If you get a property under contract at the wrong price, then you can always back out of the deal based on your financing or inspection contingency with virtually no risk.   

When doing an offer the ways you can beat out the competition besides just being the highest is to offer to close quickly, 2 weeks is standard, or offer all cash, offer to have no contingencies (Inspections, Etc).

Maybe a shorter inspection contingency?

Did you leave any contingency "outs" in your purchase agreement? 

For more information, see HUD Contingency Plan for Possible Lapse in Appropriations.

Put the properties under contract - make sure you put a contingency on the contract that allows you to back out of the deal or renegotiate if you don't have a buyer within the time period that you have the contract with the seller.5.

What suggestions do you have for contingencies on a multi-family offer aside from financing, property inspection, verifying paperwork (leases, actual expenses, work performed)? 

Even though your offer was cash, there could be a loan contingency offer with a higher net. 

Make the offer contingent on mold inspection. 

If the deal numbers are good enough to provide a contingency for them if they have to repo the property, then they will still do the deal usually.