Juho-- You wrote:
> Let's say one quota is 1000 votes, and the district populations are A:50332, > B: 1335, C:1333. > > If we allocate the seats 50-2-1, differences from accurate PR will be > 332-665-333 persons. > If we allocate the seats 51-1-1, differences from accurate PR will be > 668-335-333 persons. > > I don't think the idea of allocating the seats so that the sum of > overrepresentations and underrepresentations is as small as possible is an > arbitrary way to allocate seats. Quite natural, don't you think. Certainly. LR isn't without its own optimization. But name one person who benefits from it. Yes, it looks good on paper. But remember that, to get one thing, you give up something else. What are you giving up to get LR's optimization? You know the answer to that. When you give up SL's smaller differences in S/Q, the number of representatives for each Hare quota in the particular districts, you'd better be getting something for it. So what are you getting? Anything that counts for equal representation the way that SL's properties do? And wasn't equal representation or individuals the original goal of proportional allocation? So sure, LR minimizes the overall difference, summed over all the districts, of the difference between the districs' seats and their numbers of Hare quotas. Sounds good, but be sure to ask, to make sure that it's ok with the people to whom you're giving drastically and unnecessarily less representation per person. > Divisor methods focus on ratios of people and representatives. Why should > that be the only approach that people should use? Because equal representation for all people is the goal. Because if you give someone drastically and unnecessarily less representation, you'd better have a good justification for him/her. It's a question of how many seats a Hare quota of people in your district gets in comparison to how many seats a Hare quota of people in my district gets. In my example, where you live in district C, and I live in district B, my district is getting about twice as much representation per hare quota of people, compared to your district. Of course, even if you aren't in the shorted district, it's necessary to consider the people who are. Ask them how they like getting only half as much representation per person as the neighboring district. > > If we measure the biggest differences in persons (i.e. not in ratios like you > did), then the differences are 997 (min -332, max +665) and 1003 (min -335, > max +668) Yes, LR minimizes the difference, summed over all of the districts, of the difference between the districts' seats and their numbers of Hare quotas. See above. > One more way to read your example is to assume that district A first had > first exactly 50 quotas and 50 seats, and B had 1 quota and 1 seat. Then we > > annex some new areas to those districts (332 and 335 persons). The question > is why the 335 extra (over 1 quota) people in district B are "less > > valuable" than the extra 332 (over 50 quota) of district A? Isn't this a > valid concern, at least from one point of view? Certainly. I'm glad you brought that up, because I was going to. If those districts each had whole numbers of Hare quotas, qualifying thereby for whole numbers of seats, and we give to them those whole numbers of seats, that's quite right and fair. That, of course is what the first part of LR does. Now, new story: We find ourselves with a situation in which each of several districts has a certain number of people, and there is one seat. What should we do. Why not give the seat to the district with the most people. That makes perfect sense. In both of the two paragraphs before this, we've done the right thing, and there's no doubt about it. In other words, each half of LR makes perfect sense. But it's a fiction, because we don't really have just that 1st paragraph, or that 2nd paragraph. We don't just have one of those stories. Both stories are false, because we have something quite different. We have a situation where several districts have various non-integer numbers of Hare quotas, and we want to give to everyone the same representation per person, as nearly as possible. That means that we want the S/Q to differ as little as possible. That means that we want what SL does. And yes, it is about S/Q, because the whole original purpose was so that people would have equal representation. >> Surely no one would deny that the number of representatives that a Hare >> quota of people has is its "representation". > > I note that although you wrote these words to support Saint-Laguë, they work > also against it. Let's say we have proportions 61-13-13-13. SL allocates the > seats 2-1-1-1. The number of quotas of each district/party has is 3.05 - 0.65 > - 0.65 - 0.65. The third full quota of the largest district/party does not > get its seat. Shouldn't all quotas get their representation? Yes. Every Hare quota in my district should have as much representation as do the Hare quotas in your district. But look at what you're doing: Again, you're fragmenting the situation. ...the Hare quotas this time. Looking at a particular piece of a Hare quota and saying "This fraction of a Hare quota has no representation." But that's a fiction. Everyone in that district has representation. Their district has a certain number, S, of representatives. They aren't all seized by one fraction of the district's people, leaving the other fraction without representation. That's where the fiction comes in. In reality, the district's representatives are shared by _everyone_ in the district. If there are S representatives, and Q Hare quotas of people in the district, then there are S/Q representatives per Hare quota of people. All sorts of fallacies involve some sleight-of-hand. Don't deceive yourself in that way. Let's look at your example, above. I'll re-quote what you said: > Let's say we have proportions 61-13-13-13. SL allocates the seats 2-1-1-1. > The number of quotas of each district/party has is 3.05 - 0.65 - 0.65 - 0.65. > > The third full quota of the largest district/party does not get its seat. > Shouldn't all quotas get their representation? Ok, let's compare how many seats a Hare quota of voters has, in the different districts. Let's look at the most that that differs among the districts, with SL and with LR. In terms of subtractive difference, SL and LR are pretty similar in this example. In SL, the greatest difference is 5/6 of a seat. In LR, the greatest difference is 1 seat. But what about the greatest _factor_ by which they differ? In SL, the greatest factor is 9/4. In LR, you give 0 seats to one of the small parties (let's say that in any real election, there won't be 3 districts with an exact tie). That district has a finite number of Hare quotas, and you give to it 0 seats. Each of the other parties has infinitely many times as many seats per Hare quota as that district has. Explain LR's on-paper optimization to the people in that district. > Is this in line with "SL's optimal proportionality"? SL is one good > allocation method (for certain needs) ...like the wish to give people equal representation, as nearly as possible. You continued: > but I have hard time defining it as optimal. Optimal is a strong word, in seat allocation. For one thing, there are various different things to optimize. Bias? LR and SL are not optimal in regard to bias, because, though they're both unbiased, given a uniform probability distribution for districts, with respect to district population, they're only unbiased under that condition. That isn't optimal, because there are other seat allocation methods that are unconditionally unbiased. So, as I said, SL and LR are not optimal with regard to bias. So, is SL optimal with regard to closeness of the districts' S/Q? Again, no. Yes, SL has a transfer property with regard to subtractive difference between two districts' S/Q. In an SL allocation, if a district gives a seat to another, that will always put their S/Q farther apart, in terms of subtractive difference. That sounds pretty good, but SL doesn't guarantee that it will minimize the greatest difference in S/Q among the districts. Nor does it guarantee that it will minimize the sum of the differences in S/Q, summed over all of the district-pairs. Those optimizations can, of course be achieved, using a trial and error process that is quite feasible with a computer (but which wasn't feasible in previous centuries). Therefore, SL/Webster can't be called optimal in terms of S/Q difference. The same can be said for Hill's method (I mistakenly called it "Hall"), with regard to the ratio of districts' S/Q. It isn't optimal for the factor by which S/Q can differ among districts. Then what can we say for SL? We can say that it does a good job of keeping difference in S/Q low. After all, when it rounds off, it puts each district as close as possible to its correct equal-representation share of the seats, for the particular number of seats in the Parliament. And, when it puts all the districts' S/Q as close as possible to the same value, you know that it's also putting them close to eachother too. So then, a lot can be said for what SL does. As for bias, it counts for something that SL and LR are unbiased, given the uniform probability distribution that I spoke of above. Mike Ossipoff ---- Election-Methods mailing list - see http://electorama.com/em for list info
