Double your utility for the rest of your life compared to what? If you draw cards until you die, that sounds like it just means you have twice as much fun drawing cards as you would have without help. I guess that could be lots of fun if you’re the kind of person who gets a rush off of Russian roulette under normal circumstances, but if you’re not, you’d probably be better off flipping off Omega and watching some TV.
What if your utility would have been negative? Doesn’t doubling it make it twice as bad?
Good point. Better not draw a card if you have negative utility.
Just trust that Omega can double your utility, for the sake of argument. If you stop before you die, you get all those doublings of utility for the rest of your life.
I’d certainly draw one card. But would I stop drawing cards?
Thinking about this in commonsense terms is misleading, because we can’t imagine the difference between 8x utility and 16x utility. But we have a mathematical theory about rationality. Just apply that, and you find the results seem unsatisfactory.
Thinking about this in commonsense terms is misleading, because we can’t imagine the difference between 8x utility and 16x utility
I can’t even imagine doubling my utility once, if we’re only talking about selfish preferences. If I understand vNM utility correctly, then a doubling of my personal utility is a situation which I’d be willing to accept a 50% chance of death in order to achieve (assuming that my utility is scaled so that U(dead) = 0, and without setting a constant level, we can’t talk about doubling utility). Given my life at the moment (apartment with mortgage, two chronically ill girlfriends, decent job with unpleasantly long commute, moderate physical and mental health), and thinking about the best possible life I could have (volcano lair, catgirls), I wouldn’t be willing to take that bet. Intuition has already failed me on this one. If Omega can really deliver on his promise, then either he’s offering a lifestyle literally beyond my wildest dreams, or he’s letting me include my preferences for other people in my utility function, in which case I’ll probably have cured cancer by the tenth draw or so, and I’ll run into the same breakdown of intuition after about seventy draws, by which time everyone else in the world should have their own volcano lairs and catgirls.
With the problem as stated, any finite number of draws is the rational choice, because the proposed utility of N draws outweighs the risk of death, no matter how high N is. The probability of death is always less than 1 for a finite number of draws. I don’t think that considering the limit as N approaches infinity is valid, because every time you have to decide whether or not to draw a card, you’ve only drawn a finite number of cards so far. Certainty of death also occurs in the same limit as infinite utility, and infinite utility has its own problems, as discussed elsewhere in this thread. It might also leave you open to Pascal’s Scam—give me $5 and I’ll give you infinite utility!
But we have a mathematical theory about rationality. Just apply that, and you find the results seem unsatisfactory.
I agree—to keep drawing until you draw a skull seems wrong. However, to say that something “seems unsatisfactory” is a statement of intuition, not mathematics. Our intuition can’t weigh the value of exponentially increasing utility against the cost of an exponentionally diminishing chance of survival, so it’s no wonder that the mathematically derived answer doesn’t sit well with intuition.
Double your utility for the rest of your life compared to what? If you draw cards until you die, that sounds like it just means you have twice as much fun drawing cards as you would have without help. I guess that could be lots of fun if you’re the kind of person who gets a rush off of Russian roulette under normal circumstances, but if you’re not, you’d probably be better off flipping off Omega and watching some TV.
What if your utility would have been negative? Doesn’t doubling it make it twice as bad?
Good point. Better not draw a card if you have negative utility.
Just trust that Omega can double your utility, for the sake of argument. If you stop before you die, you get all those doublings of utility for the rest of your life.
I’d certainly draw one card. But would I stop drawing cards?
Thinking about this in commonsense terms is misleading, because we can’t imagine the difference between 8x utility and 16x utility. But we have a mathematical theory about rationality. Just apply that, and you find the results seem unsatisfactory.
I can’t even imagine doubling my utility once, if we’re only talking about selfish preferences. If I understand vNM utility correctly, then a doubling of my personal utility is a situation which I’d be willing to accept a 50% chance of death in order to achieve (assuming that my utility is scaled so that U(dead) = 0, and without setting a constant level, we can’t talk about doubling utility). Given my life at the moment (apartment with mortgage, two chronically ill girlfriends, decent job with unpleasantly long commute, moderate physical and mental health), and thinking about the best possible life I could have (volcano lair, catgirls), I wouldn’t be willing to take that bet. Intuition has already failed me on this one. If Omega can really deliver on his promise, then either he’s offering a lifestyle literally beyond my wildest dreams, or he’s letting me include my preferences for other people in my utility function, in which case I’ll probably have cured cancer by the tenth draw or so, and I’ll run into the same breakdown of intuition after about seventy draws, by which time everyone else in the world should have their own volcano lairs and catgirls.
With the problem as stated, any finite number of draws is the rational choice, because the proposed utility of N draws outweighs the risk of death, no matter how high N is. The probability of death is always less than 1 for a finite number of draws. I don’t think that considering the limit as N approaches infinity is valid, because every time you have to decide whether or not to draw a card, you’ve only drawn a finite number of cards so far. Certainty of death also occurs in the same limit as infinite utility, and infinite utility has its own problems, as discussed elsewhere in this thread. It might also leave you open to Pascal’s Scam—give me $5 and I’ll give you infinite utility!
I agree—to keep drawing until you draw a skull seems wrong. However, to say that something “seems unsatisfactory” is a statement of intuition, not mathematics. Our intuition can’t weigh the value of exponentially increasing utility against the cost of an exponentionally diminishing chance of survival, so it’s no wonder that the mathematically derived answer doesn’t sit well with intuition.