George Dantzig And John Nash: Two Pioneers In Line Essay, Research Paper

George Dantzig was born on August 14, 1914. He studied mathematics at the University of Maryland, receiving his A.B. in 1936. He received an M.A. in mathematics from the University of Michigan in 1937. Dantzig worked as a Junior Statistician in the U.S. Bureau of Labor Statistics from 1937 to 1939, then, from 1941 to 1946, he was head of the Combat Analysis Branch, U.S.A.F. Headquarters Statistical Control. He received his doctorate in mathematics from the University of California, Berkeley in 1946. In that year he was appointed Mathematical Advisor for USAF Headquarters. In 1947 Dantzig made the contribution to mathematics for which he is most famous, the simplex method of optimization. It grew out of his work with the U.S. Air Force where he became an expert on planning methods solved with desk calculators. George Dantzig mechanized the planning process by introducing “linear programming”. Linear programming is used to allocate resources, plan production, schedule workers, plan investment portfolios and formulate marketing (and military) strategies. The versatility and economic impact of linear programming in today’s industrial world is truly impressive. Dantzig became a research mathematician with the RAND Corporation in 1952, and then in 1960 he was appointed professor at Berkeley and Chairman of the Operations Research Center. While there he wrote Linear Programming and Extensions (1963). In 1966 he was appointed Professor of Operations Research and Computer Science at Stanford University.His work in a wide range of topics related to optimization and operations research over the years has been of major importance. In spite of impressive developments in computational optimization in the last 20 years, the simplex method, invented by Dantzig in 1947, has stood the test of time. Through his research in mathematical theory, computation, economic analysis, and applications to industrial problems, he has contributed more than any other researcher to the development of linear programming. Computer scientist Laszlo Lovasz said in 1980: “If one would take statistics about which mathematical problem is using up most of the computer time in the world, then (not including database handling problems like sorting and searching) the answer would probably be linear programming. In his words: “The tremendous power of the simplex method is a constant surprise to me.” The list of Dantzig’s professional accomplishments extends beyond linear programming and the simplex method to decomposition theory, sensitivity analysis, complementary pivot methods, large-scale optimization, nonlinear programming, and programming under uncertainty. His research in linear programming (and the related areas of nonlinear optimization, integer programming, and optimization under uncertainty) has had a fundamental impact on the development of operations research as a discipline. Dantzig has received many honors including the National Medal of Science (1975), the John Von Neumann Theory Prize of the Operations Research Society of America and the Institute of Management Sciences (1974), and membership in the National Academy of Sciences, the National Academy of Engineering, and the American Academy of Arts and Sciences. Since 1979, the Mathematical Programming Society has also honored Dantzig by awarding the George B. Dantzig Prize. The prize, which recognizes original research that has had a major impact on the field of mathematical programming, was awarded for the first time in 1982. Since then it has been awarded every three years. John Nash was born on June 13, 1928 in Bluefield, West Virginia. Nash’s mother, Margaret, was a Latin teacher. His father, John Sr., was an electrical engineer.In the fall of 1945, Nash enrolled at Carnegie-Mellon, then Carnegie Tech, in Pittsburgh. It was there that the label “genius” was first applied to Nash. His mathematics professor called him “a young Gauss” in class one day, referring to the great German mathematician. Nash switched from chemistry to math in his freshman year. Two years later he had a B.S. and was studying for an M.S.His graduate professor, R.J. Duffin, recalls Nash who came to him one day and described a problem he thought he had solved. Duffin realized with some astonishment that Nash, without knowing it, had independently proved Brouwer’s famed theorem. The professor’s letter of recommendation for Nash had just one line: “This man is a genius.”In 1948, the year Nash entered the doctoral program at Princeton with a fellowship, he was eager to prove himself. Nash quickly became one of the brilliant young men who performed mental pyrotechnics in the common room of Fine Hall. Soon after he arrived he invented an extremely clever game that was played with markers on hexagonal bathroom tiles. An instant fad in the common room, it was called “Nash” or “John.” Parker Brothers brought out a version a few years later called Hex.Other students found him a loner, odd as well as brilliant. He would pace around and around, following the quadrangular hallways, occasionally dashing into empty classrooms to scribble, with lightning speed, on blackboards.Nash’s Nobel-winning thesis on game theory was the product of his second year at Princeton. Game theory was the invention of Von Neumann and a Princeton economist named Oskar Morgenstern. Their 1944 book, The Theory of Games and Economic Behavior, was the first attempt to derive logical and mathematical rules about rivalries.

