I would contact the other seller and make sure that your buyer isn't bidding on that one too.
But it was a "private" auction, bidders identities protected.
I don't understand the thinking behind making it private, the only people apparently it is actually private for are other bidders, why keep other bidders from knowing who else is bidding against them?
I don't understand how he is able to see if his "private"bidder is bidding on another seller's auction.
I also don't understand all these bidders with hundreds of wins on their history that bid repeatedly in the opening days of an auction...just driving up the price. That's the eBay novice approach, not the behavior you'd expect from experienced bidders.
I do understand contract law, where if you present an offer and it is accepted without modification, then there's a deal.