Friday, November 2, 2012

Some Thoughts on Voting (UPDATED)

Suppose I invite you to bet with me on the outcome of some large set of random trials.  I'm a bit of a jerk though, so I'm offering you terms that are a tad unfair.  I'm going to make a prediction about the number of trials that turn out a certain way, and if I'm right, you'll owe me $120, while I'll only owe you $100 if I'm wrong.

If I told you that the set of random trials would be 6 million rolls of a fair die, with my bet being that the number of 6's that will be observed will be greater than 3 million, you'd be a fool to turn down the bet.  Sure, there's more in it for me if I win than there is for you, but you don't need to be a statistician to know that the odds are overwhelmingly in your favor here.

If, on the other hand, I told you that my prediction is that the number of 6's observed will be more than 1 million, you'd be well-advised to decline my bet.  If I offered you fair terms, that might be another story, but there's too much uncertainty here for the terms I've offered to be attractive.

The two scenarios I described clearly differ in that respect.  But let's look at this from another angle.  What are the odds that the very last roll of the die would have made the difference between my prediction being correct or not in the two cases?  Without going into too much technical detail, the answer is that it would be a teeny tiny bit higher in the latter case, but scarcely different from zero in either case.  We're talking 6 million random trials, after all.

What's the point of this?

If you've ever said something like "My vote doesn't count, because I live in New York", you're the type of person who makes my head hurt.  We may not know for sure how things will turn out in New Hampshire this coming Tuesday, but that doesn't mean that an individual's vote will "count" for much of anything in that state.   The fact that everyone who knows anything about politics knows how things will go in New York (or California, or Texas) doesn't make any meaningful difference to the question of whether individual votes in those states are likely to determine the outcome, just as the very last roll of the die was all but irrelevant in both of the scenarios outlined above.  Don't confuse uncertainty over the final outcome with a significant probability of a single vote determining the outcome.  Those two things are not even remotely the same.

There are two other reasons why I think less of you if you think it matters whether you live in a swing state.  (Actually, I think less of you if you think that voting in US presidential elections is anything other than a symbolic act of expressing support for one of two coalitions dedicated to making some people worse off at the expense of others, but I'll get to that later.)

First, being pivotal to the outcome of your state is not the same as being pivotal to the outcome of a presidential election.  In 1984, Ronald Reagan won the electoral votes of every state except Minnesota.  He lost Minnesota by 0.18%.  If the kind of polling and election forecasting that's done today was available then,  and everyone knew that Minnesota was going to be a squeaker, would that have meant that voters in Minnesota should have believed that they had a rare opportunity to influence the outcome of a presidential election?  Of course not, even if we set aside my previous point.  Everyone would have known that Reagan was going to win, with or without Minnesota.  Granted, the election we're about to have on Tuesday isn't predicted to look anything like that of 1984, but my point still stands.  If you live in New Hampshire, that does not mean that your vote counts more than it would if you lived in New York.  Not by any meaningful degree, anyway.  The outcome of the election is almost certainly not going to depend on New Hampshire, just as the outcome of New Hampshire is almost certainly not going to depend on your individual vote.  If you live in Ohio, the story is a little bit different.  But only a little.

Second, those who say things like "My vote doesn't count, because I live in New York" are committing the error of being internally inconsistent on top of holding logic in contempt.  If you are going to partially grant that logic is a thing and that we should value it, which you do when you acknowledge that your vote is not going to determine which candidate receives New York's electoral votes, why would you then claim that a handful of states reside in a magical universe in which the laws of probability do not apply?  If you're the type of person who says "every vote counts", I'll smile at you and think you're kinda cute, but at least you won't have contradicted yourself.  But if you're comfortable saying that your vote "counts" in swing states, why not abuse that word a little more by saying that it matters everywhere?  Is the difference between 1 in 10 million and 1 in 100 billion really so meaningful?  To my mind, both of those sound an awful lot like "zero".

In other words, I agree with Steve Levitt that if you're voting because you want to influence the outcome of the election, you're giving people a reason for judging you.  It doesn't much matter what state you live in.

Yes, I read Gelman's response to Levitt.  No, I was not persuaded.

