In the early 1980s, Rio de Janeiro mathematician Marilda Antonia de Oliveira Sotomayor, then 39, left Brazil to pursue a postdoctoral degree at the University of California, Berkeley. She dreamed of working with the American mathematician David Gale (1921–2008) and achieving scientific independence in a branch of economic mathematics known as economic growth theory, her doctoral research field. Her hopes were dashed when, upon her first meeting with Gale, he told her “I no longer work in that area.”
Disappointed, she spent the next two months at the library looking for a new problem in economic growth to research for her postdoc, until she followed a suggestion from her husband and requested another meeting with Gale to ask what he was currently working on. He told her that he was intrigued by a problem in the area of matching that he believed had a solution, but for which he hadn’t yet been able to show results. Sotomayor worked over the weekend and, on Monday, presented her result to the economist. And so she entered the field known as matching markets, a branch of game theory with applications in economics, in which agents can be matched according to their preferences. She would eventually become a world-renowned expert in the field.
Inspired by her mother’s career, Sotomayor graduated in mathematics and planned to become a higher education teacher. Encouraged by her husband, Peruvian-Brazilian Jorge Sotomayor, a mathematician like herself, she instead dove headfirst into the academic life. She has organized conferences on game theory both inside and outside Brazil and helped make the field known in Brazil and internationally. Three of these congresses took place at the University of São Paulo (USP)—where Sotomayor taught for 17 years—and saw participation from six Economics Nobel Prize winners with whom she maintained friendly relationships. With one, Alvin Roth, she wrote the book that she considers “the most important work of her career.”
Last year, in the middle of the pandemic, Sotomayor was elected a member of the American Association for the Advancement of Science (AAAS), in the social sciences category, in the area of economics. “This honor demonstrates the high regard that experts in your field and members around the country have for you,” the institution said in communicating the honor to her. The mother of a pair of twins who are now 34 years old, the mathematician gave the following interview by email, while in the midst of an extended renovation project at her home in Rio de Janeiro.
Field of expertise
Matching markets, a subfield of game theory
USP, UFRJ, and FGV-RJ
Bachelor’s degree in Mathematics from UFRJ (1967), Master’s in mathematics from IMPA (1972), PhD in mathematics at PUC-RJ and IMPA (1981), postdoctoral studies at the University of California, Berkeley
About 50 scientific articles, one book, coauthored, and four book chapters
Can you explain in a simple way what game theory is, and the field you work in, matching markets?
Game theory is a mathematical theory that studies decision-making situations in which two or more agents interact with each other according to well-defined rules. It has been applied to various areas, such as economics, biology, and computing. In economics, one of the areas in which it has been applied the most is the theory of matching markets, which originated in 1962 with the article “College admissions and the stability of marriage,” by David Gale and mathematician Lloyd Shapley (1923–2016), originally published in the journal The American Mathematical Monthly. In this work, the authors describe the market for student admissions to universities in the United States.
How are students going to university connected to game theory and market matching?
A matching market is configured as a cooperative game, from the point of view of game theory. It’s a mathematical model that represents situations that occur in an environment where participants can freely communicate with each other, making offers and counteroffers with the end purpose of forming pairs. In some situations, the same agent can be part of several pairs. If a pair is formed, the partners negotiate a contract or agreement regarding the terms that define their participation in the partnership. Naturally each participant has preferences about the possible transactions they could enter into. There are rules determining what each can and cannot do. One result of the operation of this market can be just one matching, that is, one set of pairs that do not violate market rules.
Could you give an example?
This is the case in the market for admitting students to universities. This market consists of a set of universities and another set of students. Students have preferences about which universities they would like to enter and [under the US system] universities have preferences about which students they would like to admit. These preferences can be established, for example, based on an entrance exam or other tests, as is the case in the market for admitting candidates to master’s programs in Brazil. Naturally, each university has a maximum number of students it can receive. In this market, matching is a distribution of students among universities that respects the number of vacancies in the universities in such a way that no student is assigned to more than one university. A student may be left without a school and a university might not fill all of its vacancies.
What are the key concepts to understanding these situations?
The key notion is that of stability, defined by Gale and Shapley, which captures the idea of market equilibrium. In the example of universities and students, a matching is stable if the distribution of candidates among vacancies is done in such a way that both groups are satisfied, insofar as it is not possible for any participant to obtain a more preferable partner. In other words, stability occurs if there isn’t a university or student who were not assigned to each other by the matching, such that no student prefers another university over the one to which they were assigned or prefers the university over no school at all, if they were left without a school by the matching. On the other hand, the university prefers the student over other applicants or prefers the student to having a vacancy to fill, if this occurred in the matching. Gale and Shapley showed that a stable matching always exists for the university admission market and offered a mathematical procedure to find it.
