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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