[EM] Another Approval-Condorcet compromise method

[Forest Simmons]

Martin,

Way to Go!

I think you have found a better way to use Dyadic Approval ballots than
anything I will ever come up with (unless we can find a good way to get
proportional representation out of them).

In fact, I think you may have cleared up a mystery for me.

Back ground of Mystery:

Remember my use of Dyadic Approval ballots for the purpose of simulating
Approval Runoff? It appealed to me because I didn't have to assign weights
to the various levels of inequalities as I do in my standard version of
Dyadic Approval. (Remember,on each ballot I just used the coarsest
inequality still separating the set of uneliminated candidates at each
stage of the runoff.) But I didn't like the fact that it was a non-matrix
method, i.e. it requires all of the ballots to be processed at each stage
of the runoff. So I started to wonder if it would converge to some matrix
method if iterated recursively, like runoff yields convergence to a
Condorcet winner (if there is one). 

Mystery:

What does Iterated Approval Runoff converge to after many iterations? 

I think that when there is a unique universal winner to your method that
is what the sequence of Iterated Approval Runoff winners will converge to
in a finite number of steps.

One big clue was this: If a runoff method converges under repeated
iteration, then the method to which it converges must obey at least some
weak version of Independence of Irrelevant Alternatives Criterion, since
once it has converged, the order of eliminations must more or less
stabilize.

Congratulations for your insight into the common nature of Approval and
Condorcet, and your discovery of what probably is the most natural common
generalization of the two methods. 

It may be too early for this, but I predict that in some enlightened
future world, "Harper's Method" will be the universal standard by which to
evaluate the performance of all other general purpose election methods.

Furthermore, it won't be impractical in public elections with half a dozen
or fewer candidates, because Dyadic Approval ballots are easy to vote if
the voting machine can interactively ask queries of the type, "Which
one(s) of THESE candidates would you support if THEY were the only ones
left in the election?" The voter responds by clicking check marks next to
some of THOSE candidates, and then hits ENTER.

If the voter doesn't change her mind and start over, the number of such
queries is typically one less than the number of candidates.

Congratulations!

(more later)

Forest