How to design a cost sharing auction format for collective bidding? Problem goes like this: There's one resource which can only be utilized by a single set of agents $A_i$ (at any one time) out of $n$ predefined (disjoint) sets of agents. Each agent wants to use the resource but wants to minimize its own cost (and doesn't care about the costs of others in its own set).
Example situation: Several choirs want to use the same hall for practice. Only one can use it at one time and no-one will pay more than it is worth for themselves.
Any bidding strategy is valid. The owner of the resource wants to maximize profit. 
The question is: 
Are there optimal auction formats for the owner? What are they? Are there auction formats that allow for "fair" payment for each agent (fair as measured by the utility of using the resource for that agent)?
EDIT: When I am talking about fairness I mean that each agent will try to contribute as close to their utility as possible so that the whole choir bids their reserve amount (or close to...). 
 A: You could try a system where the individual agents can bid for their groups bid.
It will be a lot of adminstration
example:
suppose an open bidding system where only the total price per choir is visuable 
Choir A  consist of Agent 1 and 2
Choir B  consist of Agent 3 and 4
The least bid is 5
agent 1  bids 5 so the lot lays now at choir A for 5 
agent 3  bids 6 so the lot lays now at choir B for 6
agent 2  bids 5 so the lot lays now at choir A for 10 (A1:5, A2:5) 
agent 4  bids 5 so the lot lays now at choir B for 11 (A3:6, A3:5)
agent 1  raises to 7 so the lot lays now at choir A for 12  (A1:7, A2:5)
agent 4  raises to 11 so the lot lays now at choir B for 17 (A3:6, A3:11)
agent 1  raises to 9 now lot still lays now at choir B for 17 (A3:6, A3:11)
agent 2  raises to 9 so the lot lays now at choir A for 18  (A1:9, A2:9)
and so on 
Problems are:
- the organiser need to keep track of all bids made by the agents, which agents are part of which choir and so on.
- what if some agents leave a choir and so. 
- how will the winning choir pay (they have all promissed to pay there bid, but what if some of them won't pay) 
- what to do with the overbid (the amount what is bid but not in the price, maybe easiest is to reduce the amount of the last agent to bid by this amount) 
- what if there is more than two choirs, then the highest bid of lowest bidding choir is not visuable, but there are ways around this.
For optimalisation:
- allow bids of the choir as Agent.
- bids don't have to raise the total bid above the winning bid.
- publish the averige bid per member of each choir 
- after each bid increase the time for the action to end (something the auction ends after 2 weeks or two days after the last agents  bid, whichever is later)
- do it like ebay, the bid is only revealed to the second highest bid plus some offset.
I don't think that there auction formats that allow for "fair" payment for each agent, maybe the only way to do this is that every member need to pay at least some part of the average bid of that choir, but what if the choirs are of unequal size, and is it fair to force some agents to pay more than that they have bid?
