Can we think about Pascal mugging the same way?
This was, as I recall, the first proposed resolution to the St. Petersburg Paradox; but as pointed out by DanielLC, it doesn't work because you can then just reformulate the statement in terms of utility, instead of dollars.
Some suggest that having a very aggressively bounded utility function is sufficient to solve the problem- if your utility for the current state of things is .9, and your maximum utility is 1, then the amount you'd gamble for a chance to get a utility of 1 is bounded at a rather low amount. But this might be throwing the baby out with the bathwater, and might be asymmetric in a way you really don't want to be. It seems like any technique that resists Pascal's Mugging, where you pay a little to avoid a tiny chance of paying a lot, should also resist paying a little to get a tiny chance of getting a lot. (Now that I say that I'm unsure, but that might be because my mind has different heuristics for blackmail, insurance, and investments.)
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.