I'm not really saying anything DanielLC hasn't said already, but perhaps it will be easier for you to take seriously when it's not just one other person disagreeing with you.
DanielLC is right and you are wrong.
Kelly says: maximise expected log bankroll. This is not the same as maximising expected bankroll, and it is not true in general that maximising E(X) and maximising E(f(X)) are equivalent when f is monotonic increasing.
If your current bankroll is $1000 and you can choose between (1) a gamble that gives you $100 with probability 3/4 and $1000 with probability 1/4 and (2) sticking with what you've got, maximising expected bankroll will tell you to choose #1 but maximising expected log bankroll will tell you to choose #2.
Kelly says: maximise expected log bankroll.
Actually, no, it does not. The Kelly Rule aims to maximize the expected bankroll (not the log of it) after many bets. It's a real-world rule and in the real world people want money, not log(money).
I think there are a couple of points of confusion here. The first is between maximizing what and maximizing how. We want to maximize the amount that you have on hand after a long series of bets. Figuring out what bets to accept and which to decline involves logs, but in the end you just want to have max(money). The s...
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.