Should I accept this unfunded offer from my dream school or wait for funded offers? I just finished the grad school application process, and I'm getting my first letters of acceptance. I got one letter from one of my bottom three schools, one from a &quot;middle of the pack&quot; school, and one from a dream school. And, at the risk of humble bragging, this is a great school with a great reputation and true luminaries in my specific research interest amongst its faculty. I literally screamed when I got the letter. However, it is not funded.
They sent me the letter of acceptance only three weeks after I submitted my application. As I understand it, that's a very quick turn-around time, especially for a prominent research institution. I assume that's a very positive thing.
I'm not sure whether to just accept the offer right away, or wait to see if some of the other dream schools respond. At least one of the other dream schools will have funding for me from the get-go if I'm accepted. I know &quot;non-funded for any amount of time&quot; is a big red-flag, but I feel the pedigree of the school that I got accepted to trumps that. Having good recommendations from any of the professors I would work under would be worth more than gold. I feel like, in this situation, you just bite the bullet and go for it.
But I also feel that if you might have basically the exact same opportunity, except with funding, at another institution or two, it might be worth waiting/holding out a bit to see how things play out.
I feel left with a few questions:

Is it stupid of me to hold out at all, given the high level of the institution I received the letter of acceptance from? I worry my ego might have got too inflated, and it's wrong to  hope that another dream school (with or without available funding) will come knocking at my door.

If I do accept the offer, but then receive an offer from another dream institution that could offer me funding, would it be bad form to pull out of my agreement of acceptance? I wouldn't want bad blood.

How long is too long to wait on accepting the offer? Will they fill the open position if I take too long to respond? I don't want to miss out on what could be the opportunity of a lifetime because I sat on my hands.
Unless you are independently wealthy, you should never take an unfunded offer for graduate school in mathematics.
Even before the pandemic, some people with a dozen good papers after two successful postdocs - a well above average publication rate - were failing to find jobs.
When I was last broadly applying for academic jobs 10 years ago, the general advice was to send 80-100 applications to jobs you were competitive for in order to have reasonable certainty of getting at least one offer.  Even before the pandemic, there weren't that many jobs in total to apply for, and each job gets more applicants than they did 10 years ago.  (This year is brutal; the only people getting research-oriented jobs will be AMS Sectional plenary lecture candidates (there are roughly 40 per year, total - that's less than the number of Berkeley+Stanford PhDs per year) or better.)
There are even fewer jobs in industry for which a math PhD is a better qualification than several years of experience.
For a more teaching-oriented academic job, the funded position at a lower-ranked graduate school is actually better preparation, because you will be teaching for your funding, and because the undergraduate population at a lower-ranked school is generally more similar to the undergraduate population at most universities than the undergraduate population at a highly-ranked school.
If you borrow money for graduate school, you could end up in a situation where you are unemployed with no means of paying back your loans.  Don't forget education loans cannot be discharged via bankruptcy.
Going to graduate school in math with funding is already a costly decision from the financial point of view, though it's fine if getting to spend a few years doing math is worth it for you (as it was for me).  Going to graduate school in math unfunded is potentially ruinous.