Maybe I just don't get it, but offering me the option AFTER you've told me that it makes no difference makes it a pointless option. I get the feeling there's a single step missing from your explanation.

From what I'm reading, there's 3 things that can happen...

You spend $100. A prophet comes to you and tells you that you will lose $10,000 in the future, and then...
1) looks at you more closely, and coughs "wait, you're not the person I was looking for."
2) tells you something that sounds plausibly true, but turns out to be false, costing you $10,000 by overpaying for your next house.
1 & 2 happen with 50% chance, if you have spent $100.
If you don't spend $100, then
3) A prophet comes to you and tells you that you will lose $10,000 in the future, and then afterwards as you sputter "why", he tells you this plausibly true thing that turns out to be false, costing you $10,000 by overpaying for your next house.

(As for what the thing is, it's either something that makes you spend $10,000 in carefully rationalizing your decision to buy a house, or $10,000 costs in overbidding)

But... you've just told me that a prophet came to me and told me I will lose $10,000 in the future. I am already on path 3. There is no going back. Time CAN create time loops but there is no cause for it to do so in your explanation. You yourself walled it off by stating the prophecy was self-fulfilling and that you could spend $100 "if the prophecy weren't immutably correct" (this in a manner implying that it is immutably correct).

You have given me a button, but the button is disabled. I can't take any actions.

Also, there's something lurking in your description which might (I really am unsure) imply that if I spend $100, the world may become inconsistent and therefore stop disappearing. Basically replace path 1 with "universe ends." Which would make spending $100 really bad, since, losing $10,000 is preferable to destroying your own universe?