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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature Notes INTRODUCTION 1. Isaac Asimov, Foundation and Earth, Doubleday, Garden City, N.Y., 1986, p. 247. 2. Isaac Asimov, Foundation’s Edge, Ballantine Books, New York, 1983 (1982), p. xi. 3. Herbert Gintis, Game Theory Evolving, Princeton University Press, Princeton, N.J., 2000, pp. xxiv–xiv. 4. Samuel Bowles, telephone interview, September 11, 2003. 5. Read Montague, interview in Houston, Tex., June 24, 2003. 6. Gintis, Game Theory Evolving, p. xxiii. 7. Asimov, Foundation and Earth, p. 132. 8. Stephen Wolfram, in his controversial book A New Kind of Science, also claims to show a network-related way of explaining quantum physics—and everything else in the universe. If he is right, game theory may someday have something to say about the universe as well. SMITH’S HAND 1. Jacob Bronowski and Bruce Mazlish, The Western Intellectual Tradition, Harper & Row, New York, 1960, p. 353. 2. David Hume, A Treatise of Human Nature, available online at http://etext.library.adelaide.edu.au/h/hume/david/h92t/introduction.html. 3. James Anson Farrer, Adam Smith, Sampson, Low, Marston, Searle and Rivington, 1881, p. 2. Available online at http://socserv2.socsci.mcmaster.ca/~econ/ugcm/3ll3/smith/farrer.html. 4. Adam Smith, The Wealth of Nations, Bantam, New York, 2003 (1776). 5. Ibid., pp. 23–24, 572. 6. Alan Krueger, Introduction, The Wealth of Nations, Bantam, New York, 2003, p. xviii.
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature 7. Ibid., p. xxiii. 8. Thomas Edward Cliffe Leslie, “The Political Economy of Adam Smith,” The Fortnightly Review, November 1, 1870. Available online at http://etext.lib.virginia.edu/modeng/modengS.browse.html 9. “Code of Nature” was most unfortunately used in the title of a work by a French communist named Morelly. He had truly wacko ideas. I don’t mean to pick on communists—they’ve had a bad enough time in recent years—but this guy really was off the charts. For one thing, he insisted that everybody had to get married whether they wanted to or not. And you had to turn 30 years old before you would be allowed to pursue an academic profession if you so desired, provided you were judged worthy. 10. Henry Maine, Ancient Law, 1861. Available online at http://www.yale.edu/lawweb/avalon/econ/maineaco.htm. Maine notes that “Jus Gentium was, in actual fact, the sum of the common ingredients in the customs of the old Italian tribes, for they were all the nations whom the Romans had the means of observing, and who sent successive swarms of immigrants to Roman soil. Whenever a particular usage was seen to be practiced by a large number of separate races in common it was set down as part of the Law common to all Nations, or Jus Gentium.” 11. Dugald Stewart, “Account of the Life and Writings of Adam Smith LL.D.,” Transactions of the Royal Society of Edinburgh, 1793. Available online at http://socserv2.socsci.mcmaster.ca/~econ/ugcm/3ll3/smith/dugald. 12. Cliffe Leslie, “Political Economy,” pp. 2, 11. 13. Roger Smith, The Norton History of the Human Sciences, W.W. Norton, New York, 1997, p. 303. 14. Colin Camerer, interview in Pasadena, Calif., March 12, 2004. 15. Nava Ashraf, Colin F. Camerer, and George Loewenstein, “Adam Smith, Behavioral Economist,” Journal of Economic Perspectives, 19 (Summer 2005): 132. 16. Stephen Jay Gould, The Structure of Evolutionary Theory, Harvard University Press, Cambridge, Mass., 2002, pp. 122–123. 17. Charles Darwin, The Origin of Species, The Modern Library, New York, 1998, p. 148. 18. Another interesting refutation of Paley comes from Stephen Wolfram, whose book A New Kind of Science generated an enormous media blitz in 2002. Wolfram makes the point that a Swiss watch—Paley’s example of complexity—is actually quite a simple, regular, predictable device. You need a designer, Wolfram said, not to produce complexity, but to ensure simplicity.A watch, after all, exhibits nothing like the complexity of life, Wolfram pointed out. Keeping time requires, above all else, absolutely regular motion to guarantee near-perfect predictability. Complexity introduces deviations from regular motion, rendering a clock worthless. And as Wolfram demonstrates throughout his book, nature—left to its own devices—produces complexity
