So, let me try again to explain why I think this is missing the point… I wrote “a single probability value fails to capture everything you know about an uncertain event.” Maybe “simple” would have been better than “single”?
The point is that you can’t solve this problem without somehow reasoning about probabilities of probabilities. You can solve it by reasoning about the expected value of different strategies. (I said so in the OP; I constructed the example to make this the obviously correct approach.) But those strategies contain reasoning about probabilities within them. So the “outer” probabilities (about strategies) are meta-probabilistic.
[Added:] Evidently, my OP was unclear and failed to communicate, since several people missed the same point in the same way. I’ll think about how to revise it to make it clearer.
So, let me try again to explain why I think this is missing the point… I wrote “a single probability value fails to capture everything you know about an uncertain event.” Maybe “simple” would have been better than “single”?
The point is that you can’t solve this problem without somehow reasoning about probabilities of probabilities. You can solve it by reasoning about the expected value of different strategies. (I said so in the OP; I constructed the example to make this the obviously correct approach.) But those strategies contain reasoning about probabilities within them. So the “outer” probabilities (about strategies) are meta-probabilistic.
[Added:] Evidently, my OP was unclear and failed to communicate, since several people missed the same point in the same way. I’ll think about how to revise it to make it clearer.