How little we know

In a recent issue of Nature there is a discussion of the history of utility theory:

Three centuries ago, in September 1713, the Swiss mathematician Nikolaus Bernoulli wrote a letter to a fellow mathematician in France, the nobleman Pierre Rémond de Montmort. In it, Bernoulli described an innocent-sounding puzzle about a lottery…The result is surprising. Each product — 1 × ½, 2 × ¼, 4 × ⅛, and so on — is a half. Because the series never ends, given that there is a real, if minute, chance of a very long run of tails before the first head is thrown, infinitely many halves must be summed. Shockingly, the expected win amounts to infinity…

In May 1728, writing from London, the 23-year-old mathematician Gabriel Cramer from Geneva weighed in. “Mathematicians value money in proportion to its quantity, and men of good sense in proportion to the usage that they may make of it.” This was a far-ranging insight. Adding a ducat to a millionaire’s account will not make him happier, Cramer reasoned. The usefulness of an extra coin is never zero, but simply less than that of the previous one — as wealth increases, so does utility, but at a decreasing rate. Assuming that utility increases with the square root of wealth, Cramer recalculated the expected win to be a little over 2.9 ducats.

Daniel encapsulated the probability scenario in a plot of utility versus monetary value, now known as a ‘utility function’ (see ‘Risky business’)… The curve’s diminishing gradient implies that it is always worth paying a premium to avoid a risk. The consequences of this simple graph are enormous. Risk aversion, as expressed in the concave shape of the utility function, tells us that people prefer to receive a smaller but certain amount of money, rather than facing a risky prospect.

It was a bit shocking to me how advanced these concepts were for the year 1700 – and how we haven’t come very far from those insights from the mathematicians of the 18th century. It’s a testament to the lack of experiment in economics that it took until the 1900s for Allais (and his “paradox”) and Kahneman and Tversky‘s theories to come about.

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