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature with wild abandon. In the biological world, such complexity is messy and unpredictable, and for that sort of thing you need no designer at all, just simple rules governing how a system evolves over time (and some of those rules might be provided by game theory). A Swiss watch, on the other hand, does not evolve over time—it just tells time. And it has no offspring, only springs. 19. Gould, Evolutionary Theory, p. 124. 20. Of course, had Einstein introduced relativity in a book, instead of writing scientific papers, the 20th century would have had a similar masterpiece. VON NEUMANN’S GAMES 1. Maria Joao Cardoso De Pina Cabral, “John von Neumann’s Contribution to Economic Science,” International Social Science Review, Fall–Winter 2004. Available online at http://www.findarticles.com/p/articles/mi_m0IMR/is_3-4_79. 2. Jeremy Bentham, An Introduction to the Principles of Morals and Legislation, Clarendon Press, Oxford, 1907 (1789), Chapters I, III. While written in 1780 and distributed privately, it wasn’t published until 1789. 3. Jeremy Bentham, A Fragment on Government, London, 1776, Preface. Available online at http://www.ecn.bris.ac.uk/het/bentham/government.htm. Although Bentham is sometimes credited with coining this phrase, a very similar expression was authored by the Irish philosopher Francis Hutcheson in 1725: “That action is best which procures the greatest happiness for the greatest numbers.” 4. Bernoulli suggested that the utility of an amount of money diminished as the logarithm of the quantity, and logarithms do increase at a diminishing rate as a quantity gets larger. But there was no other basis for determining that the logarithmic approach actually quantified anybody’s utility accurately. 5. Strictly speaking, utility theory can be used without game theory to make economic predictions, and it often is. But before game theory came along, the mathematical basis of utility was less than solid. In formulating game theory, von Neumann and Morgenstern developed a method to compute utility with mathematical rigor. Utility theory on its own can be used by individuals making solitary decisions, but when one person’s choice depends on what others are choosing, game theory is then necessary to calculate the optimum decision. 6. See Ulrich Schwalbe and Paul Walker, “Zermelo and the Early History of Game Theory,” Games and Economic Behavior, 34 (January 2001): 123–137. 7. The term “minimax” refers to the game theory principle that you should choose a strategy that minimizes the maximum loss you will suffer no matter what strategy your opponent plays and maximizes the minimum gain you can attain when choosing from each possible strategy. 8. In 1937, von Neumann published another influential paper, not specifically linked to game theory, that presented a new view on the nature of growth
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature and equilibrium in economic systems. That paper was another major element of von Neumann’s contribution to economic science. See Norman Macrae, John von Neumann, Pantheon Books, New York, 1991, pp. 247–256. 9. In a footnote, he did mention possible parallels to economic behavior. 10. In the story, Moriarty appears in Victoria station just as Holmes and Watson’s train departs for Dover, where a ferry will transport them to France. Watson believes they have successfully escaped from the villain, but Holmes points out that Moriarty will now do what Holmes himself would have done—engage a special train to speed him to Dover before the ferry departs. But anticipating this move by Moriarty, Holmes decides to get off the train in Canterbury and catch another train to Newhaven, site of another ferry to France. Sure enough, Moriarty hired a special train and went to Dover. But a game theorist would wonder why Moriarty would not have anticipated the fact that Holmes would have anticipated Moriarty’s move, etc. See Leslie Klinger, ed., The New Annotated Sherlock Holmes, Vol. 1, W.W. Norton, New York, 2005, pp. 729–734. 11. Oskar Morgenstern, “The Collaboration between Oskar Morgenstern and John von Neumann on the Theory of Game,” Journal of Economic Literature, 14 (September 1976), reprinted in John von Neumann and Oskar Morgenstern, Theory of Games and Economic Behavior, Sixtieth-Anniversary Edition, Princeton University Press, Princeton, N.J., 2004. 12. Robert J. Leonard, “From Parlor Games to Social Science: Von Neumann, Morgenstern, and the Creation of Game Theory, 1928–1944,” Journal of Economic Literature, 33 (1995): 730–761. 13. John von Neumann and Oskar Morgenstern, Theory of Games and Economic Behavior, Sixtieth-Anniversary Edition, Princeton University Press, Princeton, N.J., 2004, p. 2. 14. Ibid., p. 4. 15. Ibid., p. 2. 16. Ibid., p. 6. 17. Samuel Bowles, telephone interview, September 11, 2003. 18. Von Neumann and Morgenstern, Theory of Games, p. 11. 19. Ibid., p. 11 20. Ibid., p. 12 21. Ibid., p. 14. 22. Ibid., Theory of Games and Economic Behavior, p. 13. 23. If you really want to get technical, you have to subtract the bus fare from the winnings (or add it to the cost) when calculating the payoffs for this game. But that makes it too complicated, so let’s assume they live in a “free ride” zone. 24. Thus a mixed strategy is a “probability distribution” of pure strategies. The concept of probability distribution will become increasingly important in later chapters.
