Kristofer Munsterhjelm wrote:<br><div class="gmail_quote"><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;"><div class="im">Andy Jennings wrote:<br>
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Kristofer Munsterhjelm wrote:<br>
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Andy Jennings wrote:<br>
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Like Jameson and Toby, I have spent some time thinking about how<br>
to make a median-based PR system.<br>
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The system I came up with is similar to Jameson's, but simpler,<br>
and uses the Hare quota!<br>
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How about clustering logic? Say you have an electorate of n voters,<br>
and you want k seats. The method would be combinatorial: you'd check<br>
a prospective slate. Say the slate is {ABC...}. Then that means you<br>
make a group of n/k voters and assign A to this gorup, another group<br>
of n/k other voters and assign B to that group, and so on.<br>
The score of each slate is equal to the sum of the median scores for<br>
each assigned candidate, when considering only the voters in the<br>
assigned candidate's group. That is, A's median score when<br>
considering the voters of the first group, plus B's median score<br>
when considering the voters of the second group, and so on. The<br>
voters are moved into groups so that this sum is maximized.<br>
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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.<br>
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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).<br>
</blockquote><div><br></div><div>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.)</div>
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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?</blockquote>
<div><br></div><div>Yes, the fluid nature makes it much worse. Say there are 110 voters and we're choosing 10 winners. Here's the voter profile:</div><div>50 people love A and noone else</div><div>6 people love B and noone else</div>
<div>6 people love C and noone else</div><div>...</div><div>6 people love K and noone else</div><div><br></div><div>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:</div>
<div>6 voters who love B and 5 voters who love A</div><div>6 voters who love C and 5 voters who love A</div><div>...</div><div>6 voters who love K and 5 voters who love A</div><div><br></div><div>And then we elect B,C,..., and K, each with a perfect median in their cluster.</div>
<div><br></div><div>Clustering with the median in each cluster is way too under-determined.</div><div><br></div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;"><div class="im">
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex"> Then the last candidate is only the one with the best worst votes in<br>
the sense that there are only ten voters left.<br>
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How about using the midpoint? That is, you find the 5th voter down,<br>
not the 10th. Then when you're down to the last 10 voters, the 5th<br>
voter down is the median. Doing so would seem to reduce it to median<br>
ratings in the single-winner case, since 100/1 = 100, so you'd pick<br>
the midpoint, i.e. at the 50th voter, which is the median.<br>
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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.<br>
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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.</blockquote>
<div><br></div><div>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.)</div>
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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.<br>
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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.<br>
</blockquote><div><br></div><div>I agree that the median is more resilient in most cases.</div><div><br></div><div>- Andy</div></div>