The Kemeny variant scores the three ballot types 23 points, 27 points,
and 19 points respectively.)

I was surprised, but it's not so strange in retrospect. But it shows
that methods that are immune to DH3 don't have to pass DMTCBR. (Do there
exist methods that pass DMTCBR but can elect a universally despised
candidate with strategic voting? That, I don't know.)


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