Monday, April 9, 2012

John von Neumann:Father of Game Theory

Man and Machine 

John von Neumann, one of this century’s preeminent scientists, along with being a great mathematician and physicist, was an early pioneer in fields such as game theory, nuclear deterrence, and modern computing. He was a polymath who possessed fearsome technical prowess and is considered "the last of the great mathematicians". His was a mind comfortable in the realms of both man and machine. His kinship with the logical machine was displayed at an early age by his ability to compute the product of two eight-digit numbers in his head. His strong and lasting influence on the human world is apparent through his many friends and admirers who so often had comments as to von Neumann’s greatness as a man and a scientist.He made major contributions to the field of  set theory, functional analysis, quantum mechanics, ergodic theory, geometry, fluid dynamics, economics, linear programming, game theory, computer science, numerical analysis, hydrodynamics, nuclear physics and statistics.

Although he is often well known for his dominance of logic and rigorous mathematical science, von Neumann’s genius can be said to have grown from a comfortable and relaxed upbringing.

 

Early Life and Education in Budapest

He was born Neumann Janos on December 28, 1903, in Budapest, the capital of Hungary. He was the first born son of Neumann Miksa and Kann Margit. In Hungarian, the family name appears before the given name. So, in English, the parent’s names would be Max Neumann and Margaret Kann. Max Neumann purchased a title early in his son’s life, and so became von Neumann.

Max Neumann, born 1870, arrived in Budapest in the late 1880s. He was a non-practicing Hungarian Jew with a good education. He became a doctor of laws and then worked as a lawyer for a bank. He had a good marriage to Margaret, who came from a prosperous family.In 1903, Budapest was growing rapidly, a booming, intellectual capital. It is said that the Budapest that von Neumann was born into "was about to produce one of the most glittering single generations of scientists, writers,artists, musicians, and useful expatriate millionaires to come from one small community since the city-states of the Italian Renaissance. Indeed, John von Neumann was one of those who,through his natural genius and prosperous family, was able to excel in the elitist educational system of the time.

At a very young age, von Neumann was interested in math, the nature of numbers and the logic of the world around him. Even at age six, when his mother once stared aimlessly in front of her, he asked, "What are you calculating?" thus displaying his natural affinity for numbers. The young von Neumann was not only interested in math, though. Just as in his adult life he would claim fame in a wide range of disciplines (and be declared a genius in each one), he also had varying interests as a child. At age eight he became fascinated by history and read all forty-four volumes of the universal history, which resided in the family’s library. Even this early, von Neumann showed that he was comfortable applying his mind to both the logical and social world.His parents encouraged him in every interest, but were careful not to push their young son, as many parents are apt to do when they find they have a genius for a child. This allowed von Neumann to develop not only a powerful intellect but what many people considered a likable personality as well.

It was never in question that von Neumann would attend university, and in 1914, at the age of 10, the educational road to the university started at ,the Lutheran Gymnasium. This was one of the three best institutions of its kind in Budapest at the time and gave von Neumann the opportunity to develop his
great intellect. Before he would graduate from this high school he would be considered a colleague by most of the university mathematicians. His first paper was published in 1922, when he was 17, in the  Journal of the German Mathematical Society, dealing with the zeros of certain minimal polynomials.

 
University — Berlin, Zurich and Budapest

In 1921 von Neumann was sent to become a chemical engineer at the University of Berlin and then to Zurich two years later. Though John von Neumann had little interest in either chemistry or engineering, his father was a practical man and encouraged this path. At that time chemical engineering was a popular career that almost guaranteed a good living, in part due to the success of German chemists from 1914 to 1918. So, von Neumann set on the road planned in part by his father Max. He would spend two years in Berlin in a non-degree chemistry program. After this he would take the entrance exam for second year standing in the chemical engineering program at the prestigious Eidgennossische Technische Hochschule (ETH) in Zurich, where Einstein had failed the entrance exam in 1895 and then gained acceptance a year later.

