Hope this helps,
The single most important thing to the lender is the bottom line. Most banks have a set number that they MUST get for the property. If you present 5 offers lower than that amount the house will likely still get foreclosed on. The trick is finding out what that amount is. Then if you're lucky enough to have multiple offers, which isn't that uncommon in the short sale process, they'll get into the logistics of which offer is better. While more money is usually better, the lower cash offer (which usually goes without contigencies) can be very appealing.
But remember, the persons who approve an offer for a short sale are more often than not, a glorified machine. Meaning: Their hands are tied. They have a list of requirements made by their institution. They go down the list and make a decision. This is why it's amazing that getting a short sale approved can take so long!
In my experience, I think you'll find they typically go for the higher offer regardless of contingency unless there the cash offer is not really much lower than the financed one. What you will definitely find is that an offer with a Sale-of-Home contingency will always be sent to the bottom of the pile in a short sale.
Last night, I learned my buyer-with conventional financing - lost to a cash contract on a short sale. VA is les attractive than conventional