I hope I can be forgiven a brief digression into politics on this,
the US's 242nd Independence Day.
In response to the pending resignation of one of the nine current
Supreme Court justices, charts showing all nine, in order of their
alleged position on the supposed political spectrum, have been in
the news lately.
I've long thought that the "left" and "right" are more like random and
shifting assemblages of ideas than any sort of coherent, persistent, or
self-consistent political, legislative, legal, economic, or judicial
philosophies. I usually can't tell who is what. And people who know
me strongly disagree with each other on where on the political spectrum
I am (except that nobody thinks I'm moderate).
Can we use math to determine whether the idea of a single
one-dimensional political spectrum has any utility in explaining the
behavior of Supreme Court justices, senators, U.S. states, etc.? Can
we arrange the justices (or senators, etc.) in a line such that for
every yes-no decision they made, there's a point which everyone who
voted yes is on one side of and everyone who voted no is on the
other side of? The point can be in a different place for each such
decision, and yes and no can swap sides. But, for instance, if
justices A, B, and C are in that order, A and C should never agree
with each other except when B agrees with both of them.
The answer is almost certainly no. But how close can we get? Can we
get it to work for 99% of their decisions? For 90%?
What's an efficient algorithm for finding out? With just nine, it's
practical, with a modern computer, to try all 181440 (9!/2) possible
orders. But with 50 states, 100 senators, or 535 members of
congress, that would obviously be intractable.
Some claim that the two extremes of the spectrum are identical, or
close to it. We could test that by making the spectrum a circle, and
seeing if there's always a line that will divide everyone who voted
one way from everyone who voted the other. (A different line each
time, but with the justices (or senators, etc.) always in the same
order around the circle.)
Or we could try a two-dimensional spectrum. Place the justices on a
plane such that for every decision a line exists that divides the yes
votes from the no votes. Again, the spectrum may wrap around in one
(a cylinder) or both (a torus) dimensions.
Similarly with higher dimensions. Of course with N voters, if you
choose N-1 dimensions you can always find an N-2-plane that divides
the yes votes from the no votes, even if they're completely random.
Just as any N points can be perfectly, but uselessly, fit by some
order N-1 polynomial.
But I'm way over my pay grade, even for two dimensions. Just how
many "orderings" of N voters in a plane are there, for the purposes
of finding whether such a line exists?
Has anyone already done such work, especially with real-world data?
Also, where can I find complete machine-readable real-world voting
records for justices, senators, states (in presidential elections),
etc.? Thanks.
