Analysis of Algorithms for Combinatorial Auctions and Related Problems

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(1)UPPSALA DISSERTATIONS IN MATHEMATICS 44. Analysis of Algorithms for Combinatorial Auctions and Related Problems Kidane Asrat Ghebreamlak. Department of Mathematics Uppsala University UPPSALA 2005.

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(172) List of Papers. This thesis is based on the following papers, which are referred to in the text by their Roman numerals. I II III. IV. Ghebreamlak, K. A., Andersson, A. Caching in multi-unit combinatorial auctions. Manuscript. 1 Ghebreamlak, K. A., Andersson, A. Combinatorial versus simultaneous auctions: A comparative study. Submitted. Ghebreamlak, K. A. Analysis of a greedy algorithm for the winner determination problem in combinatorial auctions. Manuscript. Ghebreamlak, K. A. Renewal theory for combinatorial auctions. Submitted.. 1 An. extended abstract appeared in Proc. of the first international joint conference on Autonomous agents and multiagent systems (AAMAS 2002), Bologna, Italy.. v.

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(174) Contents. 1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Background . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Auction theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.1 Some Popular Auctions . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2.2 Auction Comparisons . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Combinatorial auctions (CA) . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3.1 Complementary and Substitutable items . . . . . . . . . . . . . 1.3.2 Winner Determination Problem . . . . . . . . . . . . . . . . . . . . 1.3.3 NP−hardness of the winner determination problem . . . . . 1.4 Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2 Overview of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Paper I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Paper II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Paper III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Paper IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Two Dimensional Convergence to Normality . . . . . . . . . . 2.4.2 Stopped two dimensional process . . . . . . . . . . . . . . . . . . 3 Swedish summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .. 1 1 2 2 3 3 4 4 5 7 9 9 10 11 12 13 13 17 19.

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(176) 1. Introduction. In this chapter we present some background material on auctions in general and combinatorial auctions in particular. The results of the thesis will be summarized in Chapter 2.. 1.1. Background. In auctions involving several distinguishable items, the auctioneer has to choose an auction mechanism, such as sequential, simultaneous or combinatorial auction. In a sequential auction, the items are auctioned one at a time and the winners are determined by choosing the highest bidder for each item separately. However, when multiple items are auctioned, the value of an item to a bidder may depend on which other items he wins. In particular, a bidder’s valuation function could be super-additive with respect to some of the items. Thus, when a bidder with preference for a bundle of items is submitting a bid, he should guess what other items he will receive in later auctions. This uncertainty may lead to either lower revenues or inefficient allocations. In addition, bidders face an “exposure” risk [15]. On the other hand, in a simultaneous/parallel auction, the items are auctioned simultaneously. Here, bidders will have more information about other bidders’ preferences. This minimizes the need for lookahead. But, since bidders don’t have full information about each other, still similar problems prevail. In combinatorial auctions, bidders can submit bids not only for individual items but also for combinations (packages or bundles) of items. In this case, any possible presence of synergy (superadditivity) in bidders’ valuations increases the seller’s revenue. At the same time the risk for bidders is reduced. The study of combinatorial auctions is interdisciplinary in nature. Auctions, mostly non-combinatorial ones, have been extensively studied in economics and game theory. While the advent of combinatorial bidding made operations research techniques more relevant to this topic, computer science is concerned with the computational complexity of the combinatorial problem and the expressiveness of bidding languages.. 1.

