I suppose one thing you could do here is pretend you can fit infinite rounds of the game into a finite time. Then Linda has a choice to make: she can either maximize expected wealth at tn for all finite n, or she can maximize expected wealth at tω, the timestep immediately after all finite timesteps. We can wave our hands a lot and say that making her own bets would do the former and making Logan’s bets would do the latter, though I don’t endorse the way we’re treating infinties here.
If one strategy is best for tn, it’s still going to be best at tω as t goes to infinity. Optimal strategies don’t just change like that as n goes to infinity. Sure you can argue that p(won every time) --> 0, but also that number is being multiplied by an extremely large infinity, so you can’t just say that it totals to zero (in fact, 1.2, which is her EV from a single game, raised to infinity is infinity, so I argue as n goes to infinity, her EV goes to infinity, not 0, and not a number less than the EV from Logan’s strategy).
Linda’s strategy is always optimal with respect to her own utility function, even as n goes to infinity. She’s not acting irrationally or incorrectly here.
The one world where she has won wins her $2**n, and that world exists with probability 0.6**n.
Her EV is always ($2**n)*(0.6**n), which is a larger EV (with any n) than a strategy where she doesn’t bet everything every single time. Even as n goes to infinity, and even as probability approaches 1 that she has lost everything, it’s still rational for her to have that strategy because the $2**n that she won in that one world is so massive that it balances out her EV. Some infinities are much larger than others, and ratios don’t just flip when a large n goes to infinity.
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.)
If one strategy is best for tn, it’s still going to be best at tω as t goes to infinity. Optimal strategies don’t just change like that as n goes to infinity. Sure you can argue that p(won every time) --> 0, but also that number is being multiplied by an extremely large infinity, so you can’t just say that it totals to zero (in fact, 1.2, which is her EV from a single game, raised to infinity is infinity, so I argue as n goes to infinity, her EV goes to infinity, not 0, and not a number less than the EV from Logan’s strategy).
Linda’s strategy is always optimal with respect to her own utility function, even as n goes to infinity. She’s not acting irrationally or incorrectly here.
The one world where she has won wins her $2**n, and that world exists with probability 0.6**n.
Her EV is always ($2**n)*(0.6**n), which is a larger EV (with any n) than a strategy where she doesn’t bet everything every single time. Even as n goes to infinity, and even as probability approaches 1 that she has lost everything, it’s still rational for her to have that strategy because the $2**n that she won in that one world is so massive that it balances out her EV. Some infinities are much larger than others, and ratios don’t just flip when a large n goes to infinity.
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.)