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

[Venzke Kevin]

Forest,

Thanks much for your reply.  I had to read it slowly
and I learned some math terms.

First, below is the drawing of my first example. 
Although B is Beats-All, I think an A victory is
defensible.  Under Condorcet, A could easily win,
provided that the D supporters prefer A to B, and 5 AB
voters prefer A.  The practical explanation for why A
beats B under Median is that B was more disagreeable
to D supporters than A was to CB voters, measured of
course in number of voters.  ...And of course if
Beats-All had to be elected, people would use Approval
strategy when voting.

DDDDDDDDDDDDDDDDDDD
         AAAAAAAAAAAAAAAAAAAAAA
                   BBBBBBBBBBBBBBBBBBBBBBBBBBB
                               CCCCCCCCCCCCCCC

Looking at some of your proposals for picking the
winner:

>(1) Take the team that gave up the fewest points per
game on 
>average (i.e. had the fewest points scored against
them per 
>game on average). This might be considered the team
with the 
>best defense. 
>(5) Take the team that scored the highest point
average per game. 
>(6) Take the team with the maximal minimum number of
points per game. 
>In other words, if team A scored at least 25 points
in every game, and 
>every other team scored fewer than 25 points in at
least one of their 
>games, then A should be the winner according to this
criterion. etc. 

Using these would influence the number of candidates. 
Under (1), it's best for candidates to run alongside
many clones.  Under (5) or (6), candidates are
punished for having similar support bases, like under
Plurality.

>(3) Take the team for which the maximum margin
against them was 
>minimal. This is analogous to MinMax (margins). If
there is a beats 
>all team, this method will pick it. 
>(4) Take the team for which the maximum score against
them was 
>minimal. This is analogous to MinMax (winning votes),
and also will 
>detect a beats all winner. All of these methods pick
teams based on 
>measurements of defensive prowess. But what about
offensive prowess? 

I'm a little confused.  Will these pick different
winners from Approval?

>This combination method (8) would take care of the
anomalous results 
>at the end of your message. Note for example, that
disaster associated 
>with 1000: A, 1000: B, 1: AC, and 1: BC is avoided by
taking into 
>account the offensive prowess of candidates A and B
in comparison to C 
>who is relatively strong defensively but weak
offensively. 

So you're saying that you'd compute the same
(pairwise, final) table:
     A     B     C
A    0  1001     1
B 1001     0     1
C 1000  1000     0

And the scores become (1001 - 1000) or 1 for A and B
each, but (1001 - 1) or 1000 for C.  So this is an A-B
tie.

But, there is something artificial in that in finding
the "least disapproval expressed" down each column, we
must ignore the reflexive 0 value.  And that leads to
why candidates are punished for being similar to
others.  Imagine the modified scenario:

1000: A   1: AC   1: BC   500: B   500 BD.  (I.e.,
half of B's supporters also like new candidate D.)

     A     B     C     D
A    0   1001    1    500
B  1001    0     1     0
C  1000  1000    0    500
D  1001   501    2     0

Median still gives it to C, unaffected by D's
entrance.  But using method (8) as you describe, the
scores are A 1, B 500, C 999, D 501.  Yes, C is now
crushed, but so is B, unjustifiably.  B is literally
punished for not having the exclusive support of the
BD voters.

But what do you think about "factoring in" the raw
Approval score, as the measurement of offensive
strength?  I suggested doing (Approval score / Median
score) and electing the candidate with the greatest
result.  These scores are ugly, though, and I usually
don't like the results as much.  There is more
incentive to use Approval-style strategy in voting.

A nice thing about Median's final scores is that you
can imagine what they represent: "the largest group of
voters who would oppose the given candidate, and who
are agreed on an alternative."

> Also, I believe that your median method and the 
>modified median method both satisfy the FBC.

Considering what I wrote above, it seems to me that
the "modified median" method (8) wouldn't satisfy FBC.
 I could misunderstand the issue, though.  The BD
voters get an inferior result than if they had all
just voted B.  (Well, if you remove the AC voter,
let's say, to eliminate the tie.)

Thanks much for your thoughts!

Kevin Venzke
stepjak at yahoo.fr


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