Economics Puzzle Leads to a New Understanding of a Fundamental Law of Physics

Right now, molecules in the air are moving around you in chaotic and unpredictable ways. To make sense of such systems, physicists use a law known as the Boltzmann distribution, which, rather than describe exactly where each particle is, describes the chance of finding the system in any of its possible states. This allows them to make predictions about the whole system even though the individual particle motions are random. It's like rolling a single die: Any one roll is unpredictable, but if you keep rolling it again and again, a pattern of probabilities will emerge.

Developed in the latter half of the 19th century by Ludwig Boltzmann, an Austrian physicist and mathematician, this Boltzmann distribution is used widely today to model systems in many fields, ranging from AI to economics, where it is called "multinomial logit."

Now, economists have taken a deeper look at this universal law and come up with a surprising result: The Boltzmann distribution, their mathematical proof shows, is the only law that accurately describes unrelated, or uncoupled, systems.

The research, published in the journal Mathematische Annalen, comes from two economists and mathematicians who both have backgrounds in physics: Omer Tamuz, a professor of economics and mathematics at Caltech, and Fedor Sandomirskiy, a former Caltech postdoc now serving as an assistant professor of economics at Princeton University.

"This is an example of how abstract mathematical thinking can bridge different fields—in this case, linking ideas from economic theory to physics," Tamuz says. "Caltech's interdisciplinary environment fosters discoveries like this."

To understand why a scientist would be interested in unrelated systems, consider an economist who is studying how people choose between two cereal brands. In developing a theory to describe this behavior, the scientists need to ensure that their simple model does not make nonsensical connections. For example, if the model predicted that a person's preference for a cereal brand depended on which dish soap they bought that day or what color of shirt they wore to the store, the scientists would know something was wrong with the model.

"We would rather not track extra choices that seem irrelevant, like which soap the shopper picked in another aisle," Tamuz says. "We ask the question: When would including that seemingly unrelated choice leave the model's prediction unchanged?"

While the Boltzmann distribution accurately describes such unrelated systems, Tamuz and Sandomirskiy wondered: Are there alternative theories that can do the same thing?

"Everybody uses the same theory," Tamuz said. "But which other theories have this nice property that correctly maintains the lack of connection between the unrelated behaviors? Should we use those theories instead? If there are such theories, they might be useful in both economics and physics. If there are not, then we would learn that the Boltzmann distribution is the only physical theory that is not nonsensical and that multinomial logit is the only economic model that predicts independent choices in unrelated situations."

A Throw of the Die

To find other possible theories that might work for unrelated systems, the economists developed new ways to test the underlying math. Tamuz likes to use dice to explain how they tackled the problem. Each roll of a die is random—you might get a 1, 2, 3, 4, 5, or 6—and can be thought of as the behavior of an individual person or physical system. If you roll the die many times, you'll start to see a pattern emerge—each outcome, the number 1 through 6, will occur close to one sixth of the time. This is the distribution of the single die.

If you throw two dice and record the sums of the dice, you'll get a different distribution. For example, the chance of getting a total of a 2 is 1/36 because there is only one way to roll a 2 (a 1 and a 1). But the chance of rolling an 8 is 5/36 because there are five ways to roll an 8 (a 4 and 4, 3 and 5, 5 and 3, 2 and 6, and 6 and 2). Importantly, the outcome of one die contains no information about the outcome of the other, as these are two unrelated physical systems. Going back to the economics example, one die is like the choice of cereal, and the other die is like the choice of dish soap. The random choices should not affect each other.

To understand how the researchers tested alternative theories to the Boltzmann distribution, we need to introduce a "crazy" pair of dice, such as so-called Sicherman dice, invented in 1977 by puzzle creator and math enthusiast Col. George Sicherman. Tamuz (who, in fact, keeps a pair Sicherman dice on his desk) explains that the numbers on these six-sided dice are wacky, with one die in the pair having the numbers 1, 3, 4, 5, 6, 8, and the other, 1, 2, 2, 3, 3, and 4. While each die is very different from a normal die, if you rolled both of them and only recorded the total value, you would not be able to tell them apart from regular dice. As with normal dice, the chance of rolling a sum of 2 with Sicherman dice is 1/36, and the chance of getting 8 is 5/36. In other words, the probability distribution of the sums of each type of dice is the same.

Tamuz and Sandomirskiy realized that they could use the math underlying these Sicherman dice to test alternative theories. If a theory led to the normal and crazy dice having the same probability distributions of sums, it passed the test for accurately describing unrelated systems. If the normal and crazy dice had different probability distributions of sums (which is like demonstrating the nonsensical example of soap choices affecting cereal choices), the theory failed.

The trick to testing additional alternative theories was to find more examples of crazy dice beyond the Sicherman dice. Each additional example they discovered could be used to test more theories. There are an infinite number of possible theories, which they were able to match with infinitely many theoretical pairs of crazy dice. In the end, they developed a math proof that ruled out all the alternative theories and showed that the tried-and-true Boltzmann distribution, used fervently in science for more than a century, is the only one that works.

In mathematical terms, the research boils down to polynomials, functions such as f(x)=x + 3x² + x³ which might be familiar from algebra classes. All the distributions discussed above, whether they be Boltzmann or alternative theories, can be represented by polynomials. For example, the first Sicherman die, with sides 1, 3, 4, 5, 6, 8, is represented by f(x) = x¹ + x³ + x⁴ + x⁵ + x⁶ + x⁸. The second Sicherman die, with sides 1, 2, 2, 3, 3, 4, is represented by g(x) = x¹ + 2x² + 2x³ + x⁴. The product of these polynomials, f(x) · g(x), is another polynomial that represents the distribution of the sums. This is the same as the distribution of the sums of two regular dice, each of which is represented by h(x) = x¹ + x² + x³ + x⁴ + x⁵ + x⁶. So, the product of h(x) and h(x) is the same as the product of f(x) and g(x).

This math represents the independence of the unrelated systems. The researchers' final mathematical proof required novel insights into such polynomials.

"We didn't know what to expect when we started this," Sandomirskiy says. "We were intrigued by these paradoxical predictions and wondered what it meant for a theory to not have any. In the end, we learned that it means that it has to be Boltzmann's theory. We found a new angle on a concept that has been a textbook staple for over a century."

The study titled "On the origin of the Boltzmann distribution," was funded by the National Science Foundation.