Toby Ord writes:
I think you will like this short proof that puts a lower bound on the probability that one’s vote is decisive.
It requires just one assumption (that the probability distribution over vote share is unimodal) and takes two inputs (the number of voters & the probability the underdog wins). It shows that in (single level) elections that aren’t forgone conclusions, the chance your vote is decisive can’t be much lower than 1 in the number of voters (and I show where some models that say otherwise go wrong).
Among other things, this makes it quite plausible that the moral value of voting is positive in expectation, since the aggregate value scales with n, while the probability scales with 1/n. Voting would produce net-value roughly when the value of your preferred candidate to the average citizen exceeds the cost to you of voting.
This relates to my paper with Edlin and Kaplan, “Voting as a rational choice: why and how people vote to improve the well-being of others.”
I was happy to see that Ord’s article mentioned the point made in the appendix of our 2004 paper, as it addresses a question that often arises, which is whether a vote can never be decisive because when an election is close there can be a recount.
Also, some background: it’s my impression that the p = 10^-90 crowd (that is, the people who assign ridiculously small probabilities of a single vote being decisive) are typically not big fans of the idea of democracy, so it is convenient for them to suppose that voting doesn’t matter.
I’m not saying the p = 10^-90 people are cynical, as they may sincerely believe that democracy is overrated, and then this is compounded by innumeracy. Probability is difficult!
And then there’s just the general issue that people seem to have the expectation that, when there’s any sort of debate, that all the arguments must necessarily go in their favor, so they’ll twist and turn a million ways to avoid grappling with contrary arguments; see for example this discussion thread where I tried to clarify a point but it didn’t work.
Regarding this last point, Ord writes:
I hope it is useful to have a simple formula for a completely safe lower bound for the chance a vote is decisive. Not the same as your empirically grounded versions, but nice to show people who don’t trust the data or the more complex statistical analysis.
Voting is rational, but not because one vote might be decisive. This is just the Prisoner’s Dilemma again.
David:
I don’t think voting is rational for everyone, but I do think voting is rational in high-stakes elections that are anticipated to be close, for reasons discussed in our above-linked paper, “Voting as a rational choice: why and how people vote to improve the well-being of others.”
Andrew:
I think you should withdraw your paper. It makes the same mistake that people make when using the Nash Equilibrium to analyze the Prisoner’s Dilemma. I’ve pointed out this error before. While it is interesting that you can tweak a wrong model to give the correct result in certain situations, it doesn’t change the fact that the model is wrong. While voting has its problems, people are fundamentally correct (and rational) when they say democracy will work if we all vote. (Many people have noted the connection between voting and the rationality/cooperation paradoxes.)
David
David:
I have no idea whether democracy will work if we all vote, since this never happens! It’s an interesting theoretical question which has been discussed from time to time; see for example here.
In the meantime, if you have anything specific that you think is wrong in our paper, you can feel free to explain it in blog comments or to publish an article in the journal explaining your reasoning.
Andrew,
As I already explained (several times in the past on this blog), your paper has the wrong alternatives in the cost-benefit analysis. That’s why you have to work so hard to get the correct result that voting is rational.
It is easy to see that voting is rational: Suppose it wasn’t and suppose all the rational people didn’t vote. Then the outcome would be worse than the alternative.
It is just the Prisoner’s Dilemma again. People mistakenly claim that rational prisoners should defect. But, if they defect, the outcome is worse than if they cooperate. So, they should cooperate.
If you get the wrong result, you should find the flaw in your analysis, not fiddle with the numbers to make the numbers come out the way you want.
If you think the prisoners should defect, then do you also think that rational people should litter, shoplift, not recycle, not give to charity, not vote in local elections, etc., etc.? Cooperation is rational, and is an important feature of civilized life.
A main point of political parties and get-out-the-vote efforts is to convince you that there are other like-minded people, so you aren’t alone.
I didn’t originate this analysis. I just recognized its validity when I read it. There are some references to the literature on https://www.davidmarcus.com/Paradoxes.php .
David
To put it another way, you are assuming that there is only one rational voter. Maybe that is true. But, that is a rather drastic assumption.
David:
No, I (more precisely, my collaborators and I) are not assuming anyone is a rational voter. What we’re saying is that, for a particular voter in an upcoming election, if the stakes are high and the election is forecast to be close, and this voter has other-regarding preferences, than it can be instrumentally rational for that person to vote. As we discuss in the paper, we are not assuming that voters are rational, nor are we saying this is the only reason that people vote. We do discuss the idea that utilities vary (to the extent that we can consider people, implicitly or explicitly, as being utility-maximizers) and so it makes sense, even from a purely instrumentally rational standpoint, for different people to make different decisions of whether to vote. We also discuss the motivation of campaigning; I agree with you about the motivation of get-out-the-vote efforts. Also, I don’t think prisoners should defect—or, maybe I should say, I think that prisoners in that hypothetical dilemma are free to make their own decisions within the options given to them.
