<html><head></head><body><div class="ydp36729420yahoo-style-wrap" style="font-family:Helvetica Neue, Helvetica, Arial, sans-serif;font-size:13px;"><div></div>
<div dir="ltr" data-setdir="false">One of the advantages of things like approval voting and FPTP, along with score voting and even Borda Count is that you can simply give a list of scores for all candidates, and give them as a percentage of the maximum.</div><div dir="ltr" data-setdir="false"><br></div><div dir="ltr" data-setdir="false">I quite like the idea of doing this for Condorcet methods as well, but it's really only ever going to be an academic exercise. I can't imagine them being published as part of any official results.</div><div dir="ltr" data-setdir="false"><br></div><div dir="ltr" data-setdir="false">Also, methods that don't pass LIIA presumably would still sometimes return an order as well as just the winner. Would it not be the case that ordering by plump score would sometimes contradict the order given by the method?</div><div dir="ltr" data-setdir="false"><br></div><div dir="ltr" data-setdir="false">Also if the total is set to be 100, then some candidates will inevitably get a lower percentage than the percentage of ballots where they are top ranked. This will likely not make much sense to people.</div><div dir="ltr" data-setdir="false"><br></div><div dir="ltr" data-setdir="false">Toby</div><div><br></div>
</div><div id="ydpa2f5b3ebyahoo_quoted_1630182360" class="ydpa2f5b3ebyahoo_quoted">
<div style="font-family:'Helvetica Neue', Helvetica, Arial, sans-serif;font-size:13px;color:#26282a;">
<div>
On Tuesday, 10 February 2026 at 13:03:00 GMT, Kristofer Munsterhjelm via Election-Methods <election-methods@lists.electorama.com> wrote:
</div>
<div><br></div>
<div><br></div>
<div><div dir="ltr">Suppose we want to make a method return not just who won (and the order <br></div><div dir="ltr">of finish), but how well each candidate did - how close to each other <br></div><div dir="ltr">the candidates were - by also returning a score for each.<br></div><div dir="ltr"><br></div><div dir="ltr">(See the end of the post for 2009 Burlington results :-)<br></div><div dir="ltr"><br></div><div dir="ltr">This is easy in FPTP: just count the number of first preferences and <br></div><div dir="ltr">divide by the number of voters.<br></div><div dir="ltr"><br></div><div dir="ltr">But suppose that we'd like to have less of a spoiler effect than FPTP <br></div><div dir="ltr">*and* return scores.<br></div><div dir="ltr"><br></div><div dir="ltr">Ideally, we'd like the scores to not be affected by what other <br></div><div dir="ltr">candidates are running. But that's impossible, at least for a <br></div><div dir="ltr">majoritarian method.<br></div><div dir="ltr"><br></div><div dir="ltr">Let's say that we have three candidates: A, B, and C, and it's a <br></div><div dir="ltr">Condorcet order: A beats B and C, B beats C, and C is the Condorcet <br></div><div dir="ltr">loser. Say furthermore that B's win over C is 75-25, and A's win over B <br></div><div dir="ltr">is 60-40. Then the straightforward scores if only two of them were <br></div><div dir="ltr">present would be (as pretty much every method, including FPTP, would <br></div><div dir="ltr">tell you):<br></div><div dir="ltr"><br></div><div dir="ltr">In A vs B:<br></div><div dir="ltr">A: 60%<br></div><div dir="ltr">B: 40%<br></div><div dir="ltr"><br></div><div dir="ltr">In B vs C:<br></div><div dir="ltr">B: 75%<br></div><div dir="ltr">C: 25%<br></div><div dir="ltr"><br></div><div dir="ltr">The very strictest IIA interpretation would have these scores not change <br></div><div dir="ltr">when the third candidate is introduced. (That's what cardinal methods <br></div><div dir="ltr">with absolute interpersonal comparability do.) Since there's a Condorcet <br></div><div dir="ltr">order, the *ranking* of the other candidates don't change when we <br></div><div dir="ltr">introduce a third, e.g.