I don’t have a confident opinion about the implications to longtermism, but from a purely mathematical perspective, this is an example of the following fact: the EV of the limit of an infinite sequence of policies (say yes to all bets; EV=0) doesn’t necessarily equal the limit of the EVs of each policy (no, yes no, yes yes no, …; EV goes to infinity).
In fact, either or both quantities need not converge. Suppose that bet 1 is worth -$1, bet 2 is worth +$2, bet k is worth (−1)kk and you must either accept all bets or reject all bets. The EV of rejecting all bets is zero. The limit of EV of accepting the first k bets is undefined. The EV of accepting all bets depends on the distribution of outcomes of each bet and might also diverge.
The intuition I get from this is that infinity is actually pretty weird. The idea that if you accept 1 bet, you should accept infinite identical bets should not necessarily be taken as an axiom.
I don’t have a confident opinion about the implications to longtermism, but from a purely mathematical perspective, this is an example of the following fact: the EV of the limit of an infinite sequence of policies (say yes to all bets; EV=0) doesn’t necessarily equal the limit of the EVs of each policy (no, yes no, yes yes no, …; EV goes to infinity).
In fact, either or both quantities need not converge. Suppose that bet 1 is worth -$1, bet 2 is worth +$2, bet k is worth (−1)kk and you must either accept all bets or reject all bets. The EV of rejecting all bets is zero. The limit of EV of accepting the first k bets is undefined. The EV of accepting all bets depends on the distribution of outcomes of each bet and might also diverge.
The intuition I get from this is that infinity is actually pretty weird. The idea that if you accept 1 bet, you should accept infinite identical bets should not necessarily be taken as an axiom.