Saturday, April 23, 2011

It IS worth it to vote: the math

A lot of people don’t vote, and a lot of different rationales are offered for this behavior. I want to address two rationales that I believe are incorrect; people using these reasons should reconsider. Reason 1: you are very unlikely to alter the outcome. Reason 2: it is not worth your time. I’ll give Reason 1 a rigorous treatment below. Reason 2 is tricky. What does it mean for something to be “not worth your time?” If you take 30 minutes to do something, and it provides only a dollar’s worth of benefit to you, is it worth doing? Suppose it provides a dollar’s worth of benefit to every human being alive? If you live in the US, your answer to the first question is probably “no.” You are spending your free time to earn at the rate of $2 an hour; your time would be better spent working, or doing whatever leisure activities you enjoy. Hopefully, if you have any sort of social conscience in you, your answer to the second question is “yes.” If I could work for half an hour and generate $7 billion worth of benefits, I should be happy to do so. It shouldn’t matter that the benefits are widely dispersed. If there are many opportunities like this, and people routinely take them, we will all be much better off. I’ll submit that most people choose not to vote on the rationale that IT’S NOT WORTH IT FOR ME. We should be worried about the total social benefit (or cost, if the sign comes out to be negative) of our actions.

Before I calculate any probabilities, I should also note that “the chance that you affect the outcome is small” is a bad argument. We should be willing to do things that have a small chance of a large positive outcome. If people generally take these chances, we’ll tend to be better off. Each action has an “expected benefit,” which can be thought of simply as “the probability of success multiplied by the benefit GIVEN success.” Lottery tickets are a bad bet; the expected return is negative. If the return were positive, you should be willing to buy one once in a while. If you can buy a million such “positive return” lottery tickets, you will have a good business model going. And if the benefits of the occasional winning ticket get dispersed broadly so that YOU cannot collect, your social conscience should still encourage you to drop a few bucks on these tickets once in a while. To know whether or not “voting” is a positive or negative return lottery ticket, you need to estimate a couple of things. One, the probability that you sway an election, and two, the benefit to society given that you sway the election in your favor.

I’ll use a very simple model to estimate P, the probability of swaying a vote. The conventional wisdom is that your vote only counts if there is a tie, and you are the tiebreaker. So my model only applies to a “dead heat” election, in which the prior chance of either outcome is 50/50. I’ll assume that everyone has a 50% chance of voting for either candidate, essentially determined randomly. (This assumption makes my estimate of P quite conservative; in reality a large majority of people are locked into voting for one particular candidate and a few swing-voters decide the outcome. But I’ll assert that the “each vote is a coin-flip” model is pretty good for an election that is too close to call.) With 10 voters, the chance of a tie is 0.246. With 100 voters, the chance of a tie is about 0.08. With 10,000 (a small city), it is about 0.008. With 1,000,000 voters, the chance of a tie is about 0.0008. The probability of a tie drops off like 1/Sqrt(N), where N is the number of voters. Even with 1 billion voters, the chance of breaking a tie only goes to 0.000025 (25 in 100,000).

Your chance of being the tiebreaker is a lot larger than you probably thought. (An objection to this model is that no one REALLY breaks a tie; in the event of a close election, there is always a recount. The prospect of a recount DOES NOT AFFECT the estimate of the probability estimates. I’ll explain why at the end of this post.)

Next, you must estimate the social value of the electoral outcome you favor. This is a lot trickier, but you can at least estimate some rough measure of value. I would unhesitatingly assign a value of $1 Trillion to “Ending the War on Drugs.” Perhaps another $1 Trillion to reforming our massive entitlement programs. The social benefit per person may even be small, but add up to billions or trillions when summed over all individuals affected. We’ll call the total social benefit of your favored outcome B, and observe that it is linearly proportional to N.

