Multi-Unit Auctions with Matching#

Some auctions involve sale of different types of items. Spectrum licenses in different regions, seats for a concert or event, advertising spots on the internet, assets of a company being liquidated, pieces of a procurement contract.

Connection to Gale-Shapley “Marriage Problem”#

Assignment models seeks to allocate items to people with different preferences, with each person getting \(\leq 1\) item. This sounds like a matching problem. Before we tried to assign the items without payments. Now, we assume items can have money prices.

Certain auctions function similarly to matching algorithms–they implicitly include deferred acceptance mechanisms to find efficient allocations, ideally also providing truthful (or “near-truthful”) incentives.

Assignment Model#

Sets of individuals \(\{1,\ldots,N\}\) and items \(\{1,\ldots,𝐾,\emptyset \}.\) An assignment \(k\) maps individuals to items, so \(k(i)\) is the item assigned to individual \(i\) and \(k(i) = \emptyset\) means that no item is assigned to \(i\).

Payoffs: Let \(v_{ik}\) denote individual \(i\)’s value for item \(k\). If \(i\) gets item \(k\) and pays \(p_k\), her utility is \(v_{ik} - p_k\).

Connection to the “marriage problem”: If prices were fixed, then each buyer would rank alternatives in order according to \(v_{ik} - p_k\) and Gale-Shapley could be applied. But our seller’s rank according to prices only, so we won’t hold those fixed.

Aside: Matching with Contracts?#

We have already discussed a closely related model in our study of matching with contracts, in which parties cared about a match and terms. “Terms” can include money amounts. In that theory, we assumed “strict preferences”–so, no ties. In economics, it is common to model money by a continuous variable, and this can make ties unavoidable. That changes some details of the analysis, but the main thrust is very similar. Additional connections will be made on later slides.

Example: Cubicle Assignment#

Say we are tasked with assigning cubicles to economics graduate students. To find an efficient assignment without prices or money, we can assign random numbers (RSD). The assignment is Pareto efficient for any random order, so there are many Pareto efficient assignments! Adding money changes things.


Suppose that students can trade offices for money.

  1. Is the outcome of an office draw Pareto efficient?

  2. What does a Pareto efficient outcome look like?

  3. What algorithm might identify an efficient allocation?

Pareto Efficiency#

Pareto domination#

An assignment is Pareto dominated if there is another assignment and set of cash payments that results in everyone being better off.

Pareto efficient (auctions)#

An assignment is Pareto efficient if it is not Pareto dominated.

The total value (or surplus) of the assignment \(k\) (that assigns to each \(i\) the item \(k(i)\)) is \(v_{1k(1)} + \dots + v_{Nk(N)}\).


What’s the relationship between Pareto efficiency & total value?


We have three people \(a\), \(b\), and \(c\) and two items \(x\) and \(y\). The value matrix is:

\[\begin{align*} \begin{bmatrix} 30 & 60 \\ 20 & 40 \\ 10 & 20 \end{bmatrix} \end{align*}\]

The rows from top to bottom represent the values for \(a\), \(b\), and \(c\). The left and right columns represent \(x\) and \(y\) respectively.

  1. Can it be efficient to assign \(\{(a, x), (b, y), (c, \emptyset)\}\) with some payments, so \(b\) gets his most preferred item?

  2. Identify a Pareto efficient allocation.

Efficient Assignments#

Theorem 14 (Efficient Assignments)

When payoffs are value plus cash and total transfers must be equal to zero, an assignment with transfers is Pareto efficient if and only if no other assignment has a higher total value.

Proof. Suppose an assignment achieves the maximum total value. Then any change in the assignment and any payments reduce the total value of all participants, so someone must lose from this change. Hence, such an assignment is Pareto efficient.

Conversely, suppose an assignment does NOT maximize the total value. Then there is another assignment of items that adds some amount \(\Delta > 0\) to the total value. Then, it is feasible to switch to this alternate assignment, make transfers so that everyone is indifferent about the change, and make additional payments to everyone of \(\Delta / N\). All strictly prefer that arrangement, so the original assignment is not efficient.

