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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