<div>I found a second error in my mail below.</div><div>How embarrassing.<br></div><div><br></div><div>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)."</div>
<div><br></div><div>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)."</div>
<div>In the table F(0.3) should be F(0.7) and F(0.4151) should be F(0.5849). </div><div><br></div><div>My apologies again.</div><div><br></div><div>PZ</div><div><br></div><div class="gmail_quote">On Mon, May 24, 2010 at 8:23 PM, Peter Zbornik <span dir="ltr"><<a href="mailto:pzbornik@gmail.com">pzbornik@gmail.com</a>></span> wrote:<br>
<blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex;"><div>Just a small correction to the email below:</div>
<div>I wrote (May 24, 2010): "I would rather prefer a small positive value of r, say 0.01."</div>
<div>The sentence should read "I would rather prefer r to be slighltly less than 1, say 0.99.</div>
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<div>Sorry for the error, thanks for your understanding.</div>
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<div>PZ<br><br></div></font><div><div class="h5">
<div class="gmail_quote">On Mon, May 24, 2010 at 8:19 PM, Peter Zbornik <span dir="ltr"><<a href="mailto:pzbornik@gmail.com" target="_blank">pzbornik@gmail.com</a>></span> wrote:<br>
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<div>On Mon, May 24, 2010 at 12:13 AM, Kristofer Munsterhjelm <span dir="ltr"><<a href="mailto:km-elmet@broadpark.no" target="_blank">km-elmet@broadpark.no</a>></span> wrote:<br></div></div>
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<blockquote class="gmail_quote" style="padding-left:1ex;margin:0px 0px 0px 0.8ex;border-left:#ccc 1px solid">Peter Zbornik wrote:<br>
<blockquote class="gmail_quote" style="padding-left:1ex;margin:0px 0px 0px 0.8ex;border-left:#ccc 1px solid">Dear Kristofer,<br><br>would the constant relative risk function be of any help for Approval voting?<br><br>
F=( s(1)^(r-1)+...+s(n)^(r-1) ) / (r-1).<br><br>s(i) is the number of approved council members that are elected, where 1<=i<=n, n is the number of voters<br>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.<br>
</blockquote><br>That scoring method could be used for PAV (not SAV) style optimization. One could create a whole class of PAV style methods this way:<br><br>- 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.<br>
- Voters submit approval ballots v_1 ... v_n<br>- Using brute force, find the council c so that sum(q = 1..n) f(v_q, c) is maximized.<br></blockquote></div>
<div>It would be interesting to see the performance of these functions in your chart with the pareto fronts, especially F for different values r.</div>
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<blockquote class="gmail_quote" style="padding-left:1ex;margin:0px 0px 0px 0.8ex;border-left:#ccc 1px solid"><br>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..</blockquote>
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<div>Thanks for the analysis. f can be a much more generally specified than I did.</div>
<div>I don't know much about greedy approximatioins.</div>
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<blockquote class="gmail_quote" style="padding-left:1ex;margin:0px 0px 0px 0.8ex;border-left:#ccc 1px solid"><br>
<blockquote class="gmail_quote" style="padding-left:1ex;margin:0px 0px 0px 0.8ex;border-left:#ccc 1px solid">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.<br>
</blockquote><br>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.</blockquote>
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<div>I would rather prefer a small positive value of r, say 0.01.</div>
