The Genie, Stephen and Some Bricks Part I: Disputing Standard Economic Voter Theory

Stephen Dubner and Steven Levitt proposed in 2005 that there might be four rational reasons why Americans vote:

1)      They are behaving “rationally” under incomplete or false information that their vote might “matter” in the sense of influencing the electoral outcome.

2)      They vote because it would be fun, even if very low probability, if their vote did “matter” in the sense of influencing the electoral outcome (this is compared to entering a lottery).  This assumes that there is a binary where either you do or do not influence the outcome of the election, with some low probability of the former.

3)      They vote because they believe they have an ethical duty to do so.

4)      They vote in order to receive the social returns associated from walking around for a day with an “I voted sticker”

The third reason, in my opinion, is generally not given enough credit in literature, because if it is morally imperative to discharge some duty, then it would not be “irrational” to do so.     Economists are more than happy to count agents as rational when they “do the right thing” (because they count “doing the right thing” in the benefits column); why this seems so non-important for them when it comes to analyzing voting is something I don’t understand.  I’ll ignore that issue for now (and leave it to a future post), to ask another question—does only the tie-breaking vote in a winner-takes-all system “count?”  Dubner and Levitt were writing for the popular press, as some other authors have about this, but they are making the same assumption that most economists and political scientists make about voting: voting is rational if and only if it changes a policy outcome, which is to say if and only if it breaks a tie.

One viable route to  challenge that idea would be the notion that our votes represent our beliefs, and so margin of victory does matter, even in a winner takes all system.  When politicians and the press use phrases like “having won a solid mandate from voters,” they are implicitly endorsing the idea that the larger your majority, the more you have just cause to aggressively pursue your agenda.  In that sense, any vote, for the winning candidate for any losing (including third-party candidate), is not irrational, because it changes the “mandate” of the winner by some fraction.  This isn’t a new idea of mine, so I’ll keep going.  There are some other reasons that Dubner and Levitt don’t mention about why it might be rational to vote (my sister’s suggestion was among my favorite—voting gives you the right to complain about politics).

The real reason why I think it is “rational” to vote (although I mostly think it is morally right to vote, or more directly, powerfully wrong not to vote; put alternately, we reason to ourselves that it would be wrong not to vote), still accepts the idea that the goal of voting isn’t to “express one’s will” but instead, to influence policy outcomes through the electoral process.  I’ll buy what Dubner and Levitt say about the rationality of voting—that it is only rational if you impact policy outcomes, but by disputing their understanding of cause and effect, I claim that you impact policy outcomes even when your vote does not break a tie.  I’ll call my theory the Brick Theory of Voting.

First, let’s make an observation—all citizens in the United States have equally-weighted, identical votes for most types of elections (e.g. senatorial, gubernatorial, mayoral—basically, everything except the presidential election with the electoral college system).

Let’s say you, Stephen Dubner, and eight of your other friends are walking down a path, when all of a sudden, you encounter a poorly placed fence in the way of your path.  (In case you’re wondering, Dubner is depicted from his characteristically shaggy brown hair.  You’re the one with the cool red shoes and the funky fresh scarf).

You’re all about to turn around when a magical genie appears!

The genie says “I’m not going to tear down this fence for you, but I will give you the power to tear down the fence for yourselves.    I’m giving you each a brick.  None of you is strong enough to break the fence alone, but if at least six of you throw your brick at the same time, you the fence will break.  If five or fewer throw your bricks, the fence will be unchanged.”

He continued, “To make this fun, I’m only going to give you one chance.  We’re going to do this at the count of three.”




7 people had decided to throw their bricks, so the fence broke and everyone could get by.  Notably, you decided to throw your brick, and Stephen did not.

Later, the two of you decided to talk about your decision.

“Well that was awfully dumb of you, don’t you think?” said Stephen.


Stephen: You could have just as easily not have thrown your brick, and nothing would have changed.  Your brick was wasted!  You must have known in advance that the chance your brick would have been the deciding brick would have been very small!  I will explain to you my general philosophy of voting.  It is pretty similar to my general philosophy of brick-throwing, except that the math is just very slightly different.”

