<div dir="auto">Hi Kristofer,<div dir="auto"><br></div><div dir="auto">Might the inverse of the keep value under a Warren or Meek STV count suit as well?</div><div dir="auto"><br></div><div dir="auto">Regards,</div></div><br><div class="gmail_quote gmail_quote_container"><div dir="ltr" class="gmail_attr">On Wed, 11 Feb 2026, 12:03 am Kristofer Munsterhjelm via Election-Methods, <<a href="mailto:election-methods@lists.electorama.com">election-methods@lists.electorama.com</a>> wrote:<br></div><blockquote class="gmail_quote" style="margin:0 0 0 .8ex;border-left:1px #ccc solid;padding-left:1ex">Suppose we want to make a method return not just who won (and the order <br>
of finish), but how well each candidate did - how close to each other <br>
the candidates were - by also returning a score for each.<br>
<br>
(See the end of the post for 2009 Burlington results :-)<br>
<br>
This is easy in FPTP: just count the number of first preferences and <br>
divide by the number of voters.<br>
<br>
But suppose that we'd like to have less of a spoiler effect than FPTP <br>
*and* return scores.<br>
<br>
Ideally, we'd like the scores to not be affected by what other <br>
candidates are running. But that's impossible, at least for a <br>
majoritarian method.<br>
<br>
Let's say that we have three candidates: A, B, and C, and it's a <br>
Condorcet order: A beats B and C, B beats C, and C is the Condorcet <br>
loser. Say furthermore that B's win over C is 75-25, and A's win over B <br>
is 60-40. Then the straightforward scores if only two of them were <br>
present would be (as pretty much every method, including FPTP, would <br>
tell you):<br>
<br>
In A vs B:<br>
A: 60%<br>
B: 40%<br>
<br>
In B vs C:<br>
B: 75%<br>
C: 25%<br>
<br>
The very strictest IIA interpretation would have these scores not change <br>
when the third candidate is introduced. (That's what cardinal methods <br>
with absolute interpersonal comparability do.) Since there's a Condorcet <br>
order, the *ranking* of the other candidates don't change when we <br>
introduce a third, e.g.<br>
<br>
A>B becomes A>B>C after adding C,<br>
B>C becomes A>B>C after adding A.<br>
<br>
But if the scores were to stay the same, then B's score would have to be <br>
40% and 75% at once. That's clearly impossible.<br>
<br>
So majoritarian methods' scores, if they're numbers on a scale, must to <br>
some degree be relative.<br>
<br>
I've found two ways to more or less consistently normalize the scores to <br>
the number of candidates. One is to keep the top two scorers' score the <br>
same, and the other is to make the scores sum to 100%. (For lack of a <br>
better term, I'd call the first "minmax-style" because that's what <br>
minmax does.)<br>
<br>
I've also found two ways to calculate these scores - one that's <br>
appropriate for LIIA methods, and another that should work on a much <br>
broader range of methods.<br>
<br>
So let's do the calculation types first:<br>
<br>
The LIIA style is this: Suppose that candidates are ordered x_1 > x_2 > <br>
x_3 > ... > x_n, and the pairwise victory of x_k against x_(k+1) is <br>
d(x_k, x_(k+1)). Let the score of candidate x_k be s_k. Then set<br>
<br>
s_(k+1)/s_k = d(x_k, x_(k+1))/d(x_(k+1), x_k)<br>
<br>
for k = 1..n-1.<br>
<br>
This is a set of n-1 equations with n unknowns: the normalization method <br>
fixes the last unknown.<br>
<br>
For the A>B>C example above, we'd have:<br>
s_A/s_B = 60/40<br>
s_B/s_C = 75/25.<br>
<br>
The nice thing about this approach is that the relative scores of <br>
adjacent candidates stay the same as long as the social ranking/ordering <br>
stays the same; in particular when losers or winners drop out and the <br>
method passes LIIA, the relative scores of the other candidates stay the <br>
same.<br>
<br>
The plump style that's applicable to more methods is this: For each <br>
non-winning candidate x_k, let P_k be the number of plump/bullet votes <br>
for x_k that have to be added to the election to make x_k the winner. <br>
Let P_none be the number of such votes that have to be added for a new <br>
candidate (that currently has no support) to win.<br>
<br>
Then set up a linear scale that maps 0% to P_none, and set x_1/x_2 = <br>
d(x_1, x_2)/d(x_2, x_1).[1]<br>
<br>
This again has one more unknown than equations (the rate of change of <br>
the linear scale, or equivalently, the "virtual" negative value P_1 that <br>
should be assigned to a winner, since the winner needs no plump votes to <br>
win).<br>
<br>
And again, the normalization choice determines that unknown.<br>
<br>
<br>
Now for the normalization approaches:<br>
<br>
Minmax style is simply this: Let the winner x_1's score be d(x_1, <br>
x_2)/d(x_2, x_1), so that the top two's scores stays the same no matter <br>
how many losers are removed from the election (assuming LIIA).<br>
<br>
For the A>B>C example and LIIA-style relative scores, that gives:<br>
s_A: 0.6<br>
s_A/s_B = 60/40 = 0.4<br>
s_B/s_C = 75/25 = 2/15 = 0.1333...<br>
<br>
so<br>
A: 60%<br>
B: 40%<br>
C: 13.3%<br>
<br>
Sum-to-100% is just what it says. For the A>B>C example:<br>
<br>
s_A/s_B = 60/40<br>
s_B/s_C = 75/25<br>
s_A + s_B + s_C = 100%<br>
<br>
which gives<br>
<br>
s_A = 9/17 = 52.9%<br>
s_B = 6/17 = 35.3%<br>
s_C = 2/17 = 11.8%<br>
<br>
<br>
Finally, here are the different scores for the Burlington election with <br>
RP(margins) as the base method for the plump calculations:<br>
<br>
(Minmax style)<br>
<br>
Candidate LIIA-relative % Plump-based %<br>
<br>
Montroll 53.91% 53.91%<br>
Kiss 46.09% 46.09%<br>
Wright 43.42% 43.53%<br>
Smith 41.43% 38.67%<br>
Simpson 5.36% 7.78%<br>
Write-in 0.26% 1.15%<br>
<br>
(Sums-to-100)<br>
<br>
Candidate LIIA-relative % Plump-based %<br>
<br>
Montroll 28.38% 28.21%<br>
Kiss 24.20% 24.11%<br>
Wright 22.80% 22.78%<br>
Smith 21.75% 20.23%<br>
Simpson 2.81% 4.07%<br>
Write-in 0.14% 0.60%<br>
<br>
Minmax-style is more like "approval ratings" while sums-to-100 is more <br>
like "how big a share of the total". I'd be inclined to say sums-to-100 <br>
is more natural given that ranks are relative anyway, but what do you think?<br>
<br>
-km<br>
<br>
[1] Alternatively make the algorithm accept negative ballot counts and <br>
see how many negative plump votes have to be added until adding more <br>
makes the winner lose, and fix the scale so that zero plumpers added <br>
would give 50%. "Negative plumping" like this is possible with ranked <br>
pairs, and the values and two scale points determine the system, thus <br>
making normalization unnecessary. It gives results similar to <br>
minmax-style, but slightly different (e.g. Montroll gets 54.44% instead <br>
of 53.91%)<br>
----<br>
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</blockquote></div>