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?
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