[EM] Smith,Minmax(margins) mono-add-top failure example

[Kevin Venzke]

Hi Kristofer,

Here's an attempt at a more concrete example:

A     3
ABCD  13
ACBD  1
ACDB  5
ADBC  5
BACD  16
B     3
BCDA  5
CDAB  20
DBCA  24
total 95

Margins (in order A-B A-C A-D B-C B-D C-D) are:
-1 -6 -6  40 -16  31

So final scores are: 6 16 40 31. A is not in the Smith set so B wins.

Now add 7 BA votes.

Margins are:
-8  1  1  47 -9  31

Final scores: 8 9 47 31. All candidates are in the Smith set, so A wins.

The pairwise matrices are:

 0  0  47  0  43  0  43  0
 48  1  0  0  66  1  38  1
 49  1  26  1  0  0  60  1
 49  1  54  1  29  1  0  0

 0  0  47  1  50  1  50  1
 55  1  0  0  73  1  45  1
 49  1  26  1  0  0  60  1
 49  1  54  1  29  1  0  0

Before and after respectively. The 0/1 after the vote counts indicates
the availability of a beatpath from row candidate to column candidate.

A does have two tied margins; I'm unsure if they make a difference, or
if it can be easily fixed if they do. But it doesn't seem like they
should make a difference (i.e. for checking Ext-Minmax) because those
margins are relatively weak.

At a glance it seems that this example must also work for
Smith//Minmax(margins) because A's contests are always the weakest, and
don't affect any Smith members' scores when A is outside the Smith set.
(Or to put it simply if you eliminate A in the first scenario, B will
still win because C/D do not get their scores from A.)

Let me know if I've made errors or there are any other thoughts.

Kevin