I’m under no illusion that this will convince you that our paper is not flawed. I can’t stop you from thinking that, and I’m sure there are others out there who agree with you! I just want to clear up some immediate confusions that you are expressing regarding what we say, hence the above paragraph. In any case, I appreciate your taking the time to comment.
In case this helps, I’ve expressed skepticism about a different purported application of the prisoner’s dilemma in political science; see here. As with the rational-voting issue, I’m not the only person who takes this stance; see here. This is not to say I’m necessarily correct, just that it’s not just me who is taking these positions.
Andrew,
> for a particular voter in an upcoming election, if the stakes are high
> and the election is forecast to be close, and this voter has
> other-regarding preferences, than it can be instrumentally rational for
> that person to vote.
I’m saying that is a whole lot of unnecessary hypotheses to get the conclusion. Just as the prisoners should remain silent, it is rational to vote (under some mild assumptions). If you can get the conclusion under weak assumptions, then why publish a paper where you get the conclusion under very strong assumptions?
David
Andrew: In our paper, we use approach B to show that conditions C imply D.
David: Approach B is not a valid way to show D. It can say D is false when it is really true.
Andrew: Our paper is correct because we state that we are using approach B and our calculations are what approach B says to do.
David: You have a different definition of “correct” than I do.
Andrew: Other people use approach B.
David: That’s true. But, some people use valid approach E, and show that D is true under very different conditions than C.
Meehl, P. E. (1977). The selfish voter paradox and the thrown-away vote argument. American Political Science Review, 71(1), 11-30.
https://1drv.ms/b/s!AiU3z2ipXdAWgaUJyIUMdAJ1zmvHSg
John:
I’d not heard of this article by Meehl so I went over and took a look. I found the form of the article to be exhausting to read—it’s written in the form of a hypothetical dialogue, and I’d prefer if Meehl were to have just said what he thought directly. In any case, what I got out of it was that he’s talking about sociotropic (other-directed) reasons for voting, which makes sense. Had we been aware of this article when writing our article on the topic, we would’ve cited it. I disagree with Meehl’s opinion that the probability of decisive vote doesn’t matter. He writes that the probability is less than 0.0001 so its value doesn’t matter, but I disagree. For the purpose of making a decision, there’s a big difference between probabilities of 10^-6 and 10^-9, for example. He uses phrases such as “quasi-certainty” which I guess can make sense from a psychological point of view but don’t work when a probability is multiplied by a very large number to compute expected utility.
To return to the issue of sociotropic voting: One of our key points is that “rationality” should not be taken to assume selfishness. I recommend the classic paper by Dale Miller, “The norm of self-interest.”
Regarding the connection between “economic” and “psychological” explanations for voting, I’ll refer you to section 5.1 of our paper. The short answer is that I see a relevance for calculations on the economic rationality of voting, even though I don’t think that economic rationality is the reason why people vote.
I find the Meehl piece brilliant – but, yes, exhausting to read. What I find most appealing is the rejection of the economic approach to the rationality of voting. Yes, there can be circumstances where it is economically rational (as with a small enough n or large enough potential consequences), but I think the general idea that rational voting must depend on some sort of sociotropic rationale. I think this is common with what I see as profound limitations of economic rationality. It is fairly easy to see voting as rational with many types of sociotropic reasons, and the fact that economic rationality can potentially justify voting seems mostly irrelevant to me. Why do people vote? It could be economic rationality, but it is far more likely one of these myriad other reasons. The desire to preserve economic rationality despite all this seems perverse to me.
I’ll provide what I see as a similar issue. The Endangered Species Act is a textbook example of bad economic reasoning, though I find it ethically and politically easy to justify. You can invent an economic argument why saving the snail darter (a classic example of a “worthless” species that prevented construction of a dam that had large economic benefits) is economically rational, but these arguments appear quite extreme and distorted. The snail darter might contain a cure for cancer or it might play some essential ecological function which will lead to disaster if made extinct, but these arguments seem extremely unlikely. More likely is that it is not economically rational to preserve the snail darter but easy to justify on other (e.g. sociotropic) grounds.
Similarly, it seems overly strenuous to try to defend voting, in general, on grounds of economic rationality. Isn’t it far easier to admit that most people that vote are doing so for other reasons?
