[EM] PAV and risk functions
Peter Zbornik
pzbornik at gmail.com
Mon May 24 11:19:53 PDT 2010
On Mon, May 24, 2010 at 12:13 AM, Kristofer Munsterhjelm <
km-elmet at broadpark.no> wrote:
> Peter Zbornik wrote:
>
>> Dear Kristofer,
>>
>> would the constant relative risk function be of any help for Approval
>> voting?
>>
>> F=( s(1)^(r-1)+...+s(n)^(r-1) ) / (r-1).
>>
>> s(i) is the number of approved council members that are elected, where
>> 1<=i<=n, n is the number of voters
>> r is a coefficient of risk aversion, which determines the rate at which
>> marginal utility of the voter declines with the number of council members
>> awarded to the voter.
>>
>
> That scoring method could be used for PAV (not SAV) style optimization. One
> could create a whole class of PAV style methods this way:
>
> - Define a function f(a, b) -> R, mapping pairs of candidate sets to real
> numbers, where a is the approval ballot and b is the candidate council.
> - Voters submit approval ballots v_1 ... v_n
> - Using brute force, find the council c so that sum(q = 1..n) f(v_q, c) is
> maximized.
>
It would be interesting to see the performance of these functions in your
chart with the pareto fronts, especially F for different values r.
>
> The greedy approximation can be defined in a similar generalized manner,
> but places restrictions upon the kind of f that can work. The greedy
> approximations would also be house monotone, I think, since they work by
> picking one candidate, then another, then another..
Thanks for the analysis. f can be a much more generally specified than I
did.
I don't know much about greedy approximatioins.
>
> The selection of r determines the behavior of F: If r=1, then F becomes the
>> Bernoulli-Nash social welfare function: log(s(1))+...+log(s(n)) in the
>> limit. As log(0) is minus infinity, this function requires that each voter
>> gets at least one council member, thus it over-represents minorities and
>> insures, that "everyone has their representative" in the council. This
>> function is useless, if all bullet vote for themselves.
>>
>
> Couldn't this be solved in a leximax fashion if only some voters bullet
> vote? That is, an outcome with fewer infinities win over an outcome with
> more no matter what; then if there's a tie between the number of infinities,
> the one with the greatest finite score wins.
I would rather prefer a small positive value of r, say 0.01.
I guess you could use leximax, but the method would lose its nice
mathematical properties. I think we would make up some own house mathematics
when saying that one result with infinitely low utility+3 would be better
than a result with infinitely low utility, since F measures cardinal utility
and not ordinal utility.
An other option would simply be to reward every voter one extra point of
utility from start, but that would be an ad-hoc rule (why not then add 1000
utility points or 0.01 points?).
>
> Other uses:
>> It seems that F can be used both as a proportionality index and as a
>> majoritan preference index for suitable values of r.
>
>
Just to avoid misunderstandings: My hunch was that SLI and F with r=0.5 are
more or less in a linear relation, i.e. that F(r=0.5) reaches maximum for
proportional distributions.
F(r=0.5)=( s(1)^0.5+...+s(n)^0.5 ) * 2 (exclude the last factor)
I did a calculation on the series of satisfaction from an other seat.
The series does not lie in between the d'Hondt series and sainte-lague (see
the table at the end of this mail), since F decreases slower than d'Hondt
series and Sainte-Lague.
For some valus of r, the function comes close though. I don't know if this
is a good or bad thing though, I don't know so much about d'Hondt,
Sainte-Lague and other divisor methods and I still haven't seen any
analytical proof of why Sainte-Lague is close to LR-Hare and if any divisor
method is close to the Droop quota.
The Sainte Lague series could be plugged into this function, and we would
get our PAV which gives us optimal Sainte Lague proportionality:
f=sum(1<=i<=n) sum(1<=j<=s(i)) 1/(1+((s(i)j-1)*2)), s(i) are the seats
awarded fot voter i, s(i)j is a positive integer <=s(i), n are the number of
voters. The d'Hondt method can be similarly defined for f.
>
> F can be used to explain why there are different voting systems - they
>> simply have different values of r, i.e. they have different utility
>> functions. For a suitable value of r, block-voting has higher utility and is
>> thus "better" than proportional representation, like STV.
>
>
>
> That appears to be similar to my ideas about proportionality and
> majoritarian preference being tradeoffs. For some value of r, people would
> value the latter more than the former. Thus the decision of a single value
> of r would take the shape of multiple curves overlaid on the Pareto front,
> where the curve closest to the origin that still hits a method defines the
> optimal method, somewhat like these economic planning examples:
>
> http://faculty.lebow.drexel.edu/McCainR//top/prin/txt/comsysf/compsys2.gif
> http://faculty.lebow.drexel.edu/McCainR//top/prin/txt/comsysf/compsys3.gif
>
> (Those examples have greater being better, so the front is to the upper
> right rather than lower left in my diagram.)
>
Yes, that is an excellent idea. Thus the pareto front would need two values
of r at least.
>
> F can also be used to measure which voting systems are the "best" and to
>> measure the "distance" or "similarity" between voting systems against some
>> benchmark values of r. It would be cool to see how the chart at
>> http://munsterhjelm.no/km/elections/multiwinner_tradeoffs/ would look
>> like using F with different values of r at the axis. For instance, if r=0,
>> then we would get Bayesian regret.
>> F with r=0.5 (or some other value of r between 0<r<1) could maybe be used
>> in the chart instead of the Sainte-Lague proportionality index (SLI).