During this time of practical undergraduate study, von Neumann was executing another plan that was more in tune with his interests. In the summer after his studies at Berlin and before he went to Zurich he enrolled at the Budapest University as a candidate for an advanced doctorate in mathematics. His
Ph.D. thesis was to attempt the axiomatization of set theory, developed by George Cantor. At the time, this was one of the hot topics in mathematics and had already been studied by great professors, causing a great deal of trouble to most of them. None the less, the young von Neumann, devising and executing this plan at the age of 17, was not one to shy away from great intellectual challenges.

Von Neumann breezed through his two years at Berlin and then set himself to the work on chemical engineering at the ETH and his mathematical studies in Budapest. He received excellent grades at the ETH, even for classes he almost never attended. He received a perfect mark of 6 in each of his courses during his first semester in the winter of 1923-24; courses including organic chemistry, inorganic chemistry, analytical chemistry, experimental physics, higher mathematics and French language.

From time to time he would visit Budapest University when his studies there required his presence and to visit his family. He worked on his set theory thesis in Zurich while completing classes for the ETH. After finishing his thesis he took the final exams in Budapest to receive his Ph.D. with highest honors. This was just after his graduation from the ETH, so in 1926 he had two degrees, one an undergraduate degree in chemical engineering and the other a Ph.D. in mathematics, all by the time he was twenty-two.
This von Neumann stamp, issued in Hungary in 1992, honors his contributions to mathematics and computing.

Game Theory 

Von Neumann is commonly described as a practical joker and always the life of the party. John and Klara held a party every week or so, creating a kind of salon at their house. Von Neumann used his phenomenal memory to compile an immense library of jokes which he used to liven up a conversation. Von Neumann loved games and toys, which probably contributed in great part to his work in Game Theory.

An occasional heavy drinker, Von Neumann was an aggressive and reckless driver, supposedly totaling a car every year or so. According to William Poundstone's Prisoner's Dilemma, "an intersection in Princeton was nicknamed "Von Neumann Corner" for all the auto accidents he had there."

His colleagues found it "disconcerting" that upon entering an office where a pretty secretary worked, von Neumann habitually would "bend way way over, more or less trying to look up her dress." (Steve J. Heims, John Von Neumann and Norbert Wiener: From Mathematics to the Technologies of Life and Death, 1980, quoted in Prisoner's Dilemma, p.26) Some secretaries were so bothered by Von Neumann that they put cardboard partitions at the front of their desks to block his view.Despite his personality quirks, no one could dispute that Von Neumann was brilliant. Beginning in 1927, Von Neumann applied new mathematical methods to quantum theory. His work was instrumental in subsequent "philosophical" interpretations of the theory.


For Von Neumann, the inspiration for game theory was poker, a game he played occasionally and not terribly well. Von Neumann realized that poker was not guided by probability theory alone, as an unfortunate player who would use only probability theory would find out. Von Neumann wanted to formalize the idea of "bluffing," a strategy that is meant to deceive the other players and hide information from them.
In his 1928 article, "Theory of Parlor Games," Von Neumann first approached the discussion of game theory, and proved the famous Minimax theorem. From the outset, Von Neumann knew that game theory would prove invaluable to economists. He teamed up with Oskar Morgenstern, an Austrian economist at Princeton, to develop his theory.

Their book, Theory of Games and Economic Behavior, revolutionized the field of economics. Although the work itself was intended solely for economists, its applications to psychology, sociology, politics, warfare, recreational games, and many other fields soon became apparent.Although Von Neumann appreciated Game Theory's applications to economics, he was most interested in applying his methods to politics and warfare, perhaps stemming from his favorite childhood game, Kriegspiel, a chess-like military simulation. He used his methods to model the Cold War interaction between the U.S. and the USSR, viewing them as two players in a zero-sum game.

From the very beginning of World War II, Von Neumann was confident of the Allies' victory. He sketched out a mathematical model of the conflict from which he deduced that the Allies would win, applying some of the methods of game theory to his predictions.In 1943, Von Neumann was invited to work on the Manhattan Project. Von Neumann did crucial calculations on the implosion design of the atomic bomb, allowing for a more efficient, and more deadly, weapon. Von Neumann's mathematical models were also used to plan out the path the bombers carrying the bombs would take to minimize their chances of being shot down. The mathematician helped select the location in Japan to bomb. Among the potential targets he examined was Kyoto, Yokohama, and Kokura.

