Condorcet’s Jury Theorem as an Argument against Mass Democracy

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.

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4 Responses to Condorcet’s Jury Theorem as an Argument against Mass Democracy

  1. Hi Jason,

    Assuming independence of votes and for each voter P(vote = correct) > 0.5, you must be right that there is a lot of pointless voting going on. It’s the same math that makes possible the magic of electoral predictions based on seemingly tiny samples (e.g. between 1000 and 1500 or a typical Gallup poll: so long as the samples are random.

    If votes are correlated, however, this reduces the effective “sample size,” doesn’t it? If all voters were twins, for example, and each pair thought and voted exactly alike, there would no additional information in the vote of each second twin. So one argument for mass democracy on Condorcet lines would be that correlations between votes make necessary a larger voting population. In a population of identically-minded twins, you would need twice as large a population to get the same level of accuracy, I think.

    Still, I’d guess that you’d need a very high level of correlation to need anything like 120 million voters.

    The other odd thing about sample size is that it’s entirely the size of the sample that matters, not the size of the population. So on a pure, Condorcet-accuracy model of voting, the ideal proportion of the population one would want to vote would vary with the size of the country (just because the ideal number of voters doesn’t have anything to do with the population of the country; it just has to do with the trade-off between marginal increase in accuracy and the opportunity cost of voting).

    Anyone who says “if only 100,000 people voted, they might not vote for the common good” is saying P(a vote = correct) < 0.5, aren’t they?

  2. I think that this is a worthwhile calculation.

    Here’s a case for full inclusion, based on jury theory that concedes everything you say:
    1. In actuality, some people are worse than .51 deciders based on entrenched ideologies, selective experiences, psychological frailties, etc.

    2. We cannot know how many citizens are worse than a coin flip, or that would presume we knew correct answers before asking them (at which point, why ask anyone?)

    3. Since we cannot know how many people are worse than a coin flip, and attempting to discern them would be unwise, why not just go overkill and let anyone votes who wants to?

    On the other hand, I wonder what it would do to public campaigning and the tenor of political arguments if we did just select 200,000 people at random to vote. It would be a way to possibly re-orient politics to a general good and away from “base-rallying” measures. If there is not much likelihood that “get out the vote” will turn elections, the median voter in election would more likely be close to the actual median voter, rather than a more movable object.

  3. I double-checked my math. Everything works out, except I overstated the value of the 100,001st voter. Actually, the marginal value of the 100,003rd voter (not 100,001st) is about $0.26. That is, 26 cents not 26 dollars.


    It seems like if points 1-3 hold, this means that we can’t use the Condorcet Jury Theorem to defend democracy. We are inclusive because we hope that that allowing more people in will mean that they make the right choice.

    However, this isn’t quite right. If the average voter’s probability is over .5, then adding more and more people helps. So, even if we don’t know how good the average voter is, as long as we’re sure that including everybody keeps the average above .5, and if we can’t rule out anyone as bringing down this average, then we should include everybody. (However, if we know for a fact that Bob has a p<.5, then it’s better if he didn’t vote.)

    Yet, if the average voter’s probability is <.5, then adding more people makes things worse and worse. It would be better to have one person decide on his own than to have 100 million decide. The flip side of the Condorcet Jury Theorem is that when the average voter’s p<.5, then as N increases, the probability that the electorate will make the *wrong* choice approaches 1. This is true even when p is just slightly below .5 For instance, if p = .49, then when N = 10,000, the electorate already has a 98% chance of making the wrong choice. At N= 100,000,000, the probability they will make the wrong choice is so close to 1 that my computer can’t even handle calculating it.

  4. Michael Fuerstein says:

    Hi Jason – Sorry I am so late with these comments. Thanks for your recent posts about voting, which I have found interesting. I thought I would weigh in on this since I’ve been thinking quite a lot about epistemic democracy recently.

    First: I just want to second Andrew’s point about the independence of voting. Given the extent to which voters tend to rely on common sources of information and arguments concerning their voting choices, most voting behavior seems to be massively non-independent in a way that sharply reduces the effective number of voters that should be plugged into the Condorcet formula. Encouraging universal voting might be a blunt instrument for encouraging diversity within the voting population. That is, if everyone votes, then all points of view will be represented in the voting population (though not necessarily in significant numbers). That will normally increase diversity, which is crucial for the epistemic value of a Condorcet voting process. In principle, of course, you could get diversity simply by selecting the right subset of voters. But you need a process that achieves that kind of selection reliably, and it is not clear whether there is a good alternative here to universal voting, partly because it is often unclear what the relevant diversity really is until we go out and ask everybody about their interests and concerns.

    Second: One reason you did not mention that Condorcet turns to be a weak basis for defending epistemic democracy is that voting is only the endpoint of a very long and complicated process during which the set of possible choices are reduced from some very large number to two or perhaps three. Under the right conditions, Condorcet gives us a reason to think that a voting population will be very good at choosing the best from among the options available. But whether the option chosen is any good depends on how the available options are identified. It seems to me that democracy’s epistemic benefits derive as much if not far more from the agenda-setting process that identifies issues of concern and produces concrete choices in response to them as the actual process of choosing from the available options themselves. This is relevant because encouraging universal voting is one way of encouraging the kind of widespread participation in the political process that is probably essential for successful, democratic agenda-setting.

    In general, I think that discussions of epistemic democracy tend to focus too narrowly on voting, not only at the expense of agenda-setting, but also at the expense of the many other ways in which democratic powers (e.g., liberties of speech and association) and institutions (public transparency of political decision-making and the legal process) function in the production of epistemically good governance by influencing the behavior of lawmakers.

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