On Wed, 7 Apr 2004, Forest Simmons wrote: > On Tue, 6 Apr 2004, Rob LeGrand wrote: > > > Here's a question I thought about quite a bit a while ago but never posted > > until now that there's talk of Bucklin on the list: Which candidate should > > win the following Bucklin election? > > > > 25:Brown>Jones>Davis>Smith > > 26:Davis>Smith>Brown>Jones > > 49:Jones>Smith>Brown>Davis > > > > Smith? Jones? > > I'll add my two cents worth to responses given by Kevin and Alex: > > We think of Borda as giving the win to the candidate with the highest > average rank, and we think of Bucklin as giving the win to the candidate > with the highest median rank, which normally it does, but in this case > both Smith and Jones have a median rank of two. > > Or do they? > > The answer to this question depends on how you define "median" for lumped > data. > > Suppose that I have 49 numbers in the bin marked "one", 25 numbers in the > bin marked "two", no numbers in the bin marked "three", and 26 numbers in > the bin marked "four." > > If all the numbers inside a bin are exactly equal to the mark or label on > the bin, then the median number is two. > > But suppose that the 25 numbers in the bin marked "two" are uniformly > distributed between 1.5 and 2.5. Then we can see that the median number > is halfway between 1.5+(1/25) and 1.5+(2/25) or about 1.56 , a > significantly better rank than two. > > Which median makes better sense in the context that Rob LeGrand has given > us? > > The second type would definitely make more sense if the ranks were derived > from four-slot CR ballots, since a wide range of CR values have to be > compressed into only four "bins" if you will. > > I think Jones qualifies as the highest median rank candidate, whether or > not he is the Bucklin winner. > > Here's another approach: for both Smith and Jones calculate the difference > in the number of ballots above and below rank two (the rank of their > common median ballot according to simple median). > > Jones: 49-26=23 > Smith: 0-25=-25 > > Jones beats Smith. > > I think this is the simplest way to resolve median rank "ties."
The next simplest is to use quotients instead of differences: Jones: 49/26 > 1 Smith: 0/25=0 < 1 so Jones beats Smith, again. Forest > > > > > __________________________________ > > Do you Yahoo!? > > Yahoo! Small Business $15K Web Design Giveaway > > http://promotions.yahoo.com/design_giveaway/ > > ---- > > Election-methods mailing list - see http://electorama.com/em for list info > > > > ---- > Election-methods mailing list - see http://electorama.com/em for list info > ---- Election-methods mailing list - see http://electorama.com/em for list info