Category Archives: miscellaneous

Mathematics and Concert Pianist – Eugenia Cheng mathematics and music

born under an unlucky star

via born under an unlucky star

Fourier Transformation in Data Science

via Fourier Transformation in Data Science

Fourier Transform in AI

via Fourier Transform in AI

an outlier

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John Conway, Simons Foundation, Science Lives, Mathematics, Mathematicians

via John Conway, Simons Foundation, Science Lives, Mathematics, Mathematicians

because the human brain is plastic

via because the human brain is plastic

Way to go : MIT’s PRIMES — 2020 !!!

https://www.ams.org/news?news_id=5970

The Greatest Auction Ever Held

Reference: A Beautiful Mind by Sylvia Nasar.

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Washington, D.C., December 1994:

On the afternoon of December 5, 1994, John Nash was riding in a taxi headed to Newark Airport on his way to Stockholm, where he would, in a few days time, receive from the King of Sweden the gold medal engraved with the portrait of Alfred Nobel. At around the same time, a few hundred miles in the south, in downtown Washington, D.C., Vice President Al Gore was announcing with great fanfare the opening of the “greatest auction ever.”

There was, as The New York Times would later report, no fast-talking auctioneer, no banging gavel, no Old Masters. On the auction block was thin air — airwaves that could be used for the new wireless gadgets like telephones, pagers, faxes — worth billions and billions of dollars, enough licenses for every major American city to have three competing cellular phone services. In the secret war rooms and building booths were CEOs of the world’s biggest communications conglomerates — and unlikely group of blue sky economic theoreticians who were advising them.When the auction finally closed the following March, the winning bids totaled more than $7 billion making it the biggest sale in American history of public assets and one of the most successful (and lucrative) applications of economic theory to public policy ever. Michael Rothschild, dean of Princeton’s Woodrow Wilson School, later called it “a demonstration that people thinking hard about a problem can make the world work better…a triumph of pure thought.”

The juxtaposition of Gore and Nash, the high-tech auction and the medieval pomp of the Nobel ceremony, was hardly an accident. The FCC auction was designed by young economists who were using tools created by John Nash, John Harsanyi and Reinhard Selten. Their ideas were specifically designed for analyzing rivalry and cooperation among a small number of rational players with a mix of conflicting and similar interests: people, governments, and corporations — and even animal species.

The prize itself was a long overdue acknowledgement by the Nobel committee that a sea change in economics, one that had been underway for more than a decade, had taken place. As a discipline, economics had long been dominated by Adam Smith’s brilliant metophor of the Invisible Hand. Smith’s concept of perfect competition envisions so many buyers and sellers that no single buyer or seller has to worry about the reactions of others. It is a powerful idea, one that predicted how free-market economies would evolve and gave policy-makers a guide for encouraging growth and dividing the economic pie fairly. But in the world of mega-mergers, big government, massive foreign direct investment, and whole-sale privatization, where the game is played by a handful of players, each taking into account the others’ actions, each pursuing his own best strategies, game theory has come to the fore.

After decades of resistance —- Paul Samuelson used to joke about “the swamp of n-person game theory” —- a younger generation of theorists began using game theory in areas from trade to industrial organization to public finance in the late 1970’s and early 1980’s. Game theory opened up “terrain for systematic thinking that was previously closed.” Indeed, as game theory and information economics have become increasingly entwined, markets traditionally seen as fitting the purely competitive mold have increasingly been studied using game theory assumptions. The latest generation of texts used in top graduate schools today all recast the basic theories of the firm and the consumer, the foundation of economics, in terms of strategic games. “Concepts, terminology and models from game theory have come to dominate many areas of economics,” said Avinash Dixit, an economist at Princeton who uses game theory in work on international trade and is the author of Thinking Strategically. “At last we are seeing the realization of the true potential of the revolution launched by von Neumann and Morgenstern.” And because most economic applications of game theory use the Nash equilibrium concept, “Nash is the point of departure.”

The revolution has gone far beyond research journals, experimental laboratories at Caltech and the University of Pittsburgh, and classrooms of elite business schools and universities. The current generation of economic policy-makers — including Lawrence Summers, undersecretary of the treasury, Joseph Stiglitz, chairman of the Council of Economic Advisers, and Vice-President Al Gore — are steeped in the stuff, which they say, is useful for thinking about everything from budget proposals to Federal Reserve policy to pollution cleanups.

