# In memoriam: Herbert Scarf, pioneering economist and inspiring teacher

Herbert Scarf, Sterling Professor Emeritus of Economics, died from heart failure Nov. 15, surrounded by family at his home in Sag Harbor. He was 85 years old.

Scarf was one of a handful of mathematicians who completed the transformation of economics from a ﬁeld not much more mathematical than social sciences like sociology and psychology into a discipline with the same mathematical rigor as physics and the other hard sciences. What set him apart from even his fellow mathematician-economists, his colleagues say, was the elegance of his work and its exposition. Sticking to the most fundamental questions, often raised by his friends, and searching relentlessly for the simplest answers, he created proofs and lectures whose harmony, it is said, has never been surpassed in economics.

“Herb Scarf had a great gift,” says former Yale president Richard Levin, who was one of Scarf’s colleagues in the Department of Economics. “He could convey the most complex results of his brilliant research in the simplest and clearest terms, making the diﬀicult transparent. He was a warm friend, a devoted teacher, and a loyal colleague, who will be remembered by so many.”

As a young boy who taught himself mathematics, Scarf astonished his teachers at South Philadelphia High School by ranking ﬁrst in the 1947 Statewide Mathematical Tournament. His mathematical talent earned him a scholarship to Temple University, and after placing in the top 10 nationally in the Putnam mathematics exam he obtained a scholarship to the graduate mathematics program at Princeton University, receiving his Ph.D. in 1954.

Writing about his time at Princeton, Scarf said he “was disappointed by the ultra‐pure mathematics that was the bread and butter of the department at that time,” and he hoped “that the mathematical problems I worked on would have an ultimate practical application: that life would be better for someone, or some group of people, because of the intellectual issues I was struggling with.”

Scarf had been exposed to economics at a young age when his father’s business went bankrupt in Philadelphia during the Depression. At Princeton he met a group of mathematicians who also eventually migrated into economics, including Ralph Gomory, Lloyd Shapley, and John Nash, as well as the economist and his future Yale colleague Martin Shubik.

After graduating from Princeton, rather than pursuing an academic career in mathematics, Scarf took a position at the Rand Corporation, a government-sponsored think tank set up to apply mathematical thinking to the economic, political, and strategic problems of the Cold War. There he met George Dantzig, the inventor of the simplex algorithm, the economist Kenneth Arrow, and statistician Samuel Karlin, who launched Scarf on his classic derivation of optimal inventory policy. The latter two invited Scarf to visit Stanford University, where he eventually became a faculty member in the statistics department. Soon he was recruited to Yale as a Cowles Foundation professor to replace the departing Gerard Debreu. The Cowles Foundation — devoted to the development and application of logical, mathematical, and statistical methods in economics and the social sciences — had been established by Alfred Cowles in response to the unanticipated 1929 stock market crash in order to make economics more mathematical and thus less prone to error.

In addition to inventory policy, Scarf had at least four more major breakthroughs. Perhaps his most famous discovery is the Scarf algorithm. Arrow and Debreu had proved that the equations describing economic equilibrium always have a solution when goods are divisible, but they were baﬀled by the problem of how to ﬁnd one, except in special cases. The Scarf algorithm always ﬁnds an equilibrium, no matter how complicated the economy. This gave applied economists the ability to work with much more realistic models of the economy and thus to predict the consequences of major policy reforms including NAFTA and the U.S. tax system.

Scarf was animated throughout his career by conversations he had with Gomory about the problem of how an economy copes with indivisibilities and economies of scale like a huge railway system. In economies with divisible goods, bad production plans are discarded because they fail the market test, making negative proﬁts. With indivisible goods, proﬁtability is not an appropriate guide to optimality. To answer the question “So how does society decide what to produce?” Scarf devised another algorithm for production with indivisibilities, based on his concept of a test set, which displays the minimal number of comparisons that must be made to verify that a production plan is optimal for a single ﬁrm.

Recognizing that a real-life economy might have many diﬀerent indivisible goods and many independent decision makers, Scarf turned to cooperative game theory and the concept of the core to explain what choices society might make. The core had been invented at Princeton around the time Nash was doing his now-famous work on non-cooperative game theory. Scarf characterized the class of games for which there must exist a core allocation, and he again devised an algorithm to ﬁnd such an allocation when it existed. Scarf and Shapley famously pointed out that a housing game in which *n* agents with diﬀerent tastes each have one indivisible house ﬁts the class where the core exists.

Scarf also considered the connection between core allocations and market equilibrium allocations in the standard economic model with divisible goods. Inspired by a conversation during a long walk with Shubik from Harlem to Sutton Place, he showed with Debreu that if an economy is replicated, so that every person is just like many others and therefore not unique at all, then the core allocations nearly coincide with the market equilibrium allocations. Debreu and Scarf thus showed that market equilibrium is the intersection of cooperative and non-cooperative behavior.

In addition to his groundbreaking research, Scarf was known as an inspiring teacher. “Teaching has, of course been an important part of my life,” he wrote. “I make elaborate notes that are instantly discarded when I enter the class, and I begin singing from the musical score of an opera that I have in my head.”

Undergraduates and graduate students alike said they were enchanted by the musicality of his lectures. “Herb was a wonderful mentor,” says Menahem Yaari, emeritus professor at Hebrew University and emeritus president of the Israel Academy of Sciences and Humanities. “His contributions to our ﬁeld of discourse were nothing short of breathtaking, and he was absolutely the best lecturer/expositor I had ever encountered.”

“When I was 18 years old and searching for beauty, I found Herb Scarf. Herb was my ﬁrst economics teacher, and the reason I decided to become an economist,” says John Geanakoplos, Yale economics professor and former director of the Cowles Foundation. “His graduate class in mathematical economics was the clearest and most beautiful course I have ever taken, even to this day. Where to put the silences is often the most important thing, he would say.”

Scarf was known for his grace and charm and had many close personal friendships at Yale. He frequently chaired university committees, including dean searches, and the social sciences divisional committee.

A lover of music and opera, he also enjoyed ballet and chess. As his family is fond of pointing out, he was an early feminist, supportive and proud of the achievements of his wife, Maggie, a best-selling author of psychology books, and his daughters Martha, the founder and manager of a highly successful consulting ﬁrm in Boston; Betsy, a noted psychologist in Connecticut; and Susie, a successful writer living in Sag Harbor.

He was a member of the National Academy of Sciences, a fellow of the American Academy of Arts and Sciences, a member of the American Philosophical Society, a distinguished fellow of the American Economic Association, and past president of the Econometric Society. In 1973, Scarf won the Frederick W. Lanchester Prize, which recognizes the best contribution to operations research and the management sciences. A decade later, he won the John von Neumann Theory Prize for his contributions to theory in operations research.

“Herb Scarf was one of the giants of his day in economic theory,” says Cowles Foundation director Larry Samuelson. “His work and his name will live on in the form of Scarf’s Algorithm, the Debreu-Scarf Theorem, and his many other contributions. At the same time, he was a gracious and encouraging colleague. Cowles will not be the same without him.”

A memorial service will be held on Saturday, Jan. 23 at 2 p.m. at Battell Chapel. A reception will follow at the Graduate Club.

*Yale News*, December 1, 2015