[EM] The Stable Approval Potential Criterion (SAPC)

[Forest Simmons]

In a given election it may happen that there exists at least one candidate
X such that the number of ballots on which X is ranked top (or equal top)
is at least as great as the number of ballots on which  any other fixed
candidate Y is ranked strictly above X.  We say that any such candidate X
has Stable Approval Potential.

If a method always elects a candidate with Stable Approval Potential
whenever at least one such candidate exists, then we say the method
satisfies the Stable Approval Potential Criterion (SAPC).

Note that any candidate ranked top (or equal top) on a majority of the
ballots satisfies the SAPC, and if there is a Majority Candidate (ranked
unique top on more than half of the ballots) then no other candidate has
Stable approval Potential.  Therefore the satisfaction of the SAPC entails
satisfaction of the Majority Criterion.

Note also that standard Approval satisfies the SAPC.  In fact if X is the
Approval Winner in an election using approval ballots, and Y is some other
candidate, then the number of ballots on which Y is ranked above X is no
greater than Y's approval, which is no greater than X's approval.

Chris Benham and Kevin Venzke  deserve the credit for the ideas behind this
criterion.

Now I would like to suggest three methods that not only satisfy the SAPC
criterion, but have a built-in Bucklin like fall-back process that
continues until the modified ballot set has at least one SAP candidate.

 Note that Bucklin itself is an attempt to "fall back" until a kind of
"equal top majority approval winner" is found. But Bucklin may fail in this
because even when the fall-back collapse can no longer continue without
total collapse to one level, there may not be any candidate with fifty
percent approval.

But as we saw above, when there are only two levels left (if not sooner),
at least the approval winner will be an SAP candidate.

All three methods proceed by collapsing the top two levels until there is
at least one SAP candidate.  It helps to use cardinal ratings (i.e. score
or range style ballots) to guide the collapse. At each stage the equal top
counts and pairwise opposition counts are adjusted to reflect the fall-back
collapsed state.

Method 1.  Elect the SAP candidate who is rated top on the greatest number
of fall-back ballots.

Method 2.  Elect the SAP candidate whose max pairwise opposition is least
according to the fall-back ballots.

Method 3.  Elect the SAP candidate X with the greatest difference between
the number of collapsed ballots rating X top and X's max pairwise
opposition on the fall-back ballots.

Method one is basically a proposal of Chris Benham cast in different
language.

Method two is a natural way to bring  MMPO (Min Max Pairwise Opposition)
into compliance with the Plurality Criterion.

Method three is (imho) the method that chooses the candidate with the
greatest margin of approval stability potential, i.e. the candidate most
likely to be re-approved in case of a re-vote.

I will give an example supporting this assertion in another post.

For now let me just affirm that all three of these methods satisfy the FBC
(the Favorite Betrayal Criterion), as well as the Majority Criterion,
Monotonicity, and the same kind of marginal clone independence satisfied by
Approval and other versions of Range.

Also there is an adequate Chicken Dilemma defense; if the threatening
candidate B has k supporters, and the threatened candidate A has m>k
supporters, then the A supporters should announce (and stick to it) that
they are going to rank B on k ballots (and only ask in return that the B
supporters rank A on k ballots, as well).  In other words it is a kind of
tit for tat strategy. "We'll rank your guy on as many ballots as you could
possibly rank our guy.  If you don't, then you are responsible for the
election of our common foe. "

What do you think?

Forest
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