[EM] Some thoughts on Smith and monotonicity

[Kristofer Munsterhjelm]

On 12/04/2017 11:48 AM, Markus Schulze wrote:
> Hallo,
>
>> Suppose X is a monotone method. Then Smith,X and
>> Smith//X are monotone methods.
>
> For example, Smith//Borda violates monotonicity
> because it can happen that, by ranking candidate A higher,
> some other candidate B is kicked out of the Smith set so
> that, after re-calculating the Borda scores, candidate A
> is worse off.

Whoops, sorry about that, and good catch.

Something about Borda's structure makes it hard to find a four candidate 
example of this (at least given the time I tried to do so), but here's 
one for Smith//Plurality:

1: A>B>C>D
1: A>B>D>C
2: B>D>C>A
3: C>A>B>D
2: D>A>B>C

Plurality scores: A: 2, B: 2, C: 3, D: 2, Smith set: everybody

Raise C on the BDCA ballots:

1: A>B>C>D
1: A>B>D>C
2: B>C>D>A
3: C>A>B>D
2: D>A>B>C

The Smith set is now {A, B, C} and the Plurality scores within that set 
are A: 4, B: 2, C: 3.

So C is raised and goes from winner to loser.

Also, Smith//Antiplurality:

1: A>B>C>D
1: A>B>D>C
1: B>A>C>D
2: B>D>C>A
1: C>A>D>B
1: D>C>A>B

Antiplurality scores: A: 5, B: 5, C: 6, D: 5, Smith set: everybody

Raise C on the BDCA ballots:

1: A>B>C>D
1: A>B>D>C
1: B>A>C>D
2: B>C>D>A
1: C>A>D>B
1: D>C>A>B

The Smith set is now {A, B, C} and Antiplurality scores within that set 
are A: 5, B: 5, C: 4.

Thanks :-)

Can we say anything about *when* Smith//X is monotone?