[EM] Tie-breaking for Ranked-Pairs and other methods

Eric Gorr ericgorr at cox.net
Tue Jan 14 15:14:01 PST 2003

At 2:12 PM -0500 1/14/03, Stephane Rouillon wrote:
>I do not say it is easy to compute,
>but your algorithm should try to minimize the
>sum of violated pairwise-victory within all
>the pairwise victories of the same strength.
>For this example it leads to lock all 4:2 victories
>except for D>A.

I assume you are talking about the statement:

    Keep a qualified tie defeat if it isn't in a cycle
    consisting only of itself and some combination of
    kept defeats and qualified defeats.

The way I have interpreted this statement was that it did not imply 
minimization. One of the 'some combination' would be all of the 
qualified tie defeats and the kept-defeats. Since that combination is 
in the set of all 'some combination' and since the wording is 'if it 
isn't in a cycle', they should all be thrown out.

A prior conversation with Mike seemed to confirm this conclusion.
      -- I wrote:
      If you had >= 2 qualified tie defeats, would you just test for a
      cycle with all of them added with the kept-defeats since that
      would be one combination of kept-defeats and qualified tie
      defeats? If there was a cycle, reject them all and if there
      wasn't a cycle, keep them all.
      -- Mike responded:
      True, I look for a cycle among the whole set of
      kept defeats and qualified defeats.

However, if I am wrong and it does imply minimization, then one runs 
into the additional problem (as you mentioned) of different 
combinations each consisting of the same number of defeats...there 
would be no good way to determine which combination to keep and which 
one to throw out. It is not clear that there would be a good way to 
make this determination other then random selection and I don't find 
that very compelling for an algorithm such as this (there is no good 
way to compute a genuine random number) as it should simply report a 
tie and allow the user to determine who won, whether by random 
selection or some other means.

== Eric Gorr ========= http://www.ericgorr.net ========= ICQ:9293199 ===
"Therefore the considerations of the intelligent always include both
benefit and harm." - Sun Tzu
== Insults, like violence, are the last refuge of the incompetent... ===

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