Bet A If H is true, you lose $10. Otherwise you win $15.
Bet B If H is true, you win $15. Otherwise you lose $10.
First I’m going to offer you Bet A. Immediately after you decide whether to accept Bet A, I’m going to offer you Bet B.
Can’t the sequence proposal be fixed by conditioning on the past and only considering future sequences of actions? Committing to rejecting both bets A and B is rationally impermissible if you will be offered both since it’s worse than accepting both, but after your decision on A, regardless of whether you accepted or rejected, then it could be that both accepting and rejecting B are permissible at the same time. The fact that something was my past action shouldn’t matter or prevent me from completing some particular sequence of actions that includes past actions, only my future prospects and future actions matter.
I think this makes sense for sharp probabilities, too: suppose you assign some sharp probability p≤25 to H being true, and have already rejected A, even though it had positive expected value (so this decision was irrational at the time). Then, since the expected value of B is ≤0, it’s permissible to reject B, and even required if the inequality is strict. You may be rationally required to complete a sequence of actions which was irrational before you started.
You can also apply Mogensen’s maximality rule to sequences. Given some set of plausible probability distributions, if one sequence of actions θ is better in expectation than another ϕ under at least one distribution, and not worse under any other distribution, then θ≻ϕ. If neither strict inequality holds between the two options and these are the only two options, then both are permissible. (We sacrifice the independence of irrelevant alternatives, since a third option could dominate one but not the other, only ruling out the dominated one.)
Hello,
Here is a good paper on this - https://www.princeton.edu/~adame/papers/sharp/elga-subjective-probabilities-should-be-sharp.pdf
Can’t the sequence proposal be fixed by conditioning on the past and only considering future sequences of actions? Committing to rejecting both bets A and B is rationally impermissible if you will be offered both since it’s worse than accepting both, but after your decision on A, regardless of whether you accepted or rejected, then it could be that both accepting and rejecting B are permissible at the same time. The fact that something was my past action shouldn’t matter or prevent me from completing some particular sequence of actions that includes past actions, only my future prospects and future actions matter.
I think this makes sense for sharp probabilities, too: suppose you assign some sharp probability p≤25 to H being true, and have already rejected A, even though it had positive expected value (so this decision was irrational at the time). Then, since the expected value of B is ≤0, it’s permissible to reject B, and even required if the inequality is strict. You may be rationally required to complete a sequence of actions which was irrational before you started.
You can also apply Mogensen’s maximality rule to sequences. Given some set of plausible probability distributions, if one sequence of actions θ is better in expectation than another ϕ under at least one distribution, and not worse under any other distribution, then θ≻ϕ. If neither strict inequality holds between the two options and these are the only two options, then both are permissible. (We sacrifice the independence of irrelevant alternatives, since a third option could dominate one but not the other, only ruling out the dominated one.)