"Of all of Von Neumann's postwar work, his development of the digital computer looms the largest today." (Poundstone 76) After examining the Army's ENIAC during the war, Von Neumann came up with ideas for a better computer, using his mathematical abilities to improve the computer's logic design. Once the war had ended, the U.S. Navy and other sources provided funds for Von Neumann's machine, which he claimed would be able to accurately predict weather patterns.Capable of 2,000 operations a second, the computer did not predict weather very well, but became quite useful doing a set of calculations necessary for the design of the hydrogen bomb. Von Neumann is also credited with coming up with the idea of basing computer calculations on binary numbers, having programs stored in computer's memory in coded form as opposed to punchcards, and several other crucial developments. Von Neumann's wife, Klara, became one of the first computer programmers.

Von Neumann later helped design the SAGE computer system designed to detect a Soviet nuclear attack in 1948, Von Neumann became a consultant for the RAND Corporation. RAND (Research ANd Development) was founded by defense contractors and the Air Force as a "think tank" to "think about the unthinkable." Their main focus was exploring the possibilities of nuclear war and the possible strategies for such a possibility.

Von Neumann was, at the time, a strong supporter of "preventive war." Confident even during World War II that the Russian spy network had obtained many of the details of the atom bomb design, Von Neumann knew that it was only a matter of time before the Soviet Union became a nuclear power. He predicted that were Russia allowed to build a nuclear arsenal, a war against the U.S. would be inevitable. He therefore recommended that the U.S. launch a nuclear strike at Moscow, destroying its enemy and becoming a dominant world power, so as to avoid a more destructive nuclear war later on. "With the Russians it is not a question of whether but of when," he would say. An oft-quoted remark of his is, "If you say why not bomb them tomorrow, I say why not today? If you say today at 5 o'clock, I say why not one o'clock?"
Just a few years after "preventive war" was first advocated, it became an impossibility. By 1953, the Soviets had 300-400 warheads, meaning that any nuclear strike would be effectively retaliated.
In 1954, Von Neumann was appointed to the Atomic Energy Commission. A year later, he was diagnosed with bone cancer. William Poundstone's Prisoner's Dilemma suggests that the disease resulted from the radiation Von Neumann received as a witness to the atomic tests on Bikini atoll. "A number of physicists associated with the bomb succumbed to cancer at relatively early ages.


Quantum Mechanics

Von Neumann was a creative and original thinker, but he also had the ability to take other people’s suggestions and concepts and in short order turn them into something much more complete and logical. This is in a way what he did with quantum mechanics after he went to the university in Göttingen, Germany after receiving his degrees in 1926.Quantum mechanics deals with the nature of atomic particles and the laws that govern their actions. Theories of quantum mechanics began to appear to confront the discrepancies that occurred when one used purely Newtonian physics to describe the observations of atomic particles.

One of these observations has to do with the wavelengths of light that atoms can absorb and emit. For example, hydrogen atoms absorb energy at 656.3 nm, 486.1 nm, 434.0 nm or 410.2 nm, but not the wavelengths in between.This was contrary to the principles of physics as they were at the end of the nineteenth century, which would predict that an electron orbiting the nucleus in an atom should radiate all wavelengths of light, therefore losing energy and quickly falling into the nucleus. This is obviously not what is observed, so a new theory of quanta was introduced by Berliner Max Plank in 1900 that said energy could only be emitted in certain definable packets.

This lead to two competing theories describing the nature of the atom, which could only absorb and emit energy in specific quanta. One of these, developed by Erwin Schrödinger, suggested that the electron in hydrogen is analogous to a string in a musical instrument. Like a string, which emits a
specific tone along with overtones, the electron would have a certain "tone" at which it would emit energy. Using this theory, Schrödinger developed a wave equation for the electron that correctly predicted the wavelengths of light that hydrogen would emit.