Dale,
I agree that people often don’t vote for economic reasons, but, just to be picky, the economic rationale for the dam was not that good: see https://en.wikipedia.org/wiki/Tellico_Dam
John
Point taken. I’m sure I can find a different example, but I accept your correction.
Dale:
You write, ” it seems overly strenuous to try to defend voting, in general, on grounds of economic rationality. Isn’t it far easier to admit that most people that vote are doing so for other reasons?” Again, I refer you to section 5.1 of our paper where we discuss exactly that point. You might not agree with us, but we do discuss it!
Hey—someone sent me an email and pointed out that we do refer to that Meehl article in our paper. I guess I just forgot about it. Or maybe the reference came from one of my coauthors and I never followed it up.
It continues to amaze me that on a Bayesian blog we still see arguments about “the” probability of voting being decisive, as if such a probability could be calculated without reference to the state of information that defines the probability. Like what is “the” probability I will win an arm wrestling match? Doesn’t it depend on whether I’m arm wrestling my child or a professional weightlifter? I just don’t get it.
Daniel –
> Doesn’t it depend on whether I’m arm wrestling my child or a professional weightlifter?
Not only that, the degree to which the discussion is even relevant depends entirely on context.
Am I voting with a small group on whether to order pizza or Chinese takeout? Or am I voting in a national election, in a state that has aligned with a particular party for decades. In one situation I’m voting very much because my vote might be decisive. In the other, the decisiveness of my vote is almost entirely irrelevant as to my motivation for voting. In my state, I vote not because I think my vote will be decisive, but in a sense for the opposite reason. I want my vote to be counted among those who are sending a message that there’s no question as to who will win.
Daniel:
The probability depends on the model being fit. When we speak of the probability that you, voting in California, will cast a decisive vote in a national election, that probability is based on a model fit to voting patterns in California and other states. We’re not assigning the probability in a vacuum.
Sure, but the idea that there could be some mathematical proof of a “lower bound of 1/n” on the probability is ludicrous.
Suppose I’m an NSA spook who has collected a LOT of signals intel around some very active foreign state actors. Suppose it’s the day before the election. My assessement is that about 50% of voters in Florida will vote for one candidate and 50% for the other. I finally crack the encryption key on this signals intel and I discover in this massive archive of messages that there is a massive foreign investment into hacking voting machines and paying bribes to election officials, even getting Putin’s sleeper agents installed as election officials in major Florida counties. Perhaps the entire software engineering team for the major election machine maker has been compromised by credible threats that their entire families will be hunted down and killed if they don’t hand over centralized encryption keys. My assessment that my own vote could be decisive goes from maybe 10^-7 to 0 immediately. There’s no mathematical theorem which could prove that calculation “wrong” here.
Daniel:
It all depends on context. We can expect about 15 to 20 million Californians to vote in the upcoming presidential election. Based on reasonable forecasting models, the probability that the election in California will be tied is much much less than 1 in 20 million. So 1/n is clearly not a lower bound on this probability.
> Sure, but the idea that there could be some mathematical proof of a “lower bound of 1/n” on the probability is ludicrous.
(Leaving aside the not-so-mild assumption of a unimodal distribution of vote share…)
Note that the lower bound is 1/n only when both candidates are equally likely to win.
That 1/n has to be multiplied by two times the probability of winning of the least preferred candidate.
The bound won’t be “much lower” than 1/n as long as the probability of either candidate is not “much lower” than 1/2.
Carlos, your model presumably assumes that what people vote for matters. But of course in the active foreign interference example it doesn’t. The person who wins is the one Putin tells the foreign agents to make win on the compromised voting machines.
But even in some other models, it’s reasonable to think this result is not of interest. For example, people like to take the outcome as Binomial with unknown N and p and a prior on N and p that’s unimodal around 0.5 for p. But this is a bizarre model in some ways. There’s no repetition, so saying that there’s some kind of p is already saying that there’s some probability for p*N votes in this one case. The Binomial part isn’t of interest. There’s no sampling variability that’s meaningful. So you are in essence recapitulating your prior. If your prior is something like beta(k,k) then you can say directly this prior implies a probability of integrate(beta(k,k)(p) dp) in the appropriate region. That region is of size 1/N. The density of the beta is O(1) if there’s any significant uncertainty in the vote fraction, so the probability is order 1/N…
This is kind of a trivial restatement of your prior. It’s possible to make that beta(k,k) arbitrarily sharp around 0.5 though so that the probability goes to 1, and it’s possible to shift the peak over to say 0.55 and make it sharp and the probability of interest goes to zero…
Basically 1/N * density_at(0.5) is the probability there will be a vote split. This can be arbitrarily anything between 0-1 depending on what density you assign as your prior.
