Every property and asset manager is different. I have had clients place and win offer on two exact same properties. One appraisal was accepted, the other declined. If there were a lot of offers on a popular listing, they may decline the appraised value if it was too much lower than the list price in order to accept a cash one for more. The cash offer does not require an appraisal, and the negotiated price is final. Remember, they do not disclose the number of offers or the amount of each received.
Keep in mind that Fannie Mae is also financing home purchases with their very own appraisals. They don't want other lenders denying their own borrowers with similar appraisal issues. However, they understand that the seller's goal is to market and obtain the best possible price for the property they are selling.
Good house hunting!
Florida Real Assist