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature 25. J.D. Williams, The Compleat Strategyst: Being a Primer on the Theory of Games of Strategy, McGraw-Hill, New York, 1954. 26. The actual math for calculating the optimal strategies for this game matrix is given in the Appendix. 27. In the original formulation of game theory, von Neumann insisted on treating games as if they were only one-shot affairs—no repetitions. In that case, a mixed strategy could not be implemented by choosing different strategies different percentages of the time. You could make only one choice. If your minimax solution was a mixed strategy, you had to use the random-choice device to choose which of the possible pure strategies you should play. 28. A similar version of this game is presented in a book on game theory by Morton Davis, which in turn was modified from a somewhat more complex version of “simplified” poker described by von Neumann and Morgenstern. 29. See Morton Davis, Game Theory: A Nontechnical Introduction, Dover, Mineola, N.Y., 1997 (1983), pp. 36–38. 30. Von Neumann and Morgenstern, Theory of Games, p. 43. NASH’S EQUILIBRIUM 1. Roger Myerson, “Nash Equilibrium and the History of Economic Theory,” 1999. Available online at http://home.uchicago.edu/~rmyerson/research/jelnash.pdf. 2. Paul Samuelson, “Heads I Win, Tails You Lose,” in von Neumann and Morgenstern, Theory of Games, p. 675. 3. Leonid Hurwicz, “Review: The Theory of Economic Behavior,” American Economic Review, 35 (December 1945). Reprinted in von Neumann and Morgenstern, Theory of Games, p. 664. 4. Ibid., p. 662. 5. Arthur H. Copeland, “Review,” Bulletin of the American Mathematical Society, 51 (July 1945): 498–504. Reprinted in von Neumann and Morgenstern, Theory of Games. 6. Hurwicz, “Review,” p. 647. 7. Herbert Simon, “Review,” American Journal of Sociology, 50 (May 1945). Reprinted in von Neumann and Morgenstern, Theory of Games, p. 640. 8. In the film version of A Beautiful Mind, the math is garbled beyond any resemblance to what Nash actually did. 9. John Nash, “The Bargaining Problem,” Econometrica, 18 (1950): 155– 162. Reprinted in Harold Kuhn and Sylvia Nasar, eds., The Essential John Nash, Princeton University Press, Princeton, N.J., 2002, pp. 37–46. 10. John Nash, “Non-Cooperative Games,” dissertation, May 1950. Reprinted in Kuhn and Nasar, The Essential John Nash, p. 78. 11. Ibid., p. 59.
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature 12. Erica Klarreich, “The Mathematics of Strategy,” PNAS Classics, http://www.pnas.org/misc/classics5.shtml. 13. Samuel Bowles, telephone interview, September 11, 2003. 14. Ibid. 15. John Nash, “Non-cooperative Games,” Annals of Mathematics, 54 (1951). Reprinted in Kuhn and Nasar, The Essential John Nash, p. 85. I have corrected “collaboration of communication” as printed there to “collaboration or communication”—it is clearly a typo, differing from Nash’s dissertation. 16. Kuhn, The Essential John Nash, p. 47. 17. As one reviewer of the manuscript for this book pointed out, it is not necessarily true that all economic systems converge to equilibrium, and that in some cases a chaotic physical system might be a better analogy than a chemical equilibrium system. The idea of equilibrium is nevertheless an important fundamental concept, and much of modern economics involves efforts to understand when it works and when it doesn’t. 18. This observation (in a slightly different form) has been attributed to the physicist Murray Gell-Mann. 19. Quoted in William Poundstone, Prisoner’s Dilemma, Anchor Books, New York, 1992, p. 124. 20. Mathematically, Tucker’s game was the same as one invented earlier by Merrill Flood and Melvin Dresher. Tucker devised the Prisoner’s Dilemma as a way of illustrating the payoff principles in Flood and Dresher’s game. See Poundstone, Prisoner’s Dilemma, pp. 103ff. 21. Charles Holt and Alvin Roth, “The Nash Equilibrium: A Perspective,” Proceedings of the National Academy of Sciences USA, 101 (March 23, 2004): 4000. 22. Robert Kurzban and Daniel Houser, “Experiments Investigating Cooperative Types in Humans: A Complement to Evolutionary Theory and Simulations,” Proceedings of the National Academy of Sciences USA, 102 (February 1, 2005): 1803–1807. 23. R. Duncan Luce and Howard Raiffa, Games and Decisions, John Wiley & Sons, New York, 1957, p. 10. 24. Ariel Rubenstein, Afterword, in von Neumann and Morgenstern, Theory of Games, p. 633. 25. Ibid., p. 634. 26. Ibid, p. 636. 27. Colin Camerer, Behavioral Game Theory, Princeton University Press, Princeton, N.J., 2003, p. 5. 28. Ibid., pp. 20–21. 29. The Royal Swedish Academy of Sciences, “Press Release: The Bank of Sweden Prize in Economic Sciences in Memory of Alfred Nobel 2005,” October 10, 2005.
