Because there might be more to uncertainty than subjective probability.
Let’s take a step back.
Yes, if you assume that uncertainty is entirely captured by subjective probability, then you’re completely right. But if you assume that, then you wouldn’t need the Savage axioms in the first place. The Savage axioms are one way of justifying this assumption (as well as expected utility). So, what justifies the Savage axioms?
One suggestion the original poster made was to use Dutch book arguments, or the like. But now here’s a situation where there does seem to be a qualitative difference between a random event and an uncertain event, where there is a “reasonable” thing to do that violates P2, and where nothing like a Dutch book argument seems to be available to show that it is suboptimal.
I hope that clarifies the context.
EDIT: I put “reasonable” in scare-quotes. It is reasonable, and I am prepared to defend that. But it isn’t necessary to believe it is reasonable to see why this example matters in this context.
Well, that sounds irrational. Why would you pay to switch from X to U, a change that makes no difference to the probability of you winning?
Because there might be more to uncertainty than subjective probability.
Let’s take a step back.
Yes, if you assume that uncertainty is entirely captured by subjective probability, then you’re completely right. But if you assume that, then you wouldn’t need the Savage axioms in the first place. The Savage axioms are one way of justifying this assumption (as well as expected utility). So, what justifies the Savage axioms?
One suggestion the original poster made was to use Dutch book arguments, or the like. But now here’s a situation where there does seem to be a qualitative difference between a random event and an uncertain event, where there is a “reasonable” thing to do that violates P2, and where nothing like a Dutch book argument seems to be available to show that it is suboptimal.
I hope that clarifies the context.
EDIT: I put “reasonable” in scare-quotes. It is reasonable, and I am prepared to defend that. But it isn’t necessary to believe it is reasonable to see why this example matters in this context.