Nash picked a problem for his thesis that had eluded Von Neumann. Briefly, Von Neumann only had a good theory for pure rivalries in which one side’s gain was the other’s loss. Nash focused on rivalries in which mutual gain was also possible. He showed that there were stable solutions (no player could do better given what the others were doing) for such rivalries under a wide variety of circumstances.In doing so, he turned game theory into a powerful tool that economists could use to analyze everything from business competition to trade negotiations. “It wasn’t until Nash that game theory came alive for economists,” said Robert Solow, a Nobel laureate in economics at MIT. P Ordeshook wrote: The concept of a Nash equilibrium n-tuple is perhaps the most important idea in game theory. … Whether we are analyzing candidates’ election strategies, the causes of war, agenda manipulation in legislatures, or the actions of interest groups, predictions about events reduce to a search for and description of equilibria. Put simply, equilibrium strategies are the things that we predict about people. Nash got his doctorate on his 22nd birthday, June 13, 1950. After brief interludes as an instructor at Princeton and as a consultant at the Rand Corporation, the Cold War think tank, Nash moved on to teach at MIT in 1951.He arrived itching to show that he could solve really big problems. According to one story circulating at the time, Nash was in the common room knocking, as he often did, other mathematicians’ work. An older professor is said to have challenged him to solve one of the field’s most notorious problems.The problem grew out of work done by G.F.B. Riemann, a 19th century mathematician, and was considered virtually insolvable. But Nash wound up solving it. To do so, he invented a completely new method for approaching the problem that turned out to unlock a difficulty encountered in a far larger class of problems. Most mathematicians consider this and other work Nash did in pure mathematics to be his greatest achievements, worthy of Nobels if such were given in the mathematical field. Many have said that he got his Nobel for his most trivial work.By the mid-1950s, Nash was phenomenally productive. When he got tired of mathematicians, he would wander over to the economics department to talk to Solow and another Nobel laureate, Paul Samuelson.At age 30 in the spring of 1959, Nash was committed to McLean Hospital, a psychiatric institution in Belmont, MA, connected with Harvard University. In the months leading up to his hospitalization, Nash became another person. He skipped from subject to subject. Some of his lectures no longer made sense. The months at McLean did little to arrest the disease. Nash’s paranoia intensified and he could no longer work. After resigning his MIT post, he went to Europe, wandering from city to city. He feared he was being spied on and hunted down and he tried to give up his United States citizenship. For most of the next 20 years, Nash divided his time between hospitals and Princeton. Nash became a sad, ghostly presence around Princeton and a mysterious character. Some former colleagues at Princeton and MIT tried to help with jobs on research projects, though very often Nash couldn’t accept the help. Shapley at UCLA succeeded in getting a cash mathematics prize for Nash in the 1970 s. There were other forms of kindness, like getting Nash access to university computers or remembering to invite him to seminars when old friends turned up on campus. Still, the people who stayed in regular contact with him eventually came to believe that his illness would never end.Then came a miraculous remission. And as happens, for reasons unknown, in the case of some people with schizophrenia, it was not due to any drug or treatment.During the 20-plus years of Nash’s illness, game theory flourished and it is hard to find an important article in the field that does not refer to his work. Four years ago Nash received a Noble prize in economics for the paper he wrote on game theory over a half a century ago. George Dantzig and John Nash were two of the most influential mathematicians of their time. Their contributions have had a tremendous impact on mathematics and many outside areas. ReferencesDantzig:1. www.ti.com/calc/docs/art/marcus01.htm2. www.dna.affrc.go.jp:100813. George Dantzig: Linear Programming and Extensions, Princeton University Press, Princeton, NJ, 1963Nash:1. Biography in Encyclopaedia Britannica. 2. H W Kuhn et al., The work of John F Nash, Jr. in game theory: A celebration of John F Nash, Jr. Duke Math. J. 81 (1) (1995), i-v, 1-29. 3. Madness and mathematics, The Times London, 28 Aug, 1996. 4. J Milnor, A Nobel prize for John Nash, The Mathematical Intelligencer 17 (3) (1995), 11-17. 5. The work of John Nash in game theory: Nobel Seminar, December 8, 1994, Journal of Economic Theory 69 (1) (1996), 153-185. 6. E van Damme, On the contributions of John C Harsanyi, John F Nash and Reinhard Selten, Interntional Journal of Game Theory 24 (1) (1995), 3-11. 7. Roy Weinstien, Toward a History of Game Theory, Duke University Press, London, 1992