voting

Hi everyone,

I’m pleased to announce my book The Ethics of Voting (Princeton University Press) is now published. You can read the introduction here.

(I get about $3.00 in royalties if you buy it, so here are links to Amazon, which has sold out its initial batch, and Barnes and Noble.)

The main positions I defend in the book are:

1. There’s generally no duty to vote.

2. People can exercise exemplary civic virtue and pay whatever debts they have to society (if there are such things) without participating in politics. Political participation (and knowledge) is nothing special when it comes to civic virtue.

3. If people do vote, they have strong obligations to vote for what they justifiedly believe will serve the right ends of government, or otherwise they must abstain. This holds true even though individual votes are inconsequential. (I expanded and revised my argument from”Polluting the Polls”, as, for example, I realized that it didn’t cover cases of people voting for the right things for the wrong reasons, and it didn’t handle bad fringe voting very well.)

4. It’s okay to buy, trade, or sell votes, provided you don’t violate #3.

5. If social scientific work on voter behavior is correct, then most voters probably qualify as bad voters per my theory.

I’m going to be on CBC radio (I think on Sunday Edition) this weekend discussing some of these topics in light of the likely elections in Canada.

Some of you may be familiar with Richard Tuck’s recent book Free Riding. It’s a fascinating and valuable work, but I think much of the central argument, especially about the rationality of voting, is deeply flawed. Anyways, here’s a link to my short critical note on Tuck at JESP: Tuck on the Rationality of Voting: A Critical Note.

Ok, if the mathematics discussed in my last post are right, here’s the upshot:

Condorcet’s Jury Theorem (in its original formulation) says that in an election between A and B (where A  is the right choice and B is the bad choice), for an electorate in which each voter has an independent probability p>.5 of voting for A (the right choice), then as the size of electorate increases, the probability that the electorate will elect A (the right choice) approaches 1. Even for a low value of p, such as p=.51, the probability that the electorate will choose A approaches 1 rather quickly. For instance, with 10,001 voters, the electorate already has about a 99% chance of picking A.

Some epistemic democrats defend democracy using Condorcet’s Jury Theorem. They claim that democracies are adequately modeled by the Jury Theorem, and that the average voter is more likely than not to make a good choice. There’s debate about whether democracies are well-modeled by the theorem (e.g., whether voters make statistically independent choices, and if they don’t, what impact that has). (E.g., see the chapter “The Irrelevance of the Jury Theorem” in Estlund’s [i]Democratic Authority[/i].) I’m with Jerry Gaus and Estlund–I don’t think actual democracies are adequately modeled by the theorem, so I don’t think that we can use the theorem to conclude that they tend to make good choices. (Nor can we use it to conclude that they tend to make bad choices. Note that if p>.5, then as the size of the electorate increases, the theorem says that voters are certain to choose the bad choice. And I think the evidence, if anything, points to p<.5. So, from my perspective, it’s a good thing Condorcet’s Jury Theorem doesn’t apply.)

;However, suppose you do believe that democracies are well-modeled by the theorem. If so, then it’s worth asking how many voters you really need. After all, the probability that the electorate will make the right choice shoots up near 1 pretty quickly, even when p is only slightly higher than .5. Every additional voter adds some small probability that the electorate will make the right choice. However, we get diminishing returns. The question is how rapidly the returns diminish. After all, in a high stakes election, the net value of A over B might be, let’s say, on the order of $10 trillion.

 Imagine that, just like the other voters, my p is .51. Still, suppose that my vote increases the likelihood that we’d make the right choice by 1% or even .001% Because the value of making the right choice is so high, then my additional vote counts for a lot–it has a lot of expected utility. [The expected utility of my vote in this case is the difference in value between A and B times the marginal increase in the probability that the electorate will make the right choice.] So, for example, the 1001st voter is adding only about .02% accuracy to the electorate, but that means her vote is worth $2 billion! [Note: I have the exact number at work, but I’m typing this at home. So it might a little off.] Think of the electorate as being like a machine making a choice that’s worth $10 trillion or $0. If you increase the likelihood that this machine will make the right choice by .02%, you’re increasing the expected utility of the machine’s choice by $2 billion.

So, at what N is the Nth voter only contributing a few dollars worth of accuracy? Let’s suppose that every voter has an opportunity cost of $100. That is, during the time she votes, she could have done something else worth $100 either to her personally or to promote the common good. At what N does adding additional voters become wasteful?

Now that it seems like I’ve gotten Mathematica to cooperate, it looks like for this example, where each voter has a p of .51, the net value of making the right choice is $10 trillion, and where we’re calling votes wasteful when they have an expected utility under $100, votes become wasteful at about N=100,001. (This isn’t exactly right–it’s just about the order of magnitude where the value of a vote is in the 10s. In fact, I’m calculating the value of 100,001st voter at about $26.) Note that if p is higher than .51, the net value of the right choice is lower, or if opportunity costs are higher, then N will be lower. So, N=100,000 might be a high estimate.

So, if you defend democracy using the standard formulation of Condorcet’s Jury Theorem, it seems that you should think having 120 million Americans vote is kind of a waste of time. It would be far better just to have a small number of people vote and have everyone else go about their day. 119,899,999 of these people are just adding unnecessary accuracy to an already impressively accurate machine. They should go do something else instead. We just don’t need mass democracy. It doesn’t do us that much additional good. The first 100 thousand voters contribute more than the next 100 billion. Etc.

