Are Small State Voters Overrepresented in the Presidential Election?
At first glance, it seems obvious that the answer is yes, since smaller states get more electoral vote relative to population than big states. But the question turns out to be a good deal more complicated than that.
Define a voter's voting power in an election as the probability that his vote will decide the outcome. Consider two states, one ten times the size of the other. Assume, temporarily, that the large state has ten times the electoral votes of the small. Assume, also temporarily, that the two states have the same probability distribution for the outcome, except that the distribution for the large state is proportionally stretched relative to that for the small. Assume that the chance that a state will determine the outcome of the presidential election is proportional to the number of electoral votes it casts.
On these assumptions, the analysis is straightforward. A voter in the small state has ten times as large a probability of deciding his state's outcome as a voter in the large state, but the small state has one tenth the probability of deciding the outcome of the election, so the two voters have the same voting power. Add into the model the fact that the small state has more than a tenth the electoral votes of the large and we get the obvious, and I think widely believed, conclusion—that small state voters are overrepresented.
There is, however, one more assumption we need to drop. The probability distribution for the outcome in the large state is not simply that in the small state stretched out by a factor of ten. The law of large numbers tells us that, all else being equal, the distribution in the large state will be more sharply peaked than in the small. If the peak of the distribution is at .5, that means that the probability of a one vote victory, making every voter on the winning side decisive—changing his vote would reverse the outcome—is more than a tenth as high in the larger state as in the smaller. That gives us the opposite of the previous result: Voters in smaller states are underrepresented. Since the two effects go in opposite directions, one cannot tell, on theoretical grounds, what the net effect is.
There is, however, one more complication we need to deal with. In the previous paragraph, I assumed that the peak of the probability distribution for a state's electoral outcome was at .5—that the random voter had exactly .5 probability of voting each way. That is unlikely to be true, even if we limit ourselves, as we should, to states where the vote will be close.
Suppose the typical voter has a .51 probability of voting Republican. The sharper probability distribution for the larger state increases the probability that the outcome will be 51/49. But it might, depending on how sharp the peak is, decrease the probability that the outcome will be 50/50, and so the probability that one vote will be decisive.
My conclusion is doubly indeterminate. One factor results in overrepresenting voters in small states. The other might result in over or underrepresenting them. Figuring out the actual effect would require a fairly careful examination of detailed electoral evidence. Being lazy—hence a theorist—I will leave that job to someone else.