<div><br><div class="gmail_quote">2011/7/8 Toby Pereira <span dir="ltr"><<a href="mailto:tdp201b@yahoo.co.uk">tdp201b@yahoo.co.uk</a>></span><br><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;">
<div><div style="font-family:times new roman, new york, times, serif;font-size:12pt"><div>While discussing median-based range voting - <a href="http://rangevoting.org/MedianVrange.html" target="_blank">http://rangevoting.org/MedianVrange.html</a>, Warren Smith says "Average-based range voting generalizes to a <i>multiwinner</i> <a href="http://rangevoting.org/PropRep.html" target="_blank"><font color="#000080">proportional representation</font></a> voting system called <a href="http://rangevoting.org/RRV.html" target="_blank"><font color="#000080">reweighted range voting</font></a>. (See papers 78 and 91 <a href="http://www.math.temple.edu/~wds/homepage/works.html" target="_blank"><font color="#000080">here</font></a>.) But there currently is no known way to generalize median-based range voting to do that."</div>
</div></div></blockquote><div><br></div><div><br></div><meta http-equiv="content-type" content="text/html; charset=utf-8">I've told Warren to change that, and he hasn't given me a clear criterion for what I have to do so he will. I've created a system called AT-TV which is PR and reduces to a median-based system in the single-winner case. It's Bucklin-like, in that there is a falling approval threshold, and when a candidate gets enough approvals to be elected (a Droop quota) they are, which "uses up" those votes (except for the excess). So in a one-winner case, it's based on 50th percentile (median), but in, for instance, a 3-winner case, it would be (pseudo-)maximizing the elected candidates' 75th-percentile score, not their 50th-percentile. I think this is the appropriate thing to do in the multi-winner "median" case.<div>
<br></div><div>JQ</div><div> </div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;"><div><div style="font-family:times new roman, new york, times, serif;font-size:12pt">
<div> </div>
<div>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.</div>
<div> </div>
<div>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.</div>
<div> </div>
<div>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.</div>
<div> </div>
<div>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.</div>
<div> </div>
<div>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.</div>
<div> </div>
<div>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.</div>
<div> </div>
<div>*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.)</div>
</div></div><br>----<br>
Election-Methods mailing list - see <a href="http://electorama.com/em" target="_blank">http://electorama.com/em</a> for list info<br>
<br></blockquote></div><br></div>