It does not maximize your expected bankroll. It maximizes the expected value of the log of your bankroll.
Log is a monotonic function, what maximizes one maximizes the other.
A few comments on the remainder of this subthread.
Different people are talking about different things, without making the distinctions perfectly clear. Specifically, when Lumifer says "it maximizes your expected bankroll, and maximizing your expected log bankroll is the same" I believe he means "it maximizes your expected bankroll in the long run, and maximizing your expected log bankroll in the long run is the same because in the long run you almost always get approximately your expectation". Whereas DanielLC and I have been observin
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.