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature SMITH’S STRATEGIES 1. D.G.C. Harper, “Competitive Foraging in Mallards—‘Ideal Free’ Ducks,” Animal Behaviour, 30 (1982): 575–584. 2. Of course, you could conclude that animals are in fact rational, or at least more rational than they are generally considered to be. 3. Martin Nowak, interview in Princeton, N.J., October 19, 1998. 4. Rosie Mestel, The Los Angeles Times, April 24, 2004, p. B21. 5. Maynard Smith’s first paper on evolutionary game theory was written in collaboration with Price; it appeared in Nature in 1973. The story is told in John Maynard Smith, “Evolution and the Theory of Games,” American Scientist, 64 (January–February 1976): 42. Price died in 1975. 6. John Maynard Smith, “Evolutionary Game Theory,” Physica D, 22 (1986): 44. 7. Ibid. 8. The relationship between Nash equilibria and evolutionary stable strategies can get extremely complicated, and a full discussion would include considerations of the equations governing the reproductive rate of competing species (what is known as the “replicator dynamic”). A good place to explore these issues is Herbert Gintis, Game Theory Evolving, Princeton University Press, Princeton, N.J., 2000. 9. For the calculation of the Nash equilibrium giving this ratio, see the Appendix. 10. This equivalence of a mixed population—two-thirds doves and one-third hawks—with mixed behavior of the same birds holds only in the simple case of a two-strategy game. In more complicated games, the exact math depends on whether you’re talking about mixtures of populations or mixtures of strategies. 11. Rufus Johnstone, “Eavesdropping and Animal Conflict,” Proceedings of the National Academy of Sciences USA, 98 (July 31, 2001): 9177–9180. 12. John M. McNamara and Alasdair I. Houston, “If Animals Know Their Own Fighting Ability, the Evolutionarily Stable Level of Fighting is Reduced,” Journal of Theoretical Biology, 232 (2005): 1–6. 13. Martin Nowak, interview in Princeton, October 19, 1998. 14. Ibid. 15. In all, 15 strategies participated in the round-robin tournament. Axelrod added a strategy that chose defect or cooperate at random. 16. Martin Nowak, lecture in Quincy, Mass., May 18, 2004. 17. A paper describing the results Nowak discussed in Quincy appeared the following year: Lorens A. Imhof, Drew Fudenberg, and Martin Nowak, “Evolutionary Cycles of Cooperation and Defection,” Proceedings of the National Academy of Sciences USA, 102 (August 2, 2005): 19797–10800.
