[EM] Max Power Cardinal Ratings (MPCR)

[Forest Simmons]

A Cardinal Ratings style ballot is a ballot that allows the voter to rate
each candidate on some numerical scale, for example the Olympic scale of
zero to ten.

The standard use of such a ballot is to award the win to the candidate
with the highest average rating, where the average is taken over all
ballots.

It is well known that this standard CR voting method is strategically
equivalent to Approval, i.e. the intermediate ballot slots are superfluous
for the voter who knows how to maximize the influence of his/her ballot.

An approval voter in possession of accurate probabilities could maximize
the power or likelihood of his/her vote being pivotal (in a favorable
direction) by marking the ballot in such a way that the following two
conditions are satisfied:

(1) No unapproved candidate is preferred by the voter over any approved
candidate.

(2) The likelihood of an approved candidate winning is as close as
possible to the likelihood of an unapproved candidate winning.

Ideally, in a close race with many candidates, the odds would be
fifty/fifty, i.e.  equally likely that one of your approved (versus one of
your unapproved) candidates would win.

In the case of only two viable candidates it is easy to see that if those
two candidate were tied without your vote, then (until your decisive vote
was cast) the odds of one or the other winning would be exactly
fifty/fifty.  Lumping your preferred of the two with your approved and
relegating the other among the unapproved would satisfy the above two
conditions, and your ballot would tip the scales in a (according to you)
favorable direction.

In general, when these two conditions are satisfied there is a maximum
probability that your ballot will actually tip the scales from a candidate
who would have won had you not cast your ballot to one that you prefer.

Unfortunately, in most elections it isn't likely that any voter will
actually be in possession of the necessary probabilities.

But suppose that you could start with a Cardinal Ratings ballot and
gradually convert it into an Approval ballot as the candidate viabilities
became apparent during a step-by-step process.

In particular, suppose that at each stage of the process current CR scores
for the candidates were available to you, and that you were required to
make one step of progress towards your final approval ballot by
amalgamating two adjacent slots of your CR ballot.

[Successive steps of this type ensure automatic satisfaction of condition
(1) above.]

Since your goal is to end up with viability balanced as nearly as possible
between the approved and unapproved candidates, it would make sense to
absorb the current minimum viability slot into one of its neighbors. If
you absorb it (by amalgamation) into a neighboring slot with minimal
viability, then you (hueristically) minimize the chance of regretting that
they were not on opposite sides of the final ballot.

By the way, at each stage you judge the relative viability of candidates
by their CR scores at that stage, since that is the most up-to-date
estimate available, superseding even the most accurate pre-election polls.

This whole process can easily be automated so that you don't have to worry
about the calculations or ballot modifications.

The automated version is what I call Max Power Cardinal Ratings (MPCR).

The voters fill out CR ballots.  MPCR converts these ballots into approval
ballots according to the above process.  The candidate with the most
approval on these final ballots is the method winner.

Forest