Thursday, June 20, 2013

Daston: Classical Probability in the Englightenment (1988)

I've been wanting to read Lorraine Daston's book on the early history of probability for about eight years now, but there was always something else that seemed more important. Recently, however, I finally got around to checking it out of the library along with Abraham De Moivre's The Doctrine of Chances (1718).

There are two things that I have found interesting about the narrative of the book so far: First, Daston's claim that, despite appearances, gambling in fact was not the real engine behind the emergence of probability theory; and second, her discussion of the controversy surrounding the proper definition of expectation.

Gambling or Law?

With respect to the claim about gambling, she cites contract law (and to some extent, criminal law) as a plausible interpretation of what the people of the enlightenment were really talking about when they talked about "gambling." Specifically, she cites the problem of determining a fair price for an aleatory contract (such as an insurance policy) as an important new question that gave meaning to the young subject.

As the name suggests, contracts of this kind were from the onset conceptualized as a kind of gamble. All the dice-throwing in the classical text about probability may thus have been a kind of crypto-legal pedagogy (just like a math problem about dividing a cake might really aim at teaching you something about accounting).

Reasonable Expectation

With respect to the second point, she discusses at length Daniel Bernoulli's solution to the St. Petersburg problem: Assume that the utility of money decreases linearly as a function of your capital.

This leads to a logarithmic utility function and thus aligns Bernoulli's solution with Kelly gambling. It corresponds to the intuition that bankruptcy is qualitatively different from other levels of bankroll, and that one should not adopt a gambling strategy that assigns a positive (or even high) probability to going bankrupt.

A similar and much simpler example of the same ambiguity comes up in the comparison of extreme gains that have very low probability relative to modarately large gains that have substantial probability. Gambling a lot of money on lotteries of the first kind tends to lead to bankruptcy with quite high probability.

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