>>
>
> That would be a series of 1D graphs because the tradeoff would be defined
> by F.
Yes that is the right way to put it.
> I don't see how F could be used as a proportionality measure in its own
> right since my opinion output is fractional, not binary.
> To put it another way, say there's a society where the voters vote:
> 33% for party A, 15% for party B, 27% for party C, 25% for party D,
> and the parliamentary composition is
> 41% for party A, 9% for party B, 27% for party C, 23% for party D,
>
> how would you use F to determine how proportional that is? Here, I used
> "party" instead of "opinion" for the sake of simplicity.
> The RMSE, GnI, LHI, etc, would all give a proportionality measure output
> when given those inputs.
I mixed up different concepts.
F can't be used as proportionality index in the sense that it allows for
comparing the proportionality of different elections (different
voters, ballots and candidates).
Thus, I don't know how to make F into a proportionality measure.
Thanks for pointing it out.
Find below the incremental satisfaction for a seat F(r=0.3), F(r=0.4151) and
F(r=0.5)compared to d'Hondt and Sainte-Lague (original calculations in
Excel).
Sainte-Lague<
s(i)^r - s(i-1)^r
Sainte- F(0,415) F(0,3) F(0.5)
s(i) F(r=0,3) F(r=0,4151) F(r=0,5) d'Hondt Lague
<d'Hondt
01 100,00% 100,00% 100,00% 100,00% 100,00% 1
1 1
02 23,11% 33,34% 41,42% 50,00% 33,33% 1
0 1
03 15,92% 24,44% 31,78% 33,33% 20,00% 1
0 1
04 12,53% 20,01% 26,79% 25,00% 14,29% 1
0 0
05 10,49% 17,26% 23,61% 20,00% 11,11% 1
0 0
06 09,11% 15,33% 21,34% 16,67% 09,09% 1
1 0
07 08,10% 13,90% 19,63% 14,29% 07,69% 1
1 0
08 07,33% 12,78% 18,27% 12,50% 06,67% 0
1 0
09 06,71% 11,88% 17,16% 11,11% 05,88% 0
1 0
10 06,21% 11,13% 16,23% 10,00% 05,26% 0
1 0
11 05,79% 10,50% 15,43% 09,09% 04,76% 0
1 0
12 05,43% 09,95% 14,75% 08,33% 04,35% 0
1 0
13 05,12% 09,48% 14,14% 07,69% 04,00% 0
1 0
14 04,85% 09,06% 13,61% 07,14% 03,70% 0
1 0
15 04,62% 08,69% 13,13% 06,67% 03,45% 0
1 0
16 04,41% 08,36% 12,70% 06,25% 03,23% 0
1 0
17 04,22% 08,06% 12,31% 05,88% 03,03% 0
1 0
18 04,05% 07,78% 11,95% 05,56% 02,86% 0
1 0
19 03,89% 07,53% 11,63% 05,26% 02,70% 0
1 0
20 03,75% 07,31% 11,32% 05,00% 02,56% 0
1 0
21 03,62% 07,09% 11,04% 04,76% 02,44% 0
1 0
22 03,50% 06,90% 10,78% 04,55% 02,33% 0
1 0
23 03,39% 06,72% 10,54% 04,35% 02,22% 0
1 0
24 03,29% 06,55% 10,31% 04,17% 02,13% 0
1 0
25 03,20% 06,39% 10,10% 04,00% 02,04% 0
1 0
26 03,11% 06,24% 09,90% 03,85% 01,96% 0
1 0
27 03,03% 06,11% 09,71% 03,70% 01,89% 0
1 0
28 02,95% 05,97% 09,54% 03,57% 01,82% 0
1 0
29 02,88% 05,85% 09,37% 03,45% 01,75% 0
1 0
30 02,81% 05,73% 09,21% 03,33% 01,69% 0
1 0
31 02,74% 05,62% 09,05% 03,23% 01,64% 0
1 0
32 02,68% 05,52% 08,91% 03,13% 01,59% 0
1 0
33 02,62% 05,42% 08,77% 03,03% 01,54% 0
1 0
34 02,57% 05,32% 08,64% 02,94% 01,49% 0
1 0
35 02,52% 05,23% 08,51% 02,86% 01,45% 0
1 0
36 02,47% 05,15% 08,39% 02,78% 01,41% 0
1 0
37 02,42% 05,06% 08,28% 02,70% 01,37% 0
1 0
38 02,37% 04,98% 08,17% 02,63% 01,33% 0
1 0
39 02,33% 04,91% 08,06% 02,56% 01,30% 0
1 0
40 02,29% 04,83% 07,96% 02,50% 01,27% 0
1 0
41 02,25% 04,76% 07,86% 02,44% 01,23% 0
1 0
42 02,21% 04,70% 07,76% 02,38% 01,20% 0
1 0
43 02,17% 04,63% 07,67% 02,33% 01,18% 0
1 0
44 02,14% 04,57% 07,58% 02,27% 01,15% 0
1 0
45 02,11% 04,51% 07,50% 02,22% 01,12% 0
1 0
46 02,07% 04,45% 07,41% 02,17% 01,10% 0
1 0
47 02,04% 04,39% 07,33% 02,13% 01,08% 0
1 0
48 02,01% 04,34% 07,25% 02,08% 01,05% 0
1 0
49 01,98% 04,29% 07,18% 02,04% 01,03% 0
1 0
50 01,95% 04,24% 07,11% 02,00% 01,01% 0
1 0
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