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 <[email protected]> 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 <[email protected]> wrote: > >> On Mon, May 24, 2010 at 12:13 AM, Kristofer Munsterhjelm < >> [email protected]> 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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