More a series of comments than a review...
Just going through the review posted by Fitzsimmons. Thank you so much for taking the time!
1. The longer quote of Huygens makes the point clearer, though I'm not sure it's necessary to include it in the manuscript. Imagine a lottery with two tickets. One wins $10,000 the other $20,000. The only price for buying both tickets that is reasonable both for the seller and the buyer is $30,000 because it's literally swapping $30,000 for $10,000+$20,000=$30,000 (disregarding possibilities like fraud or the seller requiring $25,000 for a life-saving operation etc).
The no-arbitrage argument is often powerful but less so when it's not possible to access all possible payouts. Whether it's a good idea to pay $15,000 for one ticket in the lottery (without being offered the other ticket too), or even $14,000 or $16,000 depends on the real-world effects of such a risk as they unfold over time for the buyer.
2. One can phrase this argument in many different ways, I'd imagine. The point made here is that E[f(x)]=f[E(x)] in practice will imply linearity of f.
3. Whether one calls Bernoulli's work an error or not is a matter of what is to be communicated. It may not be an error, in the sense that he may have wanted to put forward a theory that's very different from what we today call "expected utility theory." If one wants to communicate that Bernoulli 1738 is not what we consider EUT, it's often useful simply to call it an error. Laplace's correction of Bernoulli, or modification, suggests that he also felt it was an error and Bernoulli had meant something else [namely EUT].
4. Misunderstanding: I have no reason to believe that Sommer and Menger mis-translated Bernoulli. But Bernoulli, of course, didn't strictly write the English text.
5. Many thanks! Yes, the figure does seem to have been included in the original publication. That original figure displays the same problem: it depicts a lottery where the worst-case outcome is a net monetary gain. The figure says "I toss a coin, and for heads I give you $10, for tails $20." The text says that such a gamble should not be accepted. Bernoulli simply didn't think this figure through -- why would anyone turn down a guaranteed monetary gain?
I'm sure Bernoulli didn't mean to say that people would behave this way, but his figure illustrates that he didn't think about this carefully, and indeed it reflects what I call an "error," and what I suspect Laplace would have called an error.
6. The quote is taken out of context. My statement here is "It is possible that Bernoulli developed a different decision theory on purpose." From the context it's clear that this refers to a theory that's different from expected-utility theory.
Bernoulli's title refers to a "new theory" whereby he means a theory that's different from expected-value maximization. He couldn't have meant "new as in different from expected-utility theory" because he introducing expected utility theory in his paper (except that he makes what one may or may not call an error, in the sense that the theory he proposes is not what Laplace or von Neumann and Morgenstern call expected-utility theory).
7. Menger's error is similar to Bernoulli's. He fails to take into account that a fee has to be paid to enter the lottery. He computes the expected payout and concludes that this expectation value can "be" infinite (i.e. non-existent) even if it's measured in a different currency, like utility, if he's allowed to make possible monetary payouts infinite and utility is not bounded from above. Of course that's true.
But what's not true is that this renders the lottery St-Petersburg-like when using expected utility theory. Menger's problem only appears in Bernoulli's faulty EUT, not in EUT proper. In EUT proper, with a logarithmic utility function, the moment bankruptcy (post-lottery wealth zero) is possible, the theory says that the gamble will be turned down, even if it has an infinite expected utility payout.
The point of my comment was that for a general (concave increasing) utility function, the criteria of EUT is close to that of D. Bernoulli, and coincident after a correction factor (the probability of losing) is inserted. My last sentence, in which I remark on the (trivial) special case of linear utitlity, was beside the point.
The comment by Fitzsimmons, if I understand correctly, amounts to pointing out that Bernoulli's decision theory (BDT) is equivalent to expected utility theory (EUT) for linear utility.
This is correct and discussed in the manuscript, e.g. on p.7:
"Only a linear utility function guarantees the equality fmB = fmU. In other words, Bernoulli’s decision theory, which is often wrongly presented as equivalent to EUT, is really only equivalent to EUT under the assumption of linear utility. But that is equivalent to Huygens’s decision theory that we encountered in Sec. 2, and that was deemed an unrealistic model of human behavior."
Linear utility is equivalent to what I call Huygens's decision theory (HDT) here, namely to the maximization of expected dollar wealth (without a utility function). This leads to the following situation: Bernoulli's decision theory is only equivalent to expect utility theory in the special case where neither Bernoulli's decision theory, nor expected utility theory has any effect, namely where both are equivalent to Huygens's decision theory, which had been found to be inadequate, and whose inadequacy had motivated the development of BDT and EUT.
Corrected a typo in Eq.15 and Table 2, spotted by a twitter user
Square bracket before p_i should be after p_i or before the sum, or omitted.
A remark on the possibility of reconciling BDT and EUT.
Daniel Bernoulli’s study of 1738  is considered the beginning of expected utility theory. Here I point out that in spite of this, it is formally inconsistent with today’s standard form of expected utility theory. Bernoulli’s criterion for participating in a lottery, as written in , is not the expected change in utility.
➤ Version 1 (2019-05-25)
Ole Peters (2019). Comment on D. Bernoulli (1738). Researchers.One, https://researchers.one/articles/comment-on-d-bernoulli-1738/5f52699c36a3e45f17ae7dd2/v1.