[EM] PAV and risk functions

Peter Zbornik pzbornik at gmail.com
Mon May 24 15:50:13 PDT 2010


I found a second error in my mail below.
How embarrassing.

I wrote below (May 24, 2010 at 8:19 PM): "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)."

That was wrong, the text should have read "Find below the incremental
satisfaction for a seat F(r=0.7), F(r=0.5849) and F(r=0.5), without the
denominator (r-1) compared to d'Hondt and Sainte-Lague (original
calculations in Excel)."
In the table F(0.3) should be F(0.7) and F(0.4151) should be F(0.5849).

My apologies again.

PZ

On Mon, May 24, 2010 at 8:23 PM, Peter Zbornik <pzbornik at gmail.com> wrote:

> 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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