Re 1: looking back at the subthread, yes, I think that was the source of much confusion. I did mean maximization in the long run and for quite a while did not realize that you and DanielLC were talking about the maximization in a single iteration.
Re 4: The expectation operator is just a weighted sum (in the discrete case) or an integral (in the continuous case). I don't think it cares about the fatness of tails or whether some moments are defined or not.
Speaking generally, log(E(X)) is not the same thing as E(log(X)) (see Jensen's Inequality), but that's a different question. The question we have is that if you have some set of parameters theta that X is conditional on, does maximizing for E(X) lead you to different optimal thetas than maximizing for E(log(X))?
Re 6: Well, you have to be careful that Kelly Rule assumptions hold. It works as it works because capital growth is multiplicative, not additive, and because you expect to have many iterations of betting, for example.
The expectation operator doesn't care about fatness of tails (well, it kinda doesn't, but note that e.g. the expectation of a random variable with Cauchy distribution is undefined, precisely because of those very fat tails), but the theorem that says that in the long run your wealth is almost always close to its expectation may fail for fat-tail-related reasons.
does maximizing for E(X) lead you to different optimal thetas than maximizing for E(log(X))?
In the present case where we're looking at long-run results only, the answer might be "no" (...
A lottery ticket sometimes has positive expected value, (a $1 ticket might be expected to pay out $1.30). How many tickets should you buy?
Probably none. Informally, all but the richest players can expect to go broke before they win, despite the positive expected value of a ticket.
In more precise terms: In order to maximize the long-term growth rate of your money (or log money), you'll want to put a very small fraction of your bankroll into lotteries tickets, which will imply an "amount to invest" that is less than the cost of a single ticket, (excluding billionaires). If you put too great a proportion of your resources into a risky but positive expected value asset, the long-term growth rate of your resources can become negative. For an intuitive example, imagine Bill Gates dumping 99% percent of his wealth into a series of positive expected-value bets with single-lottery-ticket-like odds.
This article has some graphs and details on the lottery. This pdf on the Kelly criterion has some examples and general dicussion of this type of problem.
Can we think about Pascal mugging the same way?
The applicability might depend on whether we're trading resource-generating-resources for non-resource-generating assets. So if we're offered something like cash, the lottery ticket model (with payout inversely varying with estimated odds) is a decent fit. But what if we're offered utility in some direct and non-interest-bearing form?
Another limit: For a sufficiency unlikely but positive-expected-value gamble, you can expect the heat death of the universe before actually realizing any of the expected value.