[EM] Clarifying Enhanced DMC (AKA SPARR Voting)

[Ted Stern]

Consider Enhanced DMC as defined in this message from Forest Simmons,
dated July 12, 2011:

http://old.nabble.com/-EM--Enhanced-DMC-td32048790.html

I prefer the name Strong Preference Approval Round Robin (SPARR),
following from the idea that this is a form of Condorcet (Instant
Round Robin) that looks for the highest-approved candidate who is most
strongly preferred, in some sense.

Here's my restatement of the algorithm:

Find P, the set of all candidates who are not defeated pairwise by any
other higher-approved candidates.  Number the p candidates in this set
in order of approval from lowest, X_1, to highest, X_p.

If there is only one candidate in P, the SPARR winner is that
candidate, X_1.

Otherwise, initialize the Strong set U to P.

Remove P-member-covered candidates from U:

   For i from 1 to p-1,
       For j from i+1 to p,
          If all of X_j's defeats are defeated by X_i, remove X_j from U.

When finished, the Strong set U contains only those members of P who are
uncovered by other members of P.

The Strong set U always has at least one member, the DMC winner (X_1), because
by definition X_1 can never be defeated pairwise by other members of P.  The
highest approved member of U is the SPARR winner.

***** End of algorithm

Motivation:

The motivation for the SPARR method is, as Forest stated 6 months ago,
that the winner should come from the set P of candidates who are not
defeated by higher-approved candidates, which includes the Approval
winner, but should not necessarily be the least-approved member of P
just because that candidate defeats all other P-set members pairwise.

If the Approval winner X were chosen, another P-set candidate Y could
have grounds to object if Y covers X.  "Hey, I defeat you pairwise,
and everyone else you beat too!  I'm a stronger candidate than you
are."

Therefore we consider as 'strong' members of P only those candidates
who are not covered by other members of P.

The highest-approved strong candidate is the SPARR winner.

***** End of motivation

Questions:

What happens if the SPARR winner X is covered by another candidate Y
*outside* the P set?  And not only that, but the *only* reason Y is
not in P is that Y is pairwise-defeated by another non-P-member with
higher approval.  So Forest's statements about Y being defeated by Z
in P would not apply.

Here's an example of that situation: A Smith set of 6 candidates,
lettered in descending order of approval as A through F, with
the P set = {A,B,C}.

Fifteen defeats:

A > D,
B > A, B > F                 B uncovered by C, => B is SPARR winner
C > A, C > B, C > D          C covers A, eliminating A from strong set
D > B, D > E
E > A, E > B, E > C, E > F   E covers B, defeated only by non-P-member D
F > A, F > C, F > D          B's beatpath to E goes through F

B is the highest approved uncovered P-member and is the SPARR winner.

I don't think this could be considered a version of Ranked Pairs,
because even if you affirm all the High-low defeats first, you still
can't eliminate the E > B defeat by first affirming B > F > D > B,
because D has higher approval than F.

Could there be a beatpath strength formulation that applies to SPARR?

Ted
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