The decision depends on a priori probability of situations described in the thought experiments (two situations: PH and Anti-PH). The constraints force (winning in PH xor winning in Anti-PH), there are two options to choose from: (win in PH, lose in Anti-PH) and (lose in PH, win in Anti-PH). The value of either option is a weighted sum of values in PH and Anti-PH of its respective components, with weight given by their (relative) a priori probability. Since the payoffs in PH and Anti-PH are the same, the situation with higher probability calls the winning strategy overall.

More generally, for any situation described in a thought experiment, there is another thought experiment with negated payoffs. It doesn’t matter because by convention when a thought experiment is described, it’s implicitly given more a priori probability than all other related thought experiments combined.

So in case of Anti-PH given as a thought experiment, it would implicitly hold more weight than PH, thus the correct decision is to lose in PH. But in case of PH, the probabilities are the other way around, and the correct decision is to win in PH. The paradox is explained by these thought experiments not just being different possible situations, but implying different a priori distributions over all situations, including each other. They don’t exist in the same world, even though in their respective worlds the other situation is present, with lower a priori probability than in its own world.

The decision depends on a priori probability of situations described in the thought experiments (two situations: PH and Anti-PH). The constraints force (winning in PH xor winning in Anti-PH), there are two options to choose from: (win in PH, lose in Anti-PH) and (lose in PH, win in Anti-PH). The value of either option is a weighted sum of values in PH and Anti-PH of its respective components, with weight given by their (relative) a priori probability. Since the payoffs in PH and Anti-PH are the same, the situation with higher probability calls the winning strategy overall.

More generally, for any situation described in a thought experiment, there is another thought experiment with negated payoffs. It doesn’t matter because by convention when a thought experiment is described, it’s implicitly given more a priori probability than all other related thought experiments combined.

So in case of Anti-PH given as a thought experiment, it would implicitly hold more weight than PH, thus the correct decision is to lose in PH. But in case of PH, the probabilities are the other way around, and the correct decision is to win in PH. The paradox is explained by these thought experiments not just being different possible situations, but implying different a priori distributions over all situations, including each other. They don’t exist in the same world, even though in their respective worlds the other situation is present, with lower a priori probability than in its own world.