A quick and dirty strength of victory measure for LIIA methods
This came up in the Electorama call, and I thought I'd elaborate:
Here's a score/support level concept that isn't too theoretically sound,
but might be interesting. Suppose that we want to know not just who won
according to a LIIA method, or the order of finish, but also how
comfortable their victory is, or what their support level (of some kind) is.
Then if the method passes LIIA, we'd like the support values to have
some kind of preserved property so that, if we remove the loser (or the
winner), the relative scores between the remaining candidates don't change.
A simple idea I tinkered with, that should pass this property, is as
follows:
Let the social order from some LIIA method be A>B>C>D, and let the
unknown support values (strength of victory, candidate scores) be x_A,
x_B, x_C, and x_D.
Then set
x_A / x_B = (A>B)/(B>A)
x_B / x_C = (B>C)/(C>B)
x_C / x_D = (C>D)/(D>C)
||x|| = 1
An example: Using the 2009 Burlington election gives the following set
of equations:
x_Montroll / x_Kiss = M>K/K>M = 4067/3477
x_Kiss / X_Wright = K>W/W>K = 4314/4064
x_Wright / x_Smith = W>S/S>W = 3975/3793
x_Smith / x_Simpson = S>Si/Si>S = 5573/721
x_Simpson / x_WriteIn = Si>Wr/Wr>Si = 3338/165
x_Montroll + x_Kiss + x_Wright + x_Smith + x_Simpson + x_WriteIn = 1
This ends up being approximately:
x_Montroll = 28.30%
x_Kiss = 24.20%
x_Wright = 22.80%
x_Smith = 21.75%
x_Simpson = 2.81%
x_WriteIn = 0.14%
The intuition here is that A's score relative to B's should be the same
as their relative pairwise victories. (So if A beats B 10% more strongly
than B beats A, A's score should be 10% higher than B's.)
In addition, to preserve LIIA[1], the relation between A and B's scores
should only depend on pairwise contests involving A and B. Eliminating A
would not alter the order of the remaining candidates, so the social
order would simply be B>C>D; and the equation set would be reduced to
x_B / x_C = (B>C)/(C>B)
x_C / x_D = (C>D)/(D>C)
(...)
But since there are n candidates and only n-1 such relations, this only
sets the support levels up to some constant of proportionality. Here
I've chosen to make total support sum up to 100%, but that does
compromise its mathematical elegance. In addition, it's not cloneproof.
Imagine adding a bunch of clones of ay, Montroll, with the voters
ranking them in a random order. Then we get
x_M1 / x_M2 = 1
x_M2 / x_M3 = 1
...
x_M(k-1) / x_M(k) = 1
which will lower everybody's support numbers as Montroll's support is
distributed over all his clones and the other candidates still need to
have a lower score.
But it might still be interesting.
I think the most important part is that, if we're to transplant LIIA,
then the ratings/support values can only depend on the pairwise
victories along the path of the social order. I.e. we can't involve
Montroll>Wright when determining Montroll's score. The only other bit
that stays the same is the total number of voters (and only then,
sometimes). Perhaps it can be used to solve the clone dependence
problem? Or maybe one has to relax the LIIA-ish property.
-km
[1] Formally speaking, this property is: if you're basing the scores
from the social order (order of finish) of a method that passes LIIA,
and you get the scores for two overlapping candidate sets (both of which
have been selected by repeatedly eliminating losers or winners from a
larger election), then you can determine the scores for the election
containing every candidate in the union. (Ideally, I'd want the scores
to not change at all, so that if candidate A is in both elections, A has
the same score in both; but I'm not sure that's possible.)