Comment on D. Bernoulli (1738)

Probability theory emerged in the second half of the 17th century as a way to think about monetary gambles, and how to choose wisely when facing uncertainty. Early thinking suggested that people would act so as to maximise the expected change in wealth – an average over all possible outcomes. But in the early 18th century, Nicholas Bernoulli demonstrated that people don’t act this way with his well-known St. Petersburg gamble, its terms defined so that the expected wealth change is actually infinite. Wisely, supposedly, people should be willing to pay any price to take on the gamble, but real people were not. A new theory – now known as expected utility theory – emerged from Bernoulli and others who suggested that people actually maximise the expected change of a different quantity, the so-called utility of wealth. Expected utility theory introduced the plausible notion that our willingness to risk losing resources depends on the resources we already have. A gain or less of $1,000 matters much less to a billionaire than to someone struggling to put food on the table.
As LML Fellow Ole Peters explores in a new paper, Bernoulli’s ideas are often still considered as the origin of today’s expected utility theory, widely used in economics and elsewhere. In fact, however, Bernoulli’s formulation is formally inconsistent with today’s standard form of that theory, although many authors, starting with Laplace in 1814, have overlooked this difference. Peters suggests that the most likely explanation for the difference between Bernoulli’s ideas and modern expected utility theory was Bernoulli’s confusion over the early mathematical concept of the expectation value. Bernoulli tried to compute this, but instead computed something else. Of course, probability theory was in its extremely early days at the time of Bernoulli’s writing.
Unfortunately, Peters suggests, that inconsistency between Bernoulli’s decision theory and modern expected utility theory has caused immense confusion. It lies at the root, for example, of economist Karl Menger’s famous claim that proper utility functions must be bounded, a claim which cannot be founded on modern expected utility theory. Similarly, some critiques of expected utility theory – so-called “non-expected utility theories” – seem to address shortcomings that actually only exist in Bernoulli’s early work, not in Laplace’s form of the theory, which was famously axiomatized in 1944 by von Neumann and Morgenstern. Strangely, Bernoulli’s long-standing error still persists in the literature without a retraction of, or comment on, the original paper. A clear understanding of Bernoulli’s work leads to a new interpretation of expected utility theory in terms of ergodicity transformations which sheds new light on long-standing problems in economic theory, including the St. Petersburg paradox, the cooperation conundrum and the equity premium puzzle. This new interpretation is the basis of LML’s economics project.
The paper is available and currently under open review at

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