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cousin_it comments on indexical uncertainty and the Axiom of Independence - Less Wrong
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Yes, they do. In this case you just got lucky and the probabilities factored out of the calculations. The general case where they don't necessarily factor out is called evolutionary game theory: indexical probabilities correspond to replicator frequencies, utility corresponds to fitness.
I need to brush up on evolutionary game theory, but I don't see the correspondence between these two subjects yet. Can you take a standard puzzle involving indexical uncertainty, for example the Sleeping Beauty Problem, and show how to solve it using evolutionary game theory?
Hmm, I don't see any problem in that scenario. It doesn't even require game theory because the different branches don't interact. Whatever monetary rewards you assign to correct/incorrect answers, the problem will be easy to solve by simple expected utility maximization.