<br></div><div dir="ltr"><br></div><div dir="ltr">A>B becomes A>B>C after adding C,<br></div><div dir="ltr">B>C becomes A>B>C after adding A.<br></div><div dir="ltr"><br></div><div dir="ltr">But if the scores were to stay the same, then B's score would have to be <br></div><div dir="ltr">40% and 75% at once. That's clearly impossible.<br></div><div dir="ltr"><br></div><div dir="ltr">So majoritarian methods' scores, if they're numbers on a scale, must to <br></div><div dir="ltr">some degree be relative.<br></div><div dir="ltr"><br></div><div dir="ltr">I've found two ways to more or less consistently normalize the scores to <br></div><div dir="ltr">the number of candidates. One is to keep the top two scorers' score the <br></div><div dir="ltr">same, and the other is to make the scores sum to 100%. (For lack of a <br></div><div dir="ltr">better term, I'd call the first "minmax-style" because that's what <br></div><div dir="ltr">minmax does.)<br></div><div dir="ltr"><br></div><div dir="ltr">I've also found two ways to calculate these scores - one that's <br></div><div dir="ltr">appropriate for LIIA methods, and another that should work on a much <br></div><div dir="ltr">broader range of methods.<br></div><div dir="ltr"><br></div><div dir="ltr">So let's do the calculation types first:<br></div><div dir="ltr"><br></div><div dir="ltr">The LIIA style is this: Suppose that candidates are ordered x_1 > x_2 > <br></div><div dir="ltr">x_3 > ... > x_n, and the pairwise victory of x_k against x_(k+1) is <br></div><div dir="ltr">d(x_k, x_(k+1)). Let the score of candidate x_k be s_k. Then set<br></div><div dir="ltr"><br></div><div dir="ltr">s_(k+1)/s_k = d(x_k, x_(k+1))/d(x_(k+1), x_k)<br></div><div dir="ltr"><br></div><div dir="ltr">for k = 1..n-1.<br></div><div dir="ltr"><br></div><div dir="ltr">This is a set of n-1 equations with n unknowns: the normalization method <br></div><div dir="ltr">fixes the last unknown.<br></div><div dir="ltr"><br></div><div dir="ltr">For the A>B>C example above, we'd have:<br></div><div dir="ltr"> s_A/s_B = 60/40<br></div><div dir="ltr"> s_B/s_C = 75/25.<br></div><div dir="ltr"><br></div><div dir="ltr">The nice thing about this approach is that the relative scores of <br></div><div dir="ltr">adjacent candidates stay the same as long as the social ranking/ordering <br></div><div dir="ltr">stays the same; in particular when losers or winners drop out and the <br></div><div dir="ltr">method passes LIIA, the relative scores of the other candidates stay the <br></div><div dir="ltr">same.<br></div><div dir="ltr"><br></div><div dir="ltr">The plump style that's applicable to more methods is this: For each <br></div><div dir="ltr">non-winning candidate x_k, let P_k be the number of plump/bullet votes <br></div><div dir="ltr">for x_k that have to be added to the election to make x_k the winner. <br></div><div dir="ltr">Let P_none be the number of such votes that have to be added for a new <br></div><div dir="ltr">candidate (that currently has no support) to win.<br></div><div dir="ltr"><br></div><div dir="ltr">Then set up a linear scale that maps 0% to P_none, and set x_1/x_2 = <br></div><div dir="ltr">d(x_1, x_2)/d(x_2, x_1).[1]<br></div><div dir="ltr"><br></div><div dir="ltr">This again has one more unknown than equations (the rate of change of <br></div><div dir="ltr">the linear scale, or equivalently, the "virtual" negative value P_1 that <br></div><div dir="ltr">should be assigned to a winner, since the winner needs no plump votes to <br></div><div dir="ltr">win).<br></div><div dir="ltr"><br></div><div dir="ltr">And again, the normalization choice determines that unknown.