[EM] Covering Pairs

[Forest Simmons]

A subset K of candidates covers the entire set of candidates kappa iff no
member of kappa-K (the complement of K) defeats every member of K.

Let W be the matrix whose j_th entry in row i is 1 or 0 depending on
whether or not candidate i is ranked ahead of candidate j on more ballots
than not.

This W is called the pairwise win matrix.

Let C be a set of columns of W. Then AND(C) is a vector in the column space
of W formed by applying the logical AND operator component-wise to the set
of columns.

When dealing with zeros and ones, AND is the same as both muliplication and
minimization: 1×1=1, 1×0=0, min(1,1)=1, min(1,0)=0.

It is easy to see that AND(C) is the zero vector iff the set of candidates
K contributing the set of columns C covers the entire set of candidates
kappa.

In particular, the j_th and k_th columns of W will have a dot product of
zero iff candidates j and k together cover kappa.

This fact is of practical importance because ... if kappa is the set of
candidates in a public election, there will practically always be a pair of
candidates that together cover kappa.

A single candidate covers kappa iff its column in W is the zero vector ...
which means it is unbeaten pairwise.

So not every ballot set has a Condorcet Winner, but every public election
has a covering pair ... with practical certainty ... no known public ballot
set has ever required more than two candidates to cover it ... and it's not
easy to devise a plausible theoretical one.

These considerations are the practical basis for the following method:

If there is no single covering candidate (i.e. Condorcet Candidate) ...
... then within 24 hours after the votes have been tallied, each candidate
must submit a proposed covering pair.

Any proposed pair that does not cover the candidates is thrown out.

The majority winner of each confirmed covering pair is then determined.

Finally, the winner with the most votes is elected.

That's it ... except ...  legally the theoretical possibility of
non-existence of a covering pair must be "covered".

In that vein I suggest including a clause that says ... if no candidate is
able to submit a valid covering pair, an additional 24 hours is allowed for
each candidate to propose a covering triple.

Among the respective IRV winners of the confirmed covering triples, elect
the one with the most transferred votes.

-Forest
-------------- next part --------------
An HTML attachment was scrubbed...
URL: <http://lists.electorama.com/pipermail/election-methods-electorama.com/attachments/20230309/5e477d5d/attachment.htm>