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

[Toby Pereira]

Regarding this, it might be an idea to perform some sort of transformation on 
the satisfaction score for a result before using it to correspond to probability 
of coming out in the lottery. For the simple case of Proportional Approval 
Voting with D'Hondt divisors (so 1 + 1/2 + 1/3 etc), take the average 
satisfaction score for each voter under the result. Then find the inverse 
harmonic number of this. This might correspond better to level of representation 
for a voter so it could be this number that's proportional to the probability of 
this result being picked.

(Under my versions of PAV and PRV, the "representation score" of a result 
corresponds more to this this normalised version.)




________________________________
From: Toby Pereira <tdp201b at yahoo.co.uk>
To: Kristofer Munsterhjelm <km_elmet at lavabit.com>
Cc: election-methods at lists.electorama.com; fsimmons at pcc.edu
Sent: Tue, 19 July, 2011 15:46:52
Subject: Re: [EM] Correspondences between PR and lottery methods (was Centrist 
vs. non-Centrists, etc.)


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 <km_elmet at lavabit.com>
To: Toby Pereira <tdp201b at yahoo.co.uk>
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
 )
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