[EM] Approval Stable Winner

[Forest Simmons]

Here's an important tweak: The version of MaxPairwiseOpposition that we use
needs to  include power truncation, otherwise we fail the Plurality
Criterion.

In this context power truncation means a candidate truncated on a ballot is
counted as "opposed" by every other candidate on the ballot, even the other
truncated candidates

With this power truncation in place every method (one thru six) of my
previous message is equivalent to ordinary Approval when the ballots
populate only the top and bottom slots ... no in-between preferences
indicated.

As you know the same can be said for all Condorcet Compliant methods;when
there are no intermediate rankings there is always (except when the two
candidates with the most top votes have the same number of top votes) a
pairwise beats-all candidate, and that candidate is the one that is ranked
top or equal top on the greatest number of ballots.

Another possible tweak to make the method more symmetrical is to throw in
Symmetric Completion for the intermediate ranks (between top and bottom).
Put this version with method number three for measuring support S(X), and
we have the (IMHO) best zero information method of this type for ranked
preference ballots (equal rankings and truncation allowed).

To summarize this version, the ballots are ranked preference style with
truncations and equal rankings allowed. The winner is the candidate X with
the greatest ratio of S(X) to MPO(X), where S(X) is the number of ballots
on which X is ranked equal top plus half the number of ballots on which X
is ranked strictly between bottom and top, and MPO(X) is the Max Pairwise
Opposition to X with Symmetric Completion below top and double that for
bottom.

In other words, let PO(X, Y)) be the number of ballots on which Y is ranked
above X, plus half the number of ballots on which Y is ranked equal to X
plus the number of ballots on which neither X nor Y is ranked at all.  Then
MPO(X) is the max (over Y) of PO(X,Y).

As stated before this method satisfies mono-raise, mono-add, clone winner.
clone loser, Independence from Pareto Dominated Alternatives, and the
Favorite Betrayal Criterion.  And like all good EM List methods, it reduces
to Approval when the voters vote some candidates equal top and truncate the
rest.

Also the information needed for computing the winner is capable of
efficient encoding, ballot-by-ballot in an additive form.

What's not to like?

As usual, in practice this version or any of the other versions can be fit
into a Candidate Proxy or other VPR (Vote for Published Ranking) framework
for the convenience of the voters.

In my next message I will talk about the biggest obstacle to cardinal
ratings... how to rate the candidates in the zero info setting, i.e.
sincere ratings without any information about the preferences of other
voters.  What does sincere rating even mean?



On Fri, Mar 6, 2020 at 2:52 PM Forest Simmons <fsimmons at pcc.edu> wrote:

> 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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