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Sniffnoy comments on Timeless Decision Theory: Problems I Can't Solve - Less Wrong
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Strategies are probability (density) functions over choices. Behaviors are the choices themselves. Proving that two strategies are identical (by symmetry, say) doesn't license you to assume that the behaviors are the same. And it is behaviors you seem to need here. Two random variables over the same PDF are not equal.
Seldin got a Nobel for re-introducing time into game theory (with the concept of subgame perfect equilibrium as a refinement of Nash equilibrium). I think he deserved the prize. If you think that you can overturn Seldin's work with your TDT, then I say "To hell with a PhD. Write it up and go straight to Stockholm."
...unless the resulting strategies are unmixed, as will usually be the case with Prisoner's Dilemma?