Evelyn Lamb of the University of Utah has a nice piece on continued fractions in her SciAm math blog.

snip

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Let’s look at how it works for pi. John Heidemann at the Information Sciences Institute at USC has a list of all the best rational approximations (of the first kind) of pi with denominators up through about 50 million. The numbers 3/1, 13/4, 16/5, 19/6, and 22/7 are the first few fractions on this list. 13/4 is a little closer to pi than 3 is. It’s about 0.1084 away instead of 0.1416. But if we multiply the differences by the denominators, 13/4 doesn’t do so well. We get 0.1416×1=0.1416 and 0.1084×4=0.4336, so 13/4 loses pretty badly. The same is true for 16/5 and 19/6. Both of them are a little closer to pi, but they aren’t enough closer to make up for their larger denominators. Thus, 13/4, 16/5, and 19/6 are best approximations of the first kind but not of the second kind. But when we get to 22/7, things change. It’s quite close to pi. Its difference, 0.00126, is very small. If we multiply it by its denominator, we get 0.00126×7=0.00882. This beats 0.1416 pretty handily, so it’s a best approximation of the second kind. It’s also the next convergent in the continued fraction for pi.

My favorite convergent in the continued fraction for pi 355/113. It’s a really good approximation. It’s a continued fraction mic drop. The next best approximation of the first kind has the denominator 16,604. But it isn’t even that much better than 355/113. We don’t get another decimal digit of accuracy, but we have to increase our denominator by two decimal places. No thanks. We have to increase to the denominator 33,215 before we get a new approximation that’s worth bothering with.

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## on nash's contribution to economics

A nice

New Yorkerpiece by John Cassidy on John Nash's work in economics.snip

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That’s partly because Nash-influenced game theory isn’t actually a testable scientific theory at all. It is an intellectual tool—a way of organizing our thoughts systematically, applying them in a consistent manner, and ruling out errors. Like Marshallian supply-and-demand analysis or Bayesian statistics, it can be applied to many different problems, and its utility depends on the particular context. But while appealing to the Nash criteria doesn’t necessarily give the correct answer, it often rules out a lot of implausible ones, and it usually helps pin down the logic of the situation.

For these reasons, studying game theory, and learning how to recognize a Nash equilibrium, are highly worthwhile exercises. Once you learn the basics, it is amazing how broadly they can be applied. (Much of the advanced stuff can be safely skipped.) For example, in writing a book about the economics of the financial crisis and trying to figure out why so many people on Wall Street and elsewhere did things that ultimately blew up in their faces, I relied heavily on the Prisoner’s Dilemma, a simple game involving a particular type of Nash equilibrium that shows how certain incentive schemes can promote self-destructive behavior.

My experience wasn’t out of the ordinary. These days, political scientists, evolutionary biologists, and even government regulators are obliged to grasp best-response equilibria and other aspects of game theory. Whenever a government agency is considering a new rule—a set of capital requirements for banks, say, or an environmental regulation—one of the first questions it needs to ask is whether obeying the rules leads to a Nash equilibrium. If it doesn’t, the new policy measure is likely to prove a failure, because those affected will seek a way around it.

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04:46 in General Commentary, math | Permalink | Comments (0)