Being in the Right Place at the Right Time

T theme of this paper is: “Being in the right place at the right time.” The right place was the Department of Mathematics at Princeton University; the right time was the spring of 1948 and the period following. In the spring of 1948, I was a graduate student of mathematics with practically no financial resources. After finishing my military service in the United States Army in the Second World War in 1946, I had three years of financial support from the GI Bill of Rights, which allowed me to finish my undergraduate degree at Caltech in 1947 and gave me two years of graduate study at Princeton University. For the academic year of 1948–1949, I was awarded the J.S.K. Fellowship by the Mathematics Department, which paid $700 (since my tuition was paid by the GI Bill of Rights). Thus, after paying room rent and board, there was not enough left to buy a pair of shoes! That was the reason that I accepted a job moving the furniture of Dick Bellman, the father of “dynamic programming” (Bellman 1957). It happened this way. One morning at breakfast in the boarding house where I took my meals, a moving man appeared. He explained that his assistant was drunk and unable to carry furniture from a house to a moving van. He offered $10 to anyone who was willing to replace his drunk helper. The house moving seemed to be an easy one; it involved the contents of a small house in the project known as the “barracks,” temporary housing built in 1946 for military personnel returning to the university after the war. Although they were “temporary” in 1946, they are still there! From the moving man, I learned it was Dick Bellman’s furniture that I would be moving. He was leaving Princeton for an appointment as an associate professor at Stanford. It seemed like an easy way to earn $10 because the house was so small. But I was wrong! Dick Bellman’s wife had been a contestant on a television quiz show in which the prizes were household appliances. Thus, the “small house” contained two refrigerators, two stoves, two dishwashers, two television sets, etc. In short, I worked very hard for my pair of shoes. Now that you are convinced that I was an impecunious graduate student in the spring of 1948, you should understand why I visited Professor Albert Tucker at the end of the semester to ask for summer employment. It was the right time, because Tucker had had a recent meeting with George Dantzig concerning the new subject of linear programming. Dantzig had come to Princeton to ask the opinion of John von Neumann about his research. The story (Dantzig 1991) of this encounter, although interesting in its own right, is not important for our tale. The important fact is that Dantzig explained to Tucker the essential elements of linear programming and related the opinion of von Neumann that there was a close connection between Dantzig’s creation and the theory of zero-sum two-person games. Tucker, for his part, thought that there should be a connection between the transportation problem and the theory of electrical networks as modeled by Kirkhoff’s laws. In these days, just after the war had ended, it was easy to find financial support for such research, so Tucker offered me work for the summer. The third member of this project was David Gale, who had also come to Princeton as a graduate student in 1947. How did we attack this problem? None of us knew anything about the theory of games. To learn this material, we each studied different chapters of the book of von Neumann and Morgenstern (1949). Every day, one of us explained to the other two the contents of a chapter in a seminar room of the old Fine Hall. A major result of this work was the duality theorem for linear programming, which says that to each minimization problem there is associated a maximization problem constructed on the same data with a number of properties that relate them. This work was presented at the seminal meeting in Chicago, sometimes referred to as Mathematical Programming Symposium Number 0. This work is one of the chapters of Cowles Commission Monograph 13 (Gale et al. 1951), edited by Tjalling Koopmans; it was through this volume that the mathematical community learned of George Dantzig’s formulation of linear programming. In a sabbatical year at Stanford in the fall of 1949, Tucker (1985) again took up the question: What are the relations between linear programming and the theory of electrical networks as modeled by Kirkhoff and Maxwell? At this time, Tucker understood the parallel between the electrical potentials of Maxwell and Lagrange multipliers. He also identified the problem of minimizing heat loss in the network as a quadratic program. Tucker wrote to