A Popular Fix for Fuzzy Math?

By Barry A. Cipra

SIAM News, December 21, 2000

To count is a modern practice, the ancient method was to guess. ---Samuel Johnson

One of the lasting lessons of the 2000 U.S. presidential election is that counting votes is an inexact process: No matter what you do, it seems, some small percentage of ballots are bound to be miscounted. Most of the time it doesn't matter. But as the photo finish in Florida showed, a close race raises the rancorous question, Who "really" won?

Three researchers---Itai Benjamini and Oded Schramm, both of Microsoft Research, and Gil Kalai of the Hebrew University in Jerusalem---have turned the question around: How close does a race have to be in order for errors in counting to have a non-negligible chance of reversing the outcome? Their analysis indicates that a simple, nationwide popular vote would be more stable against mistakes than the beleaguered Electoral College system. Indeed, they find, straightforward majority vote is more stable than any other voting method.

Part of the analysis is obvious: The larger the voting pool, the less likely it is for mistakes to make a difference. Thus, Al Gore's popular-vote lead of 207,694 votes out of 99,268,244 ballots counted as of November 16 can be considered safe even if as many as 10% of the ballots were mistakenly counted. (This assumes, however, that errors are random, with no bias toward either candidate---which seems not to have been the case in Florida.) On the other hand, George W. Bush's 300-vote lead among Florida's 5,820,684 ballots has to be considered statistically dicey, even by Republicans.

But the Florida fiasco required two pieces of bad luck. Not only did the Florida vote have to come down to the wire, but so did the Electoral College vote: If Gore had eked out Tennessee, or if Bush had grabbed Pennsylvania, the close count in Florida would have quickly become a footnote. The non-obvious part of the analysis is that that kind of double whammy is still more likely than a single, nationwide whammy. In particular, for an extremely close race (in which, say, voters simply toss a coin in the voting booth), a mistake in counting every kth ballot leads to a roughly 1/k^{1/2} chance of reversing the "true" outcome in the popular-vote model, whereas it gives a roughly 1/k^{1/4} chance in an Electoral College-type approach.

The trio's electoral conclusions are based on theorems from their paper "Noise Sensitivity of Boolean Functions and Applications to Percolation," to appear in Publications Mathématiques of the IHES (see http://front.math.ucdavis.edu/math.PR/9811157). They note that the same mathematical conclusions apply to another vote-counting system as well: the brain. If a neuron bases its decision to fire or not on potentially unreliable input from other neurons, it may do better to get the input directly from all the sources, rather than having it passed up through a hierarchy. Could that explain the Nader vote?