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.
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.