The total benefit of your vote is then simply P*B. There is something surprising about this quantity, having to do with how it scales. B is proportional to N, and P is proportional to 1/Sqrt(N) (N again being the number of voters). The social benefit of your vote is thus proportional to Sqrt(N). In other words, the greater the count of the electorate, the greater the social benefit. I’m making a number of assumptions here, but I’ll leave it to the reader to determine how these assumptions affect my simple model. It’s possible that some assumptions make this estimate of P*B more conservative, and some make the estimate less credible. An incorrect assumption doesn’t always discount the conclusion; some incorrect assumptions bolster the conclusion, which in this case is “you should march down to the polls and vote.” I don’t believe any of my assumptions affect the scaling result: P*B IS proportional to Sqrt(N), however you slice it.

We are all slight sociopaths, so we don’t really care so much about the social benefit of our actions if we don’t personally benefit from those actions. This is especially true if we don’t even get credit for being charitable. The benefit to YOU scales as 1/Sqrt(N), so the more voters in the voting pool, the less you care about your contribution to the outcome. I’ll submit that you shouldn’t care whether the benefit is captured by you or dispersed across the population, and that you certainly shouldn’t care whether your charity is recognized or unrecognized. I realize that we aren’t built to comprehend things like “a tiny chance of a very large but dispersed benefit.” But most of us are willing to do a little charity, and most of us are willing to put a small amount of money/effort into some risky venture with a positive rate of return. There is no reason that this willingness should disappear when the action you’re contemplating is charitable AND venturesome. Somehow combining these two traits makes voting seem so useless to so many people. I realize I am suggesting that people do something they have no incentive to do (be charitable), which is much like preaching to the dead. But many of the people who offer the “voting isn’t worth your time argument” are of the blogging intellectuals type. They apparently are already willing to create some public goods in the form of reasoned arguments (their best efforts, anyway). If you’re willing to be charitable in ONE way, you should be willing to be charitable in others ways of similar magnitude. I’m not suggesting that anyone should be MORE charitable than they already are; I’m suggesting that a potential form of charity has more social value than most people believe. (An important difference between the “public good” generated by free blogging and that generated by voting: you capture some of the externalized benefits of blogging when people remark about how smart and clever you are; you are met with indifference and sometimes derision when you tell someone you voted.)

Assuming you have an estimate of P*B in mind, there is an additional reason to discount my analysis. “Discount” means “multiply by some fraction D, to reflect some diminishment in value for some particular reason.” It doesn’t mean “totally disregard.” Sometimes it doesn’t matter who you vote for because the candidates will do the same thing whoever is in office. I would suggest that there is usually at least a difference in probabilities that different candidates will do different things. Suppose McCain had, by your estimate, a 60% chance of opening up offshore drilling and Obama had a 40% chance. (These are prior probabilities, mind you. Otherwise I’d have to update Obama’s 40 to 100.) You still are shifting the odds 20% in favor of the policy you like by voting for your candidate. Apply a discount factor of 0.2 to your estimate of P*B. The social benefit of voting may still be large. There are probably a few cases where you have no idea what any of the candidates will do. In this case D = 0, and it doesn’t matter who you vote for. That would be like voting for whether or not a coin will come up heads on the next flip. (I hope that is an intuitive result which you could have guessed at a priori, and that it is a good limits test of my extremely simple model.) Don’t excuse yourself from the polls just because you aren’t sure what the candidates will do; if there is a reasonable chance that the two candidates will behave differently in office, you have more to think about.

Another perfectly good excuse to disregard my entire argument is this: I would do better to use my 30 minutes on some other charitable venture. I can work an extra half hour and loan the money on Kiva. That is fine. Your time is a finite resource. If the social benefit of “30 minute’s wages going towards microloans” exceeds “30 minutes in the poll,” feel free to pursue your best option. This would be a better objection than any I’ve heard. But if my hunch is correct, most people are dramatically underestimating the social value of their vote. If that is indeed the case, your allocations to different charities should change.

