Toby Pereira

2011-07-08 23:19:30 UTC

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.)

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.)