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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