This paper by Demange and Gale talks about two-sided markets where agents from both sides are to form partnerships involving maximum satisfaction and possibly with monetary transfers. Auction is typically one such market, and the examples are based on auctions.

The paper considers the result with respect to a basic model and then discusses a few extensions of the model. The basic model is same as the multi-item auctions in which there are a set of objects to be distributed to a set of buyers, where no buyer is interested in buying more than one object. Each buyer has a demand function, which means that given a price vector, each buyer is able to specify the set of objects they would like to have. Sellers also have a supply function, meaning at a given price vector, they are able to specify which object they are willing to sell.

An example of a demand function will be "Buyers demanding the objet(s) which maximize the excess of their (non-negative) valuations", and similarly a supply function will be "Sellers selling objects whose price exceed their valuations". Their can be other demand/supply functions as well.

Given the supply and demand functions, we talk about equilibrium prices, which are "price vectors such that every bidder can be assigned an item from his demand set, and no two bidders get the same object". This papers discusses some important properties of equilibrium prices:
1. The equilibrium price vectors form a lattice, and there exists a smallest and largest equilibrium price vector.
2. Advantages of using 'minimum price equilibrium' for allocating objects, in particular the mechanism is nonmanipulable for buyers. There is a assumption: there is no monetary transfer between agents on the same side of the market.
3. Specifying the supply functions appropriately, the sellers can force the payoff to be given by maximum rather than the minimum equilibrium price. These strategies yield a Nash Equilibrium.

The authors then suggest some general models where the above properties/results apply:
(a) Sellers might discriminate, offering better price to some buyers than others. (See *N1).
(b) Matching of students to universities, with the possibility of financial aids of different amounts.
(c) Matching brides with grooms for marriage, with possible dowries from both sides (this is a symmetric model since both direction of payments are possible).

What follows are some examples in which the above results hold: We note, in each of the cases below, a buyer is interested in buying only one item.

SINGLE ITEM AUCTIONS
In this simplest form, there is one object (and one seller of the object) and n buyers. If the ordered set (v_1, v_2, .. v_n), v_1 >= v_2 etc, are the valuations of buyers and u is the valuation of seller, then a price p will be equilibrium iff v_1 >= p >= max[v_2, u]. (This is equilibrium because this is the only price when it can be allocated to only one bidder.)
* Min. eqlbm. price: max[v_2, u] (Corresponds to English Auction). This is non-manipulable for bidders, even if they know the second highest valuation (v_2). To see that, we consider following situations:
i. The winning bidder (bidder 1) states a valuation greater than v_1: No point, as he will still win.
ii. Winning bidder states a valuation less than v_1: He may loose the item, if his valuation is less than v_2. Otherwise doesn't matter.
But, the seller can manipulate if she knows v_1. In that case, she can quote her valuation v to be arbitrarily close to v_1, and thus force the winning bidder have less profit (and thereby increasing her own profit).
* Max. eqlbm. price: v_1 (Corresponds to Dutch Auction). This is non_manipulable for sellers with same argument, while bidders can manipulate.
We note, these results don't hold if there are transfer of money between buyers, in which case the winning bidder could bribe other bidders to lower their bids below v (or bid arbitrarily close to zero if v=0), and thereby gain more profit. So, these mechanisms are not 'group-strategyproof'.

MULTIPLE (HETEROGENEOUS) ITEMS, SINGLE BUYER
Let's consider 4 different objects with sellers' and buyer's valuations (u_j and v_j) as follows:

		Object(j):
			1    2    3    4
		u_j 	$2   $3   $1   $2
		v_j 	$5   $5   $3   $1
		

MULTIPLE (HOMOGENEOUS) ITEMS, MULTIPLE BUYERS
The items here are identical, and the prices of all objects sold are equal. Consider 3 objects, with reserve price $2 each.
Case-1: 4 bidders with valuations ($7, $6, $5, $4). The 3 items will go to the first three bidders and each will be paying price p, such that $4 \le 'p' \le $5. So minimum and maximum equilibrium prices are $4 and $5 respectively. Implementation of these minimum and maximum prices are seller-manipulable and buyer-manipulable respectively, and can be shown in the same way. Case-2: 3 bidders with valuations ($7, $6, $5). In this case, the minimum equilibrium price is $2, which is equivalent to introducing a dummy bidder in the system with valuation = reserve price.

