The Condorcet Jury Theorem is often cited in support of the thesis that a group is a better truth-tracker than an individual. In other words, suppose there is a group of 
people, presented with two options 
and 
(one of which is true, while the other is false) and the probability of a single member 
(
) to make the right choice as to which option is the true one, is 
. When the previous conditions are met, the Condorcet Jury Theorem shows that the larger the size of the group, the higher is the probability of the group to hit the right option.
Several generalizations and relaxations of assumptions have been given (in particular, on the generalizability of the theorem to 
options where 
, see Christian List and Robert E. Goodin 2001) but I haven’t found much literature on the problem when the individual competence in giving the correct answer is 
. In particular I have the following observations:
In real scenarios, large groups (e.g. modern large democracies) that are called to give their vote on some option are often not independent observers of the phenomena in question and rely on a limited amount of testimonial evidence. The latter, in turn is often clustered in sectors to which different sections of the population refer (right-wing and left-wing newspapers, right-wing and left-wing talk shows, etc.). To assume that the condition of good-individual-competence is met in such situations is, I think, not realistic.
On the other hand in smaller groups such as committees of experts or scientific panels, the members are normally independent observers of some particular phenomenon, and have a wider range of independent evidence available to them when voting on what they consider to be the right option.
In order to avoid the accusation of being unrealistic, it seems correct to assume that smaller groups of experts tends to perform better than large groups of laymen, where the probability of hitting the correct answer could be driven down by an average individual competence lower than 
accuracy. There seem to be two diverging tendencies, when we talk about higher performance of groups relative to lower performance of the individuals in that same group, and the divergence seems to be implicit in the Condorcet Jury Theorem itself: on one hand, good individual competence drives the group towards the truth, on the other hand, when individual competence is low, the larger the group, the closer to the wrong solution the group is driven.
If the composition of the group is somehow dependent on the size, it might well be that group-performance does not keep increasing as the group gains in size but, assuming that larger crowds are more likely to have higher numbers of unreliable individuals, then the group-performance curve will at some point go from a positive to a negative slope. Where the zero-value of the slope is, will be determined by the particular composition of the society in which we are in. I think this would be an interesting problem to model.
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Do you know if people have been able to prove version of the theorem with relaxed independence assumptions? Because it seems to me that the phenomena that you cite to suggest that in realistic cases individual competence is often < 0.5 are also reasons to think that in realistic cases individual voters’ decisions aren’t all that independent (if you and I are getting our opinions from the same source, then the unconditional probability that you vote a certain way will be quite different from the probability conditional on my having voted that way). My impression would be that the less independence you have between voters, the less the accuracy of the final decision varies positively with the size of the group.
Hi Daniel, thanks for your reply. I haven’t looked into the results from relaxing independence though there is surely literature on the topic (just to cite a couple, Berg 1996: Condorcet’s Jury Theorem and the Reliability of Majority Voting and Estlund 2005: Opinion leaders, independence, and Condorcet’s Jury Theorem). However, I think that independence and accuracy need not be correlated in principle. Dependence may affect the results either way, if the dependence-source (or sources) is reliable then it increases the accuracy of the dependent voters, vice versa if the source is unreliable.
I do agree though that both are unrealistic assumptions. I think that in small groups (of the kind I have in mind, such as panels of experts) both independence and individual accuracy are better granted, but for different reasons, not because they are correlated. I’d like to hear more though if you think differently.
If people are getting their opinions from the same source, there might still be a kind of independence to consider. If the observer doesn’t know the biases of the source that people share, then it may be best to model them as non-independent variables. But if the observer does know about the sources, then it may be best to model the people as independent, conditional on what their source says. If Fox News, or the Huffington Post, has a particular story about some candidate, that may skew large numbers of people in the same direction in their voting practices. But given a precise enough characterization of the story, and how it tends to make people respond, it seems that knowing the actual response of one viewer/voter won’t give you any more information about how other viewer/voters will respond.
Kenny, that’s an interesting, you suggest to model agents as independent ones, conditional on whatever source is their common source. I expect the results should be the same as if modeling by assuming dependence. Another complication, a realistic one, may be when there are more sources of dependence. Again I see how one could use either method, in this case one should change the independence assumptions in order to account for the presence of more sources. I think this second scenario is much more interesting than the one with one source. Do you know if anyone has modeled that?
Fabrizio, thank you for your suggestion, I know the article although I haven’t looked into it yet, I will surely do that soon if you say it’s a must read. And, I meant “average individual competence”, I think that’s the standard interpretation.
Hi Carlo: which relaxation of the individual competence assumption do you have in mind? are you imagining a scenario in which the *average individual competence* is less than 1/2 or one in which there *may be* members of the group whose competence is less than 1/2?
One of my favorite papers on the subject, you may know it, is James Hawthorne’s “Voting in search of the public good” (available from his homepage: http://faculty-staff.ou.edu/H/James.A.Hawthorne-1/).
In giving his Jury Theorems, he does not *assume* that the average individual competence is larger that 1/2, but rather takes it as a parameter and then shows how it *constrains* group competence (for the better or for the worse, as it were).
His theorems also take as a parameter the *degree of covariance* among the voters (i.e. he does not assume independence: still the version of the theorem *with* the independence assumption follows as a special case of his theorems)
It’s a must read!