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature 18. Herbert Gintis and Samuel Bowles, “Prosocial Emotions,” Santa Fe Institute working paper 02-07-028, June 21, 2002. FREUD’S DREAM 1. Von Neumann, actually, was very interested in the brain, and his last book was a series of lectures (that he never delivered) comparing the brain to a computer. But I found no hint that he saw an explicit connection between neuroscience and game theory. 2. P. Read Montague and Gregory Berns, “Neural Economics and the Biological Substrates of Valuation,” Neuron, 36 (October 10, 2002): 265. 3. Colin Camerer, interview in Santa Monica, Calif., June 17, 2003. 4. Read Montague, interview in Houston, Tex., June 24, 2003. 5. The earliest MRI technologies were good for showing anatomical detail, but did not track changes in brain activity corresponding to behaviors. By the early 1990s, though, advances in MRI techniques led to fMRI—functional magnetic resonance imaging—which could record changes in activity over time in a functioning brain. 6. Read Montague, interview in Houston, June 24, 2003. 7. A Web page that tracks new words claimed that its first use was in the Spring 2002 issue of a publication called The Flame. 8. M.L. Platt and P.W. Glimcher, “Neural Correlates of Decision Variables in Parietal Cortex,” Nature, 400 (1999): 233–238. 9. Read Montague, interview in Houston, June 24, 2003. 10. A.G. Sanfey et al., “The Neural Basis of Economic Decision-Making in the Ultimatum Game,” Science, 300 (2003): 1756. 11. Read Montague, interview in Houston, Tex., June 24, 2003. 12. Paul Zak, interview in Claremont, Calif., August 4, 2003. 13. Aldo Rustichini, “Neuroeconomics: Present and Future,” Games and Economic Behavior, 52 (2005): 203–204. 14. James Rilling et al., “A Neural Basis for Social Cooperation,” Neuron, 35 (July 18, 2002): 395–405. 15. In this version of the game, Players A and B both get 10 “money units” and Player A chooses whether to give his 10 to Player B. If he does, the experimenter quadruples the amount to make 40, so Player B now has 50 (40 plus the original 10). Player B then chooses to give some amount back to A, or keep the whole 50. If Player A doesn’t think B returned a fair amount, Player A is given the option to “punish” B by assessing “punishment points.” Every punishment point subtracts one monetary unit from B’s payoff, but it costs A one monetary unit for every two punishment points assessed. See Dominique J.-F. de Quervain et al., “The Neural Basis of Altruistic Punishment,” Science, 305 (August 27, 2004): 1254–1258.
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature 16. Colin Camerer, interview in Santa Monica, June 17, 2003. 17. Paul Zak, interview in Claremont, Calif., August 4, 2003. SELDON’S SOLUTION 1. Werner Güth, Rolf Schmittberger, and Bernd Schwarze, “An Experimental Analysis of Ultimatum Bargaining,” Journal of Economic Behavior and Organization, 3 (December 1982): 367–388. 2. Jörgen Weibull, “Testing Game Theory,” p. 2. Available online at http://swopec.hhs.se/hastef/papers/hastef0382.pdf. 3. Ibid., p. 5. 4. Ibid., p. 17. 5. Steven Pinker, The Blank Slate, Viking, New York, 2002, p. 102. 6. Isaac Asimov, Prelude to Foundation, Bantam Books, New York, 1989, p. 10. 7. Ibid., pp. 11–12. 8. Ibid., p. 12. 9. Robert Boyd, interview in Los Angeles, Calif., April 14, 2004. 10. Some of the ultimatum game results were especially perplexing, in particular the first round of games played in Mongolia. Francisco J. Gil-White, of the University of Pennsylvania, was confused by the pattern of offers and rejections—until discovering that some players didn’t believe they would actually receive real money. In another incident, he was puzzled by the rejection of a generous offer. It turned out the player thought Gil-White was an impoverished graduate student. By rejecting all offers, the player reasoned, he would ensure all the money was given back to Gil-White. 11. Joseph Henrich, telephone interview, May 13, 2004. 12. Colin Camerer, interview in Pasadena, Calif., March 12, 2004. 13. Ibid. 14. Robert Boyd, interview in Los Angeles, Calif., April 14, 2004. 15. Colin Camerer, interview in Pasadena, Calif., March 12, 2004. 16. David J. Buller, “Evolutionary Psychology: The Emperor’s New Paradigm,” Trends in Cognitive Sciences, 9 (June 2005): 277–283. 17. Not surprisingly, evolutionary psychologists have reacted negatively to Buller’s criticisms, contending that he distorts the evidence he cites. You can find some of their counterarguments online at http://www.psych.ucsb.edu/research/cep/buller.htm. 18. Ira Black, remarks at the annual meeting of the Society for Neuroscience, Orlando, Florida, November 3, 2002. Black died in early 2006. 19. Steven Quartz and Terrence Sejnowski, Liars, Lovers, and Heroes, William Morrow, New York, 2002, pp. 41, 46.
