[Curator’s note: The following material quotes and paraphrases extensively from articles posted by the Nobel Prize Committee and The Princeton Alumni Weekly. See Sources below for details.]
Receiving the Nobel Prize
Sveriges Riksbank Prize in Economic Sciences/ Nobel Prize in Economic Sciences 2012
The Sveriges Riksbank Prize in Economic Sciences in Memory of Alfred Nobel 2012 was awarded jointly to Alvin E. Roth and Lloyd S. Shapley "for the theory of stable allocations and the practice of market design."
His work
How to bring different players together in the best possible way is a key economic problem. From the 1960s onward, Lloyd Shapley used what is known as Cooperative Game Theory to study different matching methods. Within the framework of this theory, it is especially important that a stable match is found. A stable match entails that there are no two agents who would prefer one another over their current counterparts. In collaboration with other researchers, Lloyd Shapley has succeeded in identifying methods that achieve this stability.
Early life
Lloyd Shapley *53 was born on June 2, 1923, in Cambridge, Massachusetts, into a scientific family, two astronomer parents and the fourth of five children. He grew up in the director’s residence at the Harvard Observatory, where he showed an early talent in mathematics.
I had two brilliant, straight-A students for brothers, and I guess my sister was rather bright, too. Nevertheless, we would play mathematical games sometimes around the house, play with cards and multiply them, do things like that. So I had this kind of boost from trying to out-excel my brothers, who were four and six years older than I was, and I did fairly well. So I had a family reputation of being the math whiz.
He attended Philips Exeter Academy, then attended Harvard as an undergraduate, where he did well in mathematics and less well in almost every other subject. Then in the middle of his junior year, in 1943, he was drafted, and spent nearly three years in the army.
Breaks the Soviet weather code
Corporal Shapley
In the Army Air Corps, Shapley was assigned to a secret air base in western China, a weather station which not only made weather observations, it also intercepted broadcasts. This work also involved a certain amount of cryptoanalysis, breaking the codes used in intercepted broadcasts. He was put onto that because he had a high score on the army’s mathematical aptitude test.
The bombers had to shuttle bombs and fuel over from India, and about every two weeks they had enough fuel ready and could make a raid on Japan. The planes couldn’t wait on the ground in China, since they were in range of the Japanese bombers. Because of this, they needed to know if the weather would be good three days in advance over the target area. The fronts moved down from Siberia, across China and to Japan. The weather center intercepted broadcasts from the Soviets and the Japanese, and even the US Navy in the Pacific. They needed masses of data because the quality was often very poor. So they accumulated all the data they could, plotted it out on the maps, and made long-range forecasts, and then sent it out to all the air fields within range on a particular day.
Shapley managed to break the Soviet weather code, which provided them with a lot more reliable data. For this he received a Bronze Star, promotion to corporal, and a raise of $4 a month.
He was discharged right after VJ Day, and within a few months he returned to Harvard for the spring semester. He finished up in June of 1947, though he was class of 1944 for alumni purposes.
Harvard and RAND experiences
I didn’t get my degree in ’47. I finished all the requirements for the degree, but the last semester I also failed two courses … I took four courses over the load you take, and I failed two, and maybe got a couple of A’s in the other courses. The math courses I was generally good at, and music courses sometimes. So they said, “No, you’re on probation. We can’t give a degree to someone on probation. If you survive for a year, you’ll get the degree next year” … So I wasn’t all that gung-ho about Harvard. My performance at Harvard hadn’t been that much. Of course, a great deal of education takes place in a person hanging around – of course, I’d hung around Harvard beforehand.
He had been a math major at Harvard, and done well (he was twice a member of the Putnam team). But he didn’t have any focus on any particular area of mathematics at that point. Out of the Army, out of college, he didn’t really know what he was going to do.
I ended up not poised to go to graduate school. I didn’t know what I was going to do. I was a big music lover, but I had no technical skills there. So I even went to a course at the Union Conservatory one summer before I went off.
He did send out some job applications, one of which was to RAND. And without a formal interview, he was hired. And it turned out RAND was just right for him.
I’m not all that disciplined in getting places on time or going to bed when I should so that I can get up when I should. And RAND is sort of nice – it’s open all the time, twenty-four hours. People work at night, late, and you can just go in there.
At that time, the mission of RAND was very open. It was set up by the Air Force to keep in contact with the scientific community, to get scientists to see and solve problems before a war made the solutions urgent.
So the military felt that, yes, we should keep in touch with the scientists after the war … ” So let’s just not give them an assignment. Give them some money and say, ‘You think of some problems and tell us about it.'” It was a kind of wide-open contract with the Air Force. This led to people like [John] Williams putting together a rather motley crew of people … And he hired crazy students from the math department, me.