First of all, Gelman is mostly talking right past Levitt.  Levitt didn't say that there is no reason to vote.  He even went on to say that he voted for Obama in 2008!  What he actually said was 
[w]ell, one good indicator of a person who’s not so smart is if they vote in a presidential election because they think their vote might actually decide which candidate wins...[emphasis added]
which is in no way inconsistent with anything Gelman writes after that.  Your vote is not going to decide which candidate wins.  It's not.  As Levitt says, when it gets close, they go to the courts.  You know what else is a fabulous indicator that it's a flat-out lie that every vote counts?  The fact that media outlets will announce which candidate won each state before election officials are done counting the votes.  Can you ask for more explicit evidence than that?  They literally do not count all the votes before determining the winner, and yet people say that every vote counts.  Are you kidding me?  Or do you just not know that words have meanings?  Even in those cases where it's really close, the final vote count is going to be contested, at least at the presidential level.  At that point, the outcome depends upon whose definition of a valid vote prevails, not whether you, dear reader, cast a vote for your preferred candidate.

Now, perhaps more appropriate to say that people who believe that their vote counts are simply mistaken rather than saying that they are "not so smart", since plenty of smart people believe things that are not true.  But the point still stands.  I'm no less confident saying that your vote is not going to decide the outcome of the election than I am that I will not die in a car accident tomorrow, that I will not be shot by one of my students tomorrow, or that the value of the money I have in my savings account is not likely to dramatically decline over the next few months. Yes, I'm fudging a little when I say all of those things.  What I really mean is that they are really, really, really, really, really unlikely to happen.  But I'm comfortable treating those two as if they are the same, and so are you.  If you weren't, I can only assume that you'd be a  lot more afraid of getting in a car than you are, or of setting foot on a college campus, or of keeping your money in the bank.  After all, the chance of dying in a car crash any given time you drive, or of becoming a victim in a mass shooting any given time you set foot on a college campus, or of seeing your savings plummet in value due to hyperinflation, is pretty low.  But it ain't zero.  And that's the standard Gelman applies (perhaps inadvertently) when he says "the probability of your vote being decisive in a swing state as being in the range 1 in a million to 1 in 10 million. Low, but not zero..." [emphasis added]. 

I admit, that's being a little unfair.  Gelman chose language that implies that we should treat extremely low probability events as if they are likely to occur, but that's not really what he means.  Fine.  But when he says that voting is like buying a lottery ticket to help the poor, you should not take that any more seriously than you should people who say that your vote counts provided that you live in a swing state.  We have very good reason to believe that, no matter who wins, close to half of the population will suffer an enormous loss of subjective utility on election night.  The very fact that so many people think that the outcome of the election matters is itself evidence that if your vote could determine the outcome of the election, it would not be at all like buying a lottery ticket to help the poor.  No one, not no one, is going to feel devastated if a stranger randomly gives them money for no reason.  Voting is much more akin to buying a lottery ticket in the hopes hiring a bunch of thieves to rob half of the country so that you can give the stuff they steal to the other half of the country.

Now, if you want to say that it doesn't matter that a large number of people will be profoundly upset if their candidate loses, that's fine.  Perhaps you're right!  Maybe everyone would be better off if your guy won, even those who don't believe so.  In that case, it'd be more appropriate to say that you want to hire a bunch of people to force people to follow a very specific diet and exercise plan that you've devised.  Personally, I think that anyone who really thinks that empowering their preferred party would make (most) everyone better off and (nearly) no one worse off should be disqualified from teaching political science, but maybe that's just me.  At any rate, you're either profoundly insensitive to the well-being of approximately half of the population or so confident that you know what's best for them that you'd be willing to impose an outcome on them that they expressly oppose, which still means that you're insensitive to their short-term happiness.  That may not make you stupid, but it doesn't say anything particularly flattering about you either.

I don't dispute that there are perfectly good reasons for voting.  If you want people who think that we all have a civic duty to vote to just leave you the heck alone, cool.  For many people, voting is an excellent way of making that happen, and I totally respect that.  There's pretty good evidence that the desire to avoid social sanction is a powerful motivator of turnout (see here and here).  This makes a lot of sense.  Your vote isn't going to make a difference, so it's not really like you've hired a bunch of thieves to break into half the homes in the country, or to force everyone to follow your diet and exercise plan.  If it's going to help you avoid some social sanction, and that matters to you, then there's no reason you shouldn't.

