[EM] Deterministic tie-breaking in a simple Condorcet election.

Daniel Carrera dcarrera at gmail.com
Mon Dec 20 12:15:06 PST 2021

On Mon, Dec 20, 2021 at 8:42 AM robert bristow-johnson <
rbj at audioimagination.com> wrote:

> In the language above, if two candidates are tied in a pairwise
> comparison, they are *both* marked as defeated and the plurality language
> in step (4) kicks in if the top two are tied.  I was wondering if a better
> tie breaker for a pair of candidates that otherwise beat every other
> candidate (except each other) would be to look at the *smallest* margin
> each candidate has with other candidates (the margin they have with each
> other is zero) and elect the candidate with the larger of the smallest
> margins.  If that ties, then go to the next smallest margin.
> Using the most simple language that might have a prayer of getting into
> legislation, what might be the best way of dealing with dead ties?  Really,
> the only tie that matters is the one of the "top two" candidates (whatever
> "top two" means in the Condorcet universe)

Starting from what you have, if you want to incorporate margins of defeat,
it's not too difficult to convert your text into Minimax:

   (3) If no candidate receives a majority of first preferences, a
Condorcet-consistent retabulation shall be performed by the presiding
election officer. The retabulation shall examine every possible pairing of
candidates. Given N as the number of candidates, then the number of
possible pairings of candidates is N(N-1)/2. For each possible pairing of
candidates, if the number of ballots marked ranking a selected candidate
over the other candidate does not exceed the number of ballots marked to
the contrary, then the selected candidate is designated as defeated***, and
the difference between these two numbers is the margin of defeat of the
selected candidate for that pairing. The margin of defeat can be zero***.
After all candidate pairs are examined, the candidate who remains not
designated as defeated is the Condorcet winner and is elected.

   (4) If no Condorcet winner exists in step (3), then ***the candidate
with the largest margin of defeat in any pairing is eliminated, in rounds,
until only one candidate remains. The remaining candidate is elected.***
So I added a sentence to (3) to save the margin of defeat, and that allows
me to easily code the Minimax rule in (4).

Dr. Daniel Carrera
Postdoctoral Research Associate
Iowa State University
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