Mike O.,
You wrote (Sun.May29):
I don't understand the difference between CDTT,IRV and CDTT//IRV.
With the // version, first the candidates that are not in the CDTT are
identified and then they are dropped from the ballots and the IRV count
is carried out as if though those eliminated candidates never existed.
In the comma version, IRV is applied to the *original* ballots to order
the candidates, and then the highest candidate in this order which is in
the CDTT is elected.
Or equivalently, first the members of the CDTT are identified as the
only candidates that are allowed to win. Then an IRV count proceeds on
the original ballots until all-but-one of the CDTT have been
eliminated. The one remaining CDTT member is elected.
So I don't know exactly what the CDTT is, except that it's a selection
set.
The CDTT is the set of candidates that each have a majority-strength
beatpath to any and all candidates that have a majority-strength
beatpath to them. It includes all candidates with no majority-strength
pairwise defeats.
BC:
No one should win who has a pairwise defeat that isn't the weakest
defeat in some cycle. (The strength of B's defeat by A is the number
of people voting A over B).
Beatpath Criterion (BP) on the other hand makes no mention of
majority-strength beatpaths, and seems to just be the
Defeat-Dropper(Winning Votes) method in the form of a criterion
(certainly in the three candidate case).
49: A
24: B
27: C>B
A>C 49-27
C>B 27>24
B>A 51>49
BC says that only B can win. C can't win according to BC because C's
pairwise defeat isn't the weakest in this cycle.
On the other hand the CDTT is {BC}. A majority-strength beatpath is one
in which each step/link is a majority-strength defeat.
B>A>C, but the A>C link is not majority-strength, so B has no
majority-strength beatpath to C. A is out because B does have one to A,
and A doesn't have one back to B.
CDTT,IRV (like all the other plausible deterministic CDTT methods) in
this election elects C; and so it fails the Beatpath Criterion (BC).
The point of CDTT,IRV versus CDTT//IRV is that the latter and not the
former fails Mono-add-Plump and probably also Mono-append.
This demonstration is adapted from one from Douglas Woodall that applies
to CNTT//IRV versus CNTT,IRV.
18: A>B>C>D
14: B>C>D>A
05: C
10: D>C>A>B
47 ballots
against
A B C D
for A
28
18
18
B 14
32
32
C 29
15
37
D 24
10
10
Candidates A,B,C are in a majority-strength cycle (A>B>C>A) and D
majority-strength pairwise beats A and so has majority-strength
beatpaths to each of B (D>A>B) and C (D>A>B>C).
So here all the candidates are in the CDTT and both versions are equivalent.
FPs: A18, B14, C5, D10. IRV eliminates C and then D, and A wins.
Now say four ballots that bullet-vote ("plump for")A are added.
04: A
18: A>B>C>D
14: B>C>D>A
05: C
10: D>C>A>B
51 ballots
against
A B C D
for A
32
22
22
B 14
32
32
C 29
15
37
D 24
10
10
The effect of this is to raise the majority threshold from 24 to 26, so
D's pairwise defeat of A is no longer majority-strength and
so D no longer has any majority-strength beatpaths to any of the
candidates that have them to D; so D drops out of the CDTT.
CDTT//IRV eliminates D and then proceeds with the IRV count as if D had
never existed.
FPs: A22, B14, C15. IRV eliminates B, and now C wins. Adding
bullet-votes for A changed the winner from A to C, violating Mono-add-Plump!
CDTT,IRV bars D from being allowed to win, but proceeds with the IRV
count on the original ballots (without first dropping D from the
ballots) and so A wins as before.
I hope that is all clear.
Chris Benham
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