The importance of matching theory was recognized when the Nobel Prize was awarded to Shapley and Roth in 2012
Does this idea have other applications?
In other markets, in addition to a matching, the monetary gains obtained by each participant in the conducted trades are specified. This is what happens in a labor market, formed by companies and workers. Every company wants to hire a certain number of workers and every worker wants to be employed by a company. If a company and a worker establish a relationship, then these agents must negotiate the worker’s salary based on that worker’s productivity, taking into account what other peers in the market have negotiated. Naturally, there is a reserve value for each agent [company and worker], representing the minimum monetary gain each would be willing to accept for each partnership they are able to form. For the worker, the reserve value is the lowest salary they would be willing to accept from a company; for the company, it’s the smallest net profit it would be willing to earn from a partnership with a worker. One result for this market consists of a matching that specifies who works for whom, along with workers’ wages and the firms’ net profits. Such a result is stable if all agents are receiving an amount greater than or equal to their reserve value and, furthermore, there is no company or worker not associated with each other through the matching such that the firm can pay the worker a salary greater than what she’s receiving and still obtain a net profit greater than what it’s earning with one of its current partners, which in this case means all the workers associated with their company through the matching.
Why are matchings interesting for economics?
Because they reflect the behavior of people in real-life markets. Matching theory provides mathematical models for these markets. Through these models we can understand and detect the shortcomings of these markets, which can help in organizing them better or fixing them when they break. Matching theory received recognition for its importance to economics in 2012, when the Nobel Prize in Economics was awarded to mathematician Alvin Roth and Lloyd Shapley. Shapley, along with Gale, was the founder of the theory and Roth led its application in real-life markets. Gale would also have won the award had he been alive, but he passed away in 2008.
What was your work and friendship with Gale and Roth like?
With Roth I wrote the most important work of my career, Two-sided Matching: A Study in Game-theoretic Modeling and Analysis. The book was published in 1990 and received the Lanchester Award from the Operations Research Society of America, the most sought-after prize in the field of operations research. I have a friendly relationship with him and with Robert Aumann, Eric Maskin, Roger Myerson, and Paul Milgrom—who are all Nobel laureates. I was also very good friends with Shapley and with John Nash (1928–2015) and built a very special relationship with Gale, with whom I wrote several works. My first article on matchings was coauthored with him during my postdoctoral studies at the University of California, Berkeley, in 1983. The article appeared in American Mathematical Monthly the following year.
How did you meet these academics?
With the exception of Nash, whom I first met in 1995 at a congress in Jerusalem held to honor Aumann’s 65th birthday, none of them were Nobel laureates when I met them. Except for Alvin and Gale, my first contact with them was at the Stony Brook game theory conferences, the most important international event in the field, which began in the 1980s. In 1991 I was invited to give a plenary lecture and from then on, I began attending. I was the science organizer for the congress in 2006 and the following year I organized—for the next year’s congress—a one-day event called Gale’s Feast, in honor of David Gale’s 86th birthday. He attended with his family. Almost all of these scientists came to Brazil for the game theory congresses that I organized at USP [University of São Paulo], some more than once.
What was your career like before you began working with matching?
My path from the time I graduated from UFRJ [Federal University of Rio de Janeiro] until I earned my master’s degree at IMPA [Institute of Pure and Applied Mathematics] was natural and motivated by my desire to learn more and become a professor of higher education. I heard about IMPA during the last year of my undergraduate degree. My class invited Professor Lindolpho de Carvalho Dias, then director of the UFRJ Institute of Mathematics, for an informal chat. During this conversation, I learned that he was also director of IMPA and that I could take undergraduate research courses there to complement my degree, with a view to entering their master’s degree program in mathematics. I felt really excited about the possibility of continuing my studies. The following year I started studying at IMPA with a grant from CAPES [Federal Agency for Support and Evaluation of Graduate Education] in partnership with the Ford Foundation. While working on my master’s degree in the early 1970s I was hired by the PUC [Pontifical Catholic University] in Rio, where I worked for 25 years. The intellectually stimulating work environment at the university’s Department of Mathematics, where I taught, and my marriage to a professor at IMPA who was a big enthusiast of the scientific career, influenced my decision to enter the doctoral program at the PUC Department of Mathematics, which was conducted in partnership with IMPA. I earned a PhD in mathematical sciences with a thesis in the area of mathematical economics. My interest in becoming a matching markets researcher arose while I was a postdoctoral fellow at the University of California, Berkeley.
