# Rationality of voting etc.

I was going to respond to this post by Philip Goetz (who writes that "voting kills") but I thought it would make more sense to summarize in a post of my own.  Even if you don’t care about voting, these issues–how to compute probabilities of extremely rare events–are relevant in other decision settings.

Goetz reports an estimate by Donald Redelmeier that there is an 18% increase in motor vehicle deaths on election day, corresponding to an average of 24 deaths per year and compares it to the 1 in 60 million probability of decisive vote estimated a few days before the election by Aaron Edlin, Nate Silver and myself.  (If anyone is interested in the details of our calculations, they are in this article.)

So the quick calculation goes like this:  24 out of 300 million is about five times 1 in 60 million.  So, according to these numbers, the chance of your vote making a difference is about five times, on average, as being killed in a car accident on the way to the polls.  On the other hand, people notoriously behave as to underestimate the risk of car crashes, so it’s not quite clear what to make of this.

Some other quick calculations might help make sense of this.  The probability that your vote will swing the election is essentially equal to 1/10,000 times the probability that a change of 10,000 votes will swing the outcome.  This has an average probability of about 1 in 6000, which is a little easier to grasp.

Or you can think about Congress.  There were about 20,000 elections for the House of Representatives in the twentieth century.  None were tied, but about 50 of them were within 100 votes.  (See Exercise 1.5 of Bayesian Data Analysis.)  So the probability your vote is decisive in a random congressional election is about 1 in 40,000.

Finally, Goetz comes back to the argument in my article with Edlin and Kaplan in the journal Rationality and Society on the rationality of voting: voting can be rational for people who care about others.  And, conversely, if you do vote, it is rational to choose the candidate whom you think is best for the country as a whole rather than who you think would happen to benefit you personally.

Goetz also writes, "Rationally, people should be much more interested in local elections, which they have a much greater chance of affecting."  I disagree, and this is actually something we discuss in our Rationality and Society article:  yes, the chance of your vote being decisive is much higher (on average) in a local election, but the stakes are lower.  And it is the product–the probability times the stakes–that is relevant.

None of this explains why people voted in New York (where the probability of a decisive vote was essentially zero) but it gives a reasonable justification for voting as a general habit.

Also–because this point always comes up–let me link to my note on why we can compute the probability of a decisive vote, even though the election might be decided by a recount.

Finally, in response to some of the discussion in the comments about rationality and behavior, I’d like to point you to a wonderful article, The Norm of Self-Interest, by psychologist Dale Miller, in which he argues the following:

A norm exists in Western cultures that specifies self-interest both is and ought to be a powerful determinant of behavior. This norm influences people’s actions and opinions as well as the accounts they give for their actions and opinions. In particular, it leads people to act and speak as though they care more about their material self-interest than they do.

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• http://profile.typekey.com/barrkel/ Barry Kelly

There’s another, more subtle, issue at stake. The type of people who vote are the type of people whose interests get represented, in aggregate, irrelevant of how much of an influence any of those people were individually.

It’s a little like a cultural analogue to the evolutionary reasons for emotions. Anger that results in irrationally disproportionate self-harm is still a useful trait, because the very fact that it is irrational means that one cannot count on the other’s self-interest for protection when risking their anger. Similarly, a cultural group that wields political power through voting gets its interests represented, but only if that culture integrates voting as a norm. If the longevity of interest groups is proportional to the political power they collectively wield, one would expect them to have evolved a norm of voting.

• http://profile.typekey.com/andrewgelman/ Andrew

Barry,

I agree but it’s not even necessary to set up an evolutionary model; more directly, political groups mobilize voters to turn out. Carole Uhlaner wrote a paper on this awhile ago (“Rational turnout: the neglected role of groups”).

Also, getting back to your point, James Fowler has done lab experiments that find that altruistic people are more likely to vote; see here:
http://www.stat.columbia.edu/~cook/movabletype/archives/2005/05/altruism-and-vo.html

Finally, Johathan Nagler and Jan Leighey have done some work comparing the political attitudes of voters and nonvoters:
http://www.stat.columbia.edu/~cook/movabletype/archives/2007/12/what-difference.html

• David

This is very interesting stuff.

