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.



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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