If you believe that the benefit your candidate will bring to the winners of their policies more than off-sets the harm that will be done to the losers, that's reasonable too.  I think there's a lot of self-deception that goes on when people make these claims, at least with respect to the magnitude of that effect, but I acknowledge that reasonable people can believe that the net change for the country would be positive if their candidate won.  I don't hear a lot of people say "well, in some ways, the world will be a worse place, at least for some people, but on net, it's still worth it", but I do know some.  I can't fault them for voting if that's how they justify it.

But if you want tell me that you're unambiguously making the world a better place by voting, well, I'm just not going to take you seriously.  And if you think that your vote is going to make any but the most negligible of differences in the likelihood of whatever happy outcome you envision, well, that's not improving my impression of you either.

UPDATE: For those of you interested in more technical details: Gelman's argument is essentially that pB - c > 0 will hold for altruists even though it cannot plausibly hold for narrowly self-interested voters.  (If you haven't seen this formulation before: p is the probability that your individual vote determines the outcome of the election, B is the benefit of having your preferred candidate win, and c is the cost of voting.)  He first stipulates, quite reasonably, that p is approximately 1/n, where n is the size of the electorate.  A narrowly self-interested voter is very unlikely to find the cost of voting to be worth incurring because p is so small that B provides almost no off-set to c.  For an altruistic voter, B is assumed to increase in direct proportion to N, where N is the total population.  As N goes up, n likely goes up, and p goes down, but if B goes up too, then that's no problem, and it's very likely that the cost of voting is worth bearing.

My critique is essentially this: B cannot go up anywhere near as rapidly in N as Gelman says for a true altruist, because an altruist would also care about the fact that the number of people made worse off by the election of their preferred candidate is also strictly increasing in N.  If B increases in N, but at a slower rate than p decreases with N, then the instrumental argument falls apart.  My claim is that one must either hold implausible beliefs about the extent to which politics creates both winners and losers or must be frighteningly insensitive to the well-being of the losers in order for B to increase in N rapidly enough to offset the decrease in p.  As I said, that doesn't necessarily make one stupid, as Levitt said that voting to affect the outcome does.  But it doesn't say anything particularly flattering about you either.


  1. So are you planning to vote or not?

    1. Could have been a bit clearer about that, couldn't I? :p

  2. Glad you wrote this. I sort of implicitly agreed with Gelman before I read this, but I suppose I wouldn't be voting if I lived in a swing state either.

    1. As I said in the post, I've no problem with voting per se. I just take issue with some of the arguments people offer when it comes to voting. If you're going to vote, it shouldn't be because you expect to affect the outcome. That only works if you don't understand probability very well or don't put much weight on the suffering your candidate's victory will bring to the other side. But voting to express loyalty to your team is perhaps defensible, if you're at least honest about the fact that victory for your candidate isn't an unqualified victory for all, and voting to inoculate yourself against social sanction is *perfectly* reasonable.

    2. Yea I agree with that. It would seem that people voting because it is their civic duty or because their friends/family are doing it all presume that their vote does matter. So I am not sure they can be cleanly disentangled.

  3. Hi Phil. You write: "We have very good reason to believe that, no matter who wins, close to half of the population will suffer an enormous loss of subjective utility on election night."

    I think this is true, but it cuts both ways. Suppose I think along the lines that Gelman et al describe: my vote has only an infinitesimal chance of being the difference between Obama and Romney winning, but this is outweighed by my estimate of the enormous difference in total welfare due to the different policies they would enact.

    So if I go and vote for my preferred candidate, if I do end up being pivotal, the estimated effect of my vote is a huge increase in total policy welfare, but a huge decrease in the subjective welfare of the other guy's supporters.

    But if I stay home instead, and I would have been pivotal, now there is a 50% chance that my preferred candidate will win, policy will be good, and the other party will be very sad; and a 50% chance that the other guy will win, policy will be bad, and my party will be very sad.

    So the net effect of my vote, conditional on being pivotal, is a 50% increase in the chance of good policy, a 50% increase in the chance that the other party is sad, and a 50% decrease in the chance that my party is sad. If the two parties are roughly the same size, these subjective welfare losses more or less cancel out.

    I guess what I'm trying to say is that while it's true that a pivotal vote causes a massive loss in subjective utility among supporters of one candidate, by the same token it causes a massive gain in subjective utility of supporters of the other. So I'm not convinced that the Gelman et al logic fails for this reason.

