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.