What was it like working at PUC-RJ?
Teaching has always been very absorbing and rewarding for me. When I returned from Berkeley, I needed space to develop my research area. However, game theory was not an area considered of interest by my department at PUC. Even so, by taking on a heavy workload I managed to offer a course on matching that attracted the Math Department’s best student, who completed her master’s thesis in matching markets under my supervision. I remained at PUC until 1993, when, through a competitive exam for full professor, I moved to the Department of Economics at UFRJ. When I retired there, I migrated to the Department of Economics at USP in 1997.
At what point did you become interested in the theory of matching markets?
I intended to do my doctorate in stochastic processes [processes that evolve according to random variables, in probability theory]. After two months of study in this area, my advisor, Jack Schechtman, presented me with a problem in the area of economic growth, saying, “It’s not about stochastic processes. But don’t worry, I just want to know how well you handle mathematical economics.” His economic model was ready, and I only had to deal with the mathematics. It was a generalization of Jack’s dissertation problem; he’d completed his doctorate under Gale, who was then a full professor in the math department at the University of California. And that’s how this problem generated my doctoral dissertation in 1981. It was published in the Journal of Economic Theory in 1984, with the title “On income fluctuations and capital gains.” After getting my doctorate, it happened that in order to continue working in this area I needed to gain scientific independence. Finally, the idea of doing a post-doctorate with Gale came up. I applied for a postdoctoral fellowship from the CNPq [National Council for Scientific and Technological Development] and went to Berkeley in February 1983, hoping to learn more about economic growth and gain the scientific independence I was looking for. Once there, I presented my dissertation to Gale. At the end of my presentation, he congratulated me, but then, to my disappointment he said to me in a solemn tone, “I’m not interested in that area anymore.”
Then what did you do?
I spent two months going to the library and trying to find some new problem in economic growth that interested me. One day my husband, who was also in a postdoctoral program at Berkeley, brought to my attention that I wasn’t taking advantage of the opportunity of being so close to such a remarkable mathematician as Gale. “Why don’t you ask him what subject he’s interested in now and try to learn that subject, whatever it is?” he suggested. Although the idea of learning a new area in mathematics in such a short time might have seemed absurd to me, I did what he suggested. Gale replied that he was interested in matching markets and gave me three articles and a book to read. One of the articles was the one he wrote with Shapley on students and universities. I worked to understand the definitions and demonstrations of the results. A few weeks later I returned the material Gale had lent me. I had no idea what to make of all that new knowledge, but I was hoping he might point me in some direction. Then I got another disappointment. “Well, I don’t have a single problem for you.” As I was leaving his office, he called me back with a paper in his hands: “The only problem I have is this proposition, which I’ve been trying to demonstrate for some time without success. If it’s true, it will be possible to demonstrate the theorem of Lester Dubins [1920–2010] and David Freedman [1938–2008] in three lines. It would be great to have a shorter demonstration of this result because then you could teach it in just one class,” he said.
I was always a good math student. At the age of 9, I received a prize for achieving the highest grade in our final exam
Were you able to do it?
Even without realizing the importance, both practically and theoretically, of this result, finding a shorter proof for him was a great challenge. I went home with that paper in hand and worked diligently over the weekend. It was hard to wait until Monday to show Gale my demonstration. Toward the end of my presentation, he started clapping his hands and exclaiming very excitedly: “You proved it! You proved it.” That’s when my journey began in a field of research that was virtually unexplored at that point, with only half a dozen articles published. But it opened the doors to the emergence of a mathematical theory with many applications for economics, and was to receive over the years the contributions of countless mathematicians and economists, finally winning recognition with the Nobel Prize in Economics in 2012.
Is your husband also a mathematician?
Yes. I married him in 1970, when I was a professor at PUC-RJ. He was already a renowned mathematician and provided an example for me, shared the housework with me, raised two children with me, and was the biggest supporter of my career. Now Jorge is a retired full professor at USP. He works in dynamical systems [a type of function that represents the values of a variable over time, such as the mathematical model which represents the swinging of a clock’s pendulum]. I met him at IMPA in 1970, where he was a full professor for over 20 years. He is Peruvian, and now a naturalized Brazilian. He came to Brazil in 1962 to study dynamical systems with Maurício Peixoto [1921–2019], a professor and one of the founders of IMPA. He finished his doctorate in 1964.
When did you become interested in mathematics?