The most dramatic use of game theory is by governments from Australia to Mexico to sell scarce public resources to buyers best able to develop them. The radio spectrum, T-bills, oil leases, timber, and pollution rights are now sold in auctions designed by game theorists — with far greater success than that of earlier policies.

Economists like Nobel Laureate Ronald Coase have advocated the use of auctions by government since the 1950’s. Auctions have long been used in markets where sellers of unusual items — from vintage wines to movie rights — have no idea what bidders are willing to pay. Their basic purpose is to make bidders reveal how much they value the item. But the arguments of Coase and others were stated in abstract, entirely theoretical terms, and little thought was given to how such auctions would actually be conducted. Congress remained skeptical.

Before 1994, Washington simply gave away licenses for free. Until 1982, it had been up to regulators to decide which companies deserved the licenses. Needless to say, the process was dominated by political pressures, outrageously expensive paperwork, and long delays. The pace of licensing lagged hopelessly behind market shifts and new technologies. After 1982, Washington awarded licenses using lotteries, with the winners free to resell licenses. Although the reform did speed up the granting of licenses, the process was still hugely inefficient — and unfair. Bidders with no intention of operating an actual telephone business spent millions to get into the game for the purpose of reaping a windfall. Further, although telephone companies were forced to pay the costs of obtaining licenses, Washington (and taxpayers) did not get the benefits of any revenues. There had to be a better way.

A young generation of game theorists, including Paul Milgrom, John Roberts, and Robert Wilson at the Stanford B-school, came up with that better way. Their chief contribution consisted of recognizing, as Milgrom said, that “the mere design of some auction was not enough…Getting the auction design right was also critically important.” In particular, they concluded that the most obvious auction designs —- auctioning licenses one by one in sequence using simultaneous sealed bids — was the way least likely to succeed in getting licenses into the hands of corporations that could use them best —- Washington’s stated objective. Game theorists treat an auction like a game with rules and try to evaluate how a given set of rules, taken together, is apt to affect the bidders’ behaviour. They take stock of the options the rules allow, the payoffs to the bidders associated with the options, and bidders expectations about their competitors’ likely choices.

Why did these economists conclude that traditional auction formats would not work? Mainly because the value of each individual license to a user depends — as is the case with a Rembrandt or a Picasso — on what other licenses the user is able to obtain. Some licenses are perfect substitutes for one another. That would be the case for similar spectrum bands to provide a given service. But others are complements. That would be the case for licenses to provide paging services in different parts of the country.

“To permit the efficient license assignment, an auction must allow bidders to consider various packages of licenses, combining complements and switching among substitutes during the course of the auction. Designing an auction to allow this is quite difficult,” writes Paul Milgrom., one of the economists who designed the FCC auction of which Gore was speaking.

A second source of complexity, Milgrom says, is that the purpose of the licenses is to create businesses for new services with unknown technology and unknown consumer demand. Since bidders’ opinions are bound to be wildly divergent, it is possible that license assignment would depend more on bidders’ optimism than on their ability to create a desired service. Ideally, an auction design can minimize that problem.

As Congress and the FCC inched closer to the notion of auctioning off spectrum rights, Australia and New Zealand both conducted spectrum auctions. That they proved to be costly flops and political disasters illustrated that the devil really was in the details. In New Zealand, the government ran a so-called second price auction, and newspapers were full of stories about winners who paid far below their bids. In once case, the high bid was NZ$7 million, the second bid was NZ$5000, and the winner paid the lower price. In another, an Otago University student bid NZ$1 for a television license in a small city. Nobody else bid, so he got it for one dollar. The government expected the cellular licenses to fetch NZ$240 million. The actual revenue was NZ$36 million, one-seventh of the advance estimate. In Australia, a botched auction, in which parvenu bidders pulled the wool over the government’s eyes, delayed the introduction of pay television by almost a year.

