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Web Exclusives: PAWPLUS Posted March
27, 2002: Searching
for genius, a theory, or simply patterns When I was a Princeton undergraduate, Professor John Nash *50, the subject of the movie A Beautiful Mind, was a shadow figure in a knit cap who haunted the Fine Hall math department. Nash sightings — at the Dinky train station, in Small World coffee shop, on his slowly looping bicycle rides — were a regular pastime. We called Nash the "Phantom of Fine Hall" and whispered in wonder that this aging man in goggle eyeglasses and faded Jack Purcell shoes had won a Nobel Prize in economics. A Beautiful Mind is bringing deserved mainstream recognition to Nash's long struggle to overcome schizophrenia. But moviegoers will leave the film with little sense of Nash's mathematical genius, save knowing that his contribution to modern economics can help pick up a girl at a bar. MATH GENIUS It is axiomatic in mathematics that genius blossoms early, and so students entering Princeton's math department, one of the world's best, are all seeking the next great insight. As rendered in A Beautiful Mind, when math prodigy John Nash arrived at Princeton, he searched for patterns in the pecking of pigeons. Nash found his stroke of genius at age 21 in a 27-page dissertation on game theory that would fundamentally revolutionize modern economics and eventually win him a Nobel. My classmates and I may have had genius on our mind several years back, when we entered our 9 a.m. calculus class in Fine Hall tower. At 9:05 a.m., a rumpled man in loose jeans and an untucked button down brushed aside the classroom door and flung off his sandals, which smacked against the wall. He turned barefoot to the blackboard and began writing equations, pausing only to survey his work with a mumble or wipe chalk handprints on his jeans. Fifty minutes later, Professor John Conway put down his chalk, put on his sandals and walked out. That was our first lecture ever as Princeton freshmen. Half the class had dropped out by the next lecture. GAME THEORY Before Nash, economists studied the market using sophisticated versions of Adam Smith's price theory -basic supply and demand. Smith said as buyers and sellers pursue their self-interest, the "invisible hand" of the market distributes products efficiently. But price theory can't explain the abundant real-world examples of market inefficiency. Nash approached this problem by reformulating economics as a game. To most people, a game is a way to while away a rainy afternoon. But to mathematicians, a game is not just chess or poker but any conflict situation that forces participants to develop a strategy to accomplish a goal. To mathematicians, a game is a regimented world where math is king. And a game can be a window to mathematical insight. When I was a student, Princeton math professors collected every afternoon in the third-floor lounge for tea and a round of backgammon. Professor Conway told us that he stumbled upon the discovery of a lifetime while studying Go, an ancient board game played with smooth stones. Nash's central insight, the one for which he won a Nobel Prize in 1994, was to prove that every economic game has an equilibrium point — that is, an approach to play in which no player would choose to change his strategy. If a player were to try to change his Nash equilibrium strategy, he would end up worse off than before. VON NEUMANN Nash built on previous insights, particularly those of John von Neumann, another brilliant and brash mathematician who was a professor at the Institute for Advanced Study near Princeton when Nash was a graduate student. Like the shoe-shucking Conway, von Neumann was an eccentric instructor. Instead of using the whole blackboard, von Neumann would write an equation then immediately erase it and write another as students frantically took notes. "Proof by erasure," his students called it. The same year Nash was born, 1928, von Neumann showed that in simple games like checkers or football, where one player's gain is the other's loss, rational players will tend toward a strategy that, even in a worst-case scenario, guarantees a minimum gain or caps a potential loss. That translates to conservative play — to a brand of football where teams prefer the extra point to the two-point conversion and always punt on fourth and inches. THE GAS STATION GAME Take the case of two gas stations being built in a small town made up of a single, one-mile long Main Street. Town planners agree the gas stations should be placed at one-quarter and three-quarter mile marks, so that no one in town has to drive more than a quarter mile to fill up. And since residents are distributed evenly along Main Street, both stations would share exactly half the business in town. But try explaining that to the station owners. The owner who should build at the one-quarter mile mark knows people at his end of town will never go to the competing station because it's too far away. So he'd want to build closer to the center of town to dip into his competitor's mid-town market. Of course, the other owner is equally wily, and he, too, edges his station closer to the center of town. Game theory tells us — and an astute business sense dictates — that the two gas stations will both end up on the same corner in the exact center of Main Street. The equilibrium solution of the gas station game is clearly not the most efficient. While the stations still share half the town's business, people on the edge of town have to drive farther to get gas under equilibrium than under the town planners' solution. But neither owner would have it any other way, because being located in the center of town is the most secure solution for each. This equilibrium is an example of free market inefficiency — and a critique of Adam Smith's invisible hand. NASH EQUILIBRIUM Though insightful, von Neumann's analysis of situations like the gas station game had limited real-world application because his theory couldn't accommodate even slight complications to a game. Nash, however, showed that even complicated games have an equilibrium point. He proved that participants in complex economic situations tend toward a specific, secure strategy, and he provided an elegant mathematical framework for figuring out that strategy. That apparently simple insight transformed game theory from an abstract subset of mathematics to a powerful new approach to the study of economic decision-making. THE NUCLEAR STANDOFF Nash's revolutionary insights weren't immediately accepted or even understood. When von Neumann first heard Nash's ideas, he told Nash they were "trivial" — math's harshest putdown. But over the years, even as Nash dropped math in his struggle with schizophrenia, his ideas took hold. These days, Nash-style strategic thinking can be found everywhere. Consider America's nuclear standoff with the Soviet Union. Each superpower could disarm or stockpile a nuclear arsenal. To decide, the U.S. considered what the Russians might do. If the Soviet Union armed, the U.S. needed to arm as well to defend itself. But if the Soviet Union disarmed, the U.S. would rather arm itself anyway to achieve a strategic advantage over its enemy. The Soviet Union's thinking ran the same way, and both countries settled on a policy of mutually assured destruction. Despite pacifist's complaints, a nuclear standoff was the most stable and secure solution— the Cold War's Nash equilibrium. CONWAY'S GAME Professor Conway was certainly thinking strategically that first silent lecture. An impatient genius like Conway would want to cull out the less confident and less talented, leaving a class of the curious and the quick. And students who stayed were willing to risk torture by erasure to be near the glow of genius. In later lectures, Conway threw down his chalk to digress on mathematical history, ran his hand through his hair to indulge in mathematical fascinations, ignored the course requirements to share his mathematical musings. He taught us that math is not the sawdust we learned in grade school but a passionate, and personal, subject. One class, he told us that while searching for patterns in Go,
he discovered surreal numbers: an entirely new class of numbers
able to express the infinitely small and infinitely large with an
unprecedented specificity. That semester, Professor Conway taught
us that — whether it is the pecking of pigeons or the placement
of Go stones — the thing in math is the search. For, as Nash
and Conway both prove, you never know what you might find. Daniel A. Grech is a staff writer at the Miami Herald and can be reached at dgrech@herald.com
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