Ars Mathematica

The Annals of Improbable Research, perhaps the finest general-purpose scholarly journal today, has published a satire by Yoram Bauman of Greg Mankiw’s 10 principles of economics (from his introductory economics textbook). Most of it is hit-or-miss, but his summary of Principle #2 is a brilliant satire on the concept of opportunity cost.

Opportunity cost is easy enough to understand in simple descriptive situations, but I have never seen a precise mathematical definition of it that didn’t make it seem like a contentless idea.

I’m glad to see that finally someone is putting mathematics to some good use. I present to you The social norm of leaving the toilet seat down: A game theoretic analysis. (Via Crooked Timber.)

Ariel Rubinstein has made his book, Modeling Bounded Rationality, available online. Economic models tend to assume that humans are mistake-free calculating machines; economists have tried to introduce more realistic assumptions under the banner of bounded rationality. This is a far-from-settled problem, mainly because there are more ways to be wrong than there are to be right.

A sunspot equilibrium is a market equilibrium in which prices depend on otherwise irrelevant random variable. The name is inspired by a theory of nineteenth century economist William Jevons that sunspots affected the stock market. (The theory, while wrong, isn’t quite as rediculous as it sounds. Jevons thought, by looking at the data he had on sunspots and agricultural prices, that he detected a pattern that indicated that sunspots caused crops to fail, which in turn caused recessions.)

A sunspot equilibrium would then be a self-fulfilling prophecy. If everyone expected that sunspots caused recessions, that in principle could be sufficient to cause a recession, even if the cause-and-effect existed entirely in people’s heads. Note that this outcome, while not optimal, would still be individually rational: even if you knew that sunspots didn’t really cause recessions, you would know that everyone else was expecting a recession, so you would act accordingly.

Karl Shell, one of the inventors of the concept, has a list of links to his papers on sunspot equilibria. In particular, he links to a short survey article he co-authored with Bruce Smith.

I spotted a survey article, Behavioral Economics: Past, Present, and Future, which gives a guide to this fairly-new field of economics. The subject was born from a mathematical failure. Economists had given precise axioms as to how people would take into account time and uncertainty when making decisions. The axioms allowed precise predictions that (unlike most economics) could be tested in small-scale experiments with a few test subjects. The result was almost-total failure: nearly every prediction turned out to be wrong. Instead of this being the last word on the subject, this has inspired large amounts of research into finding empirical regularities in the discrepancies between the predictions and the experimental results, and formulating a new theory that is both precise and correct. It’s interesting because the original failure could have led to a turn away from mathematical modeling altogether, but it instead has led to research in improved mathematical modeling.

Auctions have provided a real-world arena in which to apply game theory. The theory has actually been applied to design auctions; most famously, the auctions for 3G wireless spectrum were designed along the principles of the theory.

Paul Klemperer has assembled several articles on the subject into a (fairly non-technical) book, and has provided the original articles online. For a more detailed approach, see this survey.

The (sort-of) Nobel Prize in Economics has been announced. The contributions of one of the two, Robert Aumann, are almost purely mathematical work in game theory.

His most interesting idea is that of correlated equilibria. The usual definition of equilibrium in a non-cooperative game, Nash equilibrium, rules out certain kinds of cooperation, even when that cooperation is in the self-interest of each player. Correlated equilibria allow randomized strategies which rely on a random event that is known to both players. Some details about correlated equilibria can be found
here.

(As an aside, the Wikipedia entry for Aumann is unusually bad, so it’s a good candidate for updating, if anyone’s interested. There’s also no entry for correlated equilibrium.)