[EM] Ranked Rankings

[Forest Simmons]

Just as rankings allow you to order preferences without specifying a
numerical strength of preference, so ranked preferences allow one to order
the preference strengths without quantifying those strengths numerically
... for example the notation

A>B>>>C>>D>>>>E

makes clear that the strongest preference shown is D>>>>E and the weakest
is A>B, but the notation does not imply that the stronger of these is four
times as strong as the weaker.

Ranked rankings allow us to simulate multi-round Approval elimination, i.e.
to implement Instant Implicit Approval Loser Elimination (IIALE):

While more than two alternatives remain eliminate the one that is currently
(implicitly) approved on the fewest ballots.

An alternative X is currently (implicitly)  approved on a given ballot if
on that ballot there is another alternative Y such that X is ranked before
Y and there is no stronger (implied) rank relation remaining on the ballot
than the one between X and Y.

By convention no other (impliied) rank relation is stronger than the
truncation relation, "X trunc Y", meaning Y is truncated but X is not.

Example:

49 C
26 A>B [or even A >>> B]
25 B

A eliminated in the first round ... B wins.

49 C
26 A>>B>C
25 B

B eliminated in first round ... C wins.

49 C
26 A > B >> C
25 B

A eliminated, B wins.

Another possible method would calculate the Martin Harper Lottery at each
stage, and eliminate the alternatives tied for least probability.

Example:

51 A>C
49 B>C

Under IIALE alternative B is eliminated, then C is eliminated.

Under the Harper Lottery elimination, both A and B are eliminated in the
first round.

Any other ideas for using ranked ranks?

Remark: If the current definition of Universal Domain were expanded to
allow use of the ordinal information (about preference strength) provided
by ranked rankings, then .... ?

FWS
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