[EM] Criterion compliance of loser elimination and weighted positional methods

[Raph Frank]

On Sun, Feb 1, 2009 at 12:06 AM, Kristofer Munsterhjelm
<km-elmet at broadpark.no> wrote:
> The vote splitting would go like this: Say a Droop quota votes { A B C } in
> each permutation with equal probability, then a bunch of other candidates.
> The other voters vote the other candidates randomly before any of A B C.
> Then a method using Plurality with random elimination might end up
> eliminating one of A, B, or C, before it finds out that a Droop quota
> supports the set {A B C}. I don't know if this is actually possible, though.

If the coalition is solid, and A is the last candidate remaining, then
all of those voters will vote

A>(everyone else)

as the 2 other candidates have been eliminated.

A will now exceed the Droop quota and is immediately elected.

The same logic applies for B and C.  These eliminations don't have to
happen in sequential rounds.  Once then 2nd last candidate in the
coalition is eliminated, the last member automatically exceeds the
quota on the subsequent round.

> Strictly speaking, it's not that difficult to meet the DPC. Just make a
> DSC/DAC variant where you count all possible subsets. Elect the candidates
> that the DPC say must win, then do whatever you want after that. Such a
> method would be nearly useless in practice, since it would have a great
> discontinuity - it wouldn't elect near-DPC-eligible candidates any more
> often than it would elect candidates far from the DPC (unless the base
> method somehow had this property).

I am not sure how useless it would be.  I do think there could be
massive strategy issues though.

A faction would have to be careful that they vote their own candidates
as a bloc at the start.

> For STV, I think the reweighting matters in letting the method discover the
> Droop quotas. Again, I'm not sure. If elimination order doesn't matter, then
> one could make a DPC version of Random Ballot: pick the candidate/s above
> quota if there is/are any, then eliminate the candidate that ranks last on a
> random uninspected ballot. Afterwards, mark that ballot inspected. If the
> vote-splitting argument above works, elimination order *does* matter,
> though.

I don't think the order matters, so the above method would meet Droop.
 However, the candidate that actually represents the faction would be
decided by random ballot.

Depending on how you set up PR-STV methods, the intra faction election
is different.

With CPO-STV and Sculze-STV, I think the factional election ends up
being a condorcet method.

For example, if you were electing 2 seats and had the following votes

15: A1>A2>A3
10: A2>A1>A3
15: A3>A2>A1
30: C
15: B1>B2>B3
10: B2>B1>B3
15: B3>B2>B1

Standard PR-STV will go as

Round 1 and 2
A2 eliminated
B2 eliminated

Round 3 and 4
A3 eliminated
B3 eliminated

Results
A1: 35
C: 30
B1: 35

Thus B1 and A1 will win.

In effect, in each faction/party, the winner was selected by IRV.

However, using my suggestion to allow 2 different counting methods for
election and elimination, the results change.

The rule is

- equal ranks allowed
- election count -> vote divided equally
- elimination count -> vote given at true strength to all

Assume the voters decide to vote

15: A1>A2>A3
10: A2>A1>A3
15: A3=A2>A1
30: C
15: B1>B2>B3
10: B2>B1>B3
15: B3=B2>B1

(The x3 voters realise that they probably won't win anyway).

Round 1:

Election count
A1: 15
A2: 10 + 15/2 = 17.5
A3: 7.5
C: 30
B1: 15
B2: 10 + 15/2 = 17.5
B3: 7.5

=> nobody elected

Elimination count
A1: 15
A2: 10 + 15 = 25
A3: 15
C: 30
B1: 15
B2: 10 + 15 = 25
B3: 15

Hmm, 4 way tie :).  I will eliminate A1 and B3 (so both types of
elimination are considered).

Round 3:
Election count
A1: xx
A2: 10 + 15/2 + 15 = 32.5
A3: 7.5
C: 30
B1: 15
B2: 10 + 15 = 25
B3: xx

=> nobody elected

Elimination count
A1: xx
A2: 10 + 15 + 15= 40
A3: 15
C: 30
B1: 15
B2: 10 + 15 = 25
B3: xx

B1 eliminated

Round 4:

election count:
A1: xx
A2: 32.5
A3: 7.5
C: 30
B1: xx
B2: 40
B3: xx

B2 elected

Round 5
A3 eliminated

Round 6
A2 elected.

A2 and B2 are elected.  Thus the intra-faction election proceeded
using approval.