Goldfish (single-winner method)

[Blake Cretney]

--

On Thu, 13 Aug 1998 01:32:01   Mike Ositoff wrote:
>
>I don't know if Goldfish 0.2 is different from Tideman's original
>wording, which I've just received, and haven't completely
>figured out yet.
>
>When you eliminate the loser of a pairwise comparison, then,
>as far as that candidate is concerned, that's a lot like locking
>in his defeat.
>
>But the copying of his victories from his row into the row
>of the candidate who beats him seems different from Tideman,
>and original. The consequences aren't obvious to me without
>trying it out in examples.

After some thought, I think that original Tideman can be described in the same algorithm friendly way as Goldfish, and is superior to it in results.  If the following algorithm holds out as original Tideman, then I withdraw Goldfish.  Of course, there seems to be some debate about what original Tideman is.

Start by making a "victory" table.  For each row, enter the votes against each column's candidate, if the row's candidate wins pair-wise.  Otherwise enter a 0.
As well, we will mark every row as "active" initially, but some will be marked "passive" by the algorithm.

The best way to resolve ties is for a chairman, president, or random voter to enter a special ballot.  This ballot must not be truncated.

Repeat until only one row is marked "active"
    FIND:
    Find the highest value in active rows of the table. Call this cell i,j.  If more than one row share this value, choose the row that is higher in the special ballot.  
    MERGE:
    For each cell in the i row, if there is a higher value for that column in the j row, copy it over.  
Do not change the empty cells on the diagonal.
    MARK PASSIVE:
    Mark the j row as "passive".  Set cell (i,j) to be -1, so it doesn't get chosen again.

The copying of rows may seem different from Tideman, but I don't think it is.  In Tideman, when a victory is locked in,  the effect is that the pair-wise loser can no longer win over all, but can help the pair-wise winner.  The merge step accomplishes the same thing.

The difference between this algorithm and my Goldfish
algorithm is that in Goldfish, a candidate could only
be defeated once, and was eliminated.  Here, it is only marked passive, and may be defeated multiple times, as in Tideman.  The effect of this is that a decision between candidate X and Y is more likely to be determined by the majority preference between them, and less likely to be based on which of them manages to beat candidate Z by a greater amount.  This is why I now prefer original Tideman to Goldfish.

I'm interested to here what people think of using a tie-breaker ballot as described above.  I think that this allows GITC, and it seems to me the only way to accomplish GITC for all cases in a pairwise system.


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