[EM] Completion methods for Smith Sets

[Buddha Buck]

At 08:23 AM 06-18-2001 -0700, you wrote:
>I just wanted to respond to Craig before asking about Borda, Nanson, and 
>"Crosscut" methods in my next post (I'm going to call "my" method the 
>Crosscut method -- at least until I find out who was the original inventor 
>-- since it calculates from the top down and from the bottom up, then 
>tie-breaks any discrepancies)
>
>At 05:07 PM 6/18/2001 +1000, you wrote:
>>Thanks.  It's very clear.  The combined method might have some good
>>properties, although it would need a strong argument to convince people it's
>>better than some other Condorcet complaint methods.  It's big advantage is
>>that it's a good alternative/compromise.  In explaination, it sounds like an
>>IRV variation, so it might appeal to the "some preferences are more equal
>>than others" crowd, while having most of the positive properties of a
>>pairwise count method.
>
>Sounds like an IRV variation? AN IRV VARIATION?!?!? You take that back!

Well, he might, but I'm not ;-).

To me, while it does satisfy the Condorcet Criterion, it has some of the 
same drawbacks as IRV, although on a smaller scale.

Here's my understanding of  your technique (unfortunately, I'm at work, and 
my copy of your original message describing it is at home, so this is from 
memory):

A.  If there is a Condorcet winner, select it.
B.  If there isn't a Condorcet winner, strike[1] all candidates not in the 
Smith Set from the ballots
C.  Rank the remaining N candidates by:
    C1: selecting the plurality winner,
    C2: selecting the plurality winner counting ranks 1-2 as identical, 
other than already selected candidates
     ...
    CN-1: selecting the plurality winner counting ranks 1-N-1 as identical, 
other than already selected candidates
    CN: select the remaining candidate.
D. Strike the Nth candidate on the ranking found in step C from the ballots.
E: If there is more than one candidate remaining, go to step C.

[1] By 'strike from the ballots', I mean to remove an option from the 
ballots, and readjust the rankings from there.  If a ballot ranked A first, 
B second, and C third, striking A would result in a ballot that ranks B 
first and C second.

The problem is that step C requires examining every ballot at least 
once.  If this is a public election, that could require examining and 
recounting possibly millions of ballots.  With large numbers of candidates, 
the number of possible rankings grows hyper-exponentially, and even 13 
candidates under consideration would yield more possible rankings than the 
population of the Earth.  There is no way to represent ranking data in the 
aggregate; re-examining the entire collection of ballots is the only way to go.

This is a major logistical nightmare, although it isn't as bad in your 
proposal as it is in IRV, since you immediately restrict the field to just 
the Smith Set.

I've not seen many IRV supporters mention this issue, but it isn't one to 
forget.