[EM] monotonicity and summability criteria

[Daniel Bishop]

Russ Paielli wrote:

> I have an "off the wall" question that some of the math geniuses on 
> this list might find interesting.
>
> Before its recent modification, the ElectionMethods.org website had a 
> page called "Technical Evaluation of Election Methods." Two of the 
> criteria listed on that page were monotonicity and summability. Most 
> of you are familiar with the former, but the latter was my own idea. I 
> have cut some of the text from the definition of summability and 
> included it below for reference.
>
> It occurred to me a while back that the two criteria may be 
> equivalent. That is, if a method passes monotonicity, perhaps it must 
> also pass summability, and vice versa. That's just a hunch. Can anyone 
> prove (or disprove) it?

I can disprove it.

Let "Summable IRV" be the election method identical to IRV except that 
voters may only list their first 2 choices.  For n candidates, there are 
n² possible ballots, whose counts can be condensed into an nxn array.  
Therefore, Summable IRV is summable.

Consider the following vote counts with 3 candidates (A, B, C):

8:A>C
5:B>A
4:C>B

No candidate has a majority, so the plurality loser C gets eliminated.  
Then B wins.

But suppose that 2 of A's voters decide to rank A second instead of first.

6:A>C
2:C>A
5:B>A
4:C>B

This time, B gets eliminated, and A wins.

The two voters who *demoted* A on their ballots caused A to win, so 
Summable IRV fails monotonicity.

Therefore, there exists an election method that is summable but 
nonmonotonic.