In translating your opening, I'm reading that you've just bought an REO home, and you learned that 1. REO's are sold quickly; and 2. You have to make a solid offer in order to become the winning bidder.
Here are a few things that I'm seeing from your letter, and please let me know if I'm wrong.
It sounds like you're finding homes that are investor (bank) owned, and you're asking the listing agent to make offers. Either that or you have agents in many different communities looking for homes for you. Those are the only ways that I can figure out why you have so much experience with different agents.
Additionally, it seems that you may be making low offers on each of these homes, if you've made 11 offers and finally won a bid and bought a home.
That being said, I have some thoughts for you. 1. If you're looking for a home, pick a Realtor who is a Buyer's Agent (those representing the Seller cannot be a Buyer's Agent for you on the same home) and stick with that person. If you're bouncing from Realtor to Realtor, you will not find that any of htem get to know you, nor will you find that any of them will go out of their way for you. You can't expect any sort of commitment from a Realtor, if you won't commit to him/her.
2. Contrary to popular belief, when a bank owned home comes on the market, it's usually priced low enough to get many offers, which means that you will not be able to low ball the price with your offer. You will need to make the best offer you can make, and hope for the best. That being said, with the way this spring is turning out, with low inventory but many buyers, other homes are selling the same way - with multiple offers. The tide is shifting, and no longer is it a Buyer's market.
If you have already purchased your home, then I'm happy for you. If you're still looking, find a good Buyer's Agent and ask him/her to work for you. Tell them you'll work exclusively with them, then LISTEN to their advice on purchasing, and you'll have much more luck. Best of luck for you!
What is your background and experience? It sounds like you are in a job transition or able to make one. I was in the same boat. I was a professional poker player at the time and got into this business when the financial crisis hit and everyone else was fleeing the industry. I had a real strong background in corporate and start-ups on the sales, management, and marketing side. I had previously closed 49 loans as a mortgage banker, so I was not new.
If you want to talk in detail send me an email or feel free to call.
Tim Klien had a great answer though.
You will need to run and should run comparables if your search area is that large.