The FCC’s chief economist was an advocate of auctions, but no game theorists were involved in the first stage of the FCC auction design. The theorists’s phones started ringing only by accident after the FCC issued a tentative proposal for an auction format with dozens of footnotes to the theoretical literature on auctions. That was how Milgrom and his colleague Robert Wilson, leading auction theorists, got into the game. Milgrom and Wilson proposed that the FCC adopt a simultaneous, multiple round auction. In a simultaneous auction, a bunch of licenses are sold at the same time. Multiple rounds means that, after the first round of bidding prices are announced, and bidders have a chance to withdraw or raise one another’s bids. This is repeated round after round until the auction is over. The chief advantage of this format is that it allows bidders to take account of interdependencies among licenses. Just as sequential, closed-bid auctions let sellers discover what bidders are willing to pay for individual items, the simultaneous, ascending-bid auction lets them discover the market value of different groupings of items.

This early proposal —- which the FCC eventually adopted — did not cover seemingly small but critical details. Should there be deposits? Minimum bid increments? Time limits? Should the bidding system be wholly computerized or executed by hand? And so forth. Milgrom, Roberts and another game theorist, Preston McAfee, an adviser to AirTouch, provided proposals on these issues. The FCC hired another game theorist, John McMillan, of the University of Caliofornia, San Diego, to help evaluate the effect of every proposed rule. According to Milgrom, “Game theory played a central role in the analysis of the rules. Ideas of Nash equilibrium, rationalizability, backward induction, and incomplete information, though rarely named explicitly, were the real basis of daily decisions about the details of the auction process.

By late spring 1995, Washington had raised more than USD 10 billion from spectrum auction. The press and the politicians were ecstatic. Corporate bidders were largely able to protect themselves from predatory bidding and were able to assemble an economically sensible set of licenses. It was, as John McMillan said, ” a triumph for game theory.”

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PS: a triumph of pure mathematical thought !:-)

Regards,

Nalin Pithwa

 

 

 

 

 

Einstein told Banesh Hoffmann, “I am slow…!”

Picked up from Reader’s Digest: Indian Edition: March 2020 : just for the joy of sharing and learning with my students. 

January 1968: The Unforgettable Albert Einstein: A professor remembers his encounters with Albert Einstein, and pays a glowing tribute to the man’s genius and his many accomplishments.

By Banesh Hoffmann:

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It was one of the greatest scientists the world has ever known, yet if I had to convey the essence of Albert Einstein in a single word, I would choose ‘simplicity’. Perhaps, an anecdote will help. Once, caught in a downpour, he took off his hat and held it under his coat. Asked why, he explained, with admirable logic, that the rain would damage the hat, but his hair would be none the worse for its wetting. This knack for going instinctively to the heart of the matter was the secret of his major scientific discoveries — this and his extraordinary feeling for beauty.

I first met Albert Einstein in 1935, at the famous Institute for Advanced Study in Princeton, New Jersey. Einstein had been among the first to be invited to the Institute, and was offered carte blanche as to salary. To the director’s dismay, Einstein asked for an impossible sum. It was far too small !! The director had to plead with him to accept a larger salary.

I was awe in of Einstein, and hesitated before approaching him about some ideas I had been working on. My hesitation proved unwarranted. When I finally knocked on his door, a gentle voice said, “Come” — with a rising inflection that made the single word both a welcome and a question. I entered his office and found him seated on a table, calculating and smoking his pipe. Dressed in ill-fitting clothes, his hair characteristically awry, he smiled a warm welcome. His utter naturalness at once set me at ease.

As  I began to explain my ideas, he asked me to write the equations on blackboard so that he could see how they developed. Then, came the staggering — and altogether endearing —request: “Please go slowly. I do not understand things quickly.” This from Einstein ! He said it gently, and I laughed. From then on, all vestiges of fear were gone. 

BURST OF GENIUS Einstein was born in 1879 in the German city of Ulm. He had been no infant prodigy; indeed, he was so late in learning to speak that his parents feared he was a dullard. In school, though his teachers saw no special talent in him, the signs were already there. He taught himself calculus, for example, and he told me that his teachers seemed a little afraid of him because he asked questions that they could not answer. At the age of 16, he asked himself whether a light wave would seem stationary if one ran abreast of it. It seems an innocent question, but this shows Einstein going to the heart of a problem. From it there would arise, 10 years later, his theory of relativity.

