Tiebreaker Need

[Blake Cretney]

--

On Wed, 26 Aug 1998 12:06:26   Mike Ositoff wrote:
>
>To the statement that all pairwise methods need a random
>tiebreaker, I reply that all methods of any kind need one,
>because, whatever the method counts, it's possible for 2 or
>more canddidates to have the same score.
I think you're refering to my statement:
All purely pair-wise methods fail GITC.

Where I define a "purely pair-wise" method as any method that reaches a decision based purely on the pair-wise matrix, EVEN if that decision is partly random.

I don't mean this as a stinging denunciation of Condorcet.  All it means is that if a pair-wise method is to maintain GITC in the face of ties, it will have to be augmented with a non-pair-wise tie-breaker.  

So, picking a random ballot as a tie-breaker may allow GITC, because the combined method is no longer purely pair-wise.  However drawing lots betwen tied candidates does not affect the pair-wise status, and will cause a method to fail GITC.

>
>Unlikely in public elections, but some systems provide for
>drawing lots, letting the mayor break the tie etc. Fine, we'd
>leave that wording in when we propose our reform. The tie
>wording could remain unchanged. 
>
>Sure, in small committee elections, a tiebreaker is needed.
>
>A simple & decisive one is Repeated Plurality: 
>
>Pick the Plurality winner if there is one. If there isn't, then
>, from each ranking, give a vote to that ranking's 2nd choice.
>If that doesn't break the tie, use 3rd choices, etc.
>
>If that doesn't work, have a rule that the chairperson breaks
>the tie or that a randomizing device is used. For randomization,
>I like drawing a name from a bag, or flipping a coin repeatedly
>to get a random binary number, after assigning numbers from 1 to
>N to the candidates.
>
>Mike
>
I suspect Repeated Plurality fails GITC due to vote splitting.

Here's my method.
1.  Pick a random ballot and use its rankings, consider
ties as unsorted with regard to each other.
2.  Continue picking ballots.  When you find one that
orders previously unsorted candidates, use the ballot
to sort them.  Do not change the order of the already
sorted.
3.  If you go through all ballots, and some candidates
are still not sorted, order them randomly.

This provides you with a fully ranked tie-breaker ballot.  The simplest way to use this is every time you run into a tie during your decision procedure, you go to this ballot to resolve it.

This method is obviously decisive, and I believe it obeys GITC.  It looks like it would be difficult to carry out, but in fact, if the ballots are being counted by a computer, I suspect the tie-breaker ballot could be composed as the ballots are processed, so there would be no need to recount the ballots in the case of a tie.


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