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Wei_Dai comments on indexical uncertainty and the Axiom of Independence - Less Wrong
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Thanks for adding an example. Let me rephrase it:
You have been invited to take part in a game theory experiment. You are placed in an empty room with three buttons labeled "1", "2" and "my room number". Another test subject is in another room with identical buttons. You don't know your room number, or theirs, but experimenters swear they're different. If you two press buttons corresponding to different numbers, you are both awarded $100 on exit, otherwise zero.
...What was so interesting about this problem, again?
I guess it's not, unless you're already interested in figuring out the nature of indexical uncertainty. If you're not sure what's interesting about indexical uncertainty, take a look at http://www.simulation-argument.com/ and http://en.wikipedia.org/wiki/Doomsday_argument.
I think what cousin_it was asking (and I would also like to know) is: what problem with the Axiom of Independence does the indexical uncertainty in your example (or cousin_it's rephrasing) illustrate?
Let A = "I'm immunized with vaccine A", B = "I'm immunized with vaccine B", p = probability of being the original. The Axiom of Independence implies
p A + (1-p) A > p B + (1-p) A iff p A + (1-p) B > p B + (1-p) B
To see this, substitute A for C in the axiom, and then substitute B for C. This statement says that what I prefer to happen at one location doesn't depend on what happens at another location, which is false in the example. In fact, the right side of the iff statement is true while the left side is false.
Does this explanation help?
That is based on the unspoken assumption that you prefer A to B. You yourself explained that such a preference is nonsense:
If an axiom or theorem has the form "If X then Y", you should demonstrate X before invoking the axiom or theorem.
Yes, I meant to ask exactly that.