[EM] MaxMinPA

[Forest Simmons]

The following method is based on score or range style ballots.  I believe
it satisfies the FBC, Plurality, the CD, Monotonicity, Participation,  Clone
Independence, and the IPDA.  It reduces to ordinary Approval when only the
extreme ratings are used for all candidates.



I call it MinMaxPairwiseApproval or MinMaxPA for short.



It is based on a concept of “pairwise approval.”



A zero to 100% cardinal ratings ballot contributes the following amount to
the “pairwise approval of candidate X relative to candidate Y”:



The amount is either …

100% if X is rated strictly above Y, or

Zero if X is rated strictly below Y, or

Their common rating if they are rated equally.



According to this definition, the ballot’s contribution to the pairwise
approval of X relative to itself is simply the ballot’s rating of X, since
it is rated equally with itself.



The method elects the candidate whose minimum pairwise approval (relative
to all candidates including self) is maximal.



The motivation for this idea is the question, “If candidates X and Y were
the only two candidates with any significant chance of winning the
election, what is the probability that the ratings ballot voter would want
X approved (in a Designated Strategy Voting system, say)?”



If the voter rated X over Y, this probability would be 100 percent.

If the voter rated Y over X, this probability would be zero.

If the voter rated both X and Y at 100 percent, this probability would be
100 percent.

If the voter rated them both at zero, she would want neither of the
approved.

If she rated them both at 50%, then our best guess is that there is a
fifty-fifty chance that she would approve X.

Etc.



Whatever nice properties the method has depends solely on its definition,
not the motivation for the definition, so please explore it with an open
mind.



Tomorrow, when I have more time, I’ll give some examples.


Enjoy,


Forest


P.S.


The rules can be modified for ranked preference ballots:


The amount (per ballot) of approval of X relative to Y  is either ...


100 percent if X is ranked ahead of Y or equal top with Y

zero if Y is ranked ahead of X or equal bottom with X

50 percent if both are ranked equally and strictly between top and bottom.

Smith//MaxMinPA may be a nice method that trades the FBC and possibly other
nice properties for the Condorcet Criterion.
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