INFORMS News: In Memoriam - Herbert E. Scarf (1930-2015)

Herbert E. Scarf

Distinguished economist Herbert E. Scarf, Professor Emeritus of Economics at Yale University and a recipient of several of INFORMS’ most prestigious awards, passed away on Nov. 15. He was 85.

Professor Scarf was known for his seminal work in inventory theory and the computation of fixed points – a key requirement in determining economic equilibria. Among many contributions, he will be remembered for the Debreu-Scarf Theorem, showing that the set of core allocations of a competitive equilibrium shrinks to the set of competitive equilibria as the economy grows large, and for Scarf’s Theorem, showing that every balanced game has a nonempty core. Scarf’s Lemma, the combinatorial engine behind Scarf’s Theorem, has subsequently seen application in a wide variety of fields.

In a mini memoir, “My Intellectual Trajectory,” he penned for a conference presentation, Professor Scarf recalled that as a schoolboy in Philadelphia he awoke “one fine morning with the realization that I was a mathematician.”

He wrote: “I don’t mean this in a formal sense; I simply grasped what mathematics was all about. I knew that a mathematical result might have several quite distinct arguments, which could be combined in a variety of ways. I knew that a Theorem was different from a Lemma. I read the biographies of great mathematicians, and I still have my annotated copy of “Men of Mathematics” by E.T. Bell. I taught myself the calculus of several variables and the Theory of Complex Functions. I memorized the first 35 digits of Pi.

“My instructors at the South Philadelphia High School for Boys – a pretty roughhouse school – knew nothing about this passion of mine. In my eleventh grade I learned about a mathematics tournament offered by Temple University for all high school students in Pennsylvania. To the shocked surprise of my teachers and my relatives, I placed first in the tournament.”

Scarf earned a scholarship to Temple where he received his bachelor’s degree. “As a student, I had unusual habits,” he wrote. “I started to take graduate courses immediately. I rarely attended class; I would learn the material by myself and drop in to take the exams.”

Scarf went on to do graduate work at Princeton University under Salomon Bochner, studying alongside the likes of Ralph Gomory and Lloyd Shapley. John Nash, who had just earned his degree, was another notable presence. Mr. Scarf wrote his dissertation on differential operators on manifolds and applications to stochastic processes and received a Ph.D. in 1954.

Scarf’s first postdoctoral position was at the RAND Corporation where he familiarized himself with the works of George B. Dantzig and other burgeoning operations researchers. He ended up in RAND’s unit of the Economics Department involved in operations research and management sciences while working on inventory problems such as the purchase and storage of commodities whose future demands were not known with perfect certainty.

“I met Kenneth Arrow, who was himself working on the management of inventories,” Scarf wrote. “My life was changed. He invited me to spend a year with him at Stanford jointly working on inventory theory. It was a perfect time for me. The major themes of economic theory were being formulated in mathematical terms, and I fortunately had precisely the right set of skills to make serious contributions. A lovely set of apples was hanging from the tree, and I plucked them and ate them one after another with great pleasure.”

In 1957, Scarf accepted a professorial position at Stanford University where he worked alongside such O.R. leaders as Arrow and University of California-Berkeley professor Gerard Debreu, both of whom Scarf considered mentors and close friends. In 1958, he published his first major article on differential games with Lloyd Shapley of RAND in Albert Tucker and Philip Wolfe’s “Contribution to the Theory of Games.”

Professor Scarf’s major contributions to inventory policy began with his presentation of the optimality of (S,s) policies in the dynamic inventory problem at the First Stanford Symposium in 1959, building upon the work of Theodore Harris, Jacob Marschak and Arrow. That year, Professor Scarf had been a visiting associate professor at the Cowles Foundation for Research in Economics at Yale University. In 1963, after a year with Stanford’s Center for Advanced Study in the Behavioral Sciences, he accepted a full professorship at Yale where he remained throughout the rest of his remarkable career.

Professor Scarf twice served as director of the Cowles Foundation (1967-1971 and 1981-1984) and the Division of Social Sciences (1971-1974). He also kept a close relationship with the operations research community of the San Francisco Bay Area, visiting U.C. Berkeley’s Mathematical Sciences Research Institute as Research Scholar in the late 1980s and as a visiting professor to Stanford in the late 1970s.

Starting in 1963, Scarf sought a constructive procedure for finding economic equilibria without appealing to the standard fixed-point arguments. Over the next 10 years, he built upon the work of Carlton Lemke and Harold Kuhn. In 1973, Scarf and Terje Hansen (a Yale Ph.D. whose dissertation was on the subject) published “The Computation of Economic Equilibria.” The book, lauded as the first comprehensive treatment of an idea that “permits the constructive computation of approximate fixed points of continuous mappings,” resulted in Scarf receiving the 1973 Frederick W. Lanchester Prize from the Operations Research Society of America (ORSA, which merged with The Institute of Management Sciences in 1995 to create INFORMS). The publication is still celebrated as an elegant blend of theory, computational experimentation and practical application.

In 1983, Professor Scarf received the John von Neumann Theory Prize from ORSA, and he was named a member of the inaugural class of INFORMS Fellows in 2002. In addition, Professor Scarf was an elected fellow of the American Academy of Arts and Sciences, as well as the Econometric Society. Scarf was also a member of the National Academy of Sciences and a Distinguished Fellow of the American Economic Association.

Sources: INFORMS, Yale University, “My Intellectual Trajectory” by Herbert E. Scarf