A Beautiful Math

by Tom Siegfried

  Game theory, on the other hand, offers a possible rapprochement between the advocates of genetic power and the defenders of human freedom. Game theory pursues a different sort of path toward the Code of Nature. It acknowledges the power of evolution—in fact, it helps to explain evolution's ability to generate life's complexities. But game theory also explains why the belief that human nature is rooted in biology, while trivially true, is far from the whole story. Game theory poses no universal genecontrolled determinant of human social behavior, but rather requires, as Nash's math showed, a mixed strategy. It demands that people make choices from multiple possible behaviors.

  Game theory's potential scientific power is so great, I think, because it is so intellectually commodious—not narrow and confining, but capable of accommodating many seeming contradictions. That's why it can offer an explanatory structure for all the diversity in the world—a mélange of individual behaviors and personalities, the wide assortment of human cultures, the neverending list of biological species. Game theory encompasses the coexistence of selfishness and sympathy, competition and cooperation, war and peace. Game theory explains the interplay of genes and environment, heredity and culture. Game theory connects simplicity to complexity by reconciling the tension between evolutionary change and stability. Game theory ties the choices of individual people to the collective social behavior of the human race. Game theory bridges the sciences of mind and mindless matter.

  Game theory is about putting it all together. It offers a mathematical recipe for making sense of what seems to be a hopelessly messy world, providing a tangible sign that the Code of Nature is not a meaningless or impossible goal for scientists to pursue. And regardless what anyone thinks about the prospects for ultimate success, scientists are certainly pursuing that goal.

  "We want to understand human nature," says Joshua Greene, a neuroscientist and philosopher at Princeton. "That, I think, is a goal in and of itself."3

  Success may still be a long way off. But somewhere in the vision of Asimov's psychohistory lies an undoubtable truth—that all the world's multiple networks, personal and social, interact in multiple ways to generate a single future. From people to cities, corporations to governments, all of the elements of society must ultimately mesh. What appears to be the madness of crowds must have a method, and game theory's successes suggest that it's a method that science can discover.

  "The idea is really to have, in the end, a seamless understanding of the universe, from the most basic physical elements, the chemistry, the biochemistry, the neurobiology, to individual human behavior, to macroeconomic behavior—the whole gamut seamlessly integrated," says Greene. "Not in my lifetime, though."

  * * *

  Appendix

  Calculating a Nash Equilibrium

  Consider the simple game discussed in Chapter 2, where Alice and Bob compete to see how much of a debt to Alice that Bob will have to pay back. This is a zero-sum game; Alice wins exactly what Bob loses, and vice versa. The payoffs in the game matrix are the amounts Bob pays to Alice, so Bob's "payoff" in each case is the negative value of the number indicated.

  To calculate the Nash equilibrium, you must find the mixed strategies for each player that yield the best expected payoff when the other player is also choosing the best possible mixed strategy. In this example, Alice chooses Bus with probability p, and Walk with probability 1 – p (since the probabilities must add up to 1). Bob chooses Bus with probability q and Walk with probability 1 – q.

  Alice can calculate her "expected payoff" for choosing Bus or Walk as follows. Her expected payoff from Bus will be the sum of:

  Her payoff from Bus when Bob plays Bus, multiplied by the probability that Bob will play Bus, or 3 times q

  plus

  Her payoff from Bus when Bob plays Walk times the probability that Bob plays Walk, or 6 times (1 – q)

  Her expected payoff from Walk is the sum of:

  Her payoff from Walk when Bob plays Bus times the probability that Bob plays Bus, or 5 times q

  plus

  Her payoff from Walk when Bob plays Walk times the probability that Bob plays Walk, or 4 times (1 – q)

  Summarizing,

  Alice's expected payoff for Bus = 3q + 6(1 – q)

  Alice's expected payoff for Walk = 5q +4(1 – q)

  Applying similar reasoning to calculating Bob's expected payoffs yields:

  Bob expected payoff for Bus = –3p + –5(1 – p)

  Bob expected payoff for Walk = –6p + –4(1 – p)

  Now, Alice's total expected payoff for the game will be her probability of choosing Bus times her Bus expected payoff, plus her probability of choosing Walk times her Walk expected payoff. Similarly for Bob. To achieve a Nash equilibrium, their probabilities for the two choices must be such that neither would gain any advantage by changing those probabilities. In other words, the expected payoff for each choice (Bus or Walk) must be equal. (If the expected payoff was greater for one than the other, then it would be better to play that choice more often, that is, increasing the probability of playing it.)

  For Bob, his strategy should not change if

  Applying some elementary algebra skills, that equation can be recast as:

  Which, solving for p, shows that Alice's optimal probability for playing Bus is

  So Alice should choose Bus one time out of 4, and Walk 3 times out of 4.

