This is a real-life problem around me...
I, with two roommates, am going to rent a house of price $T$. The house rented has three rooms, one of them is larger and two of them are smaller. We agree that the rent of the smaller rooms are the same and less than the larger one. We want to do some "auction" process to decide who and how much to rent the larger room. Everyone has a (distinct) payoff of $p_{B,i}$, which is the payoff of the largest room, and $p_{S,i}$, which is the payoff of the smaller room, for $i=1,2,3$. If the rent of the larger house is $R$, the rent of the smaller house is $(T-R)/2$. So, the person $i$ who has the larger room has utility $p_{B,i}-R$ and has the utility of $p_{S,i}-(T/2-R/2)$ if he has a smaller room. Now we would like to design an auction method such that every person gives a price $P_i$ for $i=1,2,3$, and then select $R=R(P_1,P_2,P_3)$. We want to make this auction good if everyone bid a "good" price (that is, everyone will get a worse payoff if boasted the price from a rational estimation, say $p_{B}$ or $T-2p_{S}$ or something.) How could I achieve this?
Here are some of the objectives and rules:
We pay the rent in front, and we only pay once. We will correctly pay the amount on time, not lagging. Once paid, there is no regret. We will not consider future house prices, etc.
We will pay before living in there (but we have looked at the house before) so I hope everyone will say the truth and will not regret it afterward (and then complain why the rent is unreasonable)... that is, I want the bidding procedure to forbid the boasting or shrinking the price.
I hope the bidding is one-round bidding since my two roommates are not that patient...
We will be clear about the price, that is, at least we can say $0<p_{{S,i}}<T/3$ and $T/3<p_{{B,i}}<T$. Also, we will give a price in the range $T/3<bidding<T$.
Ignore the case when two/three of us give the same highest bid.
At least up to now, we all want the larger room.
I have learned game theory somehow and know the first-price and second-price auctions... However, this payoff is not the same as the auction. Since the other person pays more means that I can pay less... It is a kind of zero-sum (or fixed sum). This problem may be too broad but feel free to add restrictions and assumptions.
You can do it the way Germans bid in their beloved Skat game: two of you bid, and only once a winner is established, the third person enters the bidding. This won't save you money, but it will save you time and some math (if that is a good thing). In case this procedure feels biased, repeat it twice with the third man changed, and see if any of you were to change their highest bids. It shouldn't be expected, but if it does happen, you'd have improved as a group of three towards an even better solution (less rent for the auctions losers) - and wouldn't that be a great thing starting this period of life?