MULTIPLE (HETEROGENEOUS) ITEMS, MULTIPLE BUYERS
This is the case of multi-item auction. Interesting part here is how to find out the equilibrium prices(and thus, draw the lattice). Let's consider two objects and two bidders. Sellers' valuations are u_j's and buyers' valuations are v_ij

		Object: 1    2
		u_j:    $0   $0
		v_ij:
		1:      $4   $1
		2:      $2   $3
		

Clearly, 1 gets item 1 and 2 gets item 2. Minimum equilibrium price is (0,0) and maximum equilibrium price is (4,3). Other equilibrium prices can be found out by considering all possible price combinations between these two prices [eg., (0,1), (0,2), (0,3), (1,1) ...], and checking which are indeed equilibrium (this requires finding out the demand set at each price vector and check if an assignment is possible). Minimum and maximum equilibrium prices can be found out using ascending and descending auctions.
Another way is to solve the corresponding LP of the assignment problem which is:

		Max sum((i,j), (v_ij - u_j) * x_ij)
		s.t.
		sum(i, x_ij) <= 1  for all j
		sum(j, x_ij) <= 1  for all i
		x_ij >= 0  for all i,j

		For the above problem, it is:
		Max z = 4*x_11 + x_12 + 2*x_21 + 3*x_22
		s.t.
		x_11 + x_21 <= 1      	(p1)
		x_12 + x_22 <= 1	(p2)
		x_11 + x_12 <= 1	(t1)
		x_21 + x_22 <= 1	(t2)
		

Solution to this LP is (x_11=1, x_12=0, x_21=0, x_22=2, z=7) Solving this LP will give us the optimal allocation. Dual of this problem, along with the Complementarity Slackness (CS) conditions will give us all the equilibrium price, and hence the lattice. If the dual variables are as shown above, then the dual is:

		Min z'= t1 + t2 + p1 + p2
		s.t.
		p1 + t1 >= 4
		p2 + t1 >= 1
		p1 + t2 >= 2
		p2 + t2 >= 3
		

One solution to the dual LP is (p1=0, p2=0, t1=4, t2=3, z'=7). [Min CE] Another solution is: (p1=4, p2=3, t1=0, t2=0, z'=7). [Max CE]

		We note, CS conditions are:
		x_11 * (p1 + t1 - 4) = 0
		x_12 * (p2 + t1 - 1) = 0
		x_21 * (p1 + t2 - 2) = 0
		x_22 * (p2 + t2 - 3) = 0
		

Once the lattice is found out, the minimum and maximum equilibrium price can be deduced from that.

For optimal assignment, x_11 = x_22 = 1, so (p1 + t1 = 4) and (p2 + t2 = 3).

Notes.
*N1: "Sellers might discriminate, offering better price to some buyers than others." This corresponds to a model discussed in the paper 'Multi-Item Auctions'. In general, sellers can have 'discriminatory' prices being offered to each bidder. In this case, sellers will have a matrix of (reserve) prices r_ij for i-th item and j-th bidder. Bidders as usual have v_ij, valuation of i-th item of j-th bidder. This auction can be transformed to the special case described before (i.e. auction with non-discriminatory price) as follows: Consider C_ij = v_ij - r_ij. This matrix (C_ij) can be used as a 'revised valuation matrix' of j-th bidder of i-th item, with reserve prices set to zero. A price vector p_j in this 'revised' form is equivalent to (p_j + r_ij) in the actual model.

There is a one-to-one correspondence between this model and auction in job market:

		       Sellers : Workers, selling their services to employers
		       Bidders : Employers
		       j : Number of jobs
		       r_ij : Minimum salary that worker i will accept for job j
		       v_ij : Maximum salary that (the employer of) job j will offer to
		worker i
		

Now let's consider the following enhancement of the model (Debasis suggested this):
Other than the reserve price, sellers also are responsible to pay for the transportation cost to reach the item to bidder. So, other than reserve prices against each item (discriminatory or non-discriminatory), seller will have one more component which is obviously discriminatory. This can be incorporated in the model by raising the reserve prices by the transportation charge for an item-bidder pair. So if an item j has m_ij as the reserve price for j-th bidder, and the transportation charge is c_ij to reach the item to j-th bidder, then the 'revised' reserve price r_ij = m_ij + c_ij.