I had been a good math student since I was a child. At the age of 9, I received a cash prize for receiving the highest grade in the final math exam among the fourth-grade students in the neighborhood of Jacarepaguá, in Rio de Janeiro. The challenge of solving mathematical problems has always been very intellectually stimulating for me. But it was during my fourth year at the Escola Normal Carmela Dutra in Rio, where I graduated as a primary school teacher in 1961, that I discovered that I wanted to be a high school mathematics teacher. Our math teacher was much feared among the students and known for giving tough tests. I scored 100% on all the tests throughout the year, thus winning a competition with four of my colleagues, who didn’t ace the final exam.
Was your mother a math teacher?
Traditionally, in every country in the world, men have the support of family and society to realize their work ambitions and succeed, while women, in general, are still seeking their rights. A successful scientific career requires many years of dedication to research, participation in relevant congresses, visits to research centers, interaction with other researchers, etc. Women my age lived at a time when all this was very difficult for us. It was inconceivable that a woman, married or single, would travel abroad alone or have male coworkers—and if she were married, have commitments other than to her housework, children, and her husband. Women lived in the shadow of their spouse’s professional success. They lost their identity, but prided themselves on being known as Mrs. X or the wife of Dr. Y. The only profession widely regarded as appropriate for women was teaching children, because at that time there were no male teachers in primary schools. Only a minority went to university. In my case, I had the privilege of being the daughter of a high school math teacher who was ahead of her time. Noticing my love for mathematics when I was still at an early age, she guided me to attend college and obtain a degree. I entered the School of Philosophy, Sciences, and Letters, at what was then called the University of Brazil [now UFRJ], in 1964.
My election to join the American Association for the Advancement of Science in 2020 was a dream come true
Were there women in the program?
At graduation, although my class was small, women were in the majority. At that time, men who liked mathematics went into engineering, which gave them a profession with more status than teaching. A degree in mathematics was more sought after by women. Engineering was not considered a profession for women. However, when I went to IMPA only 30% of the institution’s master’s students were women. That situation has changed. Over the intervening years, women are no longer only housewives, and have been fighting to make their dreams come true. However, a woman isn’t required to go any further than taking a higher education course and attaining a doctorate. This feat alone is celebrated by the family. If she ventures into research, she’s not usually encouraged, nor is it demanded that she be successful at it. Much is still demanded of her in her domestic life, and when her spouse’s profession is outside academia, professional interests can conflict. Perhaps this explains the underrepresentation of women in scientific and research awards. The TWAS [Third World Academy of Sciences] award, considered the Nobel Prize for developing countries, which I received in 2016, is an example. Among the winners in all nine categories, I was the only woman. In Brazil, we’ve made large strides in the participation of women in universities, but in some more economically developed countries, such as Japan, being a higher education teacher is still a male profession.
How has your life been in the midst of the coronavirus pandemic?
My husband and I have been in confinement since the beginning of the pandemic. But after 20 years living in São Paulo, I became a real homebody. On the other hand, we’re lucky to live in a large, comfortable house here in Rio de Janeiro, where we can walk around the backyard and get some sun. We have a heated indoor pool, which allows us to exercise any time of year. It’s the house where I grew up. In January of last year, I started a renovation project. I’m the architect and the administrator at the same time, and these activities, along with those of being a homemaker—although both are very pleasurable—have consumed a lot of my time. I’ve discovered that I enjoy cooking and I’m collecting little scorch marks… So, I haven’t had much free time for my research, which has been very distressing. But what bothers me most about the pandemic is the fact that we can’t receive anyone here at the house who arrives by bus and can’t go to an in-person medical appointment.
What’s changed for you professionally with the pandemic?
Almost nothing. I’ve been retired since 2014 from USP, where I obtained my full professorship in economics [at the School of Economics, Administration, and Accounting, FEA]. Today I have a position at FGV-RJ [Fundação Getulio Vargas, Rio de Janeiro], where I offer an annual, one-and-a-half-month short course. As a researcher, I like working from home because I’m more productive. Last year I managed to finish three articles, which are currently being submitted for publication. I also participated in an online congress in Providence, USA, as a guest. It was a very rewarding year. In May of 2020, I was elected a member of the American Association for the Advancement of Science [AAAS], but due to the pandemic my inauguration was postponed to 2022. Anyway, being elected to the AAAS was like a good dream in the middle of a big nightmare. Still to come in 2021, I have two articles to finish and participation in two international congresses as a guest speaker. I consider myself an accomplished researcher, whose scientific work has been recognized in Brazil and abroad.