What about Monte Carlo methods? There are many problems for which Monte Carlo integration is the most efficient method available.
Monte Carlo methods can’t buy you any correctness. They are useful because they allow you to sacrifice an unnecessary bit of correctness in order to give you a result in a much shorter time on otherwise intractable problem. They are also useful to simulate the effects of real world randomness (or at least behavior you have no idea how to systematically predict).
So, for example, I used a Monte Carlo script to determine expected scale economies for print order flow in my business. Why? Because it’s simple and the behavior I am modeling is effectively random to me. I could get enough information to make a simulation that gives me 95% accuracy with a few hours of research and another few hours of programming time. Of course there is somewhere out there a non-randomized algorithm that could do a more accurate job with a faster run time, but the cost of discovering it and coding it would be far more than a day’s work, and 95% accuracy on a few dozen simulations was good enough for me to estimate more accurately than most of my competition, which is all that mattered. But Eliezer’s point stands. Randomness didn’t buy me any accuracy, it was a way of trading accuracy for development time.
How do we know that it’s improved? Isn’t it equally plausible that Franklin would be horrified because some things in our world are horrifying, and his own moral thinking was more rational than our own? Does moral thought gets more rational all on its own? It seems as though it might be difficult for moderns to know if moral thought were less rational than it used to be.
The point of the exercise (somewhat more clear in the full post) is not that every moral decision on which we differ with Ben Franklin represents a moral improvement, but that at least some do and there are many. So, there are many things about our world today that are, in fact, better than the world of the 1700s, and at least some of them would nonetheless shock or horrify someone like Ben Franklin, at least at first, even if he could ultimately be convinced wholly that they are an improvement.
So in designing any real utopia, we have to include things that are different enough to horrify us at first glance. We have to widen our scope of acceptable outcomes to include things with an argument to be better that would horrify us. And that will, in fact, potentially include outcomes that hearken back to previous times, and things that Ben Franklin (or any other rational person of the past) might consider more comforting than we would.