There is one thing I was a little unclear on after reading the paper. To drive your 10k path simulation, how are you coming up with the dispersion around your baseline percentage vote margin?

For example, if Alabama is projected to go 60.4/39.6, what drives how many standard deviations are between 60.4/39.6 and 50/50? I realize that there is a multi-state interaction effect as well, but if we can just isolate this component by examining a single state example, I’d appreciate it.

Thanks,
David

• http://michaelkirkland.org/blog Michael Kirkland

The idea that your vote swinging an election being very low meaning that voting is pointless always bugs me. For one, an election that close is essentially hung; a single vote is well below the level of noise in any sizable system. There will be lots of debatable spoiled ballots. For another, the vast majority of elections are not binary choices, but a selection from several choices. There’s also proportional representation to think of, where every vote has a definite value.

More importantly, voting is an inherently collective action. You can’t single out any particular ballot as deciding, even in a 50% + 1 situation. Futher, though it’s not as apparent in the American us vs. them style of government, most democracies have a concept of mandate; the more a candidate or party wins by, the more they have a right to pursue the agenda they ran on.

• http://profile.typekey.com/andrewgelman/ Andrew

Michael:

1. I do not say that voting is pointless. See my article with Edlin and Kaplan linked above.

2. Regarding your argument about spoiled ballots: this does not essentially affect the probability. See my link above (near the end of my post) on “why we can compute the probability of a decisive vote, even though the election might be decided by a recount.”

• Unnamed

Shouldn’t the probability of a decisive vote in a House election be (50/20,000)/200 = 1 in 80,000? You need to halve it to account for the fact that, half the time, additional votes just add to the winner’s lead.

• http://www.scissor.com/ William Pietri

I’m puzzled by the notion that voting, an essentially communal act, can be usefully looked at mainly as something we do for personal benefit.

Helping run my country, state, and city is something I do because that’s an end in itself. I do it as part of shaping the context in which I’ll accomplish my personal goals and which my descendants will grow and thrive. I do it as one of the ways I express my opinions on how the world should be. I do it because not voting is for the clueless and the parasitic. I do it because I’m a citizen, and fully embrace both my rights and duties.

I guess I’m willing to accept that those motivations are irrational in the economic sense of the term, just as long as economists are willing to accept that shoehorning everything into that framework is irrational in the common sense of the term.

• Jayson Virissimo

“I do it because not voting is for the clueless and the parasitic.” -William Pietri

William, have you ever met a principled non-voter? I assure you that they are much less clueless than the median voter. Also, when people vote in bureaucrats that direct tax collected money towards special interests and wasteful pet projects, why is it the non-voters who are parasitic? It seems to me, the opposite case can just as easily be made.

• Tim Tyler

24 out of 300 million is about five times 1 in 60 million. So, according to these numbers, the chance of your vote making a difference is about five times, on average, as being killed in a car accident on the way to the polls.

I make the car accident about five times more likely – based on the same figures.

• http://profile.typekey.com/sentience/ Eliezer Yudkowsky

If voting is primarily a collective action (which I agree that it is) then shouldn’t you only vote if you believe yourself to be smarter, more informed, or more altruistic than average? This reveals an entirely different form of voter bias.

• http://profile.typekey.com/sentience/ Eliezer Yudkowsky

(that is, smarter / more altruistic than the average voter)

• http://profile.typekey.com/andrewgelman/ Andrew

Unnamed: It depends whether you consider a possible voter who is deciding whether to show up, or a definite voter who is deciding which way to vote. Luckily these numbers are large enough in order of magnitude that we can be sloppy about factors of 2.

William: I’m not looking at voting as something people do for personal benefit. Our key point is that voting is a personal action that has a small chance of making a big difference for the whole country.

Tim: Yes, that was a typo.

Eliezer: You don’t have to feel that you’re smarter etc. than the average voter. It is enough to feel you have different interests or a different attitude than others. For example, if you favor John McCain’s economic policy (say), you don’t have to feel that you’re smarter than Paul Krugman, Tony Rezko, or even the average Democratic voter, you just have to feel that you have a different perspective on economic growth than they do. You might argue that it’s irrational for voters to have different attitudes on the economy, but given these different attitudes, if you’re voting it’s rational to decide your vote based on these beliefs. (By analogy, you might feel it’s irrational to believe in organized religion, but if you do, it can be rational behavior to spend a few minutes with the map one day figuring out the most efficient route to church.)

