Before going into the paper itself, let's start with a small discussion on 'Overdemanded Set', which the paper uses as a part of the mechanism. Let's consider a job-matching problem on a bipartite graph, with W workers and T jobs. The edges in this graph denotes the ability of workers to perform jobs. We are looking for maximum feasible allocation of jobs to workers. Let's assume, |W|=|T|. Hall's Theorem suggest that, if we can find a set of workers(A_w) such that the set of tasks that these workers can "only" perform (T(A_w)) is strictly less [ |A_w| < |T(A_w)| ], then all jobs can't be allocated to individual workers. In this case, A_w can be termed as 'Overdemanded' set of jobs. It's quite possible that a subset of A_w is also an Overdemanded set.

Now to the paper. The paper discusses two algorithms for ascending auctions in multi-item scenario. In multi-item auctions, let's assume that each bidder wants at most one item and they have valuations on each of the item. By using these mechanisms, two important properties of single-item auctions get generalized in multi-item scenario:
(1) Incentive Compatibility: Submitting the true valuation is a (weakly) dominant strategy for the bidders.
(2) The auction converges to the minimum competitive equilibrium price vector. This is the main aim of the paper.

A price vector p will be called competitive equilibrium price if every bidder can be assigned an item from his demand set, and no two bidders get the same object. Prices of unassigned objects are same as reserved price. The paper also suggests another reason to study multi-item dynamic auctions: this allow for a wider range of preferences for bidders (as has been shown by some later work).

The first mechanism, termed as Progressive Auction Mechanism (PAM) proceeds as follows: The auction starts at a price vector equal to the reserve price of each item [*N1]. Price increment is in units of transactions, as applicable. After declaration of price vector by seller, each bidder announces their demand set (the set of items they are willing to have at that price). If an assignment exist, stop. Else, the seller finds the minimum 'over demanded set' of items, and raises price of those items by unit. Seller then re-announces the current price. To emphasize the concept of 'overdemanded set', in this context it is such a set of items that the number of bidders "only" demanding items in this set is greater than the items in this set.

The following example shows this mechanism in action: Consider 4 objects with reserve prices ($1, $2, $1, $3). There are 4 bidders, with valuation matrix as shown below:

		    1    2    3    4
		-----------------------------------
		1 |    $5    $4     $2     $5
		2 |    $2    $5     $3     $6
		3 |    $3    $7     $3     $2
		4 |    $5    $7     $3     $4
		-------------------------------------

		Round 1:         	1    2    3    4
		Sellers Price:         $1    $2    $1    $3
		Demand Set of Bidders {bidder:objects}: {1:1; 2:2,4; 3:2; 4:2}
		Overdemanded Set: {2}, new price of 2=$3

		Round 2:         	1    2    3    4
		Sellers Price:         $1    $3    $1    $3
		Demand Set of Bidders {bidder:objects}: {1:1; 2:4; 3:2; 4:1,2}
		Overdemanded Set: {1,2}, new price of 1=$2, for 2=$3

		Round 3:         	1    2    3    4
		Sellers Price:      	$2    $4    $1    $3
		Demand Set of Bidders {bidder:objects}: {1:1; 2:4; 3:2; 4:1,2}
		Overdemanded Set: {1,2}, new price of 1=$3, for 2=$5

		Round 4:         	1    2    3    4
		Sellers Price:         $3    $5    $1    $3
		Demand Set of Bidders {bidder:objects}: {1:1,4; 2:4; 3:1,2,3; 4:1,2,3}
		Now the following assignment is possible: {1:1, 2:4, 3:2, 4:3}. So the
		auction ends here.
		{$3, $5, $1, $3) are the minimum competitive equilibrium prices.
		

The paper proves the important result for this mechanism: This mechanism always converges to minimum competitive equilibrium price vector.

Since the valuations and prices will not always be integer, the paper proposes a second mechanism, which can be made to reach arbitrarily close to the outcome of the first one. In this mechanism, when a bidder bids for an item, he gets committed to it, unless someone outbids him. If a new bidder wants to outbid another 'committed' bidder, he will raise the price of the item by 'x'. The auction terminates when there is no more uncommitted bidders. The outcome (final assignment) of this mechanism depends on the order in which people bid. The paper shows that the final price of any item in this mechanism will differ by at most 'kx' from PAM, where 'k' is minimum of the number of buyers and sellers.