I had an extremely similar situation when we purchased our home last year. The home had an estimated valuation of 320k but because of the ridiculousness of the US housing market bids were going for upwards of more than 20k higher than the asking prices. We put in our bid for 341,100 - and we got it. I found out later that the next highest bid was exactly 340,100. That extra 1k in our offer made the difference for us.