At RAND, a group decided to look at game theory. They set up a seminar, meeting weekly, and they would work through a chapter of Theory of Games and Economic Behavior. This book had been originally published in 1944, creating a branch of mathematics to use for economics – that is, to study and analyze situations with multiple actors.
… this was reasonably soon after the publication of Theory of Games and Economic Behavior, John von Neumann and Oskar Morgenstern’s opus, a big thing. I think it was published in ’47, a so-called second edition, which simply has an appendix added, which they didn’t have finished in time before. So it had appeared and had not made much of a splash, got big reviews and von Neumann, anyway, was well-known in mathematics already. But nothing had happened beyond that.
As a result of these seminars, Shapley spent some time working with Roger Snow, another young mathematician, on a problem raised by the book, how to find all the solutions to a matrix game.
We made some progress, and finally I guess I broke it, but Roger was also working on it. So it turned out to be Shapley-Snow … This was a work of mathematics where I had not really even read very many math papers as published and didn’t have any clear concept that I was doing anything special except solving a problem.
Von Neumann read it and became very enthusiastic, because there had been very few papers at that point which had responded to Theory of Games.
So, von Neumann, partly, I guess, for his own ego, said, “I want to encourage this work,” even though he was really not working on game theory anymore. He was working on computer ideas mostly. So he wants to encourage it. So it came back a big rave review or maybe a letter …. von Neumann was all excited about this, and he’ll publish it, and he’ll sponsor it in any journal you name, and so on.
So at that point, my stock went up, and Roger’s went up. At least these two kids – and there weren’t all that many of us around – had something enough to get a real pro like von Neumann interested. So, of course, stop everything else while we write this paper and send it off. This is my first contribution, Shapley-Snow. I call it my piece, really, though I mean it’s helpful to have Roger in the thing, but he kept saying, “You write it, and I’ll read it.”
So that was the real start of my father’s career in game theory.
Princeton mathematician and game theorist
After less than two years at RAND, Shapley headed off to graduate school at Princeton, though during the summers he worked at RAND as a consultant. The time at Princeton was vital to his development as a mathematician.
I had a lot of mathematical – I learned an awful lot of math. I just learned a lot of math at Princeton and I really got educated mathematically.
At Princeton, his dissertation was “Additive and Nonadditive Set Functions,” done with Albert Tucker. While there he published several other papers, including one called “A Value for n-person Games,” which introduced what is known as the “Shapley Value,” a solution concept in game theory, which has become a major part of the field.
One of the early works was this thing which led to the Shapley Value. It was called the Shapley Value, so my name is an adjective for that solution concept. In that case, I can sort of trace the ancestry of how did I think of it. But more specifically, one can look at the – what’s the word? Not paradigm. But anyway, the layout of the field as defined by von Neumann and Morgenstern exactly where, when they decided to do this, you could have done something else, and this would lead to the Shapley Value. So I can sort of provide a foundation for my work by following their arguments up to a certain point and then saying, “No, this is more important.” Sometimes you can use a unique answer. If you insist on uniqueness, you have to give up something else and then a different way. So it was a kind of rank and file. Sometimes it was rather neat that way.
Very soon after he developed the Shapley value, in considering applications, he worked with Martin Shubik on applying it to the measurement of power in voting situations. This led to an item that became known as the Shapley-Shubik Power Index. They, as two unknown graduate students, one in mathematics and the other in economics, had the temerity to submit this paper to the leading journal in Political Science, and much to the surprise of all concerned it was accepted in a few weeks.
Shapley found he enjoyed (and was very good at) doing research – coming up with problems, solving them, proving theorems, and writing and publishing papers. Once he had his degree, he was set for doing academic work – however, he was not really interested in teaching. Largely because of that, he returned to RAND, rather than taking a job as a mathematics professor at a university. He kept doing mathematics, proving theorems, and publishing for years.
Princeton camaraderie
In the fall of 1949, two young mathematicians interested in game theory lived on the same floor of the Graduate College and became friendly rivals — John Nash *50, who won the Nobel Prize in 1994 and became famous as the subject of the book and film A Beautiful Mind, and Lloyd Shapley *53, who was awarded the Nobel prize in 2012.
When Shapley and Nash were hotshots in Princeton’s math department, they frequently sat together in Fine Hall thrashing out ideas about game theory. Kuhn, who now is a professor emeritus in the Princeton mathematics department, said Nash and Shapley were “very much friendly rivals” who enjoyed tossing ideas back and forth or playing Go, a strategy-intensive board game played in China for thousands of years.