Another theory, developed by physicists at Göttingen including Werner Heisenberg, Max Born, and Pascual Jordan, focused on the position and momentum of an electron in an atom. They contested that these values were not directly observable (only the light emitted by the atom could be observed) and so could behave much differently from the motion of a particle in Newtonian physics. They theorized that the values of position and momentum should be described by mathematical constructs other than ordinary numbers. The calculations they used to describe the motion of the electron made use of matrices and matrix algebra.

These two systems, although apparently very different, were quickly determined to be mathematically equivalent, two forms of the same principle. The proponents of the two systems, none the less, denounced the others theories and claimed their own to be superior. It is in this environment, in 1926, that von Neumann appears on the scene and quickly went to work reconciling and advancing the theories of quantum mechanics.

Von Neumann wanted to find what the two systems, wave mechanics and matrix mechanics, had in common. Through a more rigorous mathematical approach he wanted to find a new theory, more fundamental and powerful than the other two. He abstracted the two systems using an axiomatic approach, in which each logical state is the definite consequence of the previous state. Von Neumann constructed the rules of "abstract Hilbert space" to aid in his development of a mathematical structure for quantum theory. His formalism of the subject allowed considerable advances to be made by others and even predicted strange new consequences, one that consciousness and observations alone can affect electrons in a Labaratory.



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Marriages and America

From 1927-29, after his formalization of quantum mechanics, von Neumann traveled extensively to various academic conferences and colloquia and turned out mathematical papers at the rate of one a month at times. By the end of 1929 he had 32 papers to his name, all of them in German, and each
written in a highly logical and orderly manner so that other mathematicians could easily incorporate von Neumann’s ideas into their own work.

Von Neumann was now a rising star in the academic world, lecturing on new ideas, assisting other great minds of the time with their own works, and creating an image for himself as a likable and witty young genius in his early twenties. He would often avoid arguments with the more confrontational of his colleagues by telling one of his many jokes or stories, some of which he could not tell in the presence of ladies (though there were few women at these mathematical seminars). Other times he would bring up some interesting fact from ancient history, changing the subject and making von Neumann seem surprisingly learned for his age and professional interests.
Neumann at Princeton

Near the end of 1929 he was offered a lectureship at Princeton in an America that was trying to stimulate its mathematical sciences by seeking out the best of Europe. At this same time, von Neumann decided to marry Mariette Kovesi, whom he had known since his early childhood. Their honeymoon was a cruise across the Atlantic to New York, although most of their trip was subdued
by Mariette’s unexpected seasickness.

They had a daughter, Marina, in 1935. Von Neumann was affectionate with his new daughter, but did not contribute to the care of her or to the housework, which he considered to be the job of the wife. The gap between the lively 26-year-old Mariette and the respectable 31-year-old John von Neumann
began to increase and in 1936 they broke up, Mariette going home to Budapest and von Neumann, after drifting around Europe to various engagements, went to the United States. Soon after, on a trip to Budapest, he met Klari Dan and they were married in 1938.

Although this marriage lasted longer than his first, von Neumann was often distant from his personal life, obsessed and engrossed in his thoughts and work. In this personal tradeoff of von Neumann’s the world of science profited tremendously, and much of his work changed all of our lives. Two of the
most influential and well known of von Neumann’s interests during his time in America, from 1933 (when he was appointed as one of the few original members of the Institute for Advanced Studies at Princeton) to 1957 (when he died of cancer), were the development of nuclear weapons and the invention of the modern digital computer.

Von Neumann’s Role in Nuclear Development

In the biography of a genius such as von Neumann it would be easy to overestimate his role in the development of nuclear weapons in Los Alamos in 1943. It is important to remember that there was a collection of great minds there, recruited by the American government to produce what many saw as a necessary evil. The fear that Germany would produce an atomic bomb before the US
drove the effort at Los Alamos.Von Neumann’s two main contributions to the Los Alamos project were the mathematicization of development and his contributions to the implosion bomb.

The scientists at Los Alamos were used to doing scientific experiments but it’s difficult to do many experiments when developing weapons of mass destruction. They needed some way to predict what was going to happen in these complex reactions without actually performing them. Von Neumann therefore was a member of the team that invented modern mathematical modeling. He applied his math skills at every level, from helping upper officials to make logical decisions to knocking down tough calculations for those at the bottom of the ladder.

