Russ Paielli wrote:
Folks,
On the old "Technical Evaluation" page of ElectionMethods.org, I had a criterion that I called "summability," which I defined as follows:
"Each vote should map onto a summable array, where the summation operation is associative and commutative, and the winner should be determined from the array sum for all votes cast."
The point was that plurality, Approval, and Condorcet all pass but IRV fails. Well, after further consideration, I realized that IRV actually passes too -- it just needs a much larger "array."
Rather than putting an arbitrary size limit on the array...
"Summability" is still a very useful criterion. All you need is a more precise definition. I suggest:
* An election method is "k-summable" (or "passes the k-Summability Criterion") if there exists a constant c such that in any election with n candidates, the required size of the "array" is at most c*n^k.
* An election method is "non-summable" if there is no k for which it is k-summable.
For example:
1-summable methods: Plurality, Borda, Cardinal Ratings
2-summable methods: most Condorcet methods, Bucklin, plus all 1-summable methods
3-summable methods: Iterative Ranked Approval Voting*, plus all 1-summable and 2-summable methods
non-summable methods: IRV
That's interesting. I had thought of something like that, but I did't have the mathematical background to know the appropriate terminology.
Speaking of terminology, what do you think about "first-order summable," "second-order summable," etc.? They seem a bit more appropriate for formal writing.
As for the constant c in c*n^k, do you think it might as well just be 1? Or are you aware of methods that pass for some constant other than 1 but not 1? I don't see any point throwing in an arbitrary constant if it isn't needed.
--Russ
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