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.

Hendriks, de Hoop, and de Swart in Journal of Logic, Language, and Information (2012)

In their brief introduction to a special issue on game theory and bidirectional game theory, Petra Hendriks, Helen de Hoop, and Henriëtte de Swart discuss the parallels between the two frameworks and their limits.

Handy References

The text contains the following references on bidirectional optimality theory:

• Reinhard Blutner: "Some Aspects of Optimality in Natural Language Interpretation" (1999)
• Reinhard Blutner, Helen de Hoop, Petra Hendriks: Optimal Communication (2006)
• Petra Hendriks, Helen de Hoop, Irene Krämer, Henriëtte de Swart, and Joost Zwarts: Conflicts in Interpretation (2010)

In addition, they cite the following paper as pointing out "the connection between bidirectional Optimality Theory and Game Theory" (p. 2):

• Anton Benz and Jason Mattausch: "Bidirectional Optimality Theory: An introduction" (2011)

A Theoretical Point

Besides the general introduction of the field and the players, the three authors point to a possible theoretical shortcoming of both bidirectional optimality theory and game theory.

The problem they point out is that "these frameworks generally predict a one-to-one pairing of forms and meanings," which is not empirically true (p. 2). They illustrate this with the Dutch question Wie heeft Frank vermoord?, which is ambiguous between Who did Frank kill? and Who killed Frank?

More specifically, they note that while it is true that "marked forms go with marked meanings," the reverse is not: "unmarked forms can often be used to express unmarked as well as marked meanings" (p. 3). This claim is supported by a reference to a paper in Lingua, but I don't know exactly what the example they have in mind is.

A Note On That Point

Just on the face of it, there doesn't seem to be anything wrong, from the standpoint of microeconomics, with the many-to-many relation, but it does require a slightly more sophisticated model of the "cost" of an utterance. Consider for instance:

• He is dead (+m) = "He is dead" (+m)
• He is gone (–m) = "He is dead" (+m)
• He is dead (+m) = *"He is gone" (–m)
• He is gone (–m) = "He is gone" (–m)

I haven't done the math here, but this might be explained by pragmatic effects under the right assumptions: If all messages are equally cheap, then we should indeed expect a one-to-one correspondence to emerge; however, since one of the meanings is taboo, it will tend to increase the cost of whatever message gravitates towards it. This might sustain the incentive to use analogical references instead of direct (unambiguous) references.