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature 20. E.J. Chesler, S.G. Wilson, W.R. Lariviere, S.L. Rodriguez-Zas, and J.S. Mogil, “Identification and Ranking of Genetic and Laboratory Environment Factors Influencing a Behavioral Trait, Thermal Nociception, via Computational Analysis of a Large Data Archive,” Neuroscience and Biobehavioral Reviews, 26 (2002): 907. 21. Colin Camerer, interview in Pasadena, March 12, 2004. QUETELET’S STATISTICS AND MAXWELL’S MOLECULES 1. Isaac Asimov, Foundation and Empire, Ballantine Books, New York, 1983 (1952), p. 1. 2. Ibid., p. 112. 3. Philip Ball, “The Physical Modelling of Society: A Historical Perspective,” Physica A, 314 (2002): 1. 4. Ibid., p. 7. 5. Pierre Simon Laplace, A Philosophical Essay on Probabilities, Dover, New York, 1996 (1814), p. 4. 6. Gauss was not, however, the first to devise the curve that bears his name. The French mathematician Abraham de Moivre (1667–1754) initially developed the idea in the 1730s. 7. See Frank H. Hankins, “Adolphe Quetelet as Statistician,” Studies in History, Economics and Public Law, 31 (1908): 33, 18. 8. Adolphe Quetelet, Sur L’Homme et le developpement de ses facultes, ou essai de physique sociale. The philosopher Auguste Comte also coined the term “social physics” about the same time, and had his own ideas about developing a science of society. See Roger Smith, The Norton History of the Human Sciences, Norton, New York, 1997, Chapter 12. 9. Adolphe Quetelet, Preface to Treatise on Man (1842 English edition), p. 7. Available online at http://www.maps.jcu.edu.au/course/hist/stats/quet/quetpref.htm. 10. Ibid., p. 9. 11. Ibid., p. 17. 12. Ibid., p. 14. 13. Ibid., p. 12. 14. Stephen G. Brush, “Introduction,” in Stephen G. Brush, ed., Kinetic Theory, Vol. I, The Nature of Gases and Heat, Pergamon Press, Oxford, 1965, p. 8. 15. Henry Thomas Buckle, History of Civilization in England, quoted in Ball, “Physical Modelling of Society,” p. 10. 16. Ibid., Chapter 3, “Method Employed by Metaphysicians,” pp. 119–120. Available online at http://www.perceptions.couk.com/buckle1.html. 17. Ibid., p. 120. Note also that he allowed “experiments so delicate as to isolate the phenomena,” but said that could never be done with a single mind,
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature which was always influenced by others, so that such isolation is really not possible. 18. Ibid., excerpt. Available online at http://www.d.umn.edu/~revans/PPHandouts/buckle.htm. 19. Ibid. 20. Quoted in P.M. Harman, The Natural Philosophy of James Clerk Maxwell, Cambridge University Press, Cambridge, 1998, p. 131. 21. James Clerk Maxwell, “Does the Progress of Physical Science Tend to Give Any Advantage to the Opinion of Necessity (or Determinism) over that of the Contingency of Events and the Freedom of the Will?” Reprinted in Lewis Campbell and William Garnett, The Life of James Clerk Maxwell, Macmillan and Co., London, 1882, p. 211. 22. Ignoring things like whether your opponent has a weak backhand. BACON’S LINKS 1. www.imdb.com. There are additional actors in the database who cannot be linked to Bacon because they appeared either alone or with no other actors who had appeared in any other movies including actors connected to the mainstream acting community. 2. Similar network math was developed by Anatol Rapoport, who is better known, of course, as a game theorist. 3. Duncan Watts and Steven Strogatz, “Collective Dynamics of ‘Small-World’ Networks,” Nature, 393 (June 4, 1998): 440–442. 4. Steven Strogatz, interview in Quincy, Mass., May 17, 2004. 5. These three examples were chosen because of the availability of full data on their connections; at that time, C. elegans was the only example of a nervecell network that had been completely mapped (with 302 nerve cells), the Internet Movie Data Base provided information for actor-movie links, and the power grid diagram was on public record. 6. Watts and Strogatz, “Collective Dynamics,” p. 441. 7. In fact, here’s a news bulletin: Oracle of Bacon hasn’t updated its list yet, but as of this writing its database shows that Hopper has now surpassed Rod Steiger as the most connected actor, with an average of 2.711 steps to get to another actor versus Steiger’s 2.712. Of course, these numbers continue to change as new movies are made. 8. Réka Albert and Albert-László Barabási, “Emergence of Scaling in Random Networks,” Science, 286 (15 October 1999): 509. 9. Jennifer Chayes, interview in Redmond, Wash., January 7, 2003.
