Simply put, the bank will accept the highest and best offer they can get.
I don't know where Vicki got the number $48K. Vicki, please feel free to explain your calculation, maybe I'm missing something.
On a short sale transaction, the bank usually has had an appraisal completed so they know what the comps are and will hold out for the deal which comes closest to that number, within a reasonable timeframe.
There are no fast and hard equations that banks use to authorize short sales. One of the most challenging aspects of the short sale process is the complexity and uniqueness of each transaction.
For example, If a seller is late on their mortgage 3 months, a bank will be more motivated to accept a reduced payoff (short sale) than if the seller is making payments. If a property has been on the market for a long time with no offers, the bank will be more motivated. If the market value is much lower than the underlying mortgages, the bank will be more motivated. There are also an incredible number of intangibles. How many deals is the Loss Mitigation/Asset Manager dealing with (workload), how are they compensated (motivation), etc. Put yourself in the banks shoes and you can better see how they are motivated.
Vicki raises a great point. How did you determine value? How recent are the comps you are using? Are they going to be similar to those they received from their appraiser?
If you must have the property, place your highest and best. Even then, if you miss the mark on analyzing the comparables, you stand the risk of being beat by another buyer.
You should discuss all of this with your agent.
Best of Luck.