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Good luck.
We did this once and the estate agent told me what I would have to increase my offer to on the phone after he had opened all the offers and was calling me to tell me that mine wasn't high enough. I can only assume he did the same thing to all of the people involved, I declined to increase and avoided participating in any of these from then on - there's really no reason to assume that the agent will act as an independent adjudicator.
Thanks for the reality check, folks.
We can just about afford to add 5k to our budget and still be able to pay stamp duty and whatnot. Its a nice gaff so I think we're going to at least try one last option but at this point we've mentally let it go anyway.
Plenty of other stuff out there we can actually afford.