[EM] Description of "Median," my Approval variant

Wed Mar 5 02:29:02 PST 2003

Kevin Venzke
stepjak at yahoo.fr
3/5/03

I said in my first post that I was designing an
Approval variant aimed at more often producing the
Condorcet winner.  Here it is.  I hope it's of some
interest, that it's novel, and that I don't come
across as too pretentious.

"MEDIAN" ELECTION METHOD

This is a proposal for a single-winner election method
I call "Median."  Votes are cast as under Approval,
and a method, justified by a geometry analogy, is used
to determine the "median candidate."  More precisely,
this is the candidate most likely to be supported by
the "median voter" as drawn on a political spectrum. 
The number of dimensions of the spectrum is
unimportant, and the method will not be broken if
there is in fact no spectrum behind the voters'
choices.

METHOD

The voter votes for as many candidates as he wishes. 
The total number of voters "approving" each candidate
is counted, as under Approval.  However, a
two-dimensional table has also to be maintained,
recording the number of voters that approved each pair
of candidates.

For instance, if the ballots read:
12: AB
15: BC
10: AD
9: D

After counting the votes, we would have counted not
just that A, B, C, and D's total votes received were
22, 27, 15, and 19 respectively.  We would also have
maintained the following table of the "overlap" of
support between each pair of candidates:

   A   B   C   D
A  .
B  12  .
C  0   15  .
D  10  0   0   .

Next, for each of the candidates, we determine which
of the other candidates had the most supporters who
did not also vote for the candidate we're talking
about.  This is why the overlap information is needed:
If we are determining A's supporters' disapproval for
B, for example, we must subtract the A-B overlap from
A's vote count.  For the above scenario, we could make
the following table.  Each row is the disapproval for
a candidate, while each column is the disapproval
expressed by a candidate's supporters.

   A   B   C   D     greatest
A  0   15  15  9     15 (from B or C)
B  10  0   0   19    19 (from D)
C  22  12  0   19    22 (from A)
D  12  27  15  0     27 (from B)

The "greatest" column is what matters.  It tells us
the "greatest constructive disapproval" for each
candidate.  In other words, for candidate A, there is
no group of voters larger than 15 people who did not
vote for A and who were agreed on a different
candidate.  Since 15 is the smallest "GCD," A is
judged to be the median candidate and would, by this
method, be the winner.  (A tie could be broken simply
by number of votes received.)

You can also imagine the method this way: If you
removed all of the ballots approving candidate X, how
many votes would the top-scoring candidate then have? 
Elect X if the answer is the smallest.

(You wouldn't have to use this method alone.  You
could elect the candidate who maximizes (votes
received / GCD).  Approval elects B, of course.  If
you do this division, A gets 1.467 while B gets 1.42. 
A would still win, but it's much closer.)

I won't consider the question of whether the "median
candidate," however defined, should be the winner.  I
will only use a geometrical analogy to argue that the
candidate with the smallest GCD is probably the median
candidate.

Consider the following very simple scenario:
26: A
25: AB
25: BC
27: C

Clearly we have a left, right, and center candidate. 
At least, it's possible to draw it that way.  The
Approval scores are 51 A, 50 B, 52 C, with C winning. 
Under Median, the scores are 52 A, 27 B, 51 C, with B
winning.  The spectrum basically looks like this, with
B obviously being the median:
AAAAAAAA
    BBBBBBBB
        CCCCCCCC

Here is the analogy: Think of the candidates as points
on a line or plane, and the "constructive disapproval"
of (for instance) candidate A's supporters towards
candidate B, as the distance (measured in voters) that
B would need to "travel" in order to win over A's
supporters.  (Note that the distance is not
reciprocal.  If it were, then two-candidate elections
would always be ties.)

We can think of the "greatest constructive
disapproval" for a given candidate as the greatest
distance he would have to travel in one direction to
obtain the approval of all the voters.  Candidates A
and C above would each have to win over 50 voters in a
single direction, while B is only 27 voters away from
the furthest candidate.  It's unnecessary to be able
to draw the candidates on a line or plane precisely. 
It's intuitive to suppose that the point with the
least greatest distance to another point, is the one
which is closest to the center.

STRATEGY

I think that it would be difficult to vote
strategically under Median.  This is because votes
aren't counted directly.  For instance, in order to
"hurt" a certain candidate, it's not sufficient to not
vote for him.  You need to vote for a candidate who is
likely to provide the GCD against him.  In other
words, if you want to "hurt" your second-favorite who
is similar to your favorite, you're probably out of
luck since the latter's supporters are probably not
the largest bloc of constructive disapproval against
the candidate you don't like.  A candidate's fate is
essentially decided by the voters who dislike him
sincerely.

Unlike in Approval, in Median there is reason for
centrist voters to vote for second-favorite,
off-center candidates in addition to their favorites. 
So doing can hurt a least favored candidate.  It can
never cause the favorite to lose, unless the favorite
is otherwise the "most distant" candidate from the
second-favorite.  If anyone is interested, I have a
scenario where if some centrist B voters had also
approved off-center C, then B would have been judged
the median instead of opposite off-center A.

FLUKES

Consider this:
3: A
1: AB
2: BC
3: C

Approval scores are A 4, B 3, C 5.  Median scores are
A 5, B 3, C 4.  B is the winner even though C has
majority approval.  Defensible?  Is it necessary to
claim that B was everyone's second choice, at worst? 
Or is "median voter's candidate" a valid criterion on
its own?

Two of the five C voters said B was acceptable.  If C
wins, 4 B supporters are unhappy.  If B wins, 6 people
are unhappy, but they're divided 3 and 3 on who
would've been better.  You could say that if B wins,
no more than 3 people will attend the same protest. 
You could say that only 3 people are alienated by B in
the same way.  Possibly one could claim that it's
impractical to give power to whichever majority
manages to assemble itself for a given election.

And consider this disaster:
1000: A
1000: B
1: AC
1: BC

Approval scores are A 1001, B 1001, C 2.  Median
scores are A 1001, B 1001, C 1000.  C wins with only
two votes.  Of course, this is not very likely; A and
B have to be exactly tied for C to win like this. 
Specifically, for every A-only voter we add, C needs
one more voter overlap with A in order to still win. 
Additional overlap with B will not help C in this
case.  Example:

1050: A
1: AC
1001: BC
Median scores are A 1001, B 1051, C 1050.  A is deemed
to be the pick of the median voter.

I'm interested in comments and criticism, especially
regarding philosophy, voter strategy, and election
method criteria.  I suspect that Median meets similar
criteria to Approval, although Median doesn't meet
Independence from Irrelevant Alternatives.  It is very
possible for a non-winning candidate to be
"kingmaker."  I would actually propose that, assuming
perfect information, the more candidates there are,
the "better" the result will be.

Thanks to any who may read this.

Kevin Venzke
Stepjak

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