(177) 1.2. Auction theory. An auction is the public sale of items or property, described by a set of rules that specify how the winner is determined and how much he has to pay. In addition, the rules may restrict feasible bids. Normally, there is a single seller and many potential buyers in an auction. However, the process of procurement via competitive bidding, sometimes called reverse auction, is another form of an auction. In this case, the bidders who satisfy certain rules such as quality or delivery performance compete for the right to sell their products and the person bidding the lowest price is the winner. Auctions are used in many transactions. Many items ranging from fresh flowers, fish and tobacco to antiques and real estate are sold via auctions. In the next subsection, we describe some of the most commonly used auctions. To make it easy to read, some common terms are defined in the glossary.. 1.2.1. Some Popular Auctions. The most commonly used auction is the open ascending price or English auction. In this mechanism, an auctioneer directs participants to beat the current, standing bid. New bids must increase the current bid by a predefined increment. The auction ends when no participant is willing to outbid the current standing bid. Then, the participant who placed the current bid is the winner and pays the amount he bid. The Dutch auction is the open descending price counterpart of the English auction. This is a type of first price auction in which the auctioneer begins by calling a price that is substantially higher than any bidder is likely to pay. Then, the price is gradually decreased until a bidder indicates his willingness to pay. The auction is then concluded and the winning bidder pays the stopping price. Another popular auction format is the sealed-bid auction. In the first-price sealed-bid auction, bidders simultaneously submit bids to the auctioneer without knowledge of the amount bid by other participants. The highest bidder is declared the winner and pays the amount he bid. The second-price sealed-bid auction is similar to the first-price sealed-bid except that the winner pays an amount equal to the second highest price. This type of auction is also referred to as Vickrey auction, named after William Vickrey who first described it and pointed out that bidders have a dominant strategy to bid their true values [19]. Here, it is worth to mention that the Dutch auction is strategically equivalent to the first-price sealed-bid auction. In each case, a bidder has to decide how. 2.

(178) much to bid, or at what price to indicate his willingness to buy the item. Thus they have the same strategy space. In addition, under the private values model, the English auction and the second-price sealed-bid auction have the same optimal strategy - to bid, or stay in the auction until the price reaches, the private value. Thus, although they are not strategically equivalent, they are equivalent in this weaker sense. Krishna’s book [9] offers more details on this.. 1.2.2. Auction Comparisons. One of the main issues in auction theory is the performance comparison of different auction formats. The two main criterion when choosing an auction format are revenue and efficiency. Sometimes, an auction’s vulnerability to collusion may also be a factor. For the seller, the choice of an auction format usually depends on the expected revenue it generates. However, for the society as a whole, efficiency - that the object ends up in the hands of the person who values it the most may be more important. For instance, in an auction for the sale of a publicly held asset to the private sector, the government may choose the format that allocates the object efficiently, even when a higher revenue is possible from some other less efficient auction.. 1.3. Combinatorial auctions (CA). A drawback with the above mentioned auctions is that bidders for multiple items can’t fully express their preferences. By allowing bidders to bid on packages, combinatorial auctions give them more flexibility which often increase both efficiency and revenue, while reducing “exposure” risk for bidders [15]. This inherent flexibility makes combinatorial auctions quite useful in a variety of resource allocation problems. To mention a few, they have been used for air freight contracts, telephony services, truck transport and even wooden packaging [18]. They have also been proposed for airport time slot allocation [14], and the Federal communication commission (FCC)’s multi-billion-dollar sale of radio spectrum licenses in the united states. In all these cases, the primary motivation for the use of combinatorial auctions was the presence of complementarities among the items. For a more detailed introduction to this topic, see the survey by de Vries and Vohra [3].. 3.

