I haven't read the material extensively (I've skimmed it), but here's what I think is wrong with the time-average-vs-ensemble-average argument and my attempt to steelman it.

It seems very plausible to me that you're right about the question-begging nature of Peter's version of the argument; it seems like by maximizing expected growth rate, you're maximizing log wealth.

But I also think he's trying to point at something real.

In the presentation where he uses the 1.5x/0.6x bet example, Peters shows how "expected utility over time" is an increasing line (this is the "ensemble average" -- averaging across possibilities at each time), whereas the actual payout for any player looks like a straight downward line (in log-wealth) if we zoom out over enough iterations. There's no funny business here -- yes, he's taking a log, but that's just the best way of graphing the phenomenon. It's still true that you lose almost surely if you keep playing this game longer and longer.

This is a real phenomenon. But, how do we formalize an alternative optimization criterion from it? How do we make decisions in a way which "aggregates over time rather than over ensemble"? It's natural to try to formalize something in log-wealth space since that's where we see a straight line, but as you said, that's question-begging.

Well, a (fairly general) special case of log-wealth maximization is the Kelly criterion. How do people justify that? Wikipedia's current "proof" section includes a heuristic argument which runs roughly as follows:

• Imagine you're placing bets in the same way a large number of times, N.
• By the law of large numbers, the frequency of wins and losses approximately equals their probabilities.
• Optimize total wealth at time N under the assumption that the frequencies equal the probabilities. You get the Kelly criterion.

Now, it's easy to see this derivation and think "Ah, so the Kelly criterion optimizes your wealth after a large number of steps, whereas expected utility only looks one step ahead". But, this is not at all the case. An expected money maximizer (EMM) thinking long-term will still take risky bets. Observe that (in the investment setting in which Kelly works) the EMM strategy for a single step doesn't depend on the amount of money you have -- you either put all your money in the best investment, or you keep all of your money because there are no good investments. Therefore, the payout of the EMM in a single step is some multiple C of the amount of money it begins that step with. Therefore, an EMM looking one step ahead just values its winnings at the end of the first step C more -- but this doesn't change its behavior, since multiplying everything by C doesn't change what the max-expectation strategy will be. Similarly, two-step lookahead only modifies things by $C^2$, and so on. So an EMM looking far ahead behaves just like one maximizing its holdings in the very next step.

The trick in the analysis is the way we replace a big sum over lots of possible ways things could go with a single "typical" outcome. This might initially seem like a mere computational convenience -- after all, the vast vast majority of possible sequences have approximately the expected win/loss frequencies. Here, though, it makes all the difference, because it eliminates from consideration the worlds which have the highest weight in the EMM analysis -- the worlds where things to really well and the EMM gets exponentially much money.

OK, so, is the derivation just a mistake?

I think many english-language justifications of the Kelly criterion or log-wealth maximization are misleading or outright wrong. I don't think we can justify it as an analysis of the best long-term strategy, because the analysis rules out any sequence other than those with the most probable statistics, which isn't a move motivated by long-term analysis. I don't think we can even justify it as "time average rather than ensemble average" because we're not time-averaging wealth. Indeed, the whole point is supposedly to deal with the non-ergodic cases; but non-ergodic systems don't have unique time-averaged behavior!

However, I ultimately find something convincing about the analysis: namely, from an evolutionary perspective, we expect to eventually find that only (approximate) log-wealth maximizers remain in the market (with non-negligible funds).

This conclusion is perfectly compatible with expected utility theory as embodied by the VNM axioms et cetera. It's an argument that market entities will tend to have utility=log(money), at least approximately, at least in common situations which we can expect strategies to be optimized for. More generally, there might be an argument that evolved organisms will tend to have utility=log(resources), for many notions of resources.

However, maybe Nassim Nicolas Taleb would rebuke us for this tepid and timid conclusion. In terms of pure utility theory, applying a log before taking an expectation is a distinction without a difference -- we were allowed any utility function we wanted from the start, so requiring an arbitrary transform means nothing. For example, we can "solve" the St. Petersburg paradox by claiming our utility is the log of money -- but we can then re-create the paradox by putting all the numbers in the game through an exponential function! So what's the point? We should learn from our past mistakes, and choose a framework which won't be prone to those same errors.

So, can we steelman the claims that expected utility theory is wrong? Can we find a decision procedure which is consistent with the Peters' general idea, but isn't just log-wealth maximization?

Well, let's look again at the kelly-criterion analysis. Can we make that into a general-purpose decision procedure? Can we get it to produce results incompatible with VNM? If so, is the procedure at all plausible?

As I've already mentioned, there isn't a clear way to apply the law-of-large-numbers trick in non-ergodic situations, because there is not a unique "typical" set of frequencies which emerges. Can we do anything to repair the situation, though?

I propose that we maximize median expected value. This gives a notion of "typical" which does not rely on an application of the law of large numbers, so it's fine if the statistics of our sequence don't converge to a single unique point. If they do, however, the median will evaluate things from that point. So, it's a workable generalization of the principle behind Kelly betting.

The median also relates to something mentioned in the OP:

I've felt vaguely confused for a long time about why expected value/utility is the right way to evaluate decisions; it seems like I might be more strongly interested in something like "the 99th percentile outcome for the overall utility generated over my lifetime".

The median is the 50th percentile, so there you go.

