[EM] Median-based Proportional Representation

[Toby Pereira]

While discussing median-based range voting - 
http://rangevoting.org/MedianVrange.html, Warren Smith says "Average-based range 
voting generalizes to a multiwinner proportional representation voting system 
called reweighted range voting. (See papers 78 and 91 here.) But there currently 
is no known way to generalize median-based range voting to do that."

So I was thinking about how you might get a median-based PR system, using range 
voting, or some other score system, such as Borda Count. I don't think there is 
necessarily a "perfect" method but I did come up with something (possibly 
ridiculous). You find a way to convert the scores of the candidates so that a 
candidate's median score becomes their mean score. For example, if a candidate's 
mean was 5 (out of 10) and their median was 7, their scores would undergo some 
sort of transformation so that their mean score became 7. Likewise if someone 
had a mean of 7 and a median of 5, their scores would undergo a transformation 
to reduce the mean to 5.

One way to do this is as follows: Convert the range so that it becomes 0 to 1 
(so in a 0-10 case, just divide all scores by 10). Then for each candidate you 
convert their score s to s^n where n is the number for that particular candidate 
that will make the original median score the mean of the transformed scores. For 
n over 1 the score will be reduced and for n under 1, the score will increase. 
So each candidate has their own value of n.

Once all the scores have been converted, you can just do whatever you would have 
done in your non-median-based PR system to find the winning candidates.

Obviously, this is a bit of a fudge because although we are fixing the mean for 
each candidate to what we want, the rest of the scores just end up how they end 
up. There would be different conversion systems that convert median to mean but 
give different values for the other scores.

Just looking at the median and mean here could be seen as a bit arbitrary. As 
well as converting median to mean, we would ideally also want to convert other 
percentiles accordingly. We'd want to convert the 25th percentile score to the 
25th "permeantile"*, or whatever the term is. (Is there a term?) But it would 
actually be impossible to do this properly. With repeated scores (which would 
always happen where there are more voters than possible scores), different 
percentile values will have the same score. For example, if someone's median 
score is 5, it's likely to also be 5 at the 51st percentile. But, as far as I 
understand it, the "permeantile" would not be able to have a flat gradient at 
any point, unless it's flat all the way across. So we couldn't have a "perfect" 
system that worked on this basis. So for simplicity we can just use the system 
as described.

Of course, with range voting, people might vote approval style, so many 
candidates might simply have a median of 0 or 10. In that case the only 
"reasonable" conversion would be to convert all their scores to 0 or 10 
respectively. This problem wouldn't occur to the same degree under Borda Count, 
however.

*I was thinking about how you would calculate permeantiles. In a uniform 
distribution between 0 and 1, the 25th permeantile would be 0.25. If you weight 
the averages of each side 3 to 1 in favour of the smaller side of the 
permeantile (0 to 0.25), and average these, then you get 0.25. (3*0.125 + 
1*0.625) / 4 = 0.25. So for the 10th permeantile, you have (9*0.05 + 1*0.55) / 
10 = 0.1 and so on. I imagine this would work for non-uniform distributions too. 
(Sorry for going off topic.)
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