(179) 1.3.1. Complementary and Substitutable items. When multiple items are auctioned, the value of an item to a bidder may depend on which other items he wins. In particular, a bidder’s valuation for a combination of items could be more than the sum of values for the individual item. Such items are called complementary. For instance, a bidder’s value for a set of bids on communication links forming a complete communication path may be more than the sum of his values for the individual links [13]. Similarly, in the FCC’s auction for spectrum licenses, bidders may desire licenses for several geographically adjacent regions at the same frequency. Selling such items when combinatorial bidding is not allowed may lead to lack of efficiency, and lower revenues. On the other hand, a set of items are said to be substitutable, if a bidder’s valuation for the combination of items is less than the sum of his individual values. This is typical when the items are almost identical or the bidder is interested in just one item but is indifferent between the items. Thus, in order for the bidders to be able to fully express their preferences, auction protocols should allow for the description of valuation functions involving both complementarity and substitutability. The next question is how to communicate a bidder’s preference for several packages to the auctioneer. One approach is for the auctioneer to specify a bidding (encoding) language that all the bidders must use. Along this line, Nisan [13] discusses several bidding languages and their expressive power. Alternatively, de Vries and Vohra [3] point out the possibility of using an oracle (program), in which case bidders submit oracles instead of bids.. 1.3.2. Winner Determination Problem. Once the seller received all the bids, he has to determine the set of disjoint bids that maximizes his total revenue. This problem is known as the winner determination problem (WDP). For brevity, we assume that a bidder who is interested in several bundles of items and, for any pairwise disjoint bids is willing to pay the sum of the bid prices, submits a bid for each bundle. Through out the thesis, we will use the terms “items”, “goods”, or “objects” interchangeably to refer to the distinguishable items, and the term “units” to the identical units of the same item.. Formal definition Let G = {g1 , g2 , . . . , gm } be a set of distinguishable items in the auction, and B = {b1 , b2 , . . . , bn } be a set of bids. In a multi-unit combinatorial auction, every item in the auction can have multiple identical units. In this case, a bid 4.

(180) bi is a tuple p(bi ), (ui1 , ui2 , . . . , uim ), where p(bi ) is the price offer of the bid and uik is the number of units of item k requested in bid bi . Denote by qk the total number of identical units of item k available in the auction. For each bid bi , let xi be a binary variable that indicates whether bid bi is allocated or not. Then, the resulting problem, called the multi-unit WDP, is: n. ∑ xi p(bi ). maximize:. i=1. ∑ xi uik ≤ qk ,. subject to:. 1≤k≤m. i. xi ∈ {0, 1},. 1≤i≤n. Obviously, this includes the single-unit WDP - the case where every item in the auction has exactly one unit. Note that the above formulation allows bidders to express complementarity in their valuations. But it is not general enough to handle mutually exclusive bids, and as such bidders can’t express substitutability in their valuation. One way to achieve this is to add an additional XOR (exclusive-or) constraint. Let S = {s1 , s2 , . . . , sl }, where each s j is an XOR bid, that is, a set of bids such that at most one of them should be allocated. Then, the multi-unit WDP with XOR constraints is : n. maximize:. ∑ xi p(bi ). i=1. subject to:. ∑ xi uik ≤ qk ,. 1≤k≤m. i. ∑. xi ≤ 1,. ∀ j ∈ {1, 2, . . . , l}. i|bi ∈s j. xi ∈ {0, 1},. 1≤i≤n. Fujishima et al. [4] also proposed an alternative encoding trick to handle substitutability in bidder’s valuations by introducing “dummy” items - virtual items that enforce an exclusive-or relationship. However, in case there are many substitutable items in the auction, this technique gives rise to a large number of bids.. 1.3.3. NP−hardness of the winner determination problem. Rothkopf, Pekeˇc, and Harstad [15] point out that the WDP, even in its singleunit case, is equivalent to a maximum set packing problem on a weighted hypergraph and thus NP-complete. In addition, H˚astad [5] has shown that, unless any problem in NP can be solved in probabilistic polynomial time (i.e. NP = ZPP), the size of the largest clique in a graph with n nodes is hard to approximate in polynomial time 5.

