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

[Kristofer Munsterhjelm]

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 
)