Auction Theory

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.

Nobel Prize in Economics

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

(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.)

Lévy processes

David Applebaum has a nice survey article on Lévy processes. As we’ve mentioned before, a persistent modelling problem in finance is that the variance of changes in financial time series, such as stock prices, seems to be infinite. This shows up as large jumps in price, larger than can be explained by Brownian motion. Lévy processes, a broad class of stochastic processes that generalize both Brownian motion and Poisson processes, are one candidate to model prices.

The Muddy Children Problem

Many mathematicians grew up on a diet of puzzles like those set by Raymond Smullyan and Martin Gardner. Unfortunately, ingenious and elegant as these puzzles often are, they frequently have solution methods that don’t give rise to generalisable theory.

So I was recently surprised to find that one of my favourite puzzles of this type, the Muddy Children Problem, is actually an important example that appears in courses on mathematical logic, epistemology, computer science and even quantitative finance. If you haven’t met the problem before then have a go at solving it before looking at the various papers and courses on the subject. A fairly detailed elementary treatment can be found here though there are easier to understand informal arguments in existence.

The main academic approaches to the problem are via modal logic and Kripke models.

In less politically correct days it was known as the unfaithful wives problem and Smullyan’s version of this problem involved logicians with coloured hats.

Did I mention that it’s also a drinking game?

The (Mis)Behavior of Markets

Michael of comment board fame had lent me Benoit Mandelbrot and Richard Hudson’s The (Mis)Behavior of Markets a while ago, and I finally had a chance to read it. The verdict? Still not sure.

Mandelbrot offers an eloquent critique of contemporary financial theory, and speculates on some alternatives. The limitations of the financial theory presented in textbooks is well known: rare events happen more often than predicted by a normal distribution (so-called “fat tails”), and changes in the volatility of financial time series tend to persist, so this part of Mandelbrot’s book is not original, while he does a good job of explaining it.

The part that is new is a series of alternative proposals for financial models. Unfortunately, since the book is written for a general audience, it’s thin on technical details, so I’m not really sure if they’re a good idea or not. I tracked down some links which I’ll work through as I get the chance:

Brad de Long on the “Marshallian toolkit”

Brad de Long, a Berkeley economist, has an interesting post on his weblog about limitations of current models in economics. The "Marshallian toolkit" means basically the kind of economics you find in a microeconomics textbook (in a more sophisticated form, of course). Brad is claiming that these models simply fail to explain what makes some economies grow and others stagnate.

An example of the kind of model Brad refers to is the Solow growth model, which has three basic inputs: the amount of labor, the amount of capital (which includes things like factories), and a third factor, productivity, which represents how efficiently capital and labor are used. Changes in capital and labor are explainable in terms of conventional economics, but productivity is basically a black box for technological change. When economists fit the model against the data, it turns out that truly dramatic economic growth comes from increase in productivity, the very factor that is beyond the reach of conventional economics.