(181) within a factor of n1−ε for any ε > 0. It follows that there is no polynomial time algorithm that would guarantee an approximate solution to the WDP within a factor of n1−ε from an optimal solution, where n is the number of bids in the auction. Based on a result of Halld´orsson et al. [6], Sandholm [16] also points out that, unless NP = ZPP, a bound within m1/2−ε can not be established in polynomial time for any ε > 0, where m is the number of items in the auction. Thus, the WDP can not be approximated in polynomial time within min(n1−ε , m1/2−ε ). In response to this discouraging observation, several alternative approaches to CA design have been proposed. One way is to impose restrictions on the permitted combinatorial bids so that the WDP can be solved in polynomial time. For instance, Rothkopf et al. [15] have identified the following such cases: • If, for any two bids they are either disjoint or one is a subset of the other, WDP can be solved in O(n2 ) time. • Bids include at most two items, in which case the WDP can be solved in O(n3 ) time. They also show that if bids contain up to 3 items, the problem is NP-complete. • There exist an ordering of the items, and bids request only consecutive items in that order. If the order is linear, then the problem can be solved in O(n2 ), whereas if the order is circular, it can be solved in O(n3 ) time. But, imposing restrictions on the bids may lead to inefficient allocations since this may prohibit bidders from placing bids on the combinations they prefer. Despite the inapproximability result, another line of research is to approximate the solution to the WDP; see for example [13, 16, 20, 11]. In particular, Lehmann et al. [11] show that a simple greedy algorithm establishes a bound √ m. A third approach is to use an optimal algorithm - one that finds provably optimal allocation but without guarantee that it will run in polynomial time [1, 4, 12, 16, 17]. In an attempt to streamline the search space and hence improve the efficiency of such algorithms, it seems a good idea to use a technique for caching partial search results. Yet another line of attack is to avoid the hardness of WDP by placing the computational burden on the bidders, leaving the auctioneer with a computationally easy problem. Kelly and Steinberg [8] proposed the “PAUSE” procedure that takes this approach. However, for a bidder interested in many of the items in the auction, it may be an NP-complete problem to determine whether he can make an appropriate combinatorial bid.. 6.

(182) 1.4. Glossary. 1. Auctioneer - one conducting the auction. 2. collusion - cooperative behavior among bidders. Also called bid rigging, where some bidders form a ring whose members agree not to bid against each other. 3. dominant strategy - a strategy that earns a player a larger payoff than any other, regardless of what any other players do. 4. exposure problem - is the problem that arises when combinatorial bidding is not allowed. A bidder who failed in his attempt to win a package of items may end up paying more for some subset of the items (sub-package) than they are worth. On the other hand, a bidder “unwilling to risk bidding above his individual valuations on individual items may not be able to obtain a combination for which synergies make him the efficient recipient” [15]. 5. bidder’s valuation/value for an item - the maximum amount he is willing to pay for that item. 6. private values model - each bidder knows the value of an item to himself at the time of bidding. No bidder knows the valuations of other bidders with certainty, and knowledge of other bidders’ values doesn’t affect his private value. 7. winner’s curse - a tendency for the winning bid to exceed the intrinsic value of the item being auctioned. 8. strategically equivalent games. Two games are strategically equivalent if for every strategy in one game, a player has a strategy in the other game, which results in the same outcomes.. 7.

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(184) 2. Overview of thesis. In this chapter we summarize the main results. The thesis consists of four papers, which can be viewed as two parts. The first part is more of a practical nature and contains two papers, Paper I and II. In the first paper we study the performance of a caching technique in an optimal algorithm for a multi-unit combinatorial auction. In the second paper we embark on a comparison of different auction formats in the single-unit auction. The second part contains the remaining two papers. It is more theoretical but we tried to present it in terms of the combinatorial auction problem. Here, we analyze the asymptotic performance of a greedy algorithm for a problem inspired by combinatorial auctions. In particular, we consider a special case in which every bid contains exactly 3 items, and use a Poisson process to model an auction with a random (Poisson) No. of bids. This restricted case is far from a full fledged combinatorial auction, however its winner determination problem, which is equivalent to a maximal 3-set packing on a weighted hypergraph, is still NP-hard. Although the general WDP is inapproximable within any constant, the greedy algorithm approximates this special case within a factor of 3 [2].. 2.1. Paper I. This work was motivated by a recent paper by Leyton-Brown et al. [12], in which they proposed an optimal algorithm for computing the winners in a multi-unit multi-item combinatorial auction, called CAMUS. It is a branch and bound algorithm that makes use of a caching technique to improve the running time. We present a counter-example to show that caching, as described by the authors, may fail and the algorithm may eventually give a suboptimal solution. We discuss why it fails and propose how it could be fixed and properly used in a general multi-unit combinatorial auction. In addition, to see the effectiveness of caching, we run tests on 20 problem instances, generated in the same way as described by the authors. Our experimental results show that: • successful hit in the cache is so rare that even CAMUS appears to be correct in the problem instances tested, and. 9.

