It would provide a common prior for any empirical claims. Scott Alexander puts it very well here:

A democracy provides a Schelling point, … an option which might or might not be the best, but which is not too bad and which everyone agrees on in order to stop fighting. … In the six hundred fifty years between the Norman Conquest and the neutering of the English monarchy, Wikipedia lists about twenty revolts and civil wars. … In the three hundred years since the neutering of the English monarchy and the switch to a more Parliamentary system, there have been exactly zero. … Democracy doesn’t always perform optimally, but it always performs fairly, … and that is enough to prevent people from starting civil wars.

Academia is different. Its state resembles that of pre-democratic governments, when anyone could choose a side, claim it was legitimate, and then get into endless protracted fights with the partisans of other sides. If you believe ObamaCare will destroy the economy, you will have no trouble finding a prestigious academic who agrees with you. Then all you need to do is accuse the other academics of bias, or cherry-picking, or using the wrong statistical test, or any of the other ways to discredit scientists you don’t like. …

A democratic vote among the scientific establishment is insufficient to settle these topics. The most important problem is that it gives massive power to the people who determine who gets to be part of “the scientific establishment”. … So not having any Schelling point – being hopelessly confused about the legitimacy of academic ideas – sucks. But a straight democratic vote of academics would also suck and be potentially unfair.

Prediction markets avoid these problems. There is no question of who the experts are: anyone can invest in a prediction market. There’s no question of special interests taking it over; this just distributes free money to more honest investors. Not only do they escape real bias, but more importantly they escape perceived bias. It is breathtakingly beautiful how impossible it is to rail that a prediction market is the tool of the liberal media or whatever. …

Nate Silver might do better than a prediction market, I don’t know. But Nate Silver is not a Schelling point. Nobody chose him as Official Statistics Guy via a fair process. And if someone objected to his beliefs, they could accuse him of bias and he would have no recourse until it was too late. If a prediction market is almost as good as Nate, and it is also unbiased and impossible to accuse of bias, we have our Schelling point. …

Just as democracy made it harder to fight over leadership, prediction markets make it harder to fight over beliefs. We can still fight over values, of course – if you hate teenagers having sex, and I don’t care about it, we can debate that all day long. But if we want to know whether a certain law will raise the pregnancy rate, there will be only one correct answer, and it will only be a mouse-click away.

I think this would have more positive effects than anyone anticipates. If people took it seriously, not only would the gun control debate be over in an hour, but it would end on the objectively right side, whichever side that was. If single-payer would be better than Obamacare, we could implement single-payer and anyone who tried to make up horror stories about how it would destroy health care would be laughed out of the room. And once these issues have gone away, maybe we can reach the point where half the country stops hating the other half because of disagreements which are largely over factual issues.