  Now, Alice will not want to change strategies when

  Which, solving for q, gives Bob's optimal probability for choosing Bus:

  So Bob should choose Bus half the time and Walk half the time.

  Now let's say Alice and Bob decide to play the hawk-dove game, in which the payoff structure is a little more complicated because what one player wins does not necessarily equal what the other player loses. In this game matrix, the first number in the box gives Alice's payoff; the second number gives Bob's payoff.

  Alice plays hawk with probability p and dove with probability 1 – p; Bob plays hawk with probability q and dove with probability 1 – q. Alice's expected payoff from playing hawk is –2q + 2(1 – q). Her expected payoff from dove is 0q + 1(1 – q). Bob's expected payoff from hawk is –2p + 2(1 – p); his expected payoff from dove is 0p + 1(1 – p).

  Bob will not want to change strategies when

  So p, Alice's probability of playing hawk, is 1/3.

  Alice will not want to change strategies if

  So q, Bob's probability of playing hawk, is also 1/3. Consequently the Nash equilibrium in this payoff structure is to play hawk one-third of the time and dove two-thirds of the time.

  * * *

  Further Reading

  There are dozens and dozens of books on game theory, of which a handful stand out as indispensable to grasping the theory's essential features. Those that I found most useful and illuminating:

  Camerer, Colin. Behavioral Game Theory. Princeton, N.J.: Princeton University Press, 2003.

  Gintis, Herbert. Game Theory Evolving. Princeton, N.J.: Princeton University Press, 2000.

  Kuhn, Harold W. and Sylvia Nasar, eds. The Essential John Nash. Princeton, N.J.: Princeton University Press, 2002.

  Luce, R. Duncan and Howard Raiffa. Games and Decisions. New York: John Wiley & Sons, 1957.

  Williams, J.D. The Compleat Strategyst: Being a Primer on the Theory of Games of Strategy. New York: McGraw-Hill, 1954.

  Von Neumann, John and Oskar Morgenstern. Theory of Games and Economic Behavior. Sixtieth-anniversary Edition. Princeton, N.J.: Princeton University Press, 2004.

  Two other readable books were very helpful:

  Davis, Morton D. Game Theory: A Nontechnical Introduction. Mineola, NY: Dover, 1997 (1983).

  Poundstone, William. Prisoner's Dilemma. New York: Anchor Books, 1992.

  For the rich and complex historical context of the social sciences into which game theory fits, an excellent guide is:

  Smith, Roger. The Norton History of the Human Sciences. New York: W.W. Norton, 1997.

  And for a compre
hensive account of attempts to apply physics to the social sciences:

  Ball, Philip. Critical Mass: How One Thing Leads to Another. New York: Farrar, Straus and Giroux, 2004.

  A few additional books and articles of relevance are listed here; many others addressing specific points are mentioned in the notes.

  Books

  Harman, P.M. The Natural Philosophy of James Clerk Maxwell. Cambridge: Cambridge University Press, 1998.

  Henrich, Joseph, et al., eds. Foundations of Human Sociality: Economic Experiments and Ethnographic Evidence from Fifteen Small-Scale Societies. New York: Oxford University Press, 2004.

  Macrae, Norman. John von Neumann. New York: Pantheon Books, 1991.

  Nasar, Sylvia. A Beautiful Mind. New York: Simon & Schuster, 1998.

  Watts, Duncan J. Six Degrees. New York: W.W. Norton, 2003.

  Articles

  Ashraf, Nava, Colin F. Camerer, and George Loewenstein. "Adam Smith, Behavioral Economist." Journal of Economic Perspectives, 19 (Summer 2005): 131–145.

  Ball, Philip. "The Physical Modelling of Society: A Historical Perspective." Physica A, 314 (2002): 1–14.

  Holt, Charles and Alvin Roth. "The Nash Equilibrium: A Perspective." Proceedings of the National Academy of Sciences USA, 101 (March 23, 2004): 3999–4002.

  Morgenstern, Oskar. "Game Theory." Dictionary of the History of Ideas. Available online at http://etext.virginia.edu/DicHist/dict.html.

  Myerson, Roger. "Nash Equilibrium and the History of Economic Theory." 1999. Available online at http://home.uchicago.edu/~rmyerson/research/jelnash.pdf.

  After the manuscript for this book was completed, a new review article appeared exploring the game theory-statistical mechanics relationship in depth:

  Szabó, György and Gábor Fáth. "Evolutionary Games on Graphs," http://arxiv.org/abs/cond-mat/0607344, July 13, 2006.

  * * *

  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.

  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 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.