The mugger should have some non-zero probability, p, for which zhe is indifferent between p"have $10 after fulfilling the deal" and (1-p)"have $5 now".

There is no reason to suppose this, and in fact it's unlikely, since the mugger probably isn't motivated by money (since he surely has better ways of obtaining that). In the least convenient world, he's probably just curious to know how you'll answer.

The standard solution is to have a bounded utility function, and it seems like it fully solves this reformulation as well. There may also be some other solutions that work for this, but I'm sufficiently unsure about all the notation that I'm not very confident in them.

Also, just to check, have you seen the Lifespan Dilemma?