[EM] General PR question (from Andy Jennings in 2011)

[Toby Pereira]

From: Kathy Dopp <kathy.dopp at gmail.com>

>Well, upon reflection, I think it is undesirable to remove small
>factions from the calculations.

>> I was just saying that my system deals with it "naturally" -
>> i.e. without manually taking out factions.

>How?  I don't see how? What is your formula again?  I believe squaring
>the deviations has potential to give smaller factions proportionately
>more weight, not less.

The method is:

Voters cast approval ballots. If a particular candidate receives n votes, then if this candidate is elected, each voter of that candidate will get a representation of 1/n from that candidate. Each voter who did not vote for that candidate gets 0 representation from them. So the total representation received from that candidate will be 1.

A voter's total representation is the sum of the representation they receive from each elected candidate. Assuming that each elected candidate has received at least one vote, then for c candidates, the sum of the representation of all voters will be c. The arithmetic mean will always be c/v (for v voters).

Full proportionality is achieved if every voter has representation of c/v. The measure of a set of candidates is the total of the squared deviation from c/v of the voters' representation (lower being better). But also, because the variance of x is mean (x^2) - (mean x)^2, where in this case x represents voters' representation levels, (mean x)^2 will always be the same - it will be (c/v)^2 - so we can remove it from the equation.

This means that the winning set of candidates will be the set that minimises the sum of the squares of the voters' representation levels. [end of description]

Because the best result is based on the sum of the squared of the representation levels, then if faction A has 2 seats and faction B 0 seats or they have 1 seat each, the relative proportionality of each of these will be unaffected by a small C faction.

>


>> But what I mean is that if a large faction (with say 50% of all voters) is
>> divided into two (say 25% each) because of a single controversial candidate
>> who appears on half of that faction's ballots but not the other half, then
>> if that faction receives half the candidates (and the one controversial
>> candidate is not elected), then it will be measured as unproportional
>> because each faction will have each contributed to 50% of the candidates but
>> will only be 25% of the electorate each.

>I don't see the problem. Could you possibly provide an example where
>you believe this is situation would be a problem?

Let's say we have 4 to elect and the following approval ballots:

5 voters: A, B, C, D
5 voters: A, B
10 voters: E, F

The 10 E, F voters will get both these candidates elected because they are half the voters and they get half the candidates. But for the other two candidates to be elected, if we elect AB, then each AB subfaction will have elected half of the candidates when they "should" only have elected a quarter, so this is seen as disproportional. AC would be seen as more proportional by your metric because while one subfaction would still have elected half of the elected candidates, the other would have elected a quarter, so it would be seen as more proportional overall. But this is an unbalanced allocation.

>How do you apply your method sequentially?  Many sequential methods
>I've seen are fundamentally unfair (IRV, for example that treats
>voters' votes unequally) and can tend to produce undesirable results.

Instead of finding the slate that minimises squared deviation, you could elect them one at a time. So initially the candidate that has the most approvals. Then the most proportional two-candidate set that includes the first winner. Then the most proportional three-candidate set that includes these two and so on. It would save computing time even if it doesn't find the overall most proportional candidate set.

Toby
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