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Re: Math and the electoral college's virtue
Posted:
Nov 20, 2000 1:27 AM
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In article <3A1081DB.32AC595A@math.ucla.edu>,
Mike Oliver <oliver@math.ucla.edu> writes:
|Since at the end of the day I'm more or less an (individualist) anarchist,
|I'm not going to enter into a normative discussion of what constitutes
|a "good" voting system. But *descriptively* this idea of voter
|power makes a lot of sense, and implies that almost everyone who hasn't
|specifically studied the issue has precisely the wrong notion about
|who is favored by the electoral college system.
I don't think you can get very far in trying to describe who is
"favored" by a system without considering some notion of what
constitutes a fair, equitable such distribution of power, not that
this is really the place to do it. If we want to make a nonnormative
description, it has to be just a description of what outcomes arise
and stuff like that.
|Most people who look at the issue superficially note that even the
|least populous state gets three electors, and conclude (quite wrongly)
|that the system gives disproportionate power to the small states
|because their ratio of electoral votes to population is higher.
Note that various attempts to eliminate the electoral college have
been foiled in the U.S. senate, where those small states get even
more disproportionate representations. I don't think these senators
are just being dumb; I think they have a different issue in mind. They
want a system that results in outcomes they prefer (basically).
It may be a crude model, but it's some approximation to the truth that
politicians are arranged on a left-right axis, and voters' preferences
among those available alternatives are roughly for the ones closest
to a certain point on it. (There are plenty of those who'd like for
some other issues to be considered, but that would require reorienting
the axis. Meanwhile, you get treated as if you were on the axis.)
Giving Wyoming 5 times the electoral college vote as proportional to
its size moves the point of balance point (as far as presidential
politics goes) to the right. It's no coincidence that we tend to get
presidents and Senates to the right of Houses of Representatives.
|In fact, it's easy to see that other things being equal (the "other
|things" are e.g. the closeness of the race in your state and the
|proportion of undecideds), your chance of deciding the election
|is proportional to 1/sqrt(n) where n is the number of voters.
|Since the number of electoral votes is (roughly) proportional to n,
|it follows that your power is approximately proportional to sqrt(n).
|That is, the system gives disproportionate power to voters in *large*
|states.
But the proportionality to 1/sqrt(n) (for this measure of power) is
only when the race *is* balanced. (Ha, and you thought there wasn't
going to be any math here.) Suppose all voters have .51 probability
for voting in favor of a measure. The distribution of vote totals is
roughly Gaussian with standard of deviation proportional to sqrt(n),
but falling all the way to a tie is proportional to 0.01*sqrt(n)
standard deviations from the mean. On a Gaussian, that decreases the
density much more rapidly than the 1/sqrt(n) you get from the curve
spreading out; it's going down with population like an exponential
divided by sqrt(n).
Remember also that the probability of a states' being a deciding state
in the electoral college isn't just proportional to n. It depends in
some complicated way upon how the sizes are distributed.
There's another measure of power reasonably popular among the social
scientists studying voting systems. It amounts to this (in a case
where the voters have two options only). Arrange the voters in a line
randomly, with equal probability of each permutation. Take the voter
having the property that if she or he votes the same way as everyone
on one side of them, they win. One's power is the probability of being
that voter. I seem to remember that by *that* definition, both large
and small state voters are advantaged when voting for president (with
a minimum in the middle for states of about the population of Colorado,
oh joy).
|Now you might well ask what practical difference this makes, given
|that (all the propaganda to the contrary notwithstanding) never
|has a state really been decided by a one-vote margin.
I have never seen propaganda claiming that a state has been decided
by a one-vote margin. I suspect you're just making the mistake which
mathematicians (and anarchists) are at times prone to, of substituting
a precise technical criterion for a vague assertion as if it meant the
same thing. When people say, "your vote counts" and things like that,
they do NOT normally mean "there's a substantial probability that if
all other voters had voted the same way as they actually did, but you
voted differently, that the outcome would be different". I never have
heard anyone say that, and I don't think it's a faithful precision on
what they do mean.
That would be a suitable criterion for hunting for a Nash equilibrium
of this "game", but we're not hunting for a Nash equilibrium. I don't
know of a precise way to describe what we're doing, but it's not that.
It seems like we compromise between the purist individualism of hunting
for Nash equilibrium, and the choice to act in a way that we'd be happy
to have everyone emulate. I take *some* credit and/or blame for the
net effect of people acting like I do, even if adding or subtracting
a few individuals on the margin wouldn't change the result. I care
more seriously about voting when the margin is close, and I don't
think this is genuinely irrational of me, however un-Nash-like it is.
|Well, the reason it makes a difference is that the same arguments
|hold for, say, a block of 100,000 votes. So lets say I know I'm
|soon going to be running for president, and I'm in charge of
|a committee that's assigning a pork-barrel project that I expect
|to be worth 100,000 votes in the state in which I place it, and
|my choices are California and Wyoming. If my main concern is
|my presidential ambitions, I'd be nuts to put it in Wyoming.
I don't think it would be any easier to get Wyoming to vote the other
way than it would be for California, size notwithstanding. Anyone who
is to the right of their opponent who loses Wyoming was going to lose
anyway.
If it were all a question of pork and not ideology, I think it would
be different in some complicated way; trying to form a coalition of
Western small-state voters by subsidizing the region would become a
more attractive strategy-- fewer of us to be bought off, as it were,
for the number of electoral college votes-- but people seem not to be
so purely greedy as to make this work.
Keith Ramsay
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