[EM] PR methods and Quotas

[Andy Jennings]

Kristofer Munsterhjelm wrote:

> Andy Jennings wrote:
>
>> Kristofer Munsterhjelm wrote:
>>
>>    Andy Jennings wrote:
>>
>>        Like Jameson and Toby, I have spent some time thinking about how
>>        to make a median-based PR system.
>>
>>        The system I came up with is similar to Jameson's, but simpler,
>>        and uses the Hare quota!
>>
>>
>>    How about clustering logic? Say you have an electorate of n voters,
>>    and you want k seats. The method would be combinatorial: you'd check
>>    a prospective slate. Say the slate is {ABC...}. Then that means you
>>    make a group of n/k voters and assign A to this gorup, another group
>>    of n/k other voters and assign B to that group, and so on.
>>    The score of each slate is equal to the sum of the median scores for
>>    each assigned candidate, when considering only the voters in the
>>    assigned candidate's group. That is, A's median score when
>>    considering the voters of the first group, plus B's median score
>>    when considering the voters of the second group, and so on. The
>>    voters are moved into groups so that this sum is maximized.
>>
>>
>> The median is not what you want for clustering like this, because it
>> basically ignores the scores of half the voters assigned to each candidate.
>>  That is, if I'm assigning 11 voters to each candidate, I can assign 6
>> voters who love that candidate and 5 voters who hate the candidate and still
>> have a very high median.
>>
>
> Well, yes, but the same thing holds for median ratings in general. If you
> want to find someone who represents the whole population, median ratings can
> pick someone who is loved by 51% and hated by 49%, rather than someone that
> 80% think are okay (and I think Warren have made arguments to the effect
> that this makes Range better than median).
>

Exactly.  This is why I'm questioning the median even for single-winner
elections.  Maybe you're right and we should be using the 20th percentile,
which would give us the candidate that some 80% of the population liked
best.  I tried to point out some arguments that highest minimum might be a
good method even in some single-winner environments.  It does give everyone
veto power.  But that's okay if everyone is committed to finding a solution
that's acceptable to everyone.  (In a public method, obviously, you'd have
to have some tie-breaker, like electing the candidate vetoed by the fewest
voters.)


> The question then is: what makes that logic okay when you're electing a
> single representative for the whole population, but not okay when you're
> electing one of ten representatives for 10% of the population? Is it the
> fluid nature of the clustering - that the optimizer could try to
> artificially inflate the scores by packing "hate A" voters into the A-group?


Yes, the fluid nature makes it much worse.  Say there are 110 voters and
we're choosing 10 winners.  Here's the voter profile:
50 people love A and noone else
6 people love B and noone else
6 people love C and noone else
...
6 people love K and noone else

If A were a political party, it would be entitled to at least 4 out of the
ten seats.  As a candidate, you would expect A to get a seat.  But we can
cluster the voters into:
6 voters who love B and 5 voters who love A
6 voters who love C and 5 voters who love A
...
6 voters who love K and 5 voters who love A

And then we elect B,C,..., and K, each with a perfect median in their
cluster.

Clustering with the median in each cluster is way too under-determined.

   Then the last candidate is only the one with the best worst votes in
>>    the sense that there are only ten voters left.
>>
>>    How about using the midpoint? That is, you find the 5th voter down,
>>    not the 10th. Then when you're down to the last 10 voters, the 5th
>>    voter down is the median. Doing so would seem to reduce it to median
>>    ratings in the single-winner case, since 100/1 = 100, so you'd pick
>>    the midpoint, i.e. at the 50th voter, which is the median.
>>
>>
>> True, but in filling the first seat, I don't think we should take a
>> candidate loved by 5 and hated by 95 as the first choice to represent
>> one-tenth of the population.
>>
>
> I guess you could be more gentle by placing the point at 50% (1/2) for one
> winner, 1/3 for two, 1/4 for three ... 1/11 for ten. That would be more
> Droop-like and less Hare-like. But then you can't simply eliminate those who
> contributed to the voting, I think.


Yes, it is much more Droop-like.  It seems arbitrary, though, to leave one
eleventh of the voters completely unrepresented.  (With STV, Droop is
natural, but with cardinal inputs, I see no justification for it.)


>  With any finite number of voters, the median is still the score of one
>> voter, who can change the median by changing his vote.  But you are right
>> that if the scores follow a normal distribution, then he probably can't
>> change the median very much before he crosses another voter's score and is
>> not the median vote anymore.   But that's not true for a bimodal
>> distribution.
>>
>
> He can't alter the median to an arbitrary extent, however. An outlier at
> the mean can do so by setting his score arbitrarily high (or low), and the
> max or min voter can do so, but to a limited extent, by raising his score
> (if he's max) or lowering his score (if he's min). If the median voter
> alters his score by too much, he's no longer the median voter. That may
> change the median result by some amount (unless the new median voter
> expresses the same score as the old one used to), but it's limited.
>

I agree that the median is more resilient in most cases.

- Andy
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