
Re: Math and the electoral college's virtue
Posted:
Nov 20, 2000 1:27 AM


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 leftright 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 onevote margin.
I have never seen propaganda claiming that a state has been decided by a onevote 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 unNashlike 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 porkbarrel 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 smallstate 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