Of course, you might say, “Well, if only 100,000 people voted, they might not vote for the common good but for their self-interest at the expense of the common good.” Maybe so. But if you’re saying stuff like that, I take it you don’t think democracies are well-modeled by Condorcet’s Jury Theorem.

All this hangs on my having done the mathematics correctly. So, I’ll double-check the results when I get back to work on Monday.

If the conditions of the Condorcet Jury Theorem hold, then every additional jurist/voter adds some marginal amount of accuracy to the jury as a whole.  However, this jury experiences diminishing marginal returns.  If every juror has a 51% chance of being accurate, then the jury of 101 members has about a 57% chance of being accurate, a jury of 501 members has a 67% chance of being accurate, a jury of 1001 members has a 73% chance of being accurate, a jury of 5001 members has 92% chance of being accurate, and a jury of 10,000 members has a 99.99% chance of being accurate.I’d like to know what the marginal value (in terms of her contribution to accuracy of the jury) of the Nth voter is when N is rather large.

The accuracy of a jury of N members when each juror has  a 51% chance of being accurate is given by the formula below:

Pa (probability the jury is accurate) = SUM [upper bound = N, lower bound = (N=1)/2] (N!/(N-i!)i!) * (.51^i) * (.49^(N-i))

Since that’s likely to be unclear, here’s a link to a nice print out of the formula:http://books.google.com/books?id=CdIOKZWc3oMC&lpg=PP1&dq=public%20choice%20iii&pg=PA129

It’s easy to calculate Pa using Mathematica for values where N < 6500.  After that, Mathematica and other programs can’t handle it. So, what I’d ideally like to do is find some program that can calculate Pa for higher values of N, such as N=50,000, N=500,000, N= 1,000,000, etc.

Alternatively, if there is some way to find the first derivative of this function, that might be helpful as well. Does anyone know how to do this?

What I’d really like to know is what the optimal number of jurors/voters is when the conditions of the Condorcet Jury Theorem obtain.  Even tiny increases in accuracy can have significant value if the value of being accurate is high enough.  So, for example, the marginal value of the 5007th voters is about 0.002%.  But if picking the better candidate is worth, let’s say, $1 trillion dollars, then the expected value of that vote is quite high.  But what’s the marginal impact of the 50,000th vote?  The 100,000th?  The millionth?I’m wondering if you think that democracies are adequately modeled by the Condorcet Jury Theorem (you shouldn’t, by the way), what’s the optimum number of voters?  Let’s say that the net value of being accurate is $1 trillion, and that adding additional voters is suboptimal once the marginal value of a vote goes below $1.  In that case, the optimal number of jurors (N) is given by the formula:1,000,000,000,000 * [Pa(N+1) - Pa(N)] = 1

Alas, despite trying many things over the past week or so, I have no idea how to solve this without a supercomputer.

Another alternative would be to find some upper bound and prove the the actual number is below this (already low) upper bound.  But I’ve been unsuccessful at that.

Yet another alternative is to calculate the real marginal value of votes at a bunch of Ns that Mathematica can handle, then run some regressions to find a function that models the marginal value well, and use that as substitute.  I’ve done that with a few different functions, but the problem is that these functions are of questionable accuracy for high values of N.Any ideas?

Lately, I’ve been wondering what it means to be a good citizen.  I’ve been working to develop a liberal theory of civic virtue that is, I think, properly purged of certain republican ideas.  That is, I think civic virtue for liberals is exercised primarily in non-political arenas, via activities we wouldn’t normally think of as expressing civic virtue.  More on that some other time.  As a piece of this broader project, I have a paper coming out in The Australasian Journal of Philosophy on the ethics of voting by this title.

Here’s the abstract:  Just because one has the right to vote does not mean just any vote is right.  Citizens should not vote badly.  This duty to avoid voting badly is grounded in a general duty not to engage in collectively harmful activities when the personal cost of restraint is low.  Good governance is a public good.  Bad governance is a public bad.  We should not be contributing to public bads when the benefit to ourselves is low.  Many democratic theorists agree that we shouldn’t vote badly, but that’s because they think we should vote well.  This demands too much of citizens.

So, in summary, on my view, citizens don’t in general have an obligation to vote, but if they do vote, they should vote well.  What I do in the paper is outline broadly what it means to vote badly, explain why I think you ought not to do it, and then answer various objections.

An outline of the argument is: 1.One has an obligation not to engage in collectively harmful activities when refraining from such activities does not impose significant personal costs.  2. Voting badly is to engage in a collectively harmful activity, while abstaining imposes low personal costs. 3. Therefore, one should not vote badly.

Some of the worries about this argument that I respond to are (among others): A.  If good governance is a public good as I say, shouldn’t everyone who benefits from this good contribute to it? B.  Don’t individual bad votes have incredibly low expected disutility, and if so, why bother prohibit bad voting? C. Does this position imply epistocracy (Estlund’s term, meaning the rule of those who know better) or something like it?  D. Is this view self-effacing? E.  What if citizens are good at judging character, even if they are bad at judging policies?

So, if people are interested, I’ll be writing more about this in the next few days.  Feel free to email me at Jason_brennan [at] brown.edu if you’d like a copy.  (I’ve got to make a final few revisions over the next few weeks anyways, so any comments would of course be welcome.)

[Update: I’ve added a bloggingheads video of Jason and blogger Will Wilkinson (Cato Institute) on this paper below the fold — SCM]

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