    1. Hi Brenton.

      You're exactly right that it cuts both ways and thus essentially washes out. My point is that Gelman wants us to consider the benefit and only the benefit. If we ignored the fact that a lot of people suffer when our preferred candidate wins, we can say that a very small chance of making a lot of people happy is potentially worth incurring some small cost (which is essentially what the Rationality and Society piece he published with a few co-authors says).

      If we consider the fact that voting gives you a very small chance of changing who will be happy, it's no longer clear that an altruist would find the cost of voting to be acceptable. Some people would, to be sure -- particularly those with a tribal mentality. And I don't doubt that many millions of Americans have a tribal mentality. But I do reject the argument that voting is like buying a lottery ticket to help the poor.

    2. I may have misread your post. Yes, in addition to the subjective loss of utility suffered by the losers and the subjective gain enjoyed by the winners, there's actual differences in policy.

      What I was trying to get at in my post is that *this* too cuts both ways. The policies that will be enacted by the winning candidate will objectively make some people worse off. This needs to be factored in. It's entirely possible that the net gain is still positive, I admit. But it's much harder to say that the infinitesimal chance of being pivotal is off-set by the benefit when we take seriously the idea that politics creates both winners and losers. Unless, of course, you simply decide that you don't care about the losers. In which case, as I said above, you may not be proving that you are "not so smart", but you still haven't said anything particularly flattering about yourself.

    3. C'mon Phil. This business about making some people worse off and feeling bad about losers' diminished utility is pretty weak stuff, I think.

      *Of course* an Obama victory, for ex, will probably make some extremely wealthy people worse off, both in terms of their objective net worth and their subjective utility, than they wd be under Romney. So what? Why shd I care that some billionaire somewhere can't buy another Ferrari? Esp when well over a billion people in the world still don't have access to sanitation facilities, for ex, and hundreds of millions are living in unspeakable squalor in the slums of megacities in developing countries. And more to the point, in the US, of course, there are still not insubstantial numbers of very poor people. Esp in light of this, why the **** shd I care that an Obama victory will reduce the 'utility' of some v rich people, and also the subjective short-term happiness of some (misguided) not-rich people who voted for Romney?

      I read the post quickly, but it does not acknowledge, unless I missed something, that some voters vote for ideological reasons. They support candidate X's policy views b.c they are based on underlying value commitments that they share, and they believe that X's victory wd be best, on balance, for the country as a whole and the public interest as they conceive it. I suggest you go over to, say, Robert Paul Wolff's blog to see how an intelligent philosopher w leftist commitments (and no knowledge of Monkey Cage-style pol science) views this election. Or go over to Alan Gilbert's blog Democratic Individuality. Or read for that matter any left or right-wing journal of opinion. Of course they are all partisan and ideological. That is the whole point. You don't seem to understand that some people vote b.c they have quite deeply held views about what wd be best (or least bad) for the country and even if there is a one-in-thirteen-zillion chance of their vote affecting an outcome, they wd still vote. I think this is one instance in which political scientists cd benefit from getting out of the bubble (if i can put it that way) of political science.

      P.s. Do I care about the losers of this election? If you're asking me do I stay awake at night feeling bad for, let's say, the owner of a business in Lexington KY who voted for Romney and is deeply upset that Romney lost, the answer is: no. That voter will get over it, the way a highly partisan Dem voter when Bush beat Kerry got over their disappointment. If they don't get over it, they are sick and need psychiatric help, and my feeling bad for their diminished 'utility' is not going to matter.

      P.p.s. why is there so much political debate and legal struggle over voter ID laws? is it all just symbolic? maybe. but this might nonetheless be a question worth pondering.

    4. sorry shdn't have used future tense in the opening. Obama won. Past tense. :)

    5. I think you pretty grievously misread the post, LFC. You are largely taking down a straw man. That's all I feel compelled to say at this time.

    6. I may have overreacted to some of your remarks in the comment thread and the post itself, but I have just re-read the post and I do think it is open to criticism. Perhaps I shd have expressed the criticism more temperately.

      What esp bothers me is that one has to wait for the end of the post for a grudging acknowledgment from you that "reasonable people" might think that their candidate is, on balance, the best outcome in terms of how they conceive the overall welfare of the country. This acknowledgment by you shd have come earlier and been less grudging, I think.