> Carlos, your model
His model, not mine. I was just just pointing out that he doesn’t claim to have some mathematical proof of a “lower bound of 1/n” on the probability. I agree that it’s essentially the probability of having a split vote – when that happens everyone’s vote is decisive.
This is clearly an interesting, but perhaps irrelevant, mathematical, statistical issue: “estimating the probability that your vote will be decisive.” This irrelevancy is because MAGA people have a different slant, an entirely different and paranoid mathematical, statistical model regarding this topic; the vote was rigged and Trump won in a landslide because Trump votes were not recorded but instead, were switched to Biden. Thus, “your [one] vote” is decisively irrelevant. To this very day, over three years after the January 6, 2021 uprising, many people are still actively maintaining Trump won–by far more than one vote.
Quite apart from the switching of votes, recall the Maricopa County recount and the assertion that (Communist) China was the cause of Biden winning the election in Arizona.
https://www.theguardian.com/us-news/2021/may/06/arizona-republicans-bamboo-ballots-audit-2020
“Arizona Republicans hunt for bamboo-laced China ballots in 2020 ‘audit’ effort”
I don’t know who ‘wt’ is, but that’s a pretty funny comment that suggests they are not very bright. At least they did introduce the question “am I missing something”, but they don’t seem to take it seriously. Hey, wt: yes, you are missing something.
Even in the case they’re considering — three voters, two choices of who to vote for — it is not true that each person necessarily has a 1 in 3 chance of casting the deciding vote: maybe all three of them vote for the same candidate. In that case, nobody has cast a deciding vote. A deciding vote can only happen if the race is very close. Relatedly, if the vote is 2-1 then _both_ voters on the winning side have cast deciding votes.
Furthermore, in U.S. national elections the electoral college system adds important, very non-trivial structure to the voting. If you live in a large swing state your probability of casting a deciding vote is much higher than if you live in a state that is very ‘red’ or ‘blue’. If Wyoming is evenly split (except for your vote) then the Democrat has won in a landslide no matter which way you vote. (A quick glance at Ord’s paper suggests that he does not consider anything other than majority-rules elections, so it’s not very relevant to US national elections. But this is a key point in a lot of work in this area).
Tell wt that if a lot of smart people are discussing something, there is probably something worth discussing.
“wt[you know what]?” is some kind of doubly obscure self-censorship of WTF = what the fuck
In other words Deb is expressing extreme annoyance with the subject at hand.
Hahaha very obvious, how did I miss that! I guess I am not so bright myself.
There was a famous three-way race in 2008 for Minnesota’s U.S. Senate seat. The details are here
https://en.wikipedia.org/wiki/2008_United_States_Senate_election_in_Minnesota
“When the initial count was completed on November 18, Franken was trailing Coleman by 215 votes.[2][3] The close margin triggered a mandatory recount.[4][5] After reviewing ballots that had been challenged during the recount and counting 953 wrongly rejected absentee ballots, the State Canvassing Board officially certified the recount results with Franken holding a 225-vote lead.”
With more scrutinizing in April, Franken’s lead jumped to 312. And he was declared the winner out of some 2.8 million votes cast.
———————————————————–
Not mentioned in the Wikipedia article is how politely things were conducted. I attended a few of the court hearings about the recount, and compared to today’s persistent rancor, it was like another world. Disagreement of course, but despite my usual disheveled appearance, there were no issues of security and I was able to sit close to the participants and was never urged to leave the courtroom.
From
https://thehill.com/blogs/be-blog/campaign/74073-the-truth-about-the-2008-minnesota-senate-recount-a-response-to-democratic-party-still-disenfranchising-and-oppresing-votes/
“Minnesota’s recount took 35 weeks. Coleman was represented by able lawyers throughout, including respected Washington, D.C., recount expert Ben Ginsberg, of 2000 Bush v. Gore fame. In the end, Gov. Pawlenty, now a candidate for the Republican presidential nomination, signed Franken’s election certificate.”
——————————————————————————-
As a sort of an update, Ginsberg name has hit the news somewhat recently
https://en.wikipedia.org/wiki/Benjamin_Ginsberg_(lawyer)
In September 2020 [||], The Washington Post published an op-ed by Ginsberg that criticized President Donald Trump for encouraging supporters to commit voter fraud and for making false statements claiming fraud was widespread when it is not. Ginsberg wrote, “The truth is that after decades of looking for illegal voting, there’s no proof of widespread fraud. At most, there are isolated incidents — by both Democrats and Republicans. Elections are not rigged. Absentee ballots use the same process as mail-in ballots — different states use different labels for the same process.”
