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cousin_it comments on indexical uncertainty and the Axiom of Independence - Less Wrong
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Your analysis looks correct to me. But if Wei Dai indeed meant something like your example, why did he/she say "indexical uncertainty" instead of "amnesia"? Can anyone provide an example without amnesia - a game where each player gets instantiated only once - showing the same problems? Or do people that say "indexical uncertainty" always imply "amnesia"?
Amnesia is a standard device for establishing scenarios with indexical uncertainty, to reassert the fact that your mind is in the same state in both situations (which is the essence of indexical uncertainty: a point on your map corresponds to multiple points on the territory, so whatever decision you make, it'll get implemented the same way in all those points of the territory; you can't differentiate between them, it's a pack deal).
Since the indexical uncertainty in the example just comes down to not knowing whether you are going first or second, you can run the example with someone else rather than a past / future self with amnesia as long as you don't know whether you or the other person goes first.
That's true, but that adds the complication of accounting for the probability that the other person presses Y, which of course would depend on the probability that person assigns for you to press Y, which starts an infinite recursion. There may be an interesting game here (which might illustrate another issue), but it distracts from the issue of how indexical uncertainty affects the Axiom of Independence.
Though, we could construct the game so that you and the other person are explicitly cooperating (you both get money when either of you press the button), and you have a chance to discuss strategy before the game starts. In this case, the two strategies to consider would be one person presses X and the other presses Y (which dominates both pressing X), or both press Y. The form of the analysis is still the same, for low probabilities, both pressing Y is better (the probability of two payoffs is so low it is better to optimize single payoffs), and for higher probabilities, one pressing X and one pressing Y is better (to avoid giving up the second payoff). Of course the cutoff point would be different. And the Axiom of Independence would still not apply where the indexical uncertainty makes the probabilities in the game different despite the raw probabilities of the buttons being the same under different conditions.