[EM] PR solutions

[Juho Laatu]

On 8.6.2012, at 2.07, Michael Ossipoff wrote:

> But what I don't understand, Juho, is why you say that Largest-Remainder's
> paradoxes are desirable. Saying that the paradoxes are acceptable,
> forgivable, for STV, or even for Largest Remainder (when one is advocating
> it for alleged simplicity) is one thing. Saying that the paradoxes are
> desirable is quite another thing.

Some of the paradoxes are surprising mathematical properties. Cyclic majority opinions are one such example. Even if we assume that personal preferences are linear / transitive, group preferences of those same people can be cyclic. We don't like that but we can't do anything about it since that is part of mathematics, not part of our opinions or a property of some election methods. We should not say that such cyclic majority opinions should be broken or corrected since groups actually have cyclic majority opinions. That is correct. We should however be able to find the best winner also in situations where groups have cyclic opinions.

I think typical STV problems (that are not present in CPO-STV) do not fall in tis category. Those little features don't have the positive side that would make us want them. We may use STV instead of CPO-STV because of the computational complexity of CPO-STV. The problems of multi-winner STV are unwanted but "acceptable" and "forgivable" since they are small and they allow us to make some other properties of the system good.

But in Largest Reminder the Alabama paradox can be said to be a neutral mathematical property that is linked to properties that we may want. I borrow an Alabama paradox example from the Wikipedia.

The populations or votes of three districts are (6, 6, 2). There are 10 seats. The "fair shares" are (4.286, 4.286, 1.429). If we allocate seats (4, 4, 2), two groups will get 0.286 seats too much, and one will miss 0.571 seats. That means less violation of opinions than any other allocation would give (if measured this way).

If there are 11 seats, the fair shares are (4.714, 4.714, 1.571). Seat allocation (5, 5, 1) means less less violation of opinions than any other allocation would give.

Unfortunately the last group "lost" a seat when the number of seats went up from 10 to 11. But this is a mathematical property. If we want to minimize the violation of opinions (in the sense that was used in the calculations above) we must violate another property that we may also find natural. In this case it was some kind of an idea of cumulative allocation of seats. In places where we want to allocate seats in a cumulative manner (e.g. delegates would be sent out to do their work one by one, and all the time we would like the current delegates to proportionally represent the opinions), then we should maybe use some highest average method instead of the Largest Remainder method. But if we don't have that special need, then why not allocate the seats using a formula that minimizes the violation of opinions.

Minimal violation of opinions may be our main criterion here. Since every method will have some rounding errors, people may be already be used to thinking that the third group is "lucky" if there are 10 seats, and "unlucky" if there are 11 seats, and the reverse for the other groups. The Largest Remainder method did not make any errors in the allocation. It just optimized the result based on our key criterion. We can't change mathematics, so we might just accept the fact that sometimes it gives "interesting" results.

The Alabama paradox and cyclic group opinion paradox might be undesirable in the sense that I'd be a happier man if mathematics was different. But since mathematics is what it is, I tend to think that when allocating seats I must choose if I want to minimize the violation of opinions (measured as above) (which means that I'll "desire" the associated paradoxes), or if I want to use a formula that can allocate the seats in a cumulative way, or if I want some other properties.

Juho