[EM] Tie Breaking Version of Burial Resistant Method

Forest Simmons forest.simmons21 at gmail.com
Wed Dec 29 16:00:09 PST 2021

Let's say a candidate is at the "bottom" of a set of candidates if it is
not ranked above even one member of the set.

The basic method is to elect the candidate who on the most ballots pairwise
beats every bottom candidate.

Here's the procedure Proc(Beta) with built in tie breaker:

Beta is the set of ballots serving as input for this election procedure.
Let K be the set of candidates ranked by the ballots of Beta.

For each candidate X in K let n(X) be the number of ballots B in Beta on
which X pairwise beats every bottom candidate of B.

Let T be the set argmax(n(X)), i.e.the tied winning set. If T has only one
member, elect that member. Else if T=K, i.e. all candidates are in an exact
tie, then elect by random ballot from K. Otherwise, elect Proc(Beta(T)),
i.e. apply this procedure recursively to the set of ballots restricted to T.

Without the tiebreaking extension, this method is about as susceptible to
ties as Implicit Approval.

Simple random ballot would be adequate for this single winner method, but
to get an "order of finish" method based on the function X---->n(X) the
recursive version would be very appropriate .... by transforming  n(X) into
a polynomial in powers of epsilon with an additional term for each
recursive pass through the procedure.
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