This is a bit of LML jargon that we felt is worth promoting, even though it’s terribly unfair to a great mathematician. So please, you admirers of Laplace, don’t take offense. What’s the story?

In 1738 Daniel Bernoulli wrote his famous paper that introduces expected utility theory and thereby defines the basis of neoclassical economics — macro and micro. Since you ask: this paper is famous for its treatment of the St. Petersburg paradox. The “paradox” goes like this:

- assume it is rational to evaluate gambles based on the expectation value of net cash gain.
- construct a specific example of such a gamble (with a large expected net gain).
- observe that no sensible person wants to take that gamble.
- conclude that this is paradoxical.

## Paradox — something unexpected

“Paradox” is a strong word here, unless it’s taken very literally — from + , meaning something that goes against prevailing belief (or dare I say against expectation). It’s just an inconsistency: the theory says one thing, real people do something else. Popper would have simply called it a falsification of the theory. In any case, it forces us to stop and think. Daniel Bernoulli stopped, thought, and decided he didn’t quite know why this was happening, but he could set up the mathematics so that it would resemble what people do. That bit of mathematics became known as expected utility theory, and it goes like this: apparently, people don’t evaluate gambles based on the expectation value of net cash gain (1. above is wrong). But we could replace cash with a non-linear function of cash. This is called the “utility” function because it looks a bit like it specifies how useful people seem to find cash.

Then we compute the expected net gain of utility, and we choose our utility function so that its expectation value decreases in the proposed gamble. That means people behave as if they’re evaluating gambles based on the expectation value of net utility change.

You can shoot holes into the philosophical basis of this treatment, and indeed I encourage anyone to do this. Why can we just introduce some non-linear function? What does that function mean? Is this framework purely descriptive or does it have any predictive power? What’s the purpose of this mathematical model? Are we just setting up an equation to reflect mathematically what we’ve observed? Isn’t that just like expressing our observation in French or Italian, i.e. in a different language but without adding any insight?

## No expectations, no paradox

Ergodicity economics treats the problem differently and replaces the expectation value of net cash gain with the time-average growth rate of cash, and people are found to behave as this treatment suggests. So from our perspective it seems that Bernoulli’s 1738 solution is deficient because when he wrote it down the concept of time averages hadn’t been invented yet, and people hadn’t realized that expectation values don’t generally reflect what happens over time, and that an individual has no reason to maximize expected net cash gains because those are averages over many identically prepared systems, whereas an individual is only one system.

Ok — phew — had to get that off my chest. So what about doing a Laplace?

## Bernoulli’s mathematical quirk

Daniel Bernoulli did not actually compute the expected net change in utility. Wow. I’ll say that again: in the seminal 1738 paper that defines neoclassical economics there’s a mistake. Not just the conceptual mistake I just mentioned (that expected cash changes are irrelevant to the decision maker), no: also in the mathematics.

Daniel Bernoulli did not actually compute the expected net change in utility.

If you’re interested in the details, we’ve written about it (Section IV B, p.6).

## Laplace’s well-meant re-telling of Bernoulli

Now what does Laplace have to do with this? The answer is: he wrote a textbook, the second edition is from 1814, and it became a classic. In that classic, on p.440, he re-tells Bernoulli’s treatment of the St. Petersburg paradox. But Laplace was too polite to mention that Bernoulli didn’t compute the expected net utility change. No one knows why Bernoulli didn’t do it. Maybe he made a mistake, maybe he found his idea so far-fetched that he didn’t think it mattered much and that no one would really pay attention. In any case, Laplace created the myth (apparently harmless at the time) that Bernoulli had computed the expected net change in utility. Others copied the story from Laplace — for example, Todhunter wrote another textbook in 1865, and on p.220 he tells precisely the same story, using even the same notation as Laplace. This is all understandable. Bernoulli wrote in Latin, Laplace in French (much more accessible). Incidentally, Todhunter’s book was another classic — Ken Arrow told me he had read it as a young student.

This was all well and good until Daniel Bernoulli’s paper fell into the hands of Karl Menger (not Carl, but his son — Karl [Carl had taught economics to the Austrian Crown Prince, Archduke Rudolf von Habsburg, who later committed suicide]). Karl Menger read Bernoulli very carefully, in the original, and because of the inconsistency with Laplace he got terribly confused and in his confusion concluded that only bounded utility functions are permissible. By now the story was so convoluted that no one could quite figure out what was what. Karl Menger’s study was published in 1934 in the Journal of Economics, and to this day it has not been possible to correct it (I submitted a correction to the journal after untangling all of this, but it was rejected on the grounds that, while correct, it was not considered relevant to economics).

For German-speakers: when we were working on “Evaluating gambles…” Murray Gell-Mann, asked me if a Menger is someone who sells sets.

## Laplacing it

So when someone at LML says “careful not to Laplace this one” we mean to create a terrible mess just because you’re too polite to say that something isn’t quite right. Maybe we should call it “gentlemanning it” — poor Laplace, he really doesn’t deserve this. We’ve all done it. We avoid conflict but leave the source of the conflict intact, and it will just sit there and fester and eventually break out and wreak havoc.

In Laplace’s case, more than a century after the offending politeness, several economics Nobel Laureates quoted and endorsed Menger. The most impressive quote is from Paul Samuelson [p.32]: “*Menger 1934 is a modern classic that stands above all criticism.” *With that, any hope of scientific critique was extinguished, and the modern classic remains part of the modern canon.

Unfortunately, this meant a lot of hard work was misdirected, and a lot of good work was dismissed (Kelly’s 1956 paper was dismissed on these grounds).

How do we avoid doing a Laplace? What’s the opposite? The opposite of Laplacing an issue is to address it head-on. Often that feels uncomfortable, it means nailing our colors to the mast and articulating a position. It means writing clearly what we actually believe, even if some people may not like it. It’s uncomfortable for social reasons — we don’t want to offend. It’s also uncomfortable because once we’ve said something clearly, we can be found to be wrong. (The important anecdote is that of Wolfgang Pauli walking out of someone’s seminar, shaking his head, and audibly muttering “it isn’t even wrong.” It’s not good to be wrong, but it’s even worse to be unclear — for Laplacian and many other reasons.)

Here’s a situation that’s surprisingly common, and it can indicate a danger of doing a Laplace. Say I’m discussing a draft of some paper with a colleague. The colleague asks: what does this paragraph mean, and I say “oh, yes, what I wanted to say there was X.” I try to catch myself when I do this, and then re-write things explicitly. Insert whatever “X” was, namely the thing I wanted to say. Chances are I didn’t say it clearly because I felt the discomfort of taking a clear position.

How about Keynes’ thought on probability theory?

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Good advice! I try to keep in mind the direction from E.B. White in “The Elements of Style”: “When you say something, make sure you have said it. The chances of your having said it are only fair.”

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