1 This argument only works if the bet is denominated in utils rather than in dollars. Otherwise, someone who gets diminishing marginal utility from dollars for very large sums—that would include most people—will eventually decide to stop. (If I have utility = log(dollars) and initial assets of $1M then I will stop after 25 wins, if I did the calculations right.)
1a It is not at all clear that a bet denominated in utils is even actually possible. Especially not one which, with high probability, ends up involving an astronomically large quantity of utility.
2 Even someone who doesn’t generally get diminishing marginal utility from dollars—say, an altruist who will use all those dollars for saving other people’s lives, and who cares equally about all—will find marginal utility decreasing for large enough sums, because (a) eventually the cheap problems are solved and saving the next life starts costing more, and (b) if you give me 10^15 dollars and I try to spend it all (on myself or others) then the resulting inflation will make them worth less.
3 Given that “you will eventually lose it all”, a strategy of continuing to bet does not in fact maximize expected utility.
4 The expected utility from a given choice at a given stage in the game depends on what you’d then do with the remainder of the game. For instance, if I know that my future strategy after winning this roll is going to be “keep betting for ever” then I know that my expected utility if I keep playing is zero, so I’ll choose not to do that.
5 So at most what we have (even if we assume we’ve dealt somehow with issues of diminishing marginal utility etc.) is a game where there’s an infinite “increasing” sequence of strategies but no limiting strategy that’s better than all of them. But that’s no surprise. Here’s another game with the same property: You name a positive integer N and Omega gives you $N. For any fixed N, it is best not to choose N because larger numbers are better. “Therefore” you can’t name any particular number, so you refuse to play and get nothing. If you don’t find this paradoxical—and I confess that I don’t—then I don’t think you need find the die-rolling game any worse. (Choosing N in this game <--> deciding to play for N turns in the die-rolling game.)
[EDITED to stop the LW software turning my numbered points into differently numbered and weirdly formatted points.]
[EDITED again to acknowledge that after writing all that I read on and found that others had already said more or less the same things as me. D’oh. Anyway, since apparently Qiaochu_Yuan wasn’t successful in convincing srn247, perhaps my slightly different presentation will be of some help.]
1 This argument only works if the bet is denominated in utils rather than in dollars. Otherwise, someone who gets diminishing marginal utility from dollars for very large sums—that would include most people—will eventually decide to stop. (If I have utility = log(dollars) and initial assets of $1M then I will stop after 25 wins, if I did the calculations right.)
1a It is not at all clear that a bet denominated in utils is even actually possible. Especially not one which, with high probability, ends up involving an astronomically large quantity of utility.
2 Even someone who doesn’t generally get diminishing marginal utility from dollars—say, an altruist who will use all those dollars for saving other people’s lives, and who cares equally about all—will find marginal utility decreasing for large enough sums, because (a) eventually the cheap problems are solved and saving the next life starts costing more, and (b) if you give me 10^15 dollars and I try to spend it all (on myself or others) then the resulting inflation will make them worth less.
3 Given that “you will eventually lose it all”, a strategy of continuing to bet does not in fact maximize expected utility.
4 The expected utility from a given choice at a given stage in the game depends on what you’d then do with the remainder of the game. For instance, if I know that my future strategy after winning this roll is going to be “keep betting for ever” then I know that my expected utility if I keep playing is zero, so I’ll choose not to do that.
5 So at most what we have (even if we assume we’ve dealt somehow with issues of diminishing marginal utility etc.) is a game where there’s an infinite “increasing” sequence of strategies but no limiting strategy that’s better than all of them. But that’s no surprise. Here’s another game with the same property: You name a positive integer N and Omega gives you $N. For any fixed N, it is best not to choose N because larger numbers are better. “Therefore” you can’t name any particular number, so you refuse to play and get nothing. If you don’t find this paradoxical—and I confess that I don’t—then I don’t think you need find the die-rolling game any worse. (Choosing N in this game <--> deciding to play for N turns in the die-rolling game.)
[EDITED to stop the LW software turning my numbered points into differently numbered and weirdly formatted points.]
[EDITED again to acknowledge that after writing all that I read on and found that others had already said more or less the same things as me. D’oh. Anyway, since apparently Qiaochu_Yuan wasn’t successful in convincing srn247, perhaps my slightly different presentation will be of some help.]