4 ms·

These analyses mostly ignore the obvious fact that the value of infinite money isn't, in fact, infinite, because eventually you'd run out of things to buy. Usi

These analyses mostly ignore the obvious fact that the value of infinite money isn't, in fact, infinite, because eventually you'd run out of things to buy. Using the aggregate wealth of the entire world (which is probably in the neighborhood of $250 trillion[1]) as a bankroll yields a expected value of about $50. [1] http://blogs.reuters.com/felix-salmon/2014/04/04/stop-adding-up-the-wealth-of-the-poor/ http://blogs.reuters.com/felix-salmon/2014/04/04/stop-adding...

mikeash 11y ago
That's somewhat covered in the "Expected utility theory" section of the Wikipedia article. In short, if you double your money it doesn't double the usefulness you get out of it, so you have to take that into account. However, that analysis assumes that additional money always adds some additional usefulness, and then you can make the problem reappear. To get rid of it entirely you have to declare that there is some point beyond which provides zero additional value no matter how much more money you add. And as you say, such a point surely exists.
- cperciva 11y ago
  To get rid of it entirely you have to declare that there is some point beyond which provides zero additional value no matter how much more money you add. No, you merely have to declare that there is some amount of utility which can never be obtained no matter how much money you have. (Or as the Simpsons put it: There's one thing you can't buy: A dinosaur.) In mathematical terms, if your utility function is f(x) = 1 - 1/x, every marginal dollar adds utility; but the added utility is never enough to make the gamble profitable.
  - AstralStorm 11y ago
    Oh but you can but a dinosaur - a dead one. Given enough money you could potentially clone one too. Utility function analysis is arbitrary just as the article says. Compare this to ergodic theory (used for proper dynamic system analysis), where you can clearly derive the value function over number of plays, including infinite number. What an ergodic process is you can see here: https://en.wikipedia.org/wiki/Ergodic_process https://en.wikipedia.org/wiki/Ergodic_process By computing the limit of the ensemble average you will derive the logarithmic utility function. This works for finite resources as well, but then you can also derive the time it takes to bankrupt the bank, as well as probability of that ever happening.
    - deleted 11y ago
      [deleted]
bshanks 11y ago
To rephrase your observation, the human utility function might have a maximum. Imagine the best day of your life; now imagine every day of your future life was this good, even a bit better, and that you live a rather long life. If you're altrustic, you can even imagine this number multiplied by the Earth's population. If you're a sci-fi fan/futurist, you can even imagine this number multiplied by the number of expected planets in the universe, multiplied by the amount of time until the stars burn out (as an aside, this is extremely optimistic because it's highly unlikely that even with arbitrarily large wealth that you would be able to figure out how to spend it in such a way to give every person a perfect life until the end of the universe). There's a good chance that you can't get much better than that, no matter how wealthy you became. The paradox wants you to imagine that, for very small probabilities, you can get utilities indefinitely large. The paradox wants you to say, I'll devote my entire life to earning as much money as possible and spending it all on this game because the payoff is so large that this is worth it for the tinest chance. But if there's an upper bound on the human utility function, then this doesn't work. It doesn't work because when you imagine how happy you'd be with arbitrarily large wealth, you don't imagine an arbitrarily large utility, you imagine the "max utility" situation described above where every day is like the best day of your life, which is nice but not as nice as would be required to justify almost certainly throwing your life away for a small chance of this.

mikeash 11y ago

That's somewhat covered in the "Expected utility theory" section of the Wikipedia article. In short, if you double your money it doesn't double the usefulness you get out of it, so you have to take that into account. However, that analysis assumes that additional money always adds some additional usefulness, and then you can make the problem reappear. To get rid of it entirely you have to declare that there is some point beyond which provides zero additional value no matter how much more money you add. And as you say, such a point surely exists.

cperciva 11y ago
To get rid of it entirely you have to declare that there is some point beyond which provides zero additional value no matter how much more money you add. No, you merely have to declare that there is some amount of utility which can never be obtained no matter how much money you have. (Or as the Simpsons put it: There's one thing you can't buy: A dinosaur.) In mathematical terms, if your utility function is f(x) = 1 - 1/x, every marginal dollar adds utility; but the added utility is never enough to make the gamble profitable.
- AstralStorm 11y ago
  Oh but you can but a dinosaur - a dead one. Given enough money you could potentially clone one too. Utility function analysis is arbitrary just as the article says. Compare this to ergodic theory (used for proper dynamic system analysis), where you can clearly derive the value function over number of plays, including infinite number. What an ergodic process is you can see here: https://en.wikipedia.org/wiki/Ergodic_process https://en.wikipedia.org/wiki/Ergodic_process By computing the limit of the ensemble average you will derive the logarithmic utility function. This works for finite resources as well, but then you can also derive the time it takes to bankrupt the bank, as well as probability of that ever happening.
  - deleted 11y ago
    [deleted]

bshanks 11y ago

To rephrase your observation, the human utility function might have a maximum. Imagine the best day of your life; now imagine every day of your future life was this good, even a bit better, and that you live a rather long life. If you're altrustic, you can even imagine this number multiplied by the Earth's population. If you're a sci-fi fan/futurist, you can even imagine this number multiplied by the number of expected planets in the universe, multiplied by the amount of time until the stars burn out (as an aside, this is extremely optimistic because it's highly unlikely that even with arbitrarily large wealth that you would be able to figure out how to spend it in such a way to give every person a perfect life until the end of the universe). There's a good chance that you can't get much better than that, no matter how wealthy you became. The paradox wants you to imagine that, for very small probabilities, you can get utilities indefinitely large. The paradox wants you to say, I'll devote my entire life to earning as much money as possible and spending it all on this game because the payoff is so large that this is worth it for the tinest chance. But if there's an upper bound on the human utility function, then this doesn't work. It doesn't work because when you imagine how happy you'd be with arbitrarily large wealth, you don't imagine an arbitrarily large utility, you imagine the "max utility" situation described above where every day is like the best day of your life, which is nice but not as nice as would be required to justify almost certainly throwing your life away for a small chance of this.