It sounds like you guys are straightening out the confusion, and exploring some good ideas.
----- Original Message ----- From: Toby Pereira Date: Tuesday, July 19, 2011 7:47 am Subject: Re: [EM] Correspondences between PR and lottery methods (was Centrist vs. non-Centrists, etc.) To: Kristofer Munsterhjelm Cc: [email protected], [email protected] > OK, thanks for the information. But what I meant regarding a > result (group of > winners) having a score itself is that this score is just the > total satisfaction > score for a particular result, and then it is this number that > is proportional > to the probability of that set of candidates being elected. So > rather than > looking at each candidate's chances in the lottery individually, > you could look > at whole results and the candidates are elected as one. I was > thinking that this > might be an analogue to random ballot in the single winner case. > > > > > ________________________________ > From: Kristofer Munsterhjelm > To: Toby Pereira > Cc: [email protected]; [email protected] > Sent: Tue, 19 July, 2011 15:15:15 > Subject: Re: [EM] Correspondences between PR and lottery methods > (was Centrist > vs. non-Centrists, etc.) > > Toby Pereira wrote: > > For proportional range or approval voting, if each result has > a score, you > >could make it so that the probability of that result being the > winning result is > >proportional to that score. Would that work? > > For a lottery derived from PAV or PRV, each winner has a single > score, which is > the probability that the winner would be selected in that > lottery. However, an > entire assembly (group of winners) does not have a single score > as such. > > That is, you get an output of the sort that {A: 0.15, B: 0.37, > C: 0.20, D: 0.17, > E: 0.11}, which means that in this lottery, A would win 15% of > the time. It's > relatively easy to turn this into a party list method - if party > A wins 15% of > the time, that just means that party A should get 15% of the > seats. You could > also use it in a system where each candidate has a weight, but > to my knowledge > that isn't done anywhere. > > However, if A can only occupy one seat in the assembly, it's > less obvious > whether or not A should win (or how often, if it's a > nondeterministic system) in > a two-winner election. In his reply to my question, Forest gave > some ideas on > how to figure that out. > > > Also, how is non-sequential RRV done? Forest pointed me to > this a while back - > >http://lists.electorama.com/pipermail/election-methods- > electorama.com/2010-May/026425.html > > - the bit at the bottom seems the relevant bit. Is that what > we're talking > >about? > > Very broadly, you have a function that depends on a "prospective > assembly" (list > of winners) and on the ballots. Then you try every possible > prospective assembly > and you pick the one that gives the best score. > > In proportional approval voting, each voter gets one > satisfaction point if one > of the candidates he approved is in the outcome, one plus a half > if two > candidates, one plus a half plus a third if three candidates, > and so on. The > winning assembly composition is the one that maximizes the sum > of satisfaction > points. It's also possible to make a Sainte-Laguë version where > the point > increments are 1, 1/3, 1/5... instead of 1, 1/2, 1/3 etc. > > Proportional range voting is based on the idea that you can > consider the > satisfaction function (how many points each voter gets depending > on how many > candidates in the outcome is also approved by him) is a curve > that has f(0) = 0, > f(1) = 1, f(2) = 1/2 and so on. Then you can consider ratings > other than maximum > equal to a fractional approval, so that, for instance, a voter > who rated one > candidate in the outcome at 80%, one at 100%, and another at > 30%, would have a > total satisfaction of 1 + 0.8 + 0.3 = 2.1. > > All that remains to generalize is then to pick an appropriate > continuous curve, > because the proportional approval voting function is only > defined on integer > number of approvals (1 candidate in the outcome, 2 candidates, 3 > candidates). > That's what Forest's post is about. > > (Mathematically speaking, the D'Hondt satisfaction function f(x) > is simply the > xth harmonic number. Then one can see that f(x) = integral from > 0 to 1 of (1 - > x^n)/(1-x) dx. This can be approximated by a logarithm, or > calculated by use of > the digamma function. Forest gives an integral for the > corresponding > Sainte-Laguë satisfaction function in the post you linked to, > and I give an > expression in terms of the harmonic function in reply: > http://lists.electorama.com/pipermail/election-methods- > electorama.com/2010-May/026437.html > ) ---- Election-Methods mailing list - see http://electorama.com/em for list info
