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

[fsimmons at pcc.edu]

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: fsimmons at pcc.edu, election-methods at lists.electorama.com

> 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: fsimmons at pcc.edu; election-methods at lists.electorama.com
> 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
> )