Combinatorial Auctions

In this section we deal with combinatorial auctions. The main feature of com-binatorial auctions is that they allow handling cases where a buyer’s valuation for a

bundle of goods may differ from the sum of its valuations for the individual items in that bundle.

3.1. Quantity Constraints

Consider an auction for the reservation of seats in a particular flight. Each potential buyer submits bids for each of the seats in the airplane, but restricts the total number of seats he may wish to obtain. This auction has the property that the payment of buyer i for the set of seats allocated to him, subject to his quantity constraint, is the sum of his bids for the individual seats in this set. However, this auction is a sub-additive combinatorial auction; a buyer will pay 0 for every additional seat assigned to him beyond his limit on the number of required seats.

Definition 3.1 AQuantity-constrained multi-object auctionis a sub-additive combi-natorial auction where bids are of the form(a1,p1,a2,p2, . . . ,ak,pk,q)where piis a price offer for object ai, and q is the maximal number of objects that are to be assigned.

Notice that in the above definition we used the termmulti-object auction. This is in order to emphasize that although the auction is combinatorial, it has some syntactic similarity with other types of multi-object auctions, such as constrained multi-object auctions [22], since the bids are not stated explicitly for bundles of goods.

Theorem 3.1 Quantity-constrained multi-object auctions are computationally tractable.

3.2. Binary Bids

We have seen that simple quantity constraints can be incorporated into simple multi-object auctions, while still getting tractable solutions. Previous work has tried to tackle the tractability of combinatorial auctions where bids are given for non-singleton bundles. It was shown that the case of bundles of size two is tractable, while the case of larger bundles is NP-hard. Indeed, the first use of matching techniques in the context of combinatorial auctions appears in [26]. In that paper the authors consider the case of bundles of size two. They look at an undirected graph where the nodes represent the set of goods, and edges (and edge weights) are associated with bids. The solution of the CAP can be reduced to the computation of an optimal weighted matching. Matching techniques and the more powerfulb-matching techniques can go much further. We now show that the case of bundles of size two and the case of quantity constraints can be tackled simultaneously in an efficient manner.

Definition 3.2 Aquantity-constrained multi-object auction with binary combinatorial bundlesis a sub-additive combinatorial auction that allows two types of bids: 1. The bids allowed in a quantity-constrained multi-object auction. 2. Bids of the form(a,p,b,q,l)

where p is the price offer for good a, q is the price offer for good b, and p+q−l is the price offer for the pair{a,b}, where0<l<mi n(p,q).

Given the previous seats reservation example we allow each potential buyer to express sub-additive bids for pairs of seats, and the singletons they consist of, where she can explicitly declare the rebate asked for if both seats are allocated. This is in addition to bids that allow quantity constraints.

Usingb-matching techniques it can be proven that:

Theorem 3.2 Given a quantity-constrained multi-object auction with binary combi-natorial bundles, the CAP is computationally tractable.

3.2.1. Multi-Unit Binary Combinatorial Auctions

An interesting generalization of combinatorial auctions with binary bids is related to the case where there are several available units of each of the objects. In this case, when the buyer makes a bid for the bundle{a,b}, he does not care which copies of the objectsaandbhe obtains, as long as he obtains a copy of each one of them. In order to deal with this problem formally, we will need to introduce an extension of the CAP, termed multi-unit CAP (mu-CAP).

In a multi-unit combinatorial auction set-up a seller sellsmgoods{1,2, . . . ,m}

tonpotential buyers, where there arekjunits of good j (1≤ j ≤m).

A bid of buyeri is a pair (S,p), where S is a bundle of goods and pis a non-negative real number that denotes the price offer for S.² Let X = {x1,x2, . . . ,xt}, where xi =(Si,pi) (1≤i ≤t) be a set of bids, and denote by S(xi) and P(xi) the bundle of goods and the price offer of bidxi, respectively. Themulti-unit combinatorial auction problem[mu-CAP] is to find anXo⊆X, for whichXoP(xi) is maximal, under the constraint that for every good j (1≤ j ≤m)|Xo^j| ≤kj, where Xo^j = {x∈ Xo: j∈ S(x)}. Notice that in the mu-CAP we still allow each buyer to get at most one unit of each good, but the number of available units of good jiskj≥1.

The mu-CAP fits many practical applications. For example, a retailer may wish to allow its customers to create their own desired binary bundle as part of a promotional sale, while limiting the number of units of each good available on that sale. As it turns out,b-matching allows to generalize the positive result on combinatorial auctions with binary bids [26] to the context of the mu-CAP in a quite straightforward manner.

