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Mitchell_Porter comments on Explicit Optimization of Global Strategy (Fixing a Bug in UDT1) - Less Wrong
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This was my thought process: To get the $10, my copy and I have to choose differently. I am 1, he is 2. I have to choose A or B... At some point I thought of the mapping A=1, B=2, implicitly as part of the bigger mapping (A...Z)=(1...26) I suppose. I noticed that this was a particular mapping which had spontaneously presented itself to me. So it must be a natural one for me to think of; so there is a good chance my copy will think of it as well. So I select A, hoping my copy went through an analogous process, arrived at the same mapping and selected B.
Here you are relying on omega using two ordering systems that we already find highly correlated.
What if Omega asked you to choose between a blegg and a rube instead of A and B. Along with that, Omega tells you that it did not necessarily use the same ordering of blegg and rube when posing the question to the copy.
EDIT: More thoughts: If you can't rely on an obvious correlation between the player labels and choices, why not have a strategy to make a consistent mapping from the player labels to the choices.
The key to winning this game is having both parties disagree. If both parties know the goal and have a consistent mapping process, it would be trivial for them to arrive at different choices.
A simple mapping would be alphabetize the player labels and the choice labels. Player(1) => choice(1), Player(2) => choice(2), Player(n) => choice(n).
Lexicographic ordering is indeed the most obvious one here.