[EM] Proposal

[Forest Simmons]

Here's a proposal that was buried so far inside of another thread that I
wonder if the usual suspects had a fair chance at criticizing it:

(1) Voters rank the candidates.  They may duplicate ranks and they may
skip several ranks to reflect their feelings about gaps in worthiness for
office.  Unranked candidates are considered equally ranked one rank below
the lowest ranked candidate.

(2) After the ballots have been submitted, each candidate's weight is
initialized at the value one.

(3) For J from one to 100 ...

     (3a)  On each ballot using the current candidate weights
           find the weighted average of the ranks on that ballot.

     (3b)  Use this weighted average rank as an approval cutoff rank
           (for that ballot).

     (3c)  Tally over all ballots the candidate approvals using the
           current approval cutoffs.

     (3d)  Add one to the weight of the candidate with the greatest
           approval tally.

     (3e)  Subtract .01 from each candidate's weight.

    Next J

(4) Put marbles into a bag so that each candidate corresponds to a
different color of marble, and the number or marbles for a candidate is
that candidate's final weight (the number of times that candidate won the
approval count).

(5) Draw one of the hundred marbles at random from the bag.  The candidate
to whom that marble belongs is the winner.

To convert this to a fully deterministic but pseudo random method,
eliminate steps (4) and (5) and simply take the approval winner from the
last pass through the For loop.

It would be almost impossible to distinguish the candidate winning
probabilities of the two methods based on pre-election polls.

In fact, the method was designed to estimate the approval equilibrium
candidate winning probabilities: the candidate weights after the last pass
estimate those probabilities as percentages, i.e. the number of each
candidate's marbles divided by one hundred, estimates that candidate's
equilibrium approval winning probability.

Observe that in both versions a candidate has to win at least one approval
round before having any chance of being the ultimate winner.

Note that the deterministic version fails participation in one sense:
adding ballots favorable to the winner could change the value of J for
which this winner wins to J=99, and then some other candidate wins
on the 100th pass.

However, the total weight of the winner would not be decreased by the
added favorable ballots, so the his winning probability (in the
non-deterministic version) would not decrease, and hence the prior
probability of winning in the deterministic case would not decrease
either.

So the spirit of Participation is met: you don't decrease the prior
probability of your candidate's winning by participating.

Comments?  Questions?

Forest