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature 10. Thomas Pfeiffer and Stefan Schuster, “Game-Theoretical Approaches to Studying the Evolution of Biochemical Systems,” Trends in Biochemical Sciences, 30 (January 2005): 20. 11. Ibid., pp. 23–24. 12. Suppose the drivers of two cars are caught on opposite sides of a snow-drift, both wanting to get through to go home but neither wild about shoveling snow. The “cooperator” would get out of the car and shovel through the snowdrift, while the “defector” would stay warm inside the car. If both defect, no snow gets shoveled and neither gets to go home, so they both lose. If they both shovel, they get to go home with half the work required of one shoveling alone. But if one shovels the whole thing, the other gets to go home for free. Game theory math shows that each driver’s best move depends on the other’s: If the other guy defects, your best move is to cooperate; if the other guy cooperates, your best move is to defect. This game is mathematically the same as the hawk-dove game in evolutionary game theory. 13. F.C. Santos and J.M. Pacheco, “Scale-Free Networks Provide a Unifying Framework for the Emergence of Cooperation,” Physical Review Letters, 95 (August 26, 2005). A subsequent paper by Zhi-Xi Wu and colleagues at Lanzhou University in China questions whether it is the scale-free nature of the network that is really responsible for this difference, but that’s an issue for further network/game theory research. See Zhi-Xi Wu et al., “Does the Scale-Free Topology Favor the Emergence of Cooperation?” http://arxiv.org/abs/physics/0508220, Version 2, September 1, 2005. 14. Holger Ebel and Stefan Bornholdt, “Evolutionary Games and the Emergence of Complex Networks,” http://arxiv.org/abs/cond-mat/0211666, November 28, 2002. ASIMOV’S VISION 1. In Sylvia Nasar’s book A Beautiful Mind, she suggests that Asimov’s Foundation might have been inspired by the Rand Corporation, where Nash worked on game theory in the early 1950s. But Asimov’s novel Foundation, appearing in 1951, was in fact a compilation of short stories that had begun to appear before the Rand Corporation was created in 1948. The first Foundation story appeared in 1942. 2. Serge Galam, “Sociophysics: A Personal Testimony,” Physica A, 336 (2004): 50. 3. The term “econophysics” was coined by Boston University physicist H. Eugene Stanley. 4. Serge Galam, “Contrarian Deterministic Effect: The ‘Hung Elections Scenario,’” http://arxiv.org/abs/cond-mat/0307404, July 16, 2003.
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature 5. James Clerk Maxwell, “Illustrations of the Dynamical Theory of Gases,” Philosophical Magazine (1860). Reprinted in Brush, Kinetic Theory, p. 150. 6. An important point about statistical physics is that different microstates can correspond to indistinguishable macrostates. Various possible distributions of molecular speeds, for instance, can produce identical average velocities. That is one of the reasons why statistical mechanics is so successful. Even though it makes statistical predictions, in many cases the overwhelming number of possible microstates produce similar macrostates, so the prediction of that particular macrostate has a high probability of being accurate. 7. Katarzyna Sznajd-Weron, “Sznajd Model and Its Applications,” http:// arxiv.org/abs/physics/0503239, March 31, 2005. 8. Peter Dodds and Duncan Watts, “Unusual Behavior in a Generalized Model of Contagion,” Physical Review Letters, 92 (May 28, 2004). 9. Steven Strogatz, interview in Quincy, Mass., May 17, 2004. 10. Colin Camerer, Behavioral Game Theory, Princeton University Press, Princeton, N.J., 2003, p. 465. 11. Damien Challet and Yi-Cheng Zhang, “Emergence of Cooperation and Organization in an Evolutionary Game,” Physica A, 246 (1997): 407–428. 12. Jenna Bednar and Scott Page, “Can Game(s) Theory Explain Culture? The Emergence of Cultural Behavior Within Multiple Games,” Santa Fe Institute Working Paper 04-12-039, December 20, 2004, p. 2. 13. Ibid. 14. Ibid. 15. Ibid., pp. 2–3. 16. Doyne Farmer, Eric Smith, and Martin Shubik, “Is Economics the Next Physical Science?” Physics Today, 58 (September 2005): 37. MEYER’S PENNY 1. David Meyer, interview in La Jolla, Calif., August 6, 2003. 2. You can find more on this explanation for the quantum penny game in Chiu Fan Lee and Neil F. Johnson, “Let the Quantum Games Begin,” Physics World, October 2002. 3. David A. Meyer, “Quantum Strategies,” Physical Review Letters, 82 (February 1, 1999): 1052–1055. 4. David Meyer, interview in La Jolla, Calif., August 6, 2003. 5. Lan Zhou and Le-Man Kuang, “Proposal for Optically Realizing a Quantum Game,” Physics Letters A, 315 (2003): 426–430. 6. This is a key point. You cannot use entanglement to send faster-than-light messages, because you need some other channel of communication to learn the measurement of the other particle.
