Bullet voting, pro and con

On Tuesday, Boston voters will go to the polls in the “preliminary election” for City Council. Something like a primary, the preliminary election narrows each race down to a number of candidates equal to twice the number who will be elected. Thus we will narrow the 15 candidates for at-large City Councilor down to 8, since 4 will be elected in the final election; and where there are more than two candidates for a district City Councilor we will narrow them down to two, since one is chosen in each district. (When there are only two candidates, as there are for Mayor this year, no preliminary election is needed and therefore none is held.) I say that the preliminary is something like a primary, but it’s non-partisan since practically every candidate in the city of Boston is a Democrat.

Anyway, the important mathematical issue is this: in both the preliminary and final elections, each voter is entitled to cast four votes for at-large City Councilor, but is it wise to exercise that right or should one cast only one vote, which is known as a bullet vote? (Intermediate positions — casting two or three votes — are also possible of course.) I’ll present both sides with an example from the final election, though the argument holds equally well for the preliminary except that the cut-off point is between the eighth and ninth positions rather than between the fourth and fifth:

  • Suppose you and 20 others agree that your first choice is Sam Yoon and your second choice is Matt O’Malley. And suppose, before your 21 ballots are counted, Yoon is in a tight race for fourth place, just 13 votes behind O’Malley. If the 21 of you bullet-vote for Yoon, your votes put him 8 votes ahead of O’Malley for fourth place, letting Yoon squeak in. But if the 21 of you dilute your support by voting for both of these fine candidates, Yoon remains 13 votes behind O’Malley, who then pushes Yoon out of fourth place. Net result: your second choice would be elected and your first choice defeated. Bullet voting is clearly the correct strategy.
  • On the other hand, there is also an argument against bullet voting. To understand this argument we have to consider a candidate of medium popularity whom you definitely do not want elected. Let’s call him Steve Murphy, to pick a name at random. Now suppose Murphy is 13 votes ahead of O’Malley for fourth place before the 21 ballots from you and your friends are counted. And suppose that Yoon is way ahead (or way behind, it doesn’t matter which), so your 21 votes for Yoon won’t make any difference in his election. If you bullet-vote for Yoon, what happens? Murphy gets elected by finishing in fourth place! But if you each cast two votes — one for Yoon and one for O’Malley — then O’Malley finishes 8 votes ahead of Murphy and squeaks in for a fourth-place victory! Net result of refusing to bullet-vote: your second choice would be elected and your last choice would be defeated. Bullet voting is clearly not the correct strategy!

So, to bullet-vote or not? The choice depends entirely on which scenario you think is more probable. If the race for fourth place is between your two favorite candidates, you should bullet-vote. If the race for fourth place is between your second choice and someone you strongly dislike, you should not bullet-vote. Maybe it’s not how they would handle it on Numb3rs, but math can’t give you a definite answer here. Actually, come to think of it, it is rather like last week’s Numb3rs, since it’s all a matter of Bayesian probabilities.

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