      For myself, I am under no illusion that a single vote makes any real diff in an election, and in that sense I'm prob. willing to give you your pt over Gelman. But I think the way you chose to make the pt set up some straw men of its own. Most people don't follow the kind of debates you and Gelman are having, so I'd
      suggest most people don't vote b.c of some statistical conviction that their vote might conceivably count in a close race.

      What bothers me about the post, as I tried to suggest, is that i think it mischaracterizes or ignores some people's reason for voting. Or let's say it mischaracterizes mine. Even knowing as I did that my vote was not making any real difference, I wd have felt compelled to vote in this election not b.c i wanted to make the Romney people follow a particular 'diet/exercise plan' as you put it but b.c I thought Romney's policies wd have been quite bad, even perhaps quite disastrous in some respects, for the country. Now i might have been wrong about that, but given that i held that belief, i felt compelled to vote b.c it's one of the few concrete ways i have to express a strongly held political conviction. Maybe that is irrational on my part, but irrational or not, i don't think your post really captures or makes any kind of empathic effort to understand this chain of reasoning, or if you prefer, this psychological phenomenon.

      Ok, i have had more than my say here. I will shut up now.

    7. Thanks for the follow-up. I appreciate you giving the post another read.

      You are right that I only grudgingly acknowledged that reasonable people might see their candidate is best for the overall welfare of the country. I stand by this, for reasons I'll discuss below, but I'm glad to concede that I gave the point short shrift.

      As to the other point, I still think you're being a bit unfair. I did not say that I think people vote because they think they are going to determine the outcome. The fact that I didn't spend more time talking about expressive voting is not because I live in a bubble, but because I was specifically addressing myself to an argument about instrumental voting.

      Put differently, my post can be understood thusly: Many (I dare say most) political scientists and economists say that voting for instrumental reasons makes no sense, though there may be other reasons for voting. Gelman says that there can be an instrumental basis for voting. I disagree.

      That doesn't mean that I think it's irrational to vote. I will never say such a thing. Nor does it mean that I think it silly to vote for expressive reasons. I don't think that -- though I absolutely do take issue with those who seek to endow the act of voting with moral virtue, and look down their noses at non-voters.

      With respect to the belief one's candidate (or party) is better for national welfare as a whole, I'm not at all surprised that such a view is common, but I do think it is deeply misguided, and I also think it's pretty evident that it is misguided. We have quite good evidence that the position any given person holds on any given matter of policy was chosen to minimize the probability of cognitive dissonance rather than arrived at after the individual gathered all the relevant information and carefully weighed all the pros and cons. Of course, we can't expect everyone to be an expert on every policy. There isn't enough time in the world to do that. But there's always the option of admitting that you don't know enough about climate science or evolution or economics or health policy to know what the welfare implications of each side's proposal would be. I understand why people come to believe things that are comforting to them, and it's far easier to believe that your side has all the answers than to admit that some areas of policy are insanely complex and no one in their right minds could honestly claim to know that team D or team R has got the right policies for most problems. I understand that behavior. But I don't view it as morally neutral. There are too many examples of cases where no politician will ever consider implementing policy [x], even though all the policy experts, on both sides of the aisle, agree that policy [x] would be welfare improving, because the politicians know that their core supporters have ideological reasons to believe (falsely) that policy [x] would be worse than the status quo. If people who don't understand the welfare implications of policy [x] would just admit that, I think we'd all be a lot better off. So, yes, I am dismissive of people who are confident that their team has the right answer to every policy, even though such people do not have the expertise necessary to arrive at such a conclusion.

  4. Have you cared to research the more prevalent theories of voter behavior? There's quite a lot of literature on the subject that you seem to have completely overlooked. The most prevalent hypothesis is the Rational Ignorance model—complete with it's own utility curve that makes much more sense than your random trials example.

    1. Thanks for the comment, ZDH. I'm aware there is a lot of literature on voter behavior, and I'm familiar with some of the work on the Rational Ignorance model, yes. I must confess I'm not entirely sure what you're trying to get at, besides the fact that I did not discuss some work that is on the same broad topic. The point of the random trials example was not to provide an alternative explanation for why voters behave as they do, it was to illustrate the difference between uncertainty over the outcome versus the probability of being pivotal -- which is not by any means a central concern of the RI work I've read. I'm also not sure what RI (or other approaches to explaining voter behavior) has to do with the critique of Gelman's work that was the primary focus of my post. But if I've overlooked something, I'd be grateful to you for pointing it out.