Here's another reason why I think the poorly grounded but low probabilities can (at least generally be ignored).
Imagine I think the probability is 10^-10 but I'm highly uncertain. There should be some lower probability that I'm fairly certain is equal or higher than the real probability. Maybe it's 10^-20 - then I could this lower value …
Here's another reason why I think the poorly grounded but low probabilities can (at least generally be ignored).
Imagine I think the probability is 10^-10 but I'm highly uncertain. There should be some lower probability that I'm fairly certain is equal or higher than the real probability. Maybe it's 10^-20 - then I could this lower value for EV calculations. This seems much better than rounding to literal zero.
Suppose the mugger claims that they can realize *any* amount of value that they choose. What probability should you assign to the truth of this claim? I suggest it should be closer to zero than to any other number you can express.
I would probably assign the same value I do to other very implausible religious claims.
Here's an example of rounding down to zero, leading to bad EV results. Imagine your eccentric uncle dies. He gives a sealed box to John and to a trillion other people. He told you that he flipped a coin- if it was heads, he put all his 1 million dollar fortune in John's box. If tails, he put it in someone else's box. You assume that the probability it's in John's box is 0.5 and the probability for each other box is 1 in 2 trillion. If it was tails, you're not sure how he chose what box to put it in. If you could talk to his widow you might gain valuable information- perhaps he put in the box of neighbor. The probability for each box other than John's is low and very uncertain. You round down the probability for each other box to zero. Since probabilities add up to 1, you're confident that John's box has a million dollars. You offer to but it from him for $950K. But that's absurd.
Yes, you obviously shouldn't round down in that sort of case. Again, I was identifying *necessary* not *sufficient* conditions for rounding down. In any case where you obviously shouldn't round down, my response will simply be: "I agree! I don't endorse rounding down in that case."
What I'm doing in the OP is instead providing guidance for when you SHOULDN'T round down. It does not follow from this that in all other cases you SHOULD. Rather, other cases are simply left open, so far as the argument of this post is concerned.
Here's another reason why I think the poorly grounded but low probabilities can (at least generally be ignored).
Imagine I think the probability is 10^-10 but I'm highly uncertain. There should be some lower probability that I'm fairly certain is equal or higher than the real probability. Maybe it's 10^-20 - then I could this lower value for EV calculations. This seems much better than rounding to literal zero.
Suppose the mugger claims that they can realize *any* amount of value that they choose. What probability should you assign to the truth of this claim? I suggest it should be closer to zero than to any other number you can express.
I would probably assign the same value I do to other very implausible religious claims.
Here's an example of rounding down to zero, leading to bad EV results. Imagine your eccentric uncle dies. He gives a sealed box to John and to a trillion other people. He told you that he flipped a coin- if it was heads, he put all his 1 million dollar fortune in John's box. If tails, he put it in someone else's box. You assume that the probability it's in John's box is 0.5 and the probability for each other box is 1 in 2 trillion. If it was tails, you're not sure how he chose what box to put it in. If you could talk to his widow you might gain valuable information- perhaps he put in the box of neighbor. The probability for each box other than John's is low and very uncertain. You round down the probability for each other box to zero. Since probabilities add up to 1, you're confident that John's box has a million dollars. You offer to but it from him for $950K. But that's absurd.
Yes, you obviously shouldn't round down in that sort of case. Again, I was identifying *necessary* not *sufficient* conditions for rounding down. In any case where you obviously shouldn't round down, my response will simply be: "I agree! I don't endorse rounding down in that case."
What I'm doing in the OP is instead providing guidance for when you SHOULDN'T round down. It does not follow from this that in all other cases you SHOULD. Rather, other cases are simply left open, so far as the argument of this post is concerned.