[EM] Correspondences between PR and lottery methods (was Centrist vs. non-Centrists, etc.)

[Toby Pereira]

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?

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?

(I have my own ways of course - http://www.tobypereira.co.uk/voting.html)




________________________________
From: "fsimmons at pcc.edu" <fsimmons at pcc.edu>
To: Kristofer Munsterhjelm <km_elmet at lavabit.com>
Cc: election-methods at lists.electorama.com
Sent: Tue, 19 July, 2011 2:00:40
Subject: [EM] Correspondences between PR and lottery methods (was Centrist vs. 
non-Centrists, etc.)



----- Original Message -----
From: Kristofer Munsterhjelm 
Date: Monday, July 18, 2011 1:12 pm
Subject: Re: [EM] Centrist vs. non-Centrists (was A distance based method)
To: fsimmons at pcc.edu
Cc: election-methods at lists.electorama.com

> fsimmons at pcc.edu wrote:
> > 
> > ----- Original Message -----
> > From: Kristofer Munsterhjelm 
> > Date: Wednesday, July 13, 2011 2:12 pm
> > Subject: Re: [EM] Centrist vs. non-Centrists (was A distance 
> based method)
> > To: fsimmons at pcc.edu
> > Cc: Jameson Quinn , election-methods at lists.electorama.com
> 
> >> I think you said that these are related, even: that PR 
> methods and 
> >> stochastic single-winner methods are similar, seeking 
> >> proportionality (the former in seats, the latter in time).
> >>
> > 
> > Precisely. Andy Jennings was the one who hit on the key idea for
> > constructing a lottery directly from a PR method; just do an N-
> winner> PR method for large N, and treat the candidates like we 
> treat parties
> > in a party list method; keep the candidates in the running 
> after they
> > have already won a seat. Then the number of seats won by the
> > candidate divided by the total number of seats is the candidate's
> > probability in the lottery.
> 
> How would that work with combinatorial methods like PAV -- would 
> you 
> just clone each candidate a very large number of times? (I guess 
> the 
> question is academic because running a combinatorial method with 
> a very 
> large number of candidates would take too much time anyway.)

An interesting question here is whether PAV woould give the same proportions as 
sequential PAV in the 

limit.  Also, as usual, proposed slates (with repeats allowed) could be tested 
to see which gives the 

largest PAV score.

> 
> Also, is there any way of going in the reverse direction? I can 
> see how 
> one could turn the lottery into a party list PR allocation: just 
> give 
> each party a number of seats proportional to the chance they 
> have in the 
> lottery, resolving rounding problems by apportionment algorithm 
> of 
> choice. That works when the number of seats is large.

Right.  Also if the lottery is the Ultimate Lottery, it is the lottery that 
maximizes the product of ballot 

expectations, so for apportionment you can choose the apportionment that 
maximizes the 

corresponding product under the constraint that there are n candidates and each 
gets 1/n of the 

probability. This is more of an indirect conversion based on the method of 
getting the lottery instead of 

just the lottery probabilities themselves.

>There 
> might be too 
> little information to go to individual member multiwinner 
> methods from a 
> lottery, though.
> Perhaps something to the effect of, when picking n members, just 
> spin a 
> roulette wheel with zones of size proportional to the chances in 
> the 
> lottery. If the ball lands on a zone of an already elected 
> candidate, 
> spin again, otherwise elect the candidate in question. Repeat 
> until n 
> candidates have been elected. That is nondeterministic, however.

You could make it deterministic by using the conditional probabilities, i.e. the 
probabilities that are 

conditioned on the exclusion of the candidates that have already been chosen.

Another way is to amalgamate the factions by averaging the ballots that have the 
same top choice 

(weighted average if more than one candidate rated equal top).  The lottery then 
gives a certain weight to 

each faction that may or may not be equal to the random ballot lottery.  The 
factions with probability in 

excess of the quota can pass the excess down, just as the factions with a 
deficiency pass their entire 

probability down to lower rated candidates on their amalgamated rating ballots.  
It seems like STV could 

be thought of as using the random ballot lottery probabilities in a similar way.

Andy and I were thinking mostly of Party Lists via RRV.  His question was that 
if we used RRV, either 

sequential or not, would we get the same result as the Ultimate Lottery 
Maximization.  I was able to 

show to our satisfaction, that at least in the non-sequential RRV version, the 
results would be the 

same.  It seems like the initial differences between sequential and 
non-sequential RRV would disappear 

in the limit as the number of candidates to be seated approached infinity.

Would that imply P=NP?    In other words, sequential RRV might be an efficient 
method of 

approximating a solution (for large n) of non-sequential RRV (which is 
undoubtedly NP hard).  What 

would be analogous in the Traveling Salesman Problem?  Don't hold your breath, 
but it would be 

interesting to sort out the analogy, if possible.
----
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