COMPUTING SCIENCE

# How to Count

# How Not to Count

After the fall election, when I began thinking about the failure modes of counting machines, the subject seemed like a cute mathematical diversion of no practical application. I assumed that the error rate in real hardware would be so low that a machine would almost never fall off a Hamming cliff. After all, the computer on which I simulated fallible counters is itself built out of flip-flops and similar circuits; if these devices had any appreciable error rate, my program could not possibly run.

Nevertheless, it does appear that voting machines have occasionally stumbled over Hamming cliffs. The 1990 Federal Election Commission report tells the following story: "Consider, for example, the case of the election director who a few nights before the election realized while drifting off to sleep that although she had had her custodians check her lever machine counters to ensure that they changed over from 9 to 10, she had not had them check to ensure that they changed over from 99 to 100. In ordering such a check the next day, she discovered that one in twenty counters failed the test."

Whatever the source of error, counting is not something we can always rely on doing with absolute accuracy. Philip J. Davis writes: "As we get away from trivial sums, arithmetic operations are enveloped in a smog of uncertainty. The sum 12345 + 54321 is not 66666. It is not a number. It is a probability distribution of possible answers in which 66666 is the odds-on favorite."

Mathematically savvy election boards might acknowledge the irreducible fuzziness of counting, and treat it as a statistical process. In a close election the canvassing board could count the votes multiple times, and from the resulting distribution of *R*s estimate *N* for each candidate. An alternative would be never to compile or publish vote totals with greater precision than the accuracy of the process can support. If we can only count with three significant figures, then the true outcome of an election cannot be 2,912,790 to 2,912,253. Both numbers would be more honestly expressed as 2.91 X 10^{6}. Of course this policy would increase the likelihood of ties, but that's just the point. If an election is too close to call, perhaps one should not call it.

Another option is to dispense with counting altogether. Fundamentally, in an election decided by simple majority rule, there is never a need to count ballots. It can all be done by the more primitive operations of sorting and matching. The procedure could be organized like a card game. Bring the candidates together in a room with all the ballots. Then let each candidate claim his or her ballot cards. (This is the hard part, of course, when decisions must be made about dimpled chad.) Then, once all the ballots are assigned, go around the room repeatedly, giving each candidate a turn to pitch a ballot into a discard heap. A candidate who runs out of ballots is eliminated. The last candidate with ballots in hand is the winner.

© Brian Hayes

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