Can Auction be played on a deed that is already Private?
There is an in depth discussion of costs vs. requirements by the rule team here. It boils down to this: the first sentence after the bold text &amp; colon is a cost and must be paid in order to use the card. Everything after that you try to execute, but if you are not able to do so just continue on. 

So for Auction, the cost is simply "Choose a deed in play." The only restriction in that sentence is "in play", so already Private deeds are fine. The rest of the text then executes. "It becomes Private" has no effect, but the text about gaining ghost rock executes as normal.