[EM] PAV and risk functions
Peter Zbornik
pzbornik at gmail.com
Mon May 24 11:23:57 PDT 2010
Just a small correction to the email below:
I wrote (May 24, 2010): "I would rather prefer a small positive value of r,
say 0.01."
The sentence should read "I would rather prefer r to be slighltly less than
1, say 0.99.
Sorry for the error, thanks for your understanding.
PZ
On Mon, May 24, 2010 at 8:19 PM, Peter Zbornik <pzbornik at gmail.com> wrote:
> 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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