(185) • even when it occurs, the average number of nodes pruned per successful cut is small.. 2.2. Paper II. In this paper, we compare the performance of a second-price combinatorial auction against a second-price one-shot simultaneous auction. Our analysis is made in the same setting that was previously used by Krishna and Rosenthal [10]. That is, we have two distinguishable items for sale, one “global” bidder, and two “local” bidders - one for each item. The local bidders are interested in only one of the items, while the global bidder is interested in both objects. Each bidder has a private value and the synergy parameter α , a measure of super-additivity, is publicly known. For a global bidder with private value x, his value for each item separately is x while his value for both items is 2x + α . Our first result is in contrast to an earlier conclusion by Krishna and Rosenthal. We show that, when the synergy parameter (α) is small, the strategy of the global bidder in the combinatorial mechanism is not optimal (dominant). As a result, the combinatorial auction gives a higher expected revenue than the one-shot auction for small values of α . However, in the absence of any other optimal strategy, we still use it here. We also compare the combinatorial and one-shot auction protocols under the assumption that bidders are risk-averse. Such bidders are more sensitive to financial loss (winner’s curse) that they tend to bid less aggressively, which leads to lower revenues. In this case, a global bidder has a risk aversion parameter, c, which is the slope of his payoff function; computed as follows. Whenever he wins an object, its contribution to his payoff equals the positive (resp., c times the negative) difference between his valuation and the second price on that object. Finding a closed form for the payoff function, analytically, turns out to be difficult. However, given the values of c and α , our numerical results show that if c ≥ 3.3, the combinatorial auction outperforms the one-shot simultaneous auction for all values of α whereas, if c < 3.3, the choice depends on the value of α . This shows that an auctioneer with knowledge of α and c would be able to choose the auction mechanism that maximizes his profit.. 10.

(186) 2.3. Paper III. Suppose that we have M items in the auction and, for the sake of convenience, allow the number of bids to be random. Assume also that every bid contains exactly three items. We use a Poisson process to model an auction in which the bids are submitted over a period of time. In this case, the number of bids that arrive during disjoint time intervals are independent. Then, given some distribution of bid prices, the aim is to find the asymptotic expected size of the partial allocation, and its corresponding expected total revenue from a greedy algorithm. Poisson process. Let c > 0. Assume that bids with the same combinationof3 items arrive in the interval (0, ∞) as a Poisson process with intensity 1/ M3 , independent of   other bid combinations. Thus we have M3 independent Poisson processes,   each with intensity 1/ M3 , one for each combination of 3 items. The aim is to analyze the Poisson process until time c and then increase c and M to infinity to establish asymptotic results. Note that, since the intensity of the whole process is 1, the expected number of bids in the auction at time t is t . Let f be a non-increasing price function defined on [0, ∞) with f (t) = 0, ∀t > c. That is, f (t) is the price of a bid that arrives at time t in the Poisson process. Thus, a bid with a higher price offer arrives earlier in the Poisson process and, for a given bundle of three items, the bid that arrives first dominates all other bids on that bundle. Then, the greedy algorithm works as follows. Let A be a partial allocation. At time zero, A = 0/ . As time goes on from 0 to c, when a bid arrives it is added to A if it doesn’t conflict with the bids already in A. The algorithm stops at time c. Now, let R(t) be the number  of items left unallocated when the auction is closed at time t , and r(t) = (M 3 )/(M + 6t), t ≥ 0. Whenever it is clear from the context, we use the short hand r and R instead of r(c) and R(c). Our main results are: √ Theorem 1 Suppose that Mc3 → 0 as M → ∞. Then E|R − r| = O( r). That is, on average the greedy algorithm allocates all but r items with error √ bound O( r). For instance, if c = O(M 2 ), then the number of items left unallocated is of order O(M 1/2 ). Now, let I denotes the total revenue from the greedy algorithm. Then, the following theorem, which can easily be proved using Theorem 1, gives the expected value of I for some given distribution of prices. Theorem 2 Let f(t) be the price of a bid that arrives at time t > 0. (i) Let ci , 0 ≤ i ≤ L, and a j > 0, 1 ≤ j ≤ L, be constants such that 0 = c0 < c1 < c2 < . . . < cL = c and a j−1 > a j . Furthermore, suppose that f is defined 11.

