This post is essentially my response to Pascal's mugging.
a)If an event is extremely unlikely to occur then trying to maximise expected utility is foolish unless it is repeated multiple times. In the case of the lottery, you need to buy millions of tickets to have a reasonable chance of winning once (ignoring small prizes here). In the mugging, you have to be mugged an absolutely absurd number of times before any of the muggers are even remotely likely to be telling the truth b)The hidden and secret response, which is this:
even if the lottery has positive expected utility, there are likely to be alternate uses for your money which will give better ones! If you have the money to win the lottery, you can invest elsewhere and get better returns. This is true for Pascal's mugging, where you can go spend your 10 dollars to save lives immediately via a recommendation from give well.
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.