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature 7. Adrian P. Flitney and Derek Abbott, “Introduction to Quantum Game Theory,” http://arxiv.org/abs/quant-ph/0208069, Version 2, November 19, 2002, p. 2. 8. Kay-Yut Chen, Tad Hogg, and Raymond Beausoleil, “A Practical Quantum Mechanism for the Public Goods Game,” http://arXiv.org/abs/quant-ph/0301013, January 6, 2003, p. 1. 9. Azhar Iqbal, “Impact of Entanglement on the Game-Theoretical Concept of Evolutionary Stability,” http://arXiv.org/abs/quant-ph/0508152, August 21, 2005. PASCAL’S WAGER 1. E.T. Bell, Men of Mathematics, Simon & Schuster, New York, 1937, p. 73. 2. Blaise Pascal, Pensées. Trans. W.F. Trotter, Section III. Available online at http://textfiles.com/etext/NONFICTION/pascal-pensees-569.txt. 3. Laplace, a later pioneer of probability theory, did not find Pascal’s argument very convincing. Mathematically it reduces to the suggestion that faith in a God that exists promises an infinite number of happy lives. However small the probability that God exists, multiplying it by infinity gives an infinite answer. Laplace asserts that the promise of infinite happiness is an exaggeration, literally “beyond all limits.” “This exaggeration itself enfeebles the probability of their testimony to the point of rendering it infinitely small or zero,” Laplace comments. Working out the math, he finds that multiplying the infinite happiness by the infinitely small probability cancels out the infinite happiness, “which destroys the argument of Pascal.” See Laplace, Philosophical Essay, pp. 121–122. 4. David H. Wolpert, “Information Theory—The Bridge Connecting Bounded Rational Game Theory and Statistical Physics,” http://arxiv.org/abs/cond-mat/0402508, February 19, 2004. 5. David Wolpert, interview in Quincy, Mass., May 18, 2004. 6. Wolpert, “Information Theory,” pp. 1, 2. 7. E.T. Jaynes, “Information Theory and Statistical Mechanics,” Physical Review, 106 (May 15, 1957): 620–630. 8. If you’re flipping a coin, of course, the two possibilities (heads and tails) would appear to be equally probable (although you might want to examine the coin to make sure it wasn’t weighted in some odd way). In that case, the equal probability assumption seems warranted. But it’s not so obviously a good assumption in other cases. I remember a situation many years ago when a media furor was created over a shadow on Mars that looked a little bit like a face. Some scientists actually managed to get a paper published contending that the shadow’s features were not random but actually appeared to have been constructed to look like a face! I argued against putting the picture on the front
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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature page of the paper, insisting that the probability of its being a real face (or a representation of a face) was minuscule. But a deputy managing editor replied that either it was or it wasn’t, so the odds were 50-50! I hoped he was kidding, but decided it would be wiser not to ask. 9. Jaynes, “Information Theory,” pp. 620, 621. 10. This is, of course, the basis for teachers grading on a “curve,” the bell-shaped curve or Gaussian distribution that represents equal probability of all the microstates. 11. David Wolpert, interview at NASA Ames Research Center, July 18, 2005. 12. Ibid. 13. Another decision-theory system was worked out by the statistician Abraham Wald, but the story of the similarities and differences between Wald’s and Savage’s approaches goes far beyond the scope of this discussion. If you’re interested, you might want to consult a paper exploring some of these issues: Nicola Giocoli, “Savage vs. Wald: Was Bayesian Decision Theory the Only Available Alternative for Postwar Economics?” Available online at http://www.unipa.it/aispe/papers/Giocoli.doc. 14. David Wolpert, interview at NASA Ames Research Center, July 18, 2005. 15. Strictly speaking, it’s not the temperature of an individual, it’s the temperature that the external scientist assigns to the individual, Wolpert points out. Just like in statistical physics, the temperature is a measure of what the external scientists infer concerning the molecules in a room (since a single molecule doesn’t have any particular temperature—temperature is a property of the distribution of velocities of a set of molecules). 16. Wolpert, interview at NASA Ames, July 18, 2005. EPILOGUE 1. Orson Scott Card, Ender’s Game, TOR, New York, 1991, p. 125. Thanks to my niece Marguerite Shaffer for calling Ender’s Game to my attention. 2. Ibid. p. 238. 3. Joshua Greene, interview in San Francisco, Calif., April 17, 2004.
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