“The common room at Fine Hall was the place where everyone met every afternoon, and the ideas sort of bubbled over there,” Kuhn said. “Lloyd Shapley was the best in terms of his overall accomplishments of a very bright group of people. His Nobel is long overdue.”
“Shapley is on a short list of the most important figures in game theory, many of whom were at Princeton at more or less the same time,” including Nash, David Gale *49, and Harold Kuhn *50, said Princeton economics professor Dilip Abreu. “They were huge talents, all present at the birth of the field, and that combination was quite explosive.”
Shapley returns to RAND
Though his specialty was game theory, the so-called mathematics of economics, he had never done economics. His only real exposure to the field was his collaborations with his good friend Martin Shubik, a classmate from Princeton and a long-time professor of economics at Yale.
“Shubik would talk to me for a while, then I would go and explain to him what he had just said.” Shapley also kept in contact with others of his classmates from Princeton. One of these was David Gale, who was at Brown University in 1961. They communicated by mail – long distance telephone calls were expensive and rare.
One day, a letter arrived from Gale framing a problem of choosing roommates. If you have two groups, with each individual having different preferences, is there a way to come up with a set of stable pairings of one from each group? Gale suspected there wasn’t a way to make a stable solution, but couldn’t prove it.
This solution, the deferred matching algorithm, became the paper “College Admissions and the Stability of Marriage,” published in 1962. The paper described how two reviewers rejected it before it was finally published, probably for being too simple – it’s a mathematical paper which contains no equations. As stated in the paper: "The argument is carried out not in mathematical symbols but in ordinary English; there are no obscure or technical terms. Knowledge of calculus is not presupposed. In fact, one hardly needs to know how to count. Yet any mathematician will immediately recognize the argument as mathematical …"
Yet the deferred matching algorithm has turned out to be a major tool in game theory and economics, notably for how Alvin Roth applied it to school admissions.
Even though the mission of RAND was basically military, Shapley was seldom pressured to work in that direction.
RAND was never really trying to pull me into … war-gaming things or that sort of problem, but the great thing about RAND for me is they let me do what I wanted. I succeeded in doing what I wanted … I eventually was getting National Science Foundation (NSF) grants, which were supporting me at RAND and not really drawing on RAND … I stayed at RAND for maybe ten years longer than I would have otherwise, because almost all my support came from there. NSF grants which helped me to do research on whatever it was, that attracted me at that time.
From 1952 until around the late 1970s he had a considerable collaboration with Shubik on the applications of game theory to economics beginning with a note on Solutions on N-Person Games with Ordinal Utilities. His basic work on the core of a game prompted Shubik to consider the convergence of the core to the competitive equilibrium in an exchange economy modeled as market game with side-payments. They generalized the convergence of both the core and the Shapley value to the competitive equilibrium and further work led to a generalization in the form of “Market Games” utilizing the concept of balanced sets of coalitions. This led to a deep understanding of how cooperative game theory solutions related to the competitive equilibria of exchange economies. They also applied the core concept to an economy with externalities. Their investigation of a classical economic example illustrating the emergence of competition led to The Assignment Game and the role of marginal pairs of traders.
Having completed their investigation of the relationship between cooperative game solutions with many agents and the competitive equilibria of an exchange economy, they observed that although Cournot in the 19th century had suggested the relationship between the noncooperative equilibrium and the competitive equilibrium in an open economy, this result had not been generalized to account for a closed economy. A basic economic model was suggested by Shubik, but it took much work and care to tighten this to a clean mathematical model suitable for generalization. The key observation was that the use of money arose naturally in the model in the strategic form of a game. They called the resulting general class of games “Strategic Market Games.”
Shapley was interested in several aspects of voting throughout his career. With Irwin Mann in 1962, he determined the voting power of the several states in the 51-party voting game known as the Electoral College (used to elect American presidents). He also suggested an approach to evaluating a presidential election, treating it as a game among several million voters. This eventually led to results which showed that the voters of medium-sized states have the lowest voting power in the country. Another interesting question dealt with the optimal assignments of “weights” to voters – optimal in the sense that a “correct” result is most likely to be obtained. In this case, he and Bernard Grofman in 1984 determined a formula to assign voters weights based on their likelihood of making a correct decision.
Working with Yale economist Herbert Scarf, they described a system where large, indivisible items (such as houses) could be exchanged in an optimal way, building on earlier work by David Gale on top trading cycles. This was published as “On Cores and Indivisibility” in 1974, and the system described led directly to Alvin Roth’s work on kidney exchanges.