• JH

“Even if you don’t care about voting, these issues–how to compute probabilities of extremely rare events–are relevant in other decision settings.”

It’s extremely rare that my vote will swing an election. Therefore, it doesn’t factor into my decision whether to vote or not. While more likely, it’s also extremely rare that I will die in a car accident on the way to vote. Therefore, it doesn’t factor into my decision whether to vote or not. Both events, for all intents and purposes, have a probability of 0. So, why should I care about any extremely rare events in any decision setting?

• http://profile.typekey.com/SoullessAutomaton/ a soulless automaton

Eliezer, only if you assume that the goal of voting is to select the best candidate, as opposed to giving the population what they ask for.

Many people hold self-determination as an ideal in and of itself, which is incompatible with letting smarter people choose for everyone (even if the results are arguably suboptimal).

• http://profile.typekey.com/andrewgelman/ Andrew

JH: Because (a) the event happens over and over again so the total probability is not tiny (that’s why it makes sense to care about auto safety when you drive to work this morning), or (b) the event is extremely unlikely, but if it happens, it has a huge effect (that’s why it makes sense to vote and to worry about asteroid impacts).
Or some combination of (a) and (b).

• http://profile.typekey.com/sentience/ Eliezer Yudkowsky

Andrew, sure. But that can largely be lumped in under “more altruistic” – my understanding is that the data shows that very few voters will say, “I’m voting my own interests, which are no better than anyone else’s, but they are my interests and benefit me and that’s what I’m trying to do.” So if you don’t reason like this, but rather believe that you are acting prosocially or in a fashion that is “right” in a transpersonal sense, then you’d have to believe that your prosocial values are righter than those of the average current voter.

• JH

“(a) the event happens over and over again so the total probability is not tiny (that’s why it makes sense to care about auto safety when you drive to work this morning)”

One day I will have an accident on the way to work. But, that’s not going to factor into my decision of whether or not to go to work on any given day because on any given day it’s an extremely rare event. Someone is going to die on their way to vote, but that doesn’t change my decision whether to vote or not. Someday an election will be within one vote, but that doesn’t change my decision whether to vote or not.

I’m very much an it-won’t-happen-to-me person. And, for the most part, I’m going to be right about that. “It”, whatever that is, won’t happen to me. But, I do realize that something, sometime is going to happen to me. What am I supposed to do about it, though? How is that supposed to change the choices I make. I don’t know what that something is or when that sometime is. I can take all the precautions I want on the road to prevent one extremely rare event then the next thing I know I’m shot in my own home (another extremely rare event) because I don’t own a firearm. I can take precautions on the road and arm myself in my home and the next thing I know a wildfire (extremely rare where I live) burns down my home.

“(b) the event is extremely unlikely, but if it happens, it has a huge effect (that’s why it makes sense to vote and to worry about asteroid impacts).”

I’m confused. An asteroid impact wouldn’t have a huge effect? Also, one reason not to vote is because one doesn’t think one candidate over another will have a huge effect. Sometimes, there’s just not much difference.

Extremely rare events are so unlikely that their probability might as well be 0. Even if it has a huge effect, a huge effect multiplied by 0 is still 0.

• http://zbooks.blogspot.com Zubon

JH, I advise not eating. The odds of your dying of starvation in any five minute period is very low, so why bother ever eating?

Similarly, 40,000 Americans (and 1.2 million worldwide) die in crashes each year, but your odds of crashing on any particular trip are very low. Therefore you should feel free to have a few beers and drive blindfolded, unbelted, while holding a spiked metal object in your lap. Even if that makes you 100 times more likely to crash, 100*0=0. Cancel that insurance, too.

Just because extremely rare events happen thousands of times a day is no reason for you to take any precautions against them.