Stephen continued, “It is very easy to see that in most situations, it does not make very much sense to vote. Let’s make some observations.”

Stephen continued still, “ (1) It only makes sense to vote if there is a non-trivial chance that your vote will influence the outcome of the election.  If there is only a trivial chance (a chance very very close to zero) that your vote will influence the outcome of an election, I claim that it is IRRATIONAL to vote. (2) You influence the outcome of the election in a “winner takes all” system if and only if your vote was the deciding vote (e.g. if it breaks a tie).    Otherwise, you might as well sleep in on election day, because you are meaningless and your vote has no impact. (3) In the most “generous case,” we can assume voters will vote between two candidates with a probability of .5 (e.g. they  randomly decide between two candidates).  This parameter will make it more likely to result in a tie in large populations, therefore increasingly the likelihood that your vote will break a tie.”

A genie provided a helpful table to help Stephen illustrate his point.

You say, “Woah, woah, woah…. Why is it that you only influence the outcome if your vote breaks a tie?”

Stephen says, “Isn’t it obvious?   If I wasn’t the tie-breaker and I vote, the winner still wins.  If I don’ t vote, the winner still wins.  It must mean that I did not influence the outcome of the election, and it means that I had no impact.  In small populations, like 11, there is a reasonable chance I can break a tie (almost 25%) so it makes sense for me to go vote.  But in big populations, like North Carolina, the probability becomes very small (only .03%), and that’s assuming that everyone votes randomly!  Only one Therefore, in large populations, it is irrational for me to vote. Of more than 16,000 Congressional elections in the last century, only one in the United States has been decided by a single vote; of over 40,000 state legislature elections, only seven have been.  No matter how low the costs of me voting are, because the probability of me having an impact is hopelessly small, it is irrational for me to vote.”

You say, “I think your understanding of causality may be leading to your error!”

He says, “How so?”

You: “You think the probability of me having an impact is hopelessly small.  I don’t think that’s true. Let’s return to the fence example.”

Stephen: “You must have felt really lame wasting your brick, spending all that energy on throwing it, to realize that you could have not thrown it and the universe would have been exactly the same.”

You: “But I helped break the fence.”

Stephen: “No you didn’t, the fence was going to break either way.”

You: “Which bricks broke the fence?”

Stephen: “Six of them.”

You: “Which six bricks broke the fence?”

Stephen: “Some combination of six”

You: “But which six bricks broke the fence?  How do I know that my brick didn’t break the fence?”

Stephen: “I mean, I guess its not like some particular combination of six bricks broke the fence.”

You: “So maybe a minimum of six bricks was needed to break the fence, but all of the bricks that were thrown contributing to breaking the fence?  Since all of the bricks were identical, all of the bricks impacted the outcome.  To explain this in the most didactic way possible, we know that bricks broke the fence.  We know that every brick was identical in every way.  Because every brick was identical in every way, the impacts each brick had on the fence were identical.  Because the impacts each brick had on the fence were identical, and the bricks cause the fence to break, it must have been that each brick had some impact on breaking the fence.”

Stephen: “I suppose so.”

You: “In 2010, Floyd McKissick won my district by 24,217 votes.  I could have not voted, and he still would have won the election.  As long as he had more than John Tarantino’s 14,092 votes, he was going to win.”

Stephen: “Right”

You: “So a minimum of 14,093 votes was needed for McKissick to win, but all of the votes cast for McKissick contributed to him winning.   Since all of the votes are identical, all of the votes impacted the outcome.  You think my vote never impacts outcomes.  In fact, it impacts outcomes all of the time.  My fraction of the impact is small, but nevertheless present (much like my fraction of the impact on the environment is small).   Instead of thinking that an individual voter does or does not have enormous impact with some very small probability of the former, maybe instead an individual voter always has a small impact. 

Stephen: “That is an interesting different way of thinking about cause and effect.  Maybe you are right.  I will meditate on this and get back to you.”

Look out for Part II (The ethics of voting), and Part III (The political economy of small-impact social situations)


Published by Elena Botella

Elena Botella is a Duke Undergraduate in the Class of 2013, majoring in Economics and Math.

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