Theorem 3.3 The mu-CAP with binary bids is tractable.

2This is a rather restricted version of multi-unit combinatorial auctions. In a more elaborated version a bid can ask for several units of each of the goods. For example, in such elaborated version we can give a bid for two TVs and three VCRs.

3.3. Beyond Binary Bids

As we mentioned, combinatorial auctions where bids are only for single goods or for pairs of goods are tractable [26]. However, when bids are for bundles of size greater than two, the CAP is in general intractable. Here we extend these results by considering cases of non-additive combinatorial auction, that is, where the bid for an allocated set of goods is different from the sum of bids for the singletons it consists of, and the bundles’ sizes are greater than two. We start with two such results that we believe to be of considerable importance in this regard. The first result shows that bundles of size greater than two in which the bid for an allocated set of goods is different from the sum of bids for the singletons it consists of, can be handled in polynomial time. In the second result, a general form of combinatorial auctions where bids for triplets are permitted, i.e., combinatorial auctions with symmetric bids for triplets are shown to be tractable.

Definition 3.3 Analmost-additive multi-object auctionis a combinatorial sub-additive auction where bids for non-singletons are of the form(a1,p1,a2,p2, . . . ,ak,pk,q) where pi is the price offer for object ai, the price offer for any proper subset A⊂ {a1, . . . ,ak} equals a_i∈Api, and the offer for{a1, . . . ,ak}is q; in addition, w= 1≤i≤kpi−q >0, andw < pj (1≤ j ≤k).

In an almost-additive multi-object auction a shopping list of goods is gradually built until we reach a situation where the valuations become sub-additive; sub-additivity is a result of the requirement thatw >0. The other condition onwimplies that the bid on the whole bundle is not too low with respect to the sum of bids on the single goods.

Notice that typicallyqwill be greater than the sum of any proper subset of thepi’s; our only requirement is thatqwill be lower than the sum of all thepi’s; hence, bidding on and allocation of the whole{a1,a2, . . . ,ak}bundle is a feasible and reasonable option.

Such a situation may occur if each buyer is willing to pay the sum of the costs of the goods for any strict subset of the goods, but is expecting to get some reduction if he is willing to buy the whole set.

Theorem 3.4 For an almost-additive multi-object auction, the CAP is computationally tractable.

We continue with the case of combinatorial auctions with bids for triples of goods.

The general CAP in this case is NP-hard. However, consider the following:

Definition 3.4 A combinatorial auction with sub-additive symmetric bids for triplets is a sub-additive combinatorial auction where bids are either for singletons, for pairs of goods (and the singletons they are built of), or for triplets of goods (and the correspond-ing subsets). Bids for scorrespond-ingletons and pairs of goods are as in Definition 3.2, while bids for triplets have the form(a1,p1,a2,p2,a3,p3,b1,b2): the price offer for good ai is pi, the price offer for any pair of goods{ai,aj},(1≤i,j≤3;i = j)is pi+pj−b1, and the price offer for the whole triplet{a1,a2,a3}is p1+p2+p3−b2.

Symmetric bids may be applicable to many domains. One motivation is the case where each buyer has a certain fixed cost associated with any purchase (e.g., paper work expenses, etc.), which is independent of the actual product purchased; this additional cost per product will decrease as a function of the number of products purchased (e.g., one does not need to duplicate the amount of paper work done when purchasing a pair of products rather than only one).

Theorem 3.5 Given a combinatorial auction with sub-additive symmetric bids for triplets, where each bid for triplet(a1,p1,a2,p2,a3,p3,b1,b2)has the property that b2>3b1, and pi >b2−b1(1≤i ≤3), then the CAP is computationally tractable.

The theorem makes use of the two conditions that connectb1,b2, and the bids on singletons. These conditions measure the amount of sub-additivity relative to the purely additive case where a bid for a bundle is the sum of bids for the singletons it consists of.

The first condition is that the decrease in valuation/bid for a bundle, relative to the sum of bids for the singletons it consists of, will be proportional to the bundle’s size; the second condition connects that decrease to the bids on the singletons, and requires that the above-mentioned decrease will be relatively low compared to the bids on the single goods. Both of these conditions seem quite plausible for many sub-additive auctions.

The technique for dealing with bundles of size 3 can be extended to bundles of larger size. However, the conditions on the amount of decrease in price offers as a function of the bundle size become more elaborated, which might make the result less applicable.

