fighting the hypothetical

It's established in the problem statement that the experimenter is going to destroy or falsify all records of what transpired during the game, including the fact that a game even took place, presumably to rule out cooperation motivated by reputational effects. If you want a perfectly honest and trustworthy experimenter, establish that axiomatically, or at least don't establish anything that directly contradicts.

Assuming that the other party is a clone with identical starting mind-state makes it a much more tractable problem. I don't have much idea how perfect reasoners behave; I've never met one.