In 1974, he and Robert Aumann coauthored the book Values of Non-Atomic Games; it concerns games with many players, who impinge significantly on the outcome only when they form large coalitions, but not as individuals. Examples are national elections or large economies or markets.
Another collaboration with Aumann concerned perfect equilibria in repeated games; it is a sharpening of the Folk Theorem, which was cited by the Nobel Committee when presenting Aumann with the 2005 prize.
By the late 1970s, Shapley realized that he was the only one left doing game theory at RAND. His colleagues had retired, or gone to various universities. So he quietly put the word out that he would be open to offers. He really only had two institutions on his list – UCLA and Stanford – but he got offers from all over the world. When the offer finally came from UCLA, he accepted it, and moved there in the fall of 1981.
UCLA and academic research
Teaching didn’t appeal to me at all, but academic research did … there was a big reentry on the teaching basis when I came to UCLA … The plus was I had some graduate students who I could really train and imbue with my knowledge, whatever it is. Most of my colleagues in game theory, they have generations of students of students. I had nothing like that. But the other side is I have to teach students, which I’ve learned to do, but I don’t know how to do it very well. So I’m not a big teacher, no. Research is my thing.
It was a joint appointment – in the math department to teach math to math students, and in the economics department to teach math to economics students. So despite being a professor of economics, he has never taught economics.
At a conference in June of 1987, Shapley included in his remarks,” Yesterday was my birthday. I feel like I’m a million years old.” Then he added, “Base two, of course.” The mathematicians got the joke. A year later, Alvin Roth published a book, The Shapley Value: Essays in Honor of Lloyd S. Shapley which begins by saying it is “… in honor of the 1000001st (binary) birthday of Lloyd S. Shapley.”
Though he retired from UCLA in 2001, he has tried to remain active in the field since then, collaborating with former students and attending conferences.
Game theory, I think, was made for me, because I was always messing around with great big game-like models, the sort of thing that now they call “Dungeons and Dragons.” I used to do that with my friends in school and so on. So I maybe had that mentality, too, but also the mathematical push, because this was a kind of mathematics that I had generated, quite apart from the way it’s applied. I’ve always enjoyed the mathematics of it …
The problem I get on, from whatever source, if it starts out mathematical interest, I’ll follow through to the end. If there’s simply an interesting application, well, maybe someone else can do it. I’m right in there in that close equation. The mathematical discovery is the really exciting part. You create conjectures, but you discover results. I think that’s the way. I don’t consider mathematics – you don’t create facts. Of course, the facts are always there, but you discover them.
Game theory … spreads out so quickly into so many different fields. Once you get multiple decision-makers in the same model, all kinds of things can happen. So there’s almost never been a time, after the very beginning of that seminar at RAND that first couple of months, where there was not a game theory problem ahead of me, in front of me somewhere to see. It’s also not only unexplored but unstructured, because I think my main contributions here, the big thing – someone wrote a preface in my book with papers on my sixty-fifth birthday – and he, I think correctly, said that for many years I set the agenda for the field by my work. In other words, I was sort of trying to cover the problems that I could see, so I’m building the theoretical structure as well as working on it.
Lloyd Shapley *53, a UCLA emeritus professor of economics and mathematics and co-winner of the 2012 Nobel Memorial Prize in Economic Sciences, died on March 12. He was 92 years old.
“Professor Shapley was one of the giants of game theory,” said UCLA Chancellor Gene Block. “His work in market design laid the foundation for advances in the matching of kidney donors with transplant recipients, in college admissions procedures, and in assignment of children to public schools. The entire UCLA community joins Professor Shapley’s family in mourning his passing.”
SOURCES
• The Prize in Economic Sciences 2012. NobelPrize.org. Nobel Prize Outreach AB 2022
• Lloyd S. Shapley – Facts. NobelPrize.org. Nobel Prize Outreach AB 2022
This is a biography of my father, Lloyd Shapley, based on various sources and discussions with him over the years. Most of the quotes (indented) are from an interview he did about 20 years ago concerning the RAND Corporation, where he spent much of his career. I also received assistance from Professors Martin Shubik, Robert Aumann, and Guillermo Owen, and my brother Christopher Shapley. — Peter Shapley. Interview, February 9, 1994. Joint Oral History Project on the RAND Corporation. Archives Division, National Air and Space Museum, Smithsonian Institution, Washington, DC.
• PAW: Tiger of the Week Lloyd Shapley
• PAW: Lloyd Shapley wins Nobel Prize in Economics
• Exhibition page photo: The Nobel Foundation, photo by U. Montan.
OTHER RESOURCES
• Daily Princetonian. Games Theory Shapley GS ’53 wins economics Nobel Prize
