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 nē 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.
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