• StreetWalker

On Extremely Rare Events
It’s 2006. JH is running a trading desk at UberBank. He takes the elevator down to see Kirt, a quant in a cube:

JH: Hi Geekboy! Wazzup! Here’s a file with a really great deal you gotta model for me.
Kirt: No problem, I’ll put the numbers through our customized value-at-risk model tomorrow afternoon, let it run overnight, and get your answer to you right away.
JH (shooting the monogramed French cuffs of his bespoke English shirt): Um, actually, Mr. Rocket I really need this today. Like in 30. I’ve got tickets to the game in the corporate box tonite.
Kirt (looking at the file): I see, another mortgage structure with 4 tranches.
JH: Yeah, yeah a guy I have a great relationship with from ReallyAncientInvestmentBank told me all about it at his bowling party. It’s a fantastic deal. We can make 14%!
Kirt: Bowling party?
JH: Oh it was so great. With topless Russian models serving Japanese whiskey.
Kirt: Those lower tranches with the high return carry a lot of risk, JH.
JH (checking his Blackberry): Just gimme a quickie run now, ok? I’m authorized to take on a lot more risk this year, you know the guys up top want to post 20% profits.
Kirt (starting the model): Ok, here goes. . .(time goes by) Uh-oh. Look at this. . .
JH: Wow! It looks like a snake swallowed a woolly mammoth! Look at all that safe profit in the big fat hump.
Kirt: JH, look at that bump in at the end.
JH: That’s just the head of the snake.
Kirt: No, JH, that’s a risk spike in the tail.
JH: But look at that hump! That much money would boost my bonus level and I could definitely make it on the corporate Platinum Club cruise. (dreaming) I could go to ChicBar and pick up one of those sulky Hungarian girls who doesn’t really speak English. What lovely bones. If I gave her some heroin and a Cartier bracelet, she’d go on the cruise with me. That would be great!
Kirt: JH, I don’t feel comfortable with this. Why don’t you just take the super-senior tranche and call it a day?
JH: Look, this risk is low in any year right? I mean, extremely rare?
Kirt: But this is a multi-year deal, JH. The risk builds every year.
JH: Why should I care about any extremely rare events, ever?
Kirt: Because after 3 years you have a 10% risk of losing everything. I mean it. It could be a disaster.
JH: Hey Propeller head, this deal makes money for you too. (Checks his chunky gold watch) Man, I gotta go. Are you gonna sign on this or not? TrustedRatingAgency gave this a AAA, you know.
Kirt: TrustedRatingAgency was paid by your bowling buddy at ReallyAncientInvestmentBank to put this deal together and give it a gold star, JH.
JH: No problem. All I have to do is take out a credit default swap to insure this against ReallyAncientInvestmentBank with AIG. It’s a no-brainer. Just sign it off, ok? (Exits cubicle, stage right)
Kirt (IMing his old physics pal Didi, now also a quant, at SwissGeantBank): I really miss medium-energy particle physics sometimes Didi. The jerk didn’t even stick around long enough for me to tell him that subprime default rates are in reality running at 3 times the rate our algo assumes. This job sucks.
Didi (IMing back): Think about it Kirt. Remember how you hated living like a monk in a trailer with 7 other post-docs eating Ramen noodles.
Kirt: I like Ramen noodles. Now that I’m rich, I only eat more Ramen noodles.
Didi: Those were the days. Let’s say we meet at 8 at the Vivaldi and play some Go.
Kirt: That would be great. Tiramisu for dinner.
Didi: Yeah, the Diet Coke’s on me.

• David

Andrew,

I went through your paper some more, and you are definitely doing something wrong in how you come up with the probability at 50/50

Regards,
David

• http://profile.typekey.com/andrewgelman/ Andrew

David,

The probabilities are intended to represent forecast uncertainty. For the 2008 paper I used the forecast simulations of Nate Silver which were based on variation in polls. Another version of forecast uncertainty is here:
http://www.stat.columbia.edu/~gelman/research/unpublished/electionNew3.pdf
And another version (for the 1992 election) is here:
http://www.stat.columbia.edu/~gelman/research/published/decisive.pdf

Details change with the forecasting model but the general conclusions, and the order of magnitudes of the probability calculations, don’t change.

• I Coull

In a crowd of 100 where 2 are going to arbitrarily win/lose/have an accident/die/…; the probability of 2% applies to the group. For any given individual the probability of the ‘event’ is either zero or one. Probabilistic stats are derived from and pertain to the group, not the individuals therein.