[EM] Definition attempt at "qualitatively partly wasted"

[Kristofer Munsterhjelm]

Here's a suggestion for defining when some ballots are "qualitatively
partly wasted" that I wrote some time ago but didn't get around to post.

For a set of winners, let a voter's individual winner be the one he
ranked or rated highest of those in the set. If he ranked every winner
last, then he has no individual winner.

E.g. if the winners are A and B and the voter votes
C>A>B>D
then his individual winner is A.

If the winners are A and B and the voter votes
C>D
then he has no individual winner.

====

For two sets of winners, say that a voter individually prefers the
first set if he ranks the individual winner based on the former set
higher than the individual winner based on the latter set, or if he has
an individual winner in the former but not in the latter.

For instance, if the vote is, as before,
C>A>B>D
then the voter individually prefers the winning set "A and B" to "B and
C", but has no opinion either way about "A and B" versus "A and D".
(This because no matter whether the outcome is "A and B" or "A and D",
that voter's weight would go to A.)

If the vote is
C>D
then the voter individually prefers the winning set "B and C" to "A and
B" because he has an individual winner in the former case (C), but not
in the latter.

====

Say that for an election and two sets of winners, the former set of
winners improves upon the latter if
- no voter individually prefers the latter set to the former, and
- at least one voter individually prefers the former set to the latter.

E.g. for the election
10: A>B
10: C>D

the winner set "A and D" improves upon "B and D" (because the A>B voters
individually prefer "A and D" and nobody individually prefers "B and D");
the winner set "C and B" improves upon "B and D" (same reason, but with
the C>D voters);
the winner set "A and C" improves upon every other set
(it improves upon "A and D" because the C-voters prefer "A and C" and
the A-voters don't mind, and it improves upon "C and B" because the
A-voters prefer "A and C" and the C-voters don't mind).

Let a winner set be maximal if no set improves upon it. (A and C is an
example above). If the method picks a maximal set (there may be more
than one), then it doesn't qualitatively waste any votes, otherwise it does.

And then an example for

10: A
11: C>D
12: A>B>C>D

again. I'm going to use the following notation: {X, Y} means the case
where the winners are X and Y.

{A, B} can be improved upon by {A, C} because the C-voters individually
prefer {A, C} and neither of the two other blocs prefer {A, B}. Thus {A,
B} qualitatively wastes 11 votes. (It doesn't waste the 12 because these
would all be assigned to A. In effect, if {A, B} were elected, B would
have weight zero and A would have weight 100%, in effect being a dictator.)

[A, C} can't be improved by any other set, so it is the one set of
winners that doesn't qualitatively waste any votes.

{A, D} can be improved upon by {A, C} (because the C-voters prefer {A,
C} and the others don't mind). Thus {A, D} qualitatively wastes at least
11 votes.

{B, C} can be improved upon by {A, C} (but not by {A, B} because the
C-voters would object to that). It thus qualitatively wastes 10+12 votes.

{B, D} can be improved upon by {B, C} and {A, C}. It qualitatively
wastes the whole ballot set (10+11+12 votes).