(187) on [0, ∞) by.  f (t) =. Then E|I − ∑Lk=1.  ck. ai , ci−1 ≤ t < ci , 1 ≤ i ≤ L, 0,. ak ck−1 (1+ 6t )3/2 dt| M. t ≥c = O(a1. . r(c1 )). (ii) Assume also that f (t), t ≥ 0, be a smooth (except perhaps at c) and non increasing function such that f (t) = 0, ∀t ≥ c. Then, √  E|I − 0c (1+6f (t)t )3/2 dt)| = O( f (0) M) M. As a consequence of this theorem, we have: Corollary 1 (i) Assume that the price of a bid is uniformly and independently drawn from the set {ai : 1 ≤ i ≤ L, a1 > a2 > . . . > aL−1 > aL > 0}. Let   M ck = − M3 ln 1 − Lk [1 − e(−c/( 3 )) ] , 0 ≤ k ≤ L. Then,   k ak E|I − ∑Lk=1 cck−1 r(c1 )). 6t 3/2 dt| = O(a1 (1+ M ) M −1/2 ), and let the price of a bid be (ii) Let c = 3 ln(1 − p)−1  where p =2 O(M  √  1/2 − 1]) = O( M). uniform on [0, 1]. Then, E I − ( M3 − M9c [(1 + 6c )  M. For example, if the price of a bid is 1 or 0, then the expected revenue would be equal to the number of bids allocated in Theorem 1. Similarly, if the price of a bid is uniformly and independently distributed on [0, 1] and c as given in Crollary 1, i.e c √ = O(M 5/2 ), then the expected revenue would be of order √ M 1/4 + O( M). 3 − 6M. 2.4. Paper IV. Again, we consider the Poisson process that we used to model the auction in Paper III. In our first result, we prove that after suitable normalization, both the number of bids allocated by the greedy algorithm and those submitted jointly converge in distribution to a continuous 2-dimensional Gaussian process as the number of distinguishable items in the auction, M , increases to infinity. We use this result to prove our main theorem on stopped two-dimensional process. Here, we study the case of a deterministic number of bids which is proportional to M . We stop the process when the number of bids submitted reaches some desired level, and prove that the number of bids allocated, again suitably normalized, converges to a Normal random variable as M → ∞. Now let B(s) denotes the number of bids submitted and A(s) those allocated during the time interval [0, s], s > 0. Define B by B (s) = B(s) − E(B(s)). 12.

