<html><head><style type="text/css"><!-- DIV {margin:0px;} --></style></head><body><div style="font-family:times new roman, new york, times, serif;font-size:12pt"><DIV>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?</DIV>
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<DIV>Also, how is non-sequential RRV done? Forest pointed me to this a while back - <A href="http://lists.electorama.com/pipermail/election-methods-electorama.com/2010-May/026425.html">http://lists.electorama.com/pipermail/election-methods-electorama.com/2010-May/026425.html</A> - the bit at the bottom seems the relevant bit. Is that what we're talking about?</DIV>
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<DIV>(I have my own ways of course - <A href="http://www.tobypereira.co.uk/voting.html">http://www.tobypereira.co.uk/voting.html</A>)<BR></DIV>
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<B><SPAN style="FONT-WEIGHT: bold">From:</SPAN></B> "fsimmons@pcc.edu" <fsimmons@pcc.edu><BR><B><SPAN style="FONT-WEIGHT: bold">To:</SPAN></B> Kristofer Munsterhjelm <km_elmet@lavabit.com><BR><B><SPAN style="FONT-WEIGHT: bold">Cc:</SPAN></B> election-methods@lists.electorama.com<BR><B><SPAN style="FONT-WEIGHT: bold">Sent:</SPAN></B> Tue, 19 July, 2011 2:00:40<BR><B><SPAN style="FONT-WEIGHT: bold">Subject:</SPAN></B> [EM] Correspondences between PR and lottery methods (was Centrist vs. non-Centrists, etc.)<BR></FONT><BR><BR><BR>----- Original Message -----<BR>From: Kristofer Munsterhjelm <BR>Date: Monday, July 18, 2011 1:12 pm<BR>Subject: Re: [EM] Centrist vs. non-Centrists (was A distance based method)<BR>To: <A href="mailto:fsimmons@pcc.edu" ymailto="mailto:fsimmons@pcc.edu">fsimmons@pcc.edu</A><BR>Cc: <A href="mailto:election-methods@lists.electorama.com"
ymailto="mailto:election-methods@lists.electorama.com">election-methods@lists.electorama.com</A><BR><BR>> <A href="mailto:fsimmons@pcc.edu" ymailto="mailto:fsimmons@pcc.edu">fsimmons@pcc.edu</A> wrote:<BR>> > <BR>> > ----- Original Message -----<BR>> > From: Kristofer Munsterhjelm <BR>> > Date: Wednesday, July 13, 2011 2:12 pm<BR>> > Subject: Re: [EM] Centrist vs. non-Centrists (was A distance <BR>> based method)<BR>> > To: <A href="mailto:fsimmons@pcc.edu" ymailto="mailto:fsimmons@pcc.edu">fsimmons@pcc.edu</A><BR>> > Cc: Jameson Quinn , <A href="mailto:election-methods@lists.electorama.com" ymailto="mailto:election-methods@lists.electorama.com">election-methods@lists.electorama.com</A><BR>> <BR>> >> I think you said that these are related, even: that PR <BR>> methods and <BR>> >> stochastic single-winner methods are similar, seeking <BR>> >> proportionality (the former in
seats, the latter in time).<BR>> >><BR>> > <BR>> > Precisely. Andy Jennings was the one who hit on the key idea for<BR>> > constructing a lottery directly from a PR method; just do an N-<BR>> winner> PR method for large N, and treat the candidates like we <BR>> treat parties<BR>> > in a party list method; keep the candidates in the running <BR>> after they<BR>> > have already won a seat. Then the number of seats won by the<BR>> > candidate divided by the total number of seats is the candidate's<BR>> > probability in the lottery.<BR>> <BR>> How would that work with combinatorial methods like PAV -- would <BR>> you <BR>> just clone each candidate a very large number of times? (I guess <BR>> the <BR>> question is academic because running a combinatorial method with <BR>> a very <BR>> large number of candidates would take too much time anyway.)<BR><BR>An interesting
question here is whether PAV woould give the same proportions as sequential PAV in the <BR>limit. Also, as usual, proposed slates (with repeats allowed) could be tested to see which gives the <BR>largest PAV score.<BR><BR>> <BR>> Also, is there any way of going in the reverse direction? I can <BR>> see how <BR>> one could turn the lottery into a party list PR allocation: just <BR>> give <BR>> each party a number of seats proportional to the chance they <BR>> have in the <BR>> lottery, resolving rounding problems by apportionment algorithm <BR>> of <BR>> choice. That works when the number of seats is large.<BR><BR>Right. Also if the lottery is the Ultimate Lottery, it is the lottery that maximizes the product of ballot <BR>expectations, so for apportionment you can choose the apportionment that maximizes the <BR>corresponding product under the constraint that there are n candidates and each gets 1/n of the
<BR>probability. This is more of an indirect conversion based on the method of getting the lottery instead of <BR>just the lottery probabilities themselves.<BR><BR>>There <BR>> might be too <BR>> little information to go to individual member multiwinner <BR>> methods from a <BR>> lottery, though.<BR>> Perhaps something to the effect of, when picking n members, just <BR>> spin a <BR>> roulette wheel with zones of size proportional to the chances in <BR>> the <BR>> lottery. If the ball lands on a zone of an already elected <BR>> candidate, <BR>> spin again, otherwise elect the candidate in question. Repeat <BR>> until n <BR>> candidates have been elected. That is nondeterministic, however.<BR><BR>You could make it deterministic by using the conditional probabilities, i.e. the probabilities that are <BR>conditioned on the exclusion of the candidates that have already been chosen.<BR><BR>Another way is to amalgamate
the factions by averaging the ballots that have the same top choice <BR>(weighted average if more than one candidate rated equal top). The lottery then gives a certain weight to <BR>each faction that may or may not be equal to the random ballot lottery. The factions with probability in <BR>excess of the quota can pass the excess down, just as the factions with a deficiency pass their entire <BR>probability down to lower rated candidates on their amalgamated rating ballots. It seems like STV could <BR>be thought of as using the random ballot lottery probabilities in a similar way.<BR><BR>Andy and I were thinking mostly of Party Lists via RRV. His question was that if we used RRV, either <BR>sequential or not, would we get the same result as the Ultimate Lottery Maximization. I was able to <BR>show to our satisfaction, that at least in the non-sequential RRV version, the results would be the <BR>same. It seems like the
initial differences between sequential and non-sequential RRV would disappear <BR>in the limit as the number of candidates to be seated approached infinity.<BR><BR>Would that imply P=NP? In other words, sequential RRV might be an efficient method of <BR>approximating a solution (for large n) of non-sequential RRV (which is undoubtedly NP hard). What <BR>would be analogous in the Traveling Salesman Problem? Don't hold your breath, but it would be <BR>interesting to sort out the analogy, if possible.<BR>----<BR>Election-Methods mailing list - see <A href="http://electorama.com/em" target=_blank>http://electorama.com/em</A> for list info<BR></DIV></DIV></div></body></html>