A final legitimate reason to disregard my argument is this: “I don’t know what the candidates will do, and I don’t know which policies are good for us. I have no estimate whatsoever of this silly model’s B or D.” (No excuse not knowing P; I’ve given it to you above.) That is fine; I would encourage such a person NOT to vote. I actually think a lot of people are in this position but don’t realize it.  They have never thought for a moment about the social benefits of the policies they like, and haven’t made a single effort to put a numerical value on it. Luckily such people are disproportionately likely to remove themselves from the voting pool. Many other such people create a nasty form of social pollution by voting uninformed. (And everyone makes this sort of remark about their “benighted” political opposites, so don’t anyone peg me an elitist for making this observation.) As Bryan Caplan points out in his book, “The Myth of the Rational Voter,” these voters tend to not make random errors of the sort that cancel out in aggregate. They make systematic errors in the direction of socially destructive policies. I commend people who recognize their ignorance and don’t vote. (I would commend them more if they did some reading each day and then voted, but I’m happy to take the runner-up.)

To bolster my point, consider that most people (say 90%) will vote one way or another along party lines. This means that the marginal voters are extremely powerful, and my estimates of P above are larger than I suggested. Assuming a dead-heat election, an electorate with a million people but only 100,000 swing voters is actually like an electorate of 100,000 (for which the chance of a tie is a healthy 25/10,000). If your estimate of P is higher, you should be more willing to vote.

I have heard that economics students who learn about game theory tend to play econ games more selfishly than people who don’t. That is, they tend to “defect” more often in the prisoner’s dilemma game, and take more money in the dictator game. I think these people are missing the lesson: they have a chance to create a social benefit and forego that chance. They have correctly learned that THEY benefit more if THEY cheat, no matter what their opponent does. But cooperating creates more SOCIAL benefit, no matter what the opponent does. If you can create positive social value at a small cost to yourself (say, by foregoing the rewards of a silly economics game) you should do it. Smart-asses who are literate in economic theory are far less likely to vote for the reason described in this paragraph. That is a shame, because the same people tend to have libertarian/liberal leanings. They have the potential to be a very powerful swing-vote block if they can get over their self-satisfaction and vote now and then.

*Extra credit: the recount margin. Some of you will observe that a close election is NOT won by a tie breaker. A recount is held, with the winner of the recount declared the winner. Okay, let’s model this. The recount is just as (in)accurate as the original count (which makes the whole thing foolish anyway), so here is the model: the recount has a random 50/50 chance of assigning the office to either candidate. So, where is the recount margin? Is a recount done when the count is off by 10%? By 5%? Even if the recount is not well defined (“too close to call” is randomly determined at each election) it must exist SOMEWHERE for each particular election. And it must be very near the tie, such as “within 10 votes of a tie.” Here is the “recount model”: your contribution is either to push the election into the recount margin when your choice candidate would have lost, OR to push the election OUT OF the recount margin when your choice candidate would have faced a recount. In the first case, you raise the chance of your candidate winning from 0 to 0.5; in the second case you raise the chance of him winning from 0.5 to 1.0. Your contribution to your candidate’s chance of winning is 0.5*P_Into_Margin + 0.5 *P_Out_of_Margin, where P_Into_Margin is the probability that a dead-heat election will result in a count at the low end of the recount margin, P_Out_of_Margin the high end. The probability distribution is symmetric, so these probabilities are equal to each other. The distribution is also very flat at the top, so each probability is essentially equal to P, the chance of a tie (discussed above). (If you flip a million coins, you are nearly as likely to get 500,005 heads as you are to get exactly 500,000.) So your chance of swaying an election is ~0.5*P + 0.5*P = P, the same as in the simple model above. The prospect of a recount does not significantly alter your chance of swaying the vote. I'll leave it as an exercise for the reader to determine what happens if the recount margin is not deterministic but stochastic; it still doesn't alter the result that your chance of swaying an election is large.

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