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prase comments on How much to spend on a high-variance option? - Less Wrong
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Comments (36)
Isn't there a standard reply containing utility functions? Assuming the usual diminishing marginal utility, if p(win) * U(jackpot) > U($1), then 2 p(win) * U(jackpot) > U($2) and so on. You should spend all your money! (Unless you are rich enough to buy so many tickets as to wander out of the domain of approximate linear dependece of p(win) on the number of tickets you own).
Second attempt (thanks janos):
Isn't there a standard reply containing utility functions? You should buy N tickets where N is the solution to p(win | N tickets) * U(jackpot) + U(-N dollars) is maximal. Assuming the usual diminishing marginal utility of money and that U(jackpot) * p(win | 1 ticket) > -U($-1), a unique solution should exist for N > 0.
I think you're making the wrong comparisons. If you buy $1 worth, you get p(win) * U(jackpot) + (1-p(win)) * U(-$1), which is more-or-less p(win)*U(jackpot)+U(-$1); this is a good idea if p(win) * U(jackpot) > -U(-$1). But under usual assumptions -U(-$2)>-2U(-$1). This adds up to normality; you shouldn't actually spend all your money. :)
Of course you are right, silly mistake.
(Not really important nitpick:) The dollar is spent once the ticket is bought and doesn't return even if you win, so you shoudn't have there (1-p(win)) * U(-$1), but just U(-$1).