Big Numbers

Swarms, Crowds, and PDE’s April 9, 2010

I’ve been in Austin the past couple of days, checking out the UT grad school.

Notes: the hot dogs from the carts on Sixth Street are the best I’ve ever had — the Best Wurst comes with grilled onions, sauerkraut, and curry ketchup. Prices are far better than the East Coast madness I’ve become accustomed to. The weather is ethereally lovely. The locals are friendly to strangers, and the grad students are warm and casual, though pretty damn serious about research. There are Texas stars everywhere.

There was a very interesting talk by Pierre-Louis Lions that’s of somewhat general interest. He’s using the techniques of statistical mechanics, in particular mean field theory, usually used to describe gas particles, to instead predict the behavior of large groups of people.

He’s noticed that there are certain regularities in social science data. For example, take the populations of the twenty largest cities in a country. They follow the same distribution in every country. The same is true of the distributions of the highest incomes in a country, the so-called Pareto Tails. The same is true of the distributions of the market share of the largest firms. Yet nobody organizes these systems.

He sets up a system of coupled nonlinear differential equations. It’s a “planning problem”– we have some outcome functionm across the population, we know where it starts, and we define where we want it to end. To get all the individuals in a population to behave in such a way as to get the desired final outcome, we need to offer them some kind of incentive, so we introduce a utility function u. The outcome m depends on u, and the utility u depends on the current state of the outcome function m, so these equations are coupled.

They also can’t be solved explicitly, so the interesting results are normally in special cases, where you can show the existence and uniqueness of solutions, regularity properties for the solutions, and sometimes mimic the same distributions we see in the biggest cities and biggest firms and so on.

A mean field game is a generalization of the statistical mechanics problem solved by the Hamilton-Jacobi equations, which govern a continuum of particles moving so as to optimize both their final location and minimize their cost of travel. In a mean field game, the particles also “care” about the density of the other particles around them, so the utility of each particle depends on the behavior of the other particles. This is what makes it a form of game theory.

The applications he gave were in crowd control — where do you put the exits in a theater? How should you arrange the shape of a mall? People in a crowd are very much like particles — they don’t like to bump into each other, they move somewhat randomly. You can model them the same way.

But I think the ideas here should also be suggestive for anyone who follows politics. There’s an ancient debate between the technocrats and the anti-technocrats. The technocrats think that smart people can arrange society’s structures to ensure desirable social outcomes; the anti-technocrats think that millions of individuals, all independent, acting in their own self-interest, will invariably thwart the planner, who can never achieve his desired outcome.

This research, though, ought to give would-be planners some hope. Whenever a solution exists, we can assert that there is some system of incentives that drives the crowd to behave in the desired fashion. The “moral” of the story is the same as the “moral” of mechanism design, except that instead of doing game theory with a few actors, we’re doing statistical mechanics with millions of actors.

Here’s a link to an earlier talk of Lions’ on mean field games at UCLA.
Terence Tao has a lovely and more detailed exposition on his blog.