I’d say the flip doesn’t occur when we go from fixed n to “as n tends to infinity”, it occurs when we go from that to “but what happens after the infinite sequence”.
It’s true that as n→∞, E(U(tn)) grows unboundedly. But it’s also true that the probability distribution at tn converges pointwise as n→∞ to the function that’s 1 at 0 and 0 everywhere else, which is also a valid probability distribution. This sort of thing is why I say limits aren’t always well behaved.
That said: I do think this is also a reasonable handwavey argument, but it does also require hand waving. (And it opens up questions: “what’s Linda’s expected utility from making her own bets?” Infinity. “What’s her expected utility from making Logan’s bets?” Infinity. “So why should she make her bets instead of his?” Well, one infinity is bigger than the other. “Okay, but what does that mean in this context?” I personally wouldn’t be able to give an answer that satisfies myself. That doesn’t mean no such answer exists.)
I’d say the flip doesn’t occur when we go from fixed n to “as n tends to infinity”, it occurs when we go from that to “but what happens after the infinite sequence”.
It’s true that as n→∞, E(U(tn)) grows unboundedly. But it’s also true that the probability distribution at tn converges pointwise as n→∞ to the function that’s 1 at 0 and 0 everywhere else, which is also a valid probability distribution. This sort of thing is why I say limits aren’t always well behaved.
That said: I do think this is also a reasonable handwavey argument, but it does also require hand waving. (And it opens up questions: “what’s Linda’s expected utility from making her own bets?” Infinity. “What’s her expected utility from making Logan’s bets?” Infinity. “So why should she make her bets instead of his?” Well, one infinity is bigger than the other. “Okay, but what does that mean in this context?” I personally wouldn’t be able to give an answer that satisfies myself. That doesn’t mean no such answer exists.)