I lived and prospered from the same exact scenario...
It was July 2011 we had put the condo up for sale 2 years earlier. We moved 50 miles and were renting a house in the new town while waiting for the condo to sell. It took 2 years but the condo sold. We were now renting month to month so we were very flexible since we now owned nothing. Having sold the condo, we had money, we were fully pre-approved...all we had to do was buy. In the realty world, we were "real buyers" not shoppers. We had been casually working with the same buyer's agent for a long time but now with the condo sold we were fully engaged.
We are generally not liking what we are seeing in our price range. It was a respectable price range but inventory was crap during the house crisis days. What was out there was either foreclosure or priced so high because the owed so much. The agent then come to us one day with what was generally "unscrupulous" activity that worked in our favor. Seems that another agent in his office had just taken in a super low offer on a property and the offer was being entertained.
The house had recently come down in asking price by 15%. This offer was 43% off the original price.
We had found out the price of guy #1's offer. We also knew it was contingent upon #1 selling his house (what had not yet gone up on the market.) We submitted a formal back up offer for the same price but that we could close in 20 days from acceptance.
Guy #1 scrambled. Though we never spoke with #1 we heard he was making promises like, "I'll own 2 houses at the same time." However, in those days, it didn't really matter who you were, you weren't getting 2 mortgages. So, 30 days later, guy #1 still couldn't secure financing and the house went to us, the back-up offer.
Just so happened that guy #1 was the then reigning President of the County Club. The Club which we now live 40 ft off the 18th fairway. Needless to say, we never got the "welcome postcard" from The Club like you might expect in this situation. We are not in an association and not bound to "having to" be part of the club in any way.
When I first met mey new neighbor, the one with the 5,000 sqft house, he said, "I knew someone else looking at that house." My only answer was, "I don't know, I'm not from here." That was my attempt at being both kind as well as showing, "I'm not part of your small town politics and though he may be important to you, he is an unknown to me."
Heck, I'd rather just give guy #1 a big thank you. If not for him, I would have never even looked at this house which was way out of my price range (at asking price.)
So, if you like the house, put in a purchase offer. That will be the back up-offer. Let the seller decide which offer they like better.