<br></div><div dir="ltr"><br></div><div dir="ltr"><br></div><div dir="ltr">Now for the normalization approaches:<br></div><div dir="ltr"><br></div><div dir="ltr">Minmax style is simply this: Let the winner x_1's score be d(x_1, <br></div><div dir="ltr">x_2)/d(x_2, x_1), so that the top two's scores stays the same no matter <br></div><div dir="ltr">how many losers are removed from the election (assuming LIIA).<br></div><div dir="ltr"><br></div><div dir="ltr">For the A>B>C example and LIIA-style relative scores, that gives:<br></div><div dir="ltr">s_A: 0.6<br></div><div dir="ltr">s_A/s_B = 60/40 = 0.4<br></div><div dir="ltr">s_B/s_C = 75/25 = 2/15 = 0.1333...<br></div><div dir="ltr"><br></div><div dir="ltr">so<br></div><div dir="ltr"> A: 60%<br></div><div dir="ltr"> B: 40%<br></div><div dir="ltr"> C: 13.3%<br></div><div dir="ltr"><br></div><div dir="ltr">Sum-to-100% is just what it says. For the A>B>C example:<br></div><div dir="ltr"><br></div><div dir="ltr">s_A/s_B = 60/40<br></div><div dir="ltr">s_B/s_C = 75/25<br></div><div dir="ltr">s_A + s_B + s_C = 100%<br></div><div dir="ltr"><br></div><div dir="ltr">which gives<br></div><div dir="ltr"><br></div><div dir="ltr">s_A = 9/17 = 52.9%<br></div><div dir="ltr">s_B = 6/17 = 35.3%<br></div><div dir="ltr">s_C = 2/17 = 11.8%<br></div><div dir="ltr"><br></div><div dir="ltr"><br></div><div dir="ltr">Finally, here are the different scores for the Burlington election with <br></div><div dir="ltr">RP(margins) as the base method for the plump calculations:<br></div><div dir="ltr"><br></div><div dir="ltr">(Minmax style)<br></div><div dir="ltr"><br></div><div dir="ltr">Candidate LIIA-relative % Plump-based %<br></div><div dir="ltr"><br></div><div dir="ltr">Montroll 53.91% 53.91%<br></div><div dir="ltr">Kiss 46.09% 46.09%<br></div><div dir="ltr">Wright 43.42% 43.53%<br></div><div dir="ltr">Smith 41.43% 38.67%<br></div><div dir="ltr">Simpson 5.36% 7.78%<br></div><div dir="ltr">Write-in 0.26% 1.15%<br></div><div dir="ltr"><br></div><div dir="ltr">(Sums-to-100)<br></div><div dir="ltr"><br></div><div dir="ltr">Candidate LIIA-relative % Plump-based %<br></div><div dir="ltr"><br></div><div dir="ltr">Montroll 28.38% 28.21%<br></div><div dir="ltr">Kiss 24.20% 24.11%<br></div><div dir="ltr">Wright 22.80% 22.78%<br></div><div dir="ltr">Smith 21.75% 20.23%<br></div><div dir="ltr">Simpson 2.81% 4.07%<br></div><div dir="ltr">Write-in 0.14% 0.60%<br></div><div dir="ltr"><br></div><div dir="ltr">Minmax-style is more like "approval ratings" while sums-to-100 is more <br></div><div dir="ltr">like "how big a share of the total". I'd be inclined to say sums-to-100 <br></div><div dir="ltr">is more natural given that ranks are relative anyway, but what do you think?<br></div><div dir="ltr"><br></div><div dir="ltr">-km<br></div><div dir="ltr"><br></div><div dir="ltr">[1] Alternatively make the algorithm accept negative ballot counts and <br></div><div dir="ltr">see how many negative plump votes have to be added until adding more <br></div><div dir="ltr">makes the winner lose, and fix the scale so that zero plumpers added <br></div><div dir="ltr">would give 50%. "Negative plumping" like this is possible with ranked <br></div><div dir="ltr">pairs, and the values and two scale points determine the system, thus <br></div><div dir="ltr">making normalization unnecessary. It gives results similar to <br></div><div dir="ltr">minmax-style, but slightly different (e.g. Montroll gets 54.44% instead <br></div><div dir="ltr">of 53.91%)<br></div><div dir="ltr">----<br></div><div dir="ltr">Election-Methods mailing list - see <a href="https://electorama.com/em" rel="nofollow" target="_blank">https://electorama.com/em</a> for list info<br></div></div>
</div>
</div></body></html>