penstroma
For more recent (and much more technically difficult) treatment, you can take a look at Garleanu and Pederson (2018) ("Efficiently Inefficient Markets for Asset and Asset Management").
Below are some of my thoughts after a glance through the paper:
1. The critique of the article by Doctor et al. (linked from Ben Golub's Twitter) summarizes my thoughts succinctly. I am left wondering: if everyone used "ergodicity" as the basis of their decisions, doesn't this imply that, given some scenario with risk, everyone would make the same decision? But this is certainly not true. The author mentions that different people might care about "additive growth" vs. "geometric growth", but this is equivalent to using different functional forms of utility (e.g. CARA vs CRRA). Perhaps the author is saying that his ergodicity-derived decisions are "optimal"? But in what sense? It seems to me that his maximization of the growth rate is equivalent to using log utility. This result is well-known (see the Kelly criterion).
2. Ergodicity is covered in first-year graduate econometrics (at least where I was taught). See Hayashi Chapter 2. Perhaps not everyone reads it this way, but the tone of the article seems to suggest that this topic is completely foreign to economics.
3. There is a rich economics literature in decision theory. Savage's work on subjective expected utility shows that under a certain set of axioms of rationality, the decisions that a person makes can be completely captured by a subjective probability distribution and a personal utility function. The sentence "expected utility theory implicitly assumes that individuals can interact with copies of themselves, effectively in parallel universes" is completely misguided.
4. There are a number of well-known documented violations of expected utility (e.g. the Allais paradox). These represent challenges to the axioms of rationality. Nevertheless, the point of using expected utilities is to model human behavior, even if it is only approximate. From Rubinstein's great book "Economic Fables": "I remember the moment as a student when I realized that the models in economic theory do not assume that the decision maker consciously tries to maximize his preferences, but only assume that the behavior of a decision maker can be described as if he had maximized some objective function" (emphasis mine).
5. It is true that time-separable utility (usually with constant relative risk aversion, or CRRA) is often assumed for many mainstream models for tractability reasons. However, in recent years substantial progress has been made in extending utility to recursive preferences, e.g. Epstein-Zin. In fact, Epstein-Zin is essentially as tractable as the conventional time-separable CRRA utility, and allows for the separation of elasticities between risk and time. I expect that these more sophisticated preferences will soon be the norm. There are even stranger preferences in use, for example hyperbolic discounting. A whole field (behavioral economics) is founded on the notion that humans are not rational. The point is, lots of work has been done on generalizing utility functions.
6. I am not exactly sure what point the author is trying to make with Figure 2. This appears to be nothing more than a demonstration of Ito's Lemma (the convexity adjustment necessary for solving stochastic differential equations). In fact, the author mentions as much in the last paragraph of the third section.
7. My (ungenerous) interpretation of the experiment section: "Look, experiments calibrate the coefficient of relative risk aversion to be about 1 (log utility). And there doesn't seem to be much heterogeneity in risk aversions across people. This is consistent with my ergodicity-derived decisions, which is essentially log utility. Hence my model is supported!" (For reference, the usual calibrations I encounter for CRRA range from 1 to 5.)
8. Finally, the "Outlook" section. This section definitely rubbed me the wrong way. The author patronizes the entire field of economics, boasts of the ingenuity of his invention, and claims to somehow link together everything from the equity premium of the stock market to optimal monetary policy. It sounds too overconfident of itself and dismissive of others. References to works I expect to see are absent, and self-citations are abundant.
After reading this paper, I was gently reminded of Baez's "Crackpot Index". I imagine that what I am feeling is similar to what a physics professor might feel after reading an idea for a perpetual motion machine. There is a mixture of correctness and sophistry, with a dash of illusions of grandeur. It is comprehensible enough to understand and reasonable-sounding enough to require some effort to criticize. Looking back, writing such a lengthy critique might not have been the best use of my time.
I am rather surprised this paper was published. I am not familiar with Nature Physics, but based on this article alone, my regard for it is not high. In my opinion, this paper would not be sent out for refereeing, let only pass the referees, at any reputable economics or finance journal.