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Democracy and the jury theorem

I'm reading an interesting book _Democracy and Knowledge_ on how Athens worked, and it reminded me of http://en.wikipedia.org/wiki/Condorcet%27s_jury_theorem which seems relevant to the functioning of direct democracy.

Imagine a bunch of people trying to decide on some binary decision: is X true? is Y good policy? Assume they will sincerely and independently vote for the truth or best policy as they see it. Assume they each have a slight bias toward being correct -- 51%, say, or even 0.501 probability.

Then it turns out that a vote of them can be really accurate. If p is the chance of being correct, w is the votes for the winning options, and l is the votes for the losing option, then for a particular vote gap w-l, the chance of the truth winning is C(w,l)p^w*(1-p)^l, while the truth losing is C(w,l)p^l*(1-p)^w. If you divide, it simplifies to [p/(1-p)]^(w-l)

So for p=0.51 and w-l=20, you get a ratio of 2.23, i.e. the truth is more than twice as likely to win. At w-l=100 the ratio is 54. At 1000, it's 2e17. Note this is independent of the total population size.

Even if p=0.501, you approximately just need 10x bigger gaps, e.g. w-l=1000 has a ratio of 55.

So if the assumptions hold, voting on everything should be awesome, assuming you throw out cases that win by a tiny sliver of votes. How do the assumptions hold up in the real world?

Modern juries are too small, at least for such marginal individual accuracies, though if p=0.6 it's better, ratio of 130. Worse, they're not independent -- a unanimous decision doesn't mean 12 people being convinced outright by the evidence, it means them deciding after groupthink discussion and pressure to not return a hung decision and the desire to go home. To invoke this effect you'd want Athenian juries, with 501 or 1001 or more people.

Popular opinion at large isn't independent.

The math works with everyone having a slight bias to the truth; if most people are totally clueless and there's a scattering of experts, I'm not sure if the theorem holds so well. Though Wikipedia says "One very strong version of the theorem requires only that the average of the individual competence levels of the voters (i.e. the average of their individual probabilities of deciding correctly) is slightly greater than half.[3] This version of the theorem does not require voter independence, but takes into account the degree to which votes may be correlated." No time to look at the PDFs.

Some issues like quantum mechanics and much of economics are outright counterintuitive, so people may be systematically wrong on them, with p<0.5. This could be fought with better education, though you have to get the majority to admit it's wrong and needs to be educated...

You might believe that p<0.5 on everything... though that wipes out democracy except as an empty legitimacy ritual.

Partisan voters may not really count, so the p>0.5 assumption only applies to the smaller pool of swing voters.

Votes on issues are often not on fact nor even on policy by a set criterion. E.g. the Swiss ban on minarets is bad policy for general utilitarianism or human rights, but perhaps a 'good' policy for "protecting Swiss culture" or passive-aggressively discouraging Muslim immigrants. There's no clear correctness here, just preferences.

Most 'democracies' of course don't vote on issues, just on leaders. The theorem suggests people would at least be picking the best out of two leaders. Ignoring charisma, money, and candidate height(!) as disruptive factors, and assuming people *do* elect the most expert candidate, what does that mean? If an expert is right 80% of the time, and the theorem applies, the expert would easily be outperformed by direct voting. Even a 99% correct expert would be outperformed, though the difference may not matter much.

With local representatives and government voters may well choose the apparent best legislator for their district, or leaders for their city, but this needn't aggregate to good national outcomes, as the representatives fight for local interests as the expense of global ones.

Of course, when the theorem doesn't apply for various reasons, what does that mean for representative democracy? You need the theorem to apply for evaluating the gestalt state of affairs to pick a leader, or at least to reject the current leader as failing, while being inapplicable on individual issues. Not just that the first problem is easier. If it does apply where p(reject bad leader) is 0.75, and p(decide good issues) is 0.51, then you're more likely to pick a good leader than to decide any particular issue, but per above, you may not be able to pick a leader who's as good as your collective decisions on issues. And that's leaving aside things like the principal-agent problem, of whether *any* leader will do what you want once given power...

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