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## A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature (2006) Joseph Henry Press (JHP)

### Citation Manager

Siegfried, Tom. "11 Pascal’s Wager--Games, probability, information, and ignorance." A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature. Washington, DC: The National Academies Press, 2006.

 Page 203

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A Beautiful Math: John Nash, Game Theory, and the Modern Quest for a Code of Nature

Before Maxwell, probability theory in science was mostly limited to quantifying things like measurement errors. Laplace and others showed the way to estimate how far off your measurement was likely to be from the true value for a particular degree of confidence. Laplace himself applied this approach to measuring the mass of Saturn. He concluded that there was only one chance in 11,000 that the true mass of Saturn would deviate from the then-current measurement by more than 1 percent. (As it turned out, today’s best measurement indeed differs from the one in Laplace’s day by only 0.6 percent.) Probability theory has developed into an amazingly precise way of making such estimates.

But what does probability itself really mean? If you ask people who ought to know, you’ll get different answers. The “objective” school of thought insists that the probability of an event is a property of the event. You observe in what fraction of all cases that event happens and thereby measure its objective probability. The subjective view, on the other hand, argues that probability is a belief about how likely something is to happen. Measuring how often something happens gives you a frequency, not a probability, the subjectivists maintain.

There is no point here delving into the debates about the relative merits of these two views. Dozens of books have been devoted to that controversy, which is largely irrelevant to game theory. The fact is that the prevailing view today, among physicists at least, is that the subjectivist approach contains elements that are essential for a sound assessment of scientific data.

Subjective statistics often goes under the label of Bayesian, after Thomas Bayes, the English clergyman who discussed an approach of that nature in a paper published in 1763 (two years after his death). Today a formula known as Bayes’ theorem is at the heart of practicing the subjective statistics approach (although that precise theorem was actually worked out by Laplace). In any case, the Bayesian viewpoint today comes in a variety of flavors, and there is much disagreement about how it should be interpreted and applied (perhaps because it is, after all, subjective).

From a practical point of view, though, the math of objective

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 Front Matter (R1-R8) Introduction (1-10) 1 Smith’s Hand--Searching for the Code of Nature (11-26) 2 Von Neumann’s Games--Game theory’s origins (27-50) 3 Nash’s Equilibrium--Game theory’s foundation (51-72) 4 Smith’s Strategies--Evolution, altruism, and cooperation (73-92) 5 Freud’s Dream--Games and the brain (93-109) 6 Seldon’s Solution--Game theory, culture, and human nature (110-125) 7 Quetelet’s Statistics and Maxwell’s Molecules--Statistics and society, statistics and physics (126-143) 8 Bacon’s Links--Networks, society, and games (144-163) 9 Asimov’s Vision--Psychohistory, or sociophysics? (164-181) 10 Meyer’s Penny--Quantum fun and games (182-196) 11 Pascal’s Wager--Games, probability, information, and ignorance (197-216) Epilogue (217-224) Appendix--Calculating a Nash Equilibrium (225-229) Further Reading (230-232) Notes (233-248) Index (249-264)