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<div>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.</div>
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<div>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?).</div>
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<blockquote class="gmail_quote" style="padding-left:1ex;margin:0px 0px 0px 0.8ex;border-left:#ccc 1px solid"> </blockquote>
<blockquote class="gmail_quote" style="padding-left:1ex;margin:0px 0px 0px 0.8ex;border-left:#ccc 1px solid"><br>
<blockquote class="gmail_quote" style="padding-left:1ex;margin:0px 0px 0px 0.8ex;border-left:#ccc 1px solid">Other uses:<br>It seems that F can be used both as a proportionality index and as a majoritan preference index for suitable values of r.</blockquote>
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<div>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. </div>
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<div>F(r=0.5)=( s(1)^0.5+...+s(n)^0.5 ) * 2 (exclude the last factor)</div>
<div>I did a calculation on the series of satisfaction from an other seat.</div>
<div>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.</div>
<div>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.</div>
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<div>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.</div>
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<blockquote class="gmail_quote" style="padding-left:1ex;margin:0px 0px 0px 0.8ex;border-left:#ccc 1px solid"> </blockquote>
<blockquote class="gmail_quote" style="padding-left:1ex;margin:0px 0px 0px 0.8ex;border-left:#ccc 1px solid"><br>
<blockquote class="gmail_quote" style="padding-left:1ex;margin:0px 0px 0px 0.8ex;border-left:#ccc 1px solid">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.</blockquote>
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<blockquote class="gmail_quote" style="padding-left:1ex;margin:0px 0px 0px 0.8ex;border-left:#ccc 1px solid"><br>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:<br>
<br><a href="http://faculty.lebow.drexel.edu/McCainR//top/prin/txt/comsysf/compsys2.gif" target="_blank">http://faculty.lebow.drexel.edu/McCainR//top/prin/txt/comsysf/compsys2.gif</a><br><a href="http://faculty.lebow.drexel.edu/McCainR//top/prin/txt/comsysf/compsys3.gif" target="_blank">http://faculty.lebow.drexel.edu/McCainR//top/prin/txt/comsysf/compsys3.gif</a><br>
<br>(Those examples have greater being better, so the front is to the upper right rather than lower left in my diagram.)<br></blockquote>
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<div>Yes, that is an excellent idea. Thus the pareto front would need two values of r at least. </div>
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<blockquote class="gmail_quote" style="padding-left:1ex;margin:0px 0px 0px 0.8ex;border-left:#ccc 1px solid"><br>
<blockquote class="gmail_quote" style="padding-left:1ex;margin:0px 0px 0px 0.8ex;border-left:#ccc 1px solid">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 <a href="http://munsterhjelm.no/km/elections/multiwinner_tradeoffs/" target="_blank">http://munsterhjelm.no/km/elections/multiwinner_tradeoffs/</a> would look like using F with different values of r at the axis. For instance, if r=0, then we would get Bayesian regret.<br>
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).<br></blockquote><br>That would be a series of 1D graphs because the tradeoff would be defined by F. </blockquote>
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<div>Yes that is the right way to put it.</div>
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<blockquote class="gmail_quote" style="padding-left:1ex;margin:0px 0px 0px 0.8ex;border-left:#ccc 1px solid">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.</blockquote>
