[EM] Warren: MDDA

[MIKE OSSIPOFF]

Warren--

You wrote:

The first rule of MDDA is
1. A candidate is disqualified if another candidate is ranked over him/her
by a majority of the voters.
   (Unless that rule would disqualify all the candidates, in which case no
one is disqualified.)

This is NOT a reduction to the Smith set, but rather, a reduction to the 
Condorcet
Winner or all candidates.

I reply:

No, for example there might be just one candidate who has a majority defeat, 
due to truncated ballots.

You continued:

Aside from that, what I said before goes,
and this is still bad (in fact arguably worse than a reduction to the  Smith
set) in the sense that it often leads to a lot of ties.

I reply:

No, there won't be ties in public elections, because it will be vanishingly 
rare for two or more undisqualified candidatess to be ranked on exactly the 
same number of ballots.

You continued:

So now we could discuss/invent yet another voting method, which is:

step 1.  reduce to Smith set.
step 2. the most-approved of the remaining candidates wins.

(The "Smith set" is the candidates S such that each candidate in S is 
pairwise
winner over each candidate not in S.  Different Smith - not me.)

Well - is this a better or worse voting method than the original MDDA?
Plausibly better...

I reply:

The method you refer to could be called "Smith//Approval". It fails FBC and 
SFC. It meets Smith, but I consider SFC better than Smith for the same 
reason why I said SFC is better than Condorcet's Criterion.

A good way to show the problem with Smith//Approval, BeatsAll//Approval, or 
DMC is with a truncation example. Those methods are extremely vulnerable to 
offensive truncation. In fact, of course their results are equally messed up 
by nonstrategic truncation.

Say, for simplicity, ranked candidates are counted as approved.

Sincere rankings:

40: ABC
25: BAC
35: CBA

B is the CW. A majority prefer B to A.

Actual rankings:

40: A
25: BA
35: CB

What happens in Smith//Approval?:

C beats B, B beats A and A beats C. The Smith set is {A,B,C}.

What are the Approval counts:

A: 65
B: 60
C: 35

A wins. The offensive truncation by the A voters stole the election from the 
CW, and the wishes of the sincere-voting majorilty are violated. This is an 
SFC violation because the winner is someone over whom they prefer the CW. 
The B and C voters are voting sincerely, as I define the term.

What happens with MDDA?

Only one candidate, A,  has a majorilty defeat.

Which of {B,C} has a better Approval score? B does. B wins.

Condorcet's Criterion is known to be incompatible with FBC. Anything that 
meets the Smith Criterion meets the Condorcet Criterion. Therefore 
Smith//Approval fails FBC.

Mike Ossipoff

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