As consumers, we are constantly making choices about things like what to eat at a restaurant, what clothing to buy at a store, the type of car we'd like to purchase, and so much more. Savvy producers of such goods look to economists to help estimate what products people want and how demand might shift when factors like price and availability change.
For many years, economists have relied on something called the random utility model (RUM)—a fundamental framework for rational decision-making widely used in empirical analysis in economics, marketing, and many other fields—to help predict consumer decision-making. However, there are problems using the model when the full variety of options is unknown. This issue is related to the linear ordering problem and is one that has remained unsolved since researchers across applied mathematics and mathematical psychology first started exploring it in the 1980s. Now, researchers led by Kota Saito, professor of economics at Caltech, and MIT graduate student Alec Sandroni (BS '25), have solved one of the main challenges that arises when RUM is used with incomplete datasets.
"Our findings may help improve predictions of individual choice when all the options are not observable," explains Saito, who is coauthor of a paper published in the August issue of the American Economic Review about the team's work.
RUM is the foundation of a method called discrete choice analysis, which economists use to model consumer decisions statistically to help inform pricing strategies, antitrust analysis, policy evaluation across industries, and more. It describes how a group of consumers with diverse preferences makes choices, but only when all options are clearly defined. When all possible choices aren't defined, economists often group the alternatives into an "outside option."
"We found that even if you don't observe which outside alternative gets chosen, you can get a surprising amount of information just by looking at what wasn't chosen. Some of this information might get ignored in the typical methods," says Sandroni, who worked with Saito throughout his undergraduate career as a Summer Undergraduate Research Fellowships (SURF) program student.
In fact, Sandroni, Saito, and Haruki Kono, a graduate student at MIT and a third collaborator on the study, found that the outside option tool can produce severely biased estimates. Using network flow theory—a computational method often used to model the movement of entities such as traffic and fluids—the researchers were able to clarify the limitation of the outside-option approach.
"Using our innovation, you can test for RUM with less data than we thought necessary for the RUM to work well," Sandroni says. "People can use our new technique to better evaluate the demand in a variety of industries."
Sandroni, who was a math major at Caltech, is continuing to work with Saito to investigate additional challenges in analyzing consumer choice, but he hasn't always been interested in economics.
"I thought I was going to do environmental science or physics when I first came to Caltech," he says. "But then I was looking for SURFs, I found a little math test that Kota has on his website. I think I was the only one who solved it that year, so we started working together, and I'm still studying economics today."
Although Sandroni knew little about the topic before starting the research project, Saito says he worked at a very high level on the recently published paper, generating ideas, completing rigorous proofs, and finalizing the manuscript.
"Alec quickly learned what he needed to do, and the fact that he could walk with me throughout the years is really impressive," Saito says. "The requirements of the work were very demanding for an undergraduate student, but he overcame these difficulties."
Yuexin Liao (BS '26), who is now a graduate student in mathematics and computer science at Caltech, investigated another decision-making query related to RUM last summer for a SURF project with Saito and Sandroni. In economic studies where an individual selects one option from a finite set of mutually exclusive alternatives, many datasets group choices into broad labels—for example, a car model with many trims, or the single "outside option." The main question of the project is: "When does this kind of aggregation faithfully reflect how people actually choose?"
The team showed that aggregation creates an ambiguity of composition, meaning that the grouped label may hide different underlying options for different people or markets. Under this ambiguity, the RUM has surprisingly weak testable implications. A recent working paper by the group was the first to formally study the implications of this ambiguity in the composition of aggregated alternatives. The paper was accepted by two well-recognized international conferences in economic theory; Sandroni presented the paper this summer, which is unusual as a first-year graduate student.
Saito has now recruited a new SURF student, Vivian Loh, a rising third-year student in mathematics. Loh, Saito, and Sandroni have initiated new challenges, including a proposal for a model that can describe human choice more generally through simple conditions called monotonicity. They also aim to apply economic decision theory to machine learning.
"I enjoy working with undergraduates because they are often open to very difficult questions and bring new approaches to the problems," Saito says.
The American Economic Review paper is titled "Random Utility with Unobservable Alternatives." The research was supported by the National Science Foundation and Caltech's SURF program.