Have you ever bought or sold a home? Unless the buyer is buying in cash, or at least has large cash reserves to give to you at closing, they can only afford up to whatever the bank approves them for, and lenders don't hand out $400k loans for properties valued at $350k just because "the seller made a hard bargain and the buyer agreed". If they did, they'd be upside down the second after closing because a foreclosure means the lender/bank can only sell the house for $350k less than what the buyer owes (and they don't tend to stake their foreclosure sales on whether or not they find another oblivious buyer willing to pay way over market for a house that they could go across the street and get for much cheaper).
Even if you do that, do you assume they have the extra money to pay you in savings? Because the lender for sure isn't increasing their liability for a house they'd lose money on if the buyer skips the first month. Also, this is illegal if you already have a PSA, so I assume you're negotiating before any sale goes pending, in which you'd probably be better off with a regular two-week bidding window and bidding war anyways.
