How I Learned to Stop Worrying and Love the Anthropic Trilemma

My impression was that the Anthropic Trilemma was Eliezer uncharacteristically confusing himself when reality itself didn't need to be.

After step one, you have a 50% chance of finding yourself the original; there is nothing controversial about this much. If you are the original, you have a 50% chance of finding yourself still so after step two, and so on. That means after step 99, your subjective probability of still being the original is 0.5^99, in other words as close to zero as makes no difference.

Joe is just confused about the math. Multiplying the 0.5s together like that is the wrong thing to do. The subjective expectation I would have after being copied for the 99th time is that I have 50% chance of being told I am a copy.

Perhaps Joe's confusion is not fully specifying what his expectation is before the process starts. He expects that after he is cloned:

• p(there is an original) = 1
• p(there is a clone) = 1
• p(after one clone is made the original will expect to be told that he is a clone) = 0.5
• p(after one clone is made that clone will expect to be told that he is a clone) = 0.5
• p(after two clones are made there will be an original) = 1
• p(after two clones are made there will be a first clone) = 1
• p(after two clones are made there will be a second clone) = 1
• p(after two clones have been made the original clone will expect to be told that he is the second clone) = 0.5
• p(after two clones have been made the first clone will expect to be told that he is the second clone) = 0 (The first clone doesn't expect to be told anything more.)
• p(after two clones have been made the second clone will expect to be told that he is the second clone) = 0.5

Now, consider multiplying

• p(after two clones have been made the second clone will expect to be told that he is the second clone) * p(after one clone is made that clone will expect to be told that he is a clone).

Why would I do that? That makes no sense. That would give me 0.25 but so what? It certainly doesn't mean that p(after two clones have been made the second clone will expect to be told that he is the second clone | after one clone is made that clone will expect to be told that he is a clone) = 0.25. If Joe thinks you can do that he is just wrong and so will end up making a decision that doesn't optimally reflect his preferences.