Maximizing the median indeed violates VNM:

• It's discontinuous. Small differences in probability can change the median outcome by a lot. Maybe this isn't so bad -- who really cares about continuity, anyway? Yeah, seemingly small differences in probability create "unjustified" large differences in perceived quality of a plan, but only in circumstances where outcomes are sparse enough that the median is not very "informed".
• It violates independence, in a more obviously concerning way. A median-maximizer doesn't care about "outlier" outcomes. It's indifferent between the following two plans, which seems utterly wrong:
• A plan with 100% probability of getting you $100
• A plan with 60% probability of getting you $100, and 40% probability of getting you killed.

Both of these concerns become negligible as we take a long-term view. The longer into the future we look, the more outcomes there will be, making the median more robust to shifting probabilities. Similarly, a median-maximizer is indifferent between the two options above, but if you consider the iterated game, it will strongly prefer the global strategy of always selecting the first option.

Still, I would certainly not prefer to optimize median value myself, or create AGI which optimizes median value. What if there's a one-shot situation which is similar to the 40%-death example? I think I similarly don't want to maximize the 99th percentile outcome, although this is less clearly terrible.

Can we give an evolutionary argument for median utility, as a generalization of the evolutionary argument for log utility? I don't think so. The evolutionary argument relies on the law of large numbers, to say that we'll almost surely end up in a world where log-maximizers prosper. There's no similar argument that we almost surely end up in the "median world".

So, all told:

• I don't think there's a good argument against expectation-maximization here.
• But I do think those who think there is should consider median-maximization, as it's an alternative to expectation-maximization which is consistent with much of the discussion here.
• I basically buy the argument that utility should be log of money.
• I don't think it's right to describe the whole thing as "time-average vs ensemble-average", and suspect some of the "derivations" are question-begging.
• I do think there's an evolutionary argument which can be understood from some of the derivations, however.

Thanks for taking the time to delve into this!

You note that expected utility with a risk-averse utility function is sufficient to make appropriate choices [in those particular scenarios].

This is a slight tangent, but I'm curious to what extent you think people actually follow something that approximates this utility function in real life? It seems like some gamblers instinctively use a strategy of this nature (e.g. playing with house money) or explicitly run the numbers (e.g. the Kelly criterion). And I doubt that anyone is dumb enough to keep betting their entire bankroll on a positive EV bet until they inevitably go bust.

But in other cases (like retirement planning, as you mentioned) a lot of people really do seem to make the mistake of relying on ensemble-average probabilities. Some of them will get burned, with much more serious consequences than merely making a silly bet at the casino.

I guess what I'm asking is: even if Peters et al are wrong about expected utility, do you think they're right about the dangers of failing to understand ergodicity?

(I've only spent several hours thinking about this, so I'm not confident in what I say below. I think Ole Peters is saying something interesting, although he might not be phrasing things in the best way.)

Time-average wealth maximization and utility=log(wealth) give the same answers for multiplicative dynamics, but for additive dynamics they can prescribe different strategies. For example, consider a game where the player starts out with $30, and a coin is flipped. If heads, the player gains $15, and if tails, the player loses $11. This is an additive process since the winnings are added to the total wealth, rather than calculated as a percentage of the player's wealth (as in the 1.5x/0.6x game). Time-average wealth maximization asks whether $(15-11)/2>0$, and takes the bet. The agent with utility=log(wealth) asks whether $\left(\log \right(30+15)+\log (30-11\left)\right)/2>\log 30$, and refuses the bet.

What happens when this game is repeatedly played? That depends on what happens when a player reaches negative wealth. If debt is allowed, the time-average wealth maximizer racks up a lot of money in almost all worlds, whereas the utility=log(wealth) agent stays at $30 because it refuses the bet each time. If debt is not allowed, and instead the player "dies" or is refused the game once they hit negative wealth, then with probability at least 1/8, the time-average wealth maximizer dies (if it gets tails on the first three tosses), but when it doesn't manage to die, it still racks up a lot of money.

In a world where this was the "game of life", the utility=log(wealth) organisms would soon be out-competed by the time-average wealth maximizers that happened to survive the early rounds. So the organisms that tend to evolve in this environment will have utility linear in wealth.

So I understand Ole Peters to be saying that time-average wealth maximization adapts to the game being played, in the sense that organisms which follow its prescriptions will tend to out-compete other kinds of organisms.

Tangentially: reading about the history of gambling theory (the "unfinished game" problem, etc.) is pretty interesting.

Imagine how weird it was when people basically didn't understand expected value at all! Did casinos even know what they were doing, or did they somewhat routinely fail after picking the wrong game design? Did they only settle on profitable designs by accident? Are blackjack, roulette, and other very old games still with us because they happened not to bankrupt casinos that ran them, and were only later analyzed with tools capable of identifying whether the house had the edge?

Peters' December 2019 Nature Physics paper (https://www.nature.com/articles/s41567-019-0732-0 ) provides some perspective on 0.6/1.5x coin flip example and other conclusions of the above discussion. (If Peters' claims have changed along the way, I wouldn't know.)

In my reading, there Peters' basic claim is not that ergodicity economics can solve the coin flip game in a way that classical economics can not (because it can, by switching to expected log wealth utility instead of expected wealth), but the utility functions as originally presented are a clutch that misinforms us on people's psychological motives in doing economic decisions. So, while the mathematics of many parts stays the same, the underlying phenomena can be more saliently reasoned about by looking at the individual growth rates in context of whether the associated wealth "process" is additive or multiplicative or something else. Thus there is also less need to use lingo where people may have an (innate, weirdly) "risk-averse utility function" (as compared to some other less risk-averse theoretical utility function).