3.4. A Super-Additive Combinatorial Auction

In the previous sub-sections we have presented solutions for some non-trivial sub-additive combinatorial auctions. Here we show an instance of super-additive com-binatorial auctions that can be solved by similar techniques.

Definition 3.5 A combinatorial auction with super-additive symmetric bids for triplets is a super-additive combinatorial auction where each buyer submits a bid for a triplet of goods and its corresponding subsets, and is guaranteed to obtain at least one good.

The bids for triplets have the form(a1,p1,a2,p2,a3,p3,b1,b2): pi is the price offer for good ai, the price offer for any pair of goods{ai,aj}(1≤i,j ≤3;i= j)is pi+

pj+b1, and the price offer for the whole triplet{a1,a2,a3}is p1+p2+p3+b2. Notice that the major additional restriction we have here is that the auction pro-cedure must allocate at least a single good to each buyer. Of course, in practice the auction will have a reserve price, which will make this assumption less bothering for several applications. This is since it will require each buyer to pay at least the reserve price for the good it gets. For example, assume a car manufacturer wishes to promote a new set of car models by selling a set of cars in an auction to his buyers; it can decide

to guarantee that each participating buyer will be allocated at least one car but assign a reserve price; a car will not be sold for a price which is below that reserve price.

Theorem 3.6 For combinatorial auctions with super-additive symmetric bids for triplets such that1.5b1≤b2≤2b1, the CAP is computationally tractable.

3.5. Combinatorial Network Auctions

Linear goods refer to a set of goods where there is some linear order on them, i.e., they can be put on a line with a clear ordering among them. An example may be spots along a seashore. Linear goods turned out to be a very interesting setting for multi-object auctions. The assumption is that bids will refer only to whole intervals.

For example, bids will refer only to a set of spots along the seashore that define an interval (with no “holes”). Auctions for linear goods are also a useful case of tractable combinatorial auctions (see [26, 20, 13]). Consider for example the case of discrete time scheduling of one resource (e.g., allocation of time slots in a conference room), or for the allocation of one-dimensional space (e.g., allocation of slots on a seashore), etc. When each request refers to a complete sequence of time slots, then by referring to each time slot as a good we get the case of auctions with linear goods. Another real-life example that fits also into this type of problem is that of radio frequency auctions, such as the FCC auction. For more on the FCC auction, see [18], [8] and [25]. In an auction for linear goods we have an ordered list ofmgoods,g1, . . . ,gm, and bids should refer to bundles of the formgi,gi+1,gi+2, . . . ,gj−1,gjwhere j ≥i, i.e., there are no

“holes” in the bundle. The combinatorial auction problem, where bids are submitted on intervals is the same as auctions for linear goods, and is known as theInterval Auction Problem. This problem was first discussed by Rothkopf et al. in [26]. It was also studied by van Hoesel & M¨uller in [13]. A wide extension of the result on the tractability of auctions for linear goods is the following combinatorial network auctions problem.

Definition 3.6 A network of goods is a network G(O)=(V(O),E(O)), where the set of nodes, V(O), is isomorphic to the set of goods O = {g1, . . . ,gm}. Acombinatorial network auctionwith respect to the set of goods O and the network G(O), is a combi-natorial auction where bids can be submitted only for bundles associated with paths in G(O).

Combinatorial network auction whereG(O) is a tree is termed acombinatorial tree auction. Combinatorial tree auction may be applicable in communication networks, where the underline graph is a tree. This is a reasonable assumption since in many cases the backbone of the network is a spanning tree. In such a case, goods are the edges of the tree and any message that should be delivered from vertexi to vertex jcan be transmitted along the single path fromito jin the tree.

It is clear that combinatorial auctions for linear goods are simple instances of combinatorial tree auctions, where the tree is a simple path. Using, yet again, matching techniques, we can show:

Theorem 3.7 Given a combinatorial tree auction problem, the CAP is computationally tractable.

As trees are undirected graphs with no cycles, one might hope that the tractability of the CAP could be extended to acyclic directed graphs (DAG). However, this does not hold since the combinatorial network auction problem is NP-complete if G(O) is a DAG. This NP-completeness result is obtained by using the following simple transformation to the general combinatorial auction problem. Consider a complete graph onnvertices. Assume the edges are oriented such that for any two verticesiand j, the edge between them is directed fromi to j iffi< j. Then, it is easy to see that any bundle of goods can be represented by a path on the DAG, and vice versa, each path corresponds to the bundle of goods that consists of the goods represented by the vertices of the path.