The atomic bombs that were eventually dropped were of two kinds, one using uranium-235 as its fissionable material, the other using plutonium. An atomic chain reaction occurs when the fissionable material present in the bomb reaches a critical mass, or density. In the uranium-235 bomb, this was done using the gun method. A large mass of uranium-235, still under the critical mass, would have another mass of uranium-235 shot into a cavity. The combined masses would then reach critical mass, where an uncontrolled nuclear fission reaction would occur. This process was known to work and was a relatively simple procedure. The difficult part was obtaining the uranium-235, which has to be separated from other isotopes of uranium, which are chemically identical.Plutonium, on the other hand, can be separated using chemical means, and so production of plutonium based bombs could progress more quickly. The problem here was that plutonium bombs could not use the gun method. The plutonium would need to reach critical mass through another technique, implosion. Here, a mass of plutonium is completely surrounded by high explosives that are ignited simultaneously to cause the plutonium mass to compress to supercritical levels and explode.

Although von Neumann did not arrive first at the implosion technique for plutonium, he was the one who made it work, developing the "implosion lens" of high explosives that would correctly compress the plutonium.This is just one more example of von Neumann’s ability to pick up an idea and advance it where others had gotten stuck.



Development of Modern Computing
Von Neumann with first super Computer


Just like the project at Los Alamos, the development of the modern computer was a collaborative effort including the ideas and effort of many great scientists. Also like the development of nuclear weaponry, there have been many volumes written about the development of modern computer. With so much involved in the process and von Neumann himself being involved in so much of it,only a few contributions can be covered here.

A von Neumann language is any of those programming languages that are high-level abstract isomorphic copies of von Neumann architectures. As of 2009, most current programming languages fit into this description, likely as a consequence of the extensive domination of the von Neumann computer architecture during the past 50 years[

Von Neumann’s experience with mathematical modeling at Los Alamos, and the computational tools he used there, gave him the experience he needed to push the development of the computer. Also, because of his far reaching and influential connections, through the IAS, Los Alamos, a number of Universities and his reputation as a mathematical genius, he was in a position to secure
funding and resources to help develop the modern computer. In 1947 this was a very difficult task because computing was not yet a respected science. Most people saw computing only as making a bigger and faster calculator. Von Neumann on the other hand saw bigger possibilities.

Von Neumann wanted computers to be put to use in all fields of science, bringing a more logical and precise nature to those fields as he had tried to do. With his contributions to the architecture of the computer, which describe how logical operations are represented by numbers that can then be read
and processed, many von Neumann’s dreams have come true. Today we have extremely powerful computing machines used in scores of scientific fields, as well many more non-scientific fields.

In von Neumann’s later years, however, he worked and dreamed of applications for computers that have not yet been realized. He drew from his many other interests and imagined powerful combinations of the computer’s ability to perform logically and quickly with our brain’s unique ability to solve ill defined problems with little data, or life’s ability to self-reproduce and evolve.In this vein, von Neumann developed a theory of artificial automata. Von Neumann believed that life was ultimately based on logic, and so any construct that supports logic should be able to support life. Artificial automata, like their natural counter parts, process information and proceed in their actions based on data received from their environment in light of rules and instructions they hold internally. Cellular automata are a class of automata that exist in an infinite plane that is covered by square cells, much like a sheet of graph paper. Each of these cells can rest in a number of states. The whole plane of cells will go through time steps, where the new state of each cell is determined by its own state and the state of the cells neighboring it. In these simple actions there lies a great complexity and the basis for life like actions.

Untimely End

Perhaps all deaths can be considered to come too early; John von Neumann’s own death came far too early. He died on February 8, 1957, 18 months after he was diagnosed with cancer.He never finished his work on automata theory, although he worked as long as he possibly could. He attended ceremonies held in his honor using a wheelchair, and tried to keep up appearances with his family and friends. Though he had accomplished so much in his years he could not accept death, could not consider a world that existed without his mind constantly thinking and solving. But today, his ideas live on and affect our lives in more ways than the few examples given here can demonstrate.

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