We’re getting this infinity as a limit though, which means that we can approach the infinity case by perfectly reasonable cases. In the case of the St. Petersburg lottery, suppose that the lottery stops after N coin flips, but you get to choose N. In that case, you can still get your payout arbitrarily large by choosing N sufficiently high. “Arbitrarily large” seems like a well-behaved analogue of infinity.
In the case of the OP, I’m sure that if TimFreeman were a god, he would be reasonably accommodating about special requests such as “here’s $1, but please, if you’re a god, don’t flip the coin more than N times.” Suddenly, there’s no infinity, but by choosing N sufficiently high, you can make the arbitrarily large payout in the unlikely case that TimFreeman is a god counterbalance the certain loss of $1.
Okay, that is definitely more reasonable. It’s now essentially become analogous to a Pascal’s mugging, where a guy comes up to me in the street and says that if I give him £5 then he will give me whatever I ask in the unlikely event that he is God. So why waste time with a lottery, why not just say that?
I don’t have a really convincing answer, Pascal’s Mugging is a problem that needs to be solved, but I suspect I can find a decision-theory answer without needing to give up on what I want just because its not convenient.
The best I can manage right now is that there is a limit to how much I can specify in my lifetime, and the probability of Tim being God multiplied by that limit is too low to be worthwhile.
The reason the lottery is there is that you don’t have to specify N. Sure, if you do, it makes the scary infinities go away, but it seems natural that you shouldn’t improve your expected outcome by adding a limit on how much you can win, so it seems that the outcome you get is at least as good as any outcome you could specify by specifying N.
True, “seems natural” isn’t a good guideline, and in any case it’s obvious that there’s something fishy going on with our intuitions. However, if I had to point to something that’s probably wrong, it probably wouldn’t be the intuition that the infinite lottery is at least as good as any finite version.
We’re getting this infinity as a limit though, which means that we can approach the infinity case by perfectly reasonable cases. In the case of the St. Petersburg lottery, suppose that the lottery stops after N coin flips, but you get to choose N. In that case, you can still get your payout arbitrarily large by choosing N sufficiently high. “Arbitrarily large” seems like a well-behaved analogue of infinity.
In the case of the OP, I’m sure that if TimFreeman were a god, he would be reasonably accommodating about special requests such as “here’s $1, but please, if you’re a god, don’t flip the coin more than N times.” Suddenly, there’s no infinity, but by choosing N sufficiently high, you can make the arbitrarily large payout in the unlikely case that TimFreeman is a god counterbalance the certain loss of $1.
Okay, that is definitely more reasonable. It’s now essentially become analogous to a Pascal’s mugging, where a guy comes up to me in the street and says that if I give him £5 then he will give me whatever I ask in the unlikely event that he is God. So why waste time with a lottery, why not just say that?
I don’t have a really convincing answer, Pascal’s Mugging is a problem that needs to be solved, but I suspect I can find a decision-theory answer without needing to give up on what I want just because its not convenient.
The best I can manage right now is that there is a limit to how much I can specify in my lifetime, and the probability of Tim being God multiplied by that limit is too low to be worthwhile.
The reason the lottery is there is that you don’t have to specify N. Sure, if you do, it makes the scary infinities go away, but it seems natural that you shouldn’t improve your expected outcome by adding a limit on how much you can win, so it seems that the outcome you get is at least as good as any outcome you could specify by specifying N.
True, “seems natural” isn’t a good guideline, and in any case it’s obvious that there’s something fishy going on with our intuitions. However, if I had to point to something that’s probably wrong, it probably wouldn’t be the intuition that the infinite lottery is at least as good as any finite version.