[EM] Approval Stable Winner

[Forest Simmons]

Remember the "approval stable winner" is supposed to be the candidate that
stands the best chance of still being approved even when the polls and
pundits raise him/her up to be the target candidate to beat during the lead
up to voting night, as we saw in the lead up to Super  Tuesday, for example.

This is a class of methods in which the candidate X with the highest ratio
of S(X) over MPO(X) is elected, where MPO(X) is the Max Pairwise Opposition
against X, and S(X) is the estimated sincere zero informatinsupport for X.
It is this S(X) support estimates that distinguish the different methods or
versions of the method, if you will.

Why zero information?  Because we know from experience that the pollsters
and pundits have their agendas and biases that can turn "information" into
disinformation that is worse than zero information. The purpose of a
Designated Strategy Voting (DSV) style method is to trust the information
from the ballots themselves over the biased polls.  Voters can take the
polls with a grain of salt if they know that honest ballot information will
be used fairly.  There is no strategy free method, but if the DSV method is
trustworthy it will make better strategy decisions for the voters than the
pollsters and pundits will.

Here are some possibilities for S(X) in no particular order:

1. Let S(X) be the number of ballots that rank or rate X above bottom.

2. Let S(X) be the number of ballots that explicitly approve X.

3. Let S(X) be the number of ballots that rank or rate X equal top plus
half the number of ballots that rank X strictly between Top and Bottom.

4.  Assuming Range style ballots, let S(X) be the sum of the scores of X
over all of the ballots.

5. Let S(X) be the number of ballots on which X is rated at least as high
as the midrange of the possible ratings.

6. Assuming the ratings are in the range from zero to one, let S(X) be the
number of ballots on which the number of candidates rated strictly above X
is strictly greater than the total of all ratings on that ballot.

This S(X) is what I call the strategic zero information approval total, for
reasons that I have explained elsewhere, and I will sketch now for easy
reference:

It is a consequence of a general principle (the corner point principle)
of linear programing that there is always a corner point of the feasible
region of decisions where the linear objective function is optimized  when
that region has piecewise linear boundaries.

In range voting the feasible decision region for each voter is an
n-dimensional hypercube where n is the number of candidates.  A corner of
such a cube is a point where all of the candidates are voted at the
extremes, i.e. approval style strategy is optimal.

So here is the question we are faced with: how do we convert sincere
ratings into optimal approval ballots in a zero information setting?

It has often been observed that from a statistical point of view, if each
voter were to approve with probability p percent every candidate rated on
herr ballot p percent of the way between min range and max range, then
provided the electorate were sufficiently large, the range election outcome
would not be affected.

The question comes up ... what is the expected number of approvals on your
ballot if you were to use this method to convert your ballot from
fractional score o approval.  The answer from elementary probability theory
is very simple: it is merely the sum of the candidate ratings on your
ballot after they have been normalized between zero and one.

This result allows us to determine how many candidates to approve without
any need to flip coins or spin spinners.

The method has its determinacy restored after our brief excursion into a
Monte Carlo thought experiment.

If the sum of the normalized ratings rounds to n, then approve your top n
favorite candidates.

This is what I call "strategic zero information approval."

I have suggested six estimates for S(X).  Have I overlooked any goods
ones?  Any tweaks?  Other comments?

I should mention that the resulting methods based on any of these six
possible definitions of S(X) result in monotonic, clone free methiods that
satisfy Independence from Pareto Dominated Alternatives (IDPA) and the
Favorite Betrayal Criterion (FBC).

If you would rather trade in the FBC for the CC (Condorcet Criterion) you
can use "covering enhancement" on the S(X)/MPO(X) ratio order to "upgrade"
to an uncovered winner without sacrificing any of the mentioned criteria
except the FBC.

Thanks,

Forest
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