<blockquote class="gmail_quote" style="padding-left:1ex;margin:0px 0px 0px 0.8ex;border-left:#ccc 1px solid"><br>To put it another way, say there's a society where the voters vote:<br> 33% for party A, 15% for party B, 27% for party C, 25% for party D,<br>
and the parliamentary composition is<br> 41% for party A, 9% for party B, 27% for party C, 23% for party D,<br><br>how would you use F to determine how proportional that is? Here, I used "party" instead of "opinion" for the sake of simplicity.<br>
The RMSE, GnI, LHI, etc, would all give a proportionality measure output when given those inputs.</blockquote>
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<div>I mixed up different concepts.</div>
<div>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). </div>
<div>Thus, I don't know how to make F into a proportionality measure.</div>
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<div>Thanks for pointing it out.</div>
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<div>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).</div>
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<div> Sainte-Lague<<br> s(i)^r - s(i-1)^r Sainte- F(0,415) F(0,3) F(0.5) <br>
s(i) F(r=0,3) F(r=0,4151) F(r=0,5) d'Hondt Lague <d'Hondt<br>01 100,00% 100,00% 100,00% 100,00% 100,00% 1 1 1<br>02 23,11% 33,34% 41,42% 50,00% 33,33% 1 0 1<br>
03 15,92% 24,44% 31,78% 33,33% 20,00% 1 0 1<br>04 12,53% 20,01% 26,79% 25,00% 14,29% 1 0 0<br>05 10,49% 17,26% 23,61% 20,00% 11,11% 1 0 0<br>
06 09,11% 15,33% 21,34% 16,67% 09,09% 1 1 0<br>07 08,10% 13,90% 19,63% 14,29% 07,69% 1 1 0<br>08 07,33% 12,78% 18,27% 12,50% 06,67% 0 1 0<br>
09 06,71% 11,88% 17,16% 11,11% 05,88% 0 1 0<br>10 06,21% 11,13% 16,23% 10,00% 05,26% 0 1 0<br>11 05,79% 10,50% 15,43% 09,09% 04,76% 0 1 0<br>
12 05,43% 09,95% 14,75% 08,33% 04,35% 0 1 0<br>13 05,12% 09,48% 14,14% 07,69% 04,00% 0 1 0<br>14 04,85% 09,06% 13,61% 07,14% 03,70% 0 1 0<br>
15 04,62% 08,69% 13,13% 06,67% 03,45% 0 1 0<br>16 04,41% 08,36% 12,70% 06,25% 03,23% 0 1 0<br>17 04,22% 08,06% 12,31% 05,88% 03,03% 0 1 0<br>
18 04,05% 07,78% 11,95% 05,56% 02,86% 0 1 0<br>19 03,89% 07,53% 11,63% 05,26% 02,70% 0 1 0<br>20 03,75% 07,31% 11,32% 05,00% 02,56% 0 1 0<br>
21 03,62% 07,09% 11,04% 04,76% 02,44% 0 1 0<br>22 03,50% 06,90% 10,78% 04,55% 02,33% 0 1 0<br>23 03,39% 06,72% 10,54% 04,35% 02,22% 0 1 0<br>
24 03,29% 06,55% 10,31% 04,17% 02,13% 0 1 0<br>25 03,20% 06,39% 10,10% 04,00% 02,04% 0 1 0<br>26 03,11% 06,24% 09,90% 03,85% 01,96% 0 1 0<br>
27 03,03% 06,11% 09,71% 03,70% 01,89% 0 1 0<br>28 02,95% 05,97% 09,54% 03,57% 01,82% 0 1 0<br>29 02,88% 05,85% 09,37% 03,45% 01,75% 0 1 0<br>
30 02,81% 05,73% 09,21% 03,33% 01,69% 0 1 0<br>31 02,74% 05,62% 09,05% 03,23% 01,64% 0 1 0<br>32 02,68% 05,52% 08,91% 03,13% 01,59% 0 1 0<br>
33 02,62% 05,42% 08,77% 03,03% 01,54% 0 1 0<br>34 02,57% 05,32% 08,64% 02,94% 01,49% 0 1 0<br>35 02,52% 05,23% 08,51% 02,86% 01,45% 0 1 0<br>
36 02,47% 05,15% 08,39% 02,78% 01,41% 0 1 0<br>37 02,42% 05,06% 08,28% 02,70% 01,37% 0 1 0<br>38 02,37% 04,98% 08,17% 02,63% 01,33% 0 1 0<br>
39 02,33% 04,91% 08,06% 02,56% 01,30% 0 1 0<br>40 02,29% 04,83% 07,96% 02,50% 01,27% 0 1 0<br>41 02,25% 04,76% 07,86% 02,44% 01,23% 0 1 0<br>
42 02,21% 04,70% 07,76% 02,38% 01,20% 0 1 0<br>43 02,17% 04,63% 07,67% 02,33% 01,18% 0 1 0<br>44 02,14% 04,57% 07,58% 02,27% 01,15% 0 1 0<br>
45 02,11% 04,51% 07,50% 02,22% 01,12% 0 1 0<br>46 02,07% 04,45% 07,41% 02,17% 01,10% 0 1 0<br>47 02,04% 04,39% 07,33% 02,13% 01,08% 0 1 0<br>
48 02,01% 04,34% 07,25% 02,08% 01,05% 0 1 0<br>49 01,98% 04,29% 07,18% 02,04% 01,03% 0 1 0<br>50 01,95% 04,24% 07,11% 02,00% 01,01% 0 1 0</div>
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