Algorithm to Allocate Efficiently?#

Suppose items are initially owned by a seller that has zero value for the items, and suppose the seller wants to allocate the items efficiently. A seller might also care about selling for high prices, but we won’t focus on that concern today. Let’s consider an auction mechanism in which prices rise until all offers are accepted: a kind of deferred acceptance mechanism.

Ascending (Clock) Auction#

Algorithm 15 (Ascending Auction)

Seller has a price clock for each item.

  1. Price of each item starts at \(0\).

  2. Buyers demand their most preferred (acceptable) item-price combination.

  3. If any good has “excess demand” (i.e., more than one bids), increase its price “slightly” and return to step \(1\). Otherwise, end.


  • At any given moment, each buyer bids for the item it wants most at the current clock prices (“truthful bidding”).

  • Prices rise continuously rather than “jumping” discretely.

Example 1#

Three bidders \(a, b\) and \(c\). Two items \(x\) and \(y\).

\[\begin{align*} \begin{bmatrix} 30 & 60 \\ 20 & 40 \\ 10 & 20 \end{bmatrix} \end{align*}\]

What is the efficient assignment?


\(\{(a, y), (b, x)\}\)

The Magic of the Market#

Theorem 15

In the assignment market setting, a (“continuous”) simultaneous ascending auction with truthful bidding will both

  • finish at the lowest (buyer best!) market clearing prices, and

  • result in an efficient (value-maximizing) assignment .

The auction works like a deferred acceptance algorithm to find market clearing prices – an efficient assignment and market-clearing prices.

Summary of Results#

Assignment model: \(N\) bidders, \(K\) items. Each bidder wants at most one item. Bidder \(i\)’s payoff if it pays \(p_k\) for item \(k\) is \(v_{ik} - p_k\).

Key Results:

  • There is an assignment that maximizes total value and is efficient. For almost all values, this assignment is unique.

  • There are always market clearing prices for the items, and (item-by-item) minimal (“buyer best”) market clearing prices.

  • These buyer-best prices can be found using an ascending auction – assuming truthful bidding.

Connection to Matching#

Think of each bidder as forming a preference list that factors in both item and money preferences (think of prices as being in discrete dollar increments)

Example 11

First choice is to pay zero for item 1, second choice is to pay one dollar for item 1, third choice is to pay zero dollars for item 2, fourth choice is to pay two dollars for item 1, etc.

Items prefer more money, but don’t care who offers it.

Deferred Acceptance?#

Each bidder submits a preference list.

  1. Seller runs deferred acceptance algorithm

  2. Bidders “propose” to the items.

  3. Items reject all but the highest offer.

  4. Rejected bidders move to the next item on their preference list, if any. (They “raise their bids” as the algorithm proceeds). Algorithm will eventually terminate.

Matching Algorithm

“Auction” Algorithm


Men make offers to most preferred remaining acceptable woman.

Bidders offer most preferred remaining acceptable purchase.


Women hold best man, reject others.

“Sellers/items” hold best offer, reject others.


Rejected man strikes the woman from his/her list.

Rejected bidder strikes offer from his/her list.


Process continues until no new offers or rejections.

Process continues until no new offers or rejections.


Implement last held allocation.

Implement last held allocation.

Deferred Acceptance “Auction”#

What we know from matching theory:

  • DA algorithm will converge to a “stable” allocation.

  • Bidder-offering DA gives stable allocation preferred by bidders, and is strategy-proof for the bidders.

Stability: each bidder prefers the item they get, at the price they pay, to any other item at the price it receives. So at the final item prices, demand equals supply!

Completing the auction/matching link:


  • A stable allocation is a competitive equilibrium

  • Bidder-proposing DA gives the lowest market-clearing prices.

One can also show that the lowest market-clearing prices in the assignment model are exactly the Vickrey prices, and that truthful bidding in the assignment model is a Nash equilibrium.


  • Assignment model captures settings where bidders with diverse preferences must be assigned to a diverse set of goods, and payments are allowed.

  • Competitive equilibrium is a natural candidate for a “good” outcome, especially with lowest market-clearing prices.

  • A well-designed auction can elicit willingness-to-pay from bidders and identify market clearing prices.

    • Simultaneous ascending auction

    • Sealed bid assignment auction (Vickrey pricing)

  • There is a close connection to matching theory, and a version of the DA can work as an ascending auction.