If I draw cards until I die, my expected utility is positive infinity. Though I will almost surely die and end up with utility 0, it is logically possible that I will never die, and end up with a utility of positive infinity. In this case, 10 + 0(positive infinity) = positive infinity.
The next paragraph requires that you assume our initial utility is 1.
If you want, warp the problem into an isomorphic problem where the probabilities are different and all utilities are finite. (Isn’t it cool how you can do that?) In the original problem, there’s always a 5⁄6 chance of utility doubling and a 1⁄6 chance of it going to 1⁄2. (Being dead isn’t THAT bad, I guess.) Let’s say that where your utility function was U(w), it is now f(U(w)), where f(x) = 1 − 1/(2 + log_2 x). In this case, the utilities 1⁄2, 1, 2, 4, 8, 16, . . . become 0, 1⁄2, 2⁄3, 3⁄4, 4⁄5, 5⁄6, . . . . So, your initial utility is 1⁄2, and Omega will either lower your utility to 0 or raise it by applying the function U’ = U/(U + 1). Your expected utility after drawing once was previously U’ = 5⁄3U + 1⁄2; it’s now… okay, my math-stamina has run out. But if you calculate expected utility, and then calculate the probability that results in that expected utility, I’m betting that you’ll end up with a 1⁄2 probability of ever* dying.
(The above paragraph surrounding a nut: any universe can be interpreted as one where the probabilities are different and the utility function has been changed to match… often, probably.)
If I draw cards until I die, my expected utility is positive infinity. Though I will almost surely die and end up with utility 0, it is logically possible that I will never die, and end up with a utility of positive infinity. In this case, 10 + 0(positive infinity) = positive infinity.
The next paragraph requires that you assume our initial utility is 1.
If you want, warp the problem into an isomorphic problem where the probabilities are different and all utilities are finite. (Isn’t it cool how you can do that?) In the original problem, there’s always a 5⁄6 chance of utility doubling and a 1⁄6 chance of it going to 1⁄2. (Being dead isn’t THAT bad, I guess.) Let’s say that where your utility function was U(w), it is now f(U(w)), where f(x) = 1 − 1/(2 + log_2 x). In this case, the utilities 1⁄2, 1, 2, 4, 8, 16, . . . become 0, 1⁄2, 2⁄3, 3⁄4, 4⁄5, 5⁄6, . . . . So, your initial utility is 1⁄2, and Omega will either lower your utility to 0 or raise it by applying the function U’ = U/(U + 1). Your expected utility after drawing once was previously U’ = 5⁄3U + 1⁄2; it’s now… okay, my math-stamina has run out. But if you calculate expected utility, and then calculate the probability that results in that expected utility, I’m betting that you’ll end up with a 1⁄2 probability of ever* dying.
(The above paragraph surrounding a nut: any universe can be interpreted as one where the probabilities are different and the utility function has been changed to match… often, probably.)