If you buy a house at "auction," can you put it on any property?
The rules are unclear on the exact procedure:
. Building Shortages

... If there are a limited number of houses and hotels available and two or more players wish to buy more than the Bank has, the houses or hotels must be sold at auction to the highest bidder.

My interpretation is that everyone entering the auction must announce, on entry, which Colour Group they are bidding for; and upon winning an auction must place the acquired house on that Colour Group.
Further, if an auction is being held for more than one house it is likely best that the players agree to hold separate auctions for each one.  Some interesting game theoretic concepts might arise, in this circumstance, when an auction is being held for multiple houses at a single occurrence; and it would be a shame for these to not come to light.

Update
To be clear, my proposal is:

A house auction begins when players have stated intention to buy more houses than there are houses available.

To have stated such intent, each player has, per force announced the Colour Group for which they are buying a house; and must have the capability of making such purchase from cash on hand (House Rule: and by mortgaging additional properties) but not by selling houses back to the bank. An invalid entry comprises Bankruptcy to the Bank - know your limits.

Auction price always begins at $1, regardless of what combination of Colour Groups have been announced by the auction Entrants for this House(s).

The number of houses at auction is the amount by which house demand exceeds supply. If there are 9 houses and four players announce intent to buy (for a maximum of 1 Colour Group each) 13 houses. than an auction begins for 13-9 = 4 houses.

A bid by a valid entrant (above) is final once made. If an Entrant bids an amount greater than they can immediately pay from cash and from mortgaging undeveloped properties on undeveloped Colour Groups that entrant is Bankrupt to the Bank and out of the game.

The Player winning each House Auction must place all houses won on any properties of the one Colour Group announced by that player as destination when entering the Auction; in accordance with the normal placement rules for houses on a Colour Group.


There are two deliberate strategic considerations explicitly available to the players here:

The last Player(s) into an Auction have increased ability to control the size of that Auction.

It is possible, and allowed, to attempt acquisition of houses more cheaply than normal through strategic auctions. The Players have an obligation to protect themselves against such activity by being in a position to compete in Auctions. this is no different from acquiring Properties cheaply in Property Auctions.