[EM] "summability"

[Daniel Bishop]

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

* The IRAV method uses ranked ballots to simulate multiple rounds of 
Approval Voting.  In each round, a ballot is counted as an approval vote 
for the voter's favorite of the top two candidates, and for all 
candidates the voter prefers over both of them.  Counting stops when the 
vote count reaches the same vote totals as an earlier round.  The votes 
can be represented as a 2-dimensional array of 1-dimensional arrays in 
which the indices are the two front runners and the element arrays are 
the number of votes for each candidate.