comingstorm
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@michael e sullivan: re "Monte Carlo methods can't buy you any correctness" -- actually, they can. If you have an exact closed-form solution (or a rapidly-converging series, or whatever) for your numbers, you really want to use it. However not all problems have such a thing; generally, you either simplify (giving a precise, incorrect number that is readily computable and hopefully close), or you can do a numerical evaluation, which might approach the correct solution arbitrarily closely based on how much computation power you devote to it.
Quadrature (the straightforward way to do numerical integration using regularly-spaced samples) is a numeric evaluation method which is efficient for smooth, low-dimensional problems. However, for higher-dimensional problems, the number of samples becomes impractical. For such difficult problems, Monte Carlo integration actually converges faster, and can sometimes be the only feasible method.
Somewhat ironically, one field where Monte Carlo buys you correctness is numeric evaluation of Bayesian statistical models!
What about Monte Carlo methods? There are many problems for which Monte Carlo integration is the most efficient method available.
(you are of course free to suggest and to suspect anything you like; I will, however, point out that suspicion is no substitute for building something that actually works...)
This "perfectly rational" game-theoretic solution seems to be fragile, in that the threshold of "irrationality" necessary to avoid N out of N rounds of defection seems to be shaved successively thinner as N increases from 1.
Also, though I don't remember the details, I believe that slight perturbations in the exact rules may also cause the exact game-theoretic solution to change to something more interesting. Note that adding uncertainty in the exact number of rounds has the effect of removing your induction premise: e.g., a 1% chance of ending the iteration each round has the effect of making the hanging genuinely unexpected.
Anyway, the iterated prisoner's dilemma is a better approximation of our social... (read more)
Tallying these, it looks like roughly one in six have actually come true. Another one in six seems likely to come true in the readily-forseeable future (say, five to eight years). Note that many of these depend on what you're willing to call a "computer". I contend that just because something has a microcontroller running it doesn't make it count as a computer; e.g., a traffic light doesn't qualify. But, should a cheap-ass dumb cellphone count? I think a certain amount of user-mediated flexibility should be a requirement, but ultimately it's a semantic argument anyway...
One weakness is pretty clear -- excessive optimism in the speed of development/adoption.... (read more)