Einstein failed his entrance examinations at the Swiss Federal Polytechnic School in Zurich, but was admitted a year later. There he went beyond his regular work to study the masterworks of physics on his own. Rejected when he applied for academic positions, he ultimately found work, in 1902, as a patent examiner in Berne, and there, in 1905, his genius burst into fabulous flower.

Among the extraordinary things he produced in that memorable year were his theory of relativity, with its famous offshoot E = mc^{2} (energy equals mass times the speed of light squared), and his quantum theory of light. These two theories were not only revolutionary but seemingly self-contradictory as well: the former was intimately linked to the theory that light consists of waves, while the latter said that it consists of somehow of particles. Yet this unknown young man boldy proposed both at once — and he was right in both cases, though how he could possibly have been is far too complex a story to tell here.

MENTAL MAGIC

Collaborating with Einstein was an unforgettable experience. In 1937, the Polish physicist Leopold Infeld and I asked if we could work with him. He was pleased with the proposal, since he had an idea about gravitation waiting to be worked out in detail. Thus, we got to know not merely the man and the friend, but also the professional.

The intensity and depth of his concentration were fantastic. When battling a recalcitrant problem, he worried it as an animal worries its prey. Often, when we found ourselves up against a seemingly insuperable difficulty, he would stand up, put his pipe on the table, and say in his quaint English, “I will a little tink” (he could not pronounce “th”). Then, he would pace up and down, twirling a lock of his long greying hair around his forefinger.

A dreamy, faraway yet inward look would come over his face. There was no appearance of concentration, no furrowing of his brow — only a placid inner communion. The minutes would pass, and then suddenly Einstein would stop pacing as his face relaxed into a gentle smile. He has found the solution to the problem. Sometimes it was so simple that Infeld and I could have kicked ourselves for not having thought of it. But the magic had been performed invisibly in the depths of Einstein’s mind, by a process we could not fathom.

When his wife died, he was deeply shaken, but insisted that now more than ever was the time to be working hard. I vividly remember going to his house to work with him during that sad time. His face was haggard and grief-lined but he put forth a great effort to concentrate. Seeking to help him, I steered the discussion away from routine matters into more difficult theoretical problems, and Einstein gradually became absorbed in the discussion. We kept at it for some two hours, and at the end his eyes were no longer sad. As I left, he thanked me with moving sincerity, but the words he found sounded almost incongruous. “It was a fun,” he said. He had a moment of surcease from grief, and these groping words expressed a deep emotion.

IDEAS FROM GOD

Although Einstein felt no need for religious ritual and belonged to no formal religious group, he was the most deeply religious man I have known. He once said to me, “ideas come from God,” and one could hear the capital ‘G’ in the reverence with which he pronounced the word. On the marble fireplace in the mathematics building at Princeton University is carved, in the original German, what one might call his scientific credo: God is subtle, but He is not malicious.” By this Einstein meant that scientists could expect to find their task difficult, but not hopeless. The Universe was a Universe of law, and God was not confusing with deliberate paradoxes and contradictions.

Einstein was an accomplished amateur musician. We used at play duets; he at the violin, I at the piano. One day he surprised me by saying that Mozart was the greatest composer of all. Beethoven, he said, “created” his music but the music of Mozart was of such purity and beauty that one felt he merely “found” it — that it had always existed as part of the inner beauty of the Universe, waiting to be revealed.

It was this very Mozartian simplicity that most characterized Einstein’s methods. His 1905 theory of relativity, for example, was built on two simple assumptions. One is the so-called principle of relativity, which means, roughly speaking, that we cannot tell whether we are at rest or moving smoothly. The other assumption is that the speed of light is the same, no matter what the speed of the object that produces it. You can see how reasonable this is if you think of agitating a stick in a lake to create waves. Whether you wiggle the stick from a stationary pier. or from a rushing speedboat, the waves once generated are on their own, and their speed has nothing to do with that of the stick.

Each of these assumptions, by itself, was so plausible as to seem primitively obvious. But. together they were in such violent conflict that a lesser man would have dropped one or the other and fled in panic. Einstein daringly kept both — and by doing so he revolutionized physics. For he demonstrated that they could after all, exist peacefully side by side, provided we give up cherished beliefs about the nature of time.

