[EM] "summability"

[Russ Paielli]

Daniel Bishop dbishop-at-neo.tamu.edu |EMlist| wrote:
> 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