Apparently a random individual vote is worth at least $10 to the presidential campaigns: https://www.newsweek.com/joe-biden-billion-dollar-campaign-spent-13-every-vote-1553058
I never really thought about it like this. The backers of the campaigns spend only $1 billion for access to $6 trillion dollar budget. Even if you can only redirect 1% to your backers (through regulations, subsidies, contracts, jobs, etc) that is still 6:1 payout for what is close to 50/50 odds. And its possible you can get maybe 10x more.
People are giving out their votes for way too cheap. Make them spend $100-$1000 to get your vote and they would break even.
It’s even more extreme than you suggest! Some large percentage of voters are not swayable by anything that might reasonably happen in the campaign — I include myself in this group, for the upcoming presidential election. Perhaps 90% of voters are like that. Moreover, the campaigns should really only care about swing voters in swing votes: there’s no point wasting money to try to rack up a big victory in a highly partisan state, when all you need is a little one. I’m sure either campaign would gladly spend hundreds of dollars to pick up one additional voter in a swing state.
Yea, I was thinking about the swing state subset.* Also, I was off by 10x (1% of $6 trillion is $60 B, not $6 B). And the winnings can flow for 4 years of budget, rather than only one.
So rather than 6:1 payout (assuming you can redirect 1% of the budget), it is 240:1.
So financially backing a campaign pays out 240:1 with ~50% chance of winning. This must be the funnest horserace ever devised for the big donors. This actually makes sense, on tv we are watching people screaming at the racetrack, hoping to affect the results.
* If swingstate voters are ~20% of the total, their votes are worth around $10k each, but theyre giving them away for ~ $10. And thats assuming *only* 1% of the budget can be directed into the right pockets. Its just like facebook, people are utterly underestimating the value of their info/vote.
Losing votes don’t die — do they? — the way losing lottery tickets die. Can anybody recommend discussions of the effect of a vote in the current election on the outcomes of future elections?
(Joshua, you’re my best hope among the commenters here.)
William –
You might be in bad shape if I’m a best hope… but… I find your comment interesting.
Indeed, it helped me to realize that I don’t think casting a vote in an election is really a discrete act. It’s part of a continuum of actions and necessarily connects to actions that come before and after. As such, it does seem to me that trying to decide if casting a vote is rational or not is too circumscribed.
Why do people vote?
Why do people answer political polls?
More than 50% do the first and less than 10% do the second. I may well be wrong, but my subjective judgement is that my action is more likely to have an impact on results if I respond to the political poll than if I vote (I can see all sorts of modeling issues distinguishing between the two, but my gut reaction is that this assessment is correct for me, yet I vote and don’t respond to political polls).
Then, what do these observations say about rationality? I’d say that voting but not responding to polls is completely rational – but it has little or nothing to do with the probability of influencing the results. Almost any sort of sociotropic motivation is much easier for me to buy.
Dale:
We discuss this very issue of survey response rates in section 3.2 of our paper. I highly recommend you read the paper; it’s readable, and it addresses many of the issues discussed by you and others in comments. It’s funny to see people keep bringing up questions that we directly talked about in our article. We also discuss sociotropic motivations; indeed, that’s the main theme of the article.
OK, thanks. Yes you do address it. I don’t buy the argument entirely – it is a sort of comparative statics argument. Because polling is so much more common, we would expect to see declining response rates. I agree, but that is only one change among many. There are also reasons why an actual vote is less likely to influence the outcome of an election, so we would also expect declining rates of voting. I would still say that under current conditions, responding to a poll is more likely to actually influence the election results than voting – but that is certainly debatable.
I don’t think we disagree about the myriad reasons why people engage in political behaviors – voting, polling, contributing. I think we agree that such behavior is mostly rational. If there is disagreement, it may be concerning what we mean by the term “rational.” I see attempts (and this is not necessarily what you are doing) to define rationality in terms of self-interested payoffs as less compelling that motivations that involve principles, ethics, and social relationships. These can of course be characterized as particular utility functions, but I don’t see that as a productive exercise. Utility easily becomes tautological – people always act in their self interest once you define self interest as what they decide to do.
Now that I’ve looked your paper more, I don’t think you are doing what I object to. But the whole debate about the probability of influencing election results seems off-point to me. And the article you point to in the original post focuses on a lower bound for the probability of influencing the election results – which I find relatively uninteresting.