Science is like a house of cards, with concepts like time and space at the lowest level. Tampering with time brought most of the house tumbling down, and it was this made Einstein’s work so important —- and so controversial. At a conference in Princeton in honour of his 70th birthday, one of the speakers, a Nobel prize winner, tried to convey the magical quality of Einstein’s achievement. Words failed him, and with a shrug of helplessness he pointed to his wrist-watch, and said in tones of awed amazement, “It all came from this.” His very ineloquence made this the most eloquent tribute I have heard to Einstein’s genius.

SAND SENSE

Although fame had little effect on Einstein as a person, he could not escape it; he was, of course, instantly recognizable. One autumn Saturday, I was walking with him in Princeton discussing some technical matters. Parents and almuni were streaming excitedly toward the stadium, their minds on the coming football game. As they approached us, they paused in sudden recognition, and a momentary air of solemnity came over them as if they had been reminded of a world far removed from the thrills of football. Yet Einstein seemed totally unaware of the effect he was having on them, and went on with the discussion as though they were not there.

We think of Einstein as one concerned only with the deepest aspects of science. But he saw scientific principles in every day things to which most of us would give barely a second thought.He once asked me if I had ever wondered why a man’s feet will sink into either dry or completely submerged sand, while sand that is merely damp provides a firm surface. When I could not answer, he offered a simple explanation. It depends, he pointed out, on surface tension, the elastic-skin effect of a liquid surface. This is what holds a drop together, or causes two small raindrops on a window pane to pull into one big drop the moment their surfaces touch.

When sand is damp, Einstein explained, there are tiny amounts of water between the grains. The surface tensions of these tiny amounts of water pull all the grains together, and friction then makes them hard to budge. When the sand is dry, there is obviously no water between grains. If the sand is fully immersed, there is water between grains, but there is no water surface between them to pull them together. This is not as important as relativity; yet as his youthful question, about running abreast of a light wave showed, there is no telling what seeming trifle will lead an Einstein to a major discovery. And, the puzzle of the sand gives us an inkling of the power and elegance of Einstein’s mind.

COSMIC SIMPLICITY:

Einstein’s work, performed with pencil and paper, seemed remote from the turmoil of everyday life. But his ideas were so revolutionary that they caused violent controversy and irrational anger. Indeed, in order to be able to award him a belated Nobel Prize, the selection committee had to avoid mentioning relativity, and pretend that his prize was primarily due to his work on quantum theory. Political events upset the serenity of his life even more. When the Nazis came to power in Germany, his theories were officially declared false because they had been formulated by a Jew. His property was confiscated, and it is said that a price was put on his head.

When scientists in the United States fearful that the Nazis might develop an atomic bomb, sought to alert American authorities to that danger, they were scarcely heeded. In desperation, they drafted a letter, which Einstein signed and sent directly to President Roosevelt. It was this act that led to the fateful decision to go all-out on the production of an atomic bomb — an endeavour in which Einstein took no active part. When he heard of the agony and destruction that his E=mc^{2} had wrought, he was dismayed beyond measure and from then on there was a look of ineffable sadness in his eyes.

There was something elusively whimsical about Einstein. It is illustrated by my favourite anecdote about him. In his first year in Princeton, on Christmas Eve, so the story goes, some children sang carols outside his house. Having finished, they knocked on his door and explained that they were collecting money to buy Christmas presents. Einstein listened then said, “Wait a moment.” He put on his scarf and overcoat, and took his violin from its case. Then, joining the children, he accompanied their singing of “Silent Night” on his violin.

How shall I sum up what it meant to have known Einstein and his work? Like the Nobel prize winner who pointed helplessly at his watch, I can find no adequate words. It was akin to the revelation of the great art that lets one see what was formerly hidden. And, when for example, I walk on the sand of a lonely beach, I am reminded of his ceaseless search for cosmic simplicity —- and the scene takes on a deeper, sadder beauty.

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Regards,

Nalin Pithwa.

PS: Thinking takes time, practice, perseverance and solitude. The reward of an intellectual discovery, mathematical or other, is far richer and complete than instant gratification. :-))