(188) 2.4.1. Two Dimensional Convergence to Normality. Our results in this section depend on a martingale convergence theorem by Jacod and Shiryaev [7, Theorem VIII.3.11]. In addition, we stop the process at Mt instead of c, t > 0 fixed.   1 1 = 13 (1 − α −1 (t)), For each t > 0, let α(t) = (1 + 6t) 2 , g(t) = 13 1 − √1+6t  M    3i 3 / 3 = 1− M λi = M−3i + O(i/M 2 ), i = 0, 1, 2, . . . , M3 , and 3 A(Mt)−1. F(t) = ∑i=0 1/λi , with F(t) = 0 if A(Mt) = 0. Note that, F(t) is the expected arrival time of the last bid to arrive before time Mt .  √ √  A(Mt) B(Mt) √ √ − Mg(t), − Mt conOur first result states that the process M M verges in distribution to a continuous two dimensional Gaussian process with mean zero. To prove this, we need the following theorem.. Let σ11 (t) =. 1 5 15 (α (t) − 1),. and σ12 (t) = σ21 (t) = σ22 (t) = t . Then, p. Theorem 3 (i) For every t > 0, F(t) − t, as M → ∞. M →   (Mt) d F(t)−Mt B √ , √M → − G(t), t ≥ 0, where G(t) = (G1 (t), G2 (t)) is a contin(ii) M uous two dimensional Gaussian process with mean zero and covariance function E(Gi (s)G j (t)) = σi j (s), 0 ≤ s ≤ t < ∞, i, j = 1, 2.. The second part of this theorem follows from the theorem by Jacod and Shiryaev. Simplifying slightly, the proof of the first part is to stop the Poisson process at time τ at which most, but not all, items are sold. Then show that, p − t . Finally, we apply the theorem by Jacod and in the stopped process, F(t) M → Shiryaev. Now let t > 0 be fixed and 0 < s ≤ t . Define the functions 1 δ11 (s,t) = 15 (α −1 (s) − α −6 (s)), δ12 (s,t) = sα −3 (s), δ21 (s,t) = sα −3 (t), and δ22 (s,t) = s.   √ A(Mt) F(t)−Mt −3 √ √ Note that, M − Mg(t) = α (t) + o p (1). Then, using TheoM rem 3 and the fact that any linear transformation of a Gaussian process yields another Gaussian process, we have  √ √  d B(Mt) ¯ √ √ Theorem 4 For each fixed t > 0, A(Mt) − Mg(t), − Mt → − G(t), M M where G¯ is a continuous two dimensional Gaussian process with mean zero and covariance function E[G¯ i (s)G¯ j (t)] = δi j (s,t), 0 ≤ s ≤ t < ∞, i, j = 1, 2.. 2.4.2. Stopped two dimensional process. In this section we present our result on the stopped two-dimensional process (A(Mt), B(Mt)). We stop the process when the number of bids submitted reaches some level, say N , which is proportional to the number of items in the auction. The interest is then to compute the limiting distribution of the 13.

(189) number of bids allocated when the process stops. Simplifying slightly, consider an auction with M distinguishable items and N bids such that N/M → γ as M → ∞, γ costant. Then, what proportion of the items would the greedy algorithm asymptotically allocate in this case ? min{t : B(t) = N}, i.e., Let AN be the number of winning bids and τN = we stop the Poisson process at time τN . Recall r(t) = (M 3 )/(M + 6t), t ≥ 0. p p p Note that B(Mt) − t . Using this, it easily follows that τNN → − 1, and τMN → − γ . That M → M is, asymptotically τN is roughly γM and r(τN ) is √1+6γ . Then, by Theorem 1,  M E(R) = r(τN ) + O( r(τN )) = √1+6γ + O(M 1/2 ). Since AN = A(τN ) = 13 (M − R), the proportion of items allocated by the algorithm, on average, is E( 3AMN ) = 1−(1+6γ)−1/2 +O(M −1/2 ). For instance, if γ = 104, then 1 − (1 + 6γ)−1/2 = 0.96, i.e, asymptotically about 96% of the items will be allocated. Formally, we have √    ⇒ N(0, σγ2 ) as M → ∞, Theorem 5 M AMN − 13 1 − (1 + 6N/M)−1/2 where σγ2 =. (1+6γ)5/2 −15γ−1 . 15(1+6γ)3. Note that, σγ2 has a maximum ≈ 0.735 at γ ≈ 0.51.. 14.

(190) Acknowledgments. First of all, I would like to express my gratitude to my advisors, Prof. Svante Janson and Prof. Arne Andersson for their support, guidance, and for sharing their vast scientific knowledge. Special thanks also goes to Prof. Sten Kaijser for his support and guidance at the begining of my studies here in Uppsala. I also take this opportunity to thank Leif Abrahamsson for being a good friend, Christian Nygaard and Carl Edstr¨om for computer related assistance, and others I forgot to mention. I would like to thank the International Science Programme for financial support, as well as administrative assistance of its staff members. I would especially like to thank Prof. Roland Tellgren for extending the financial support. I am grateful to all my friends outside the department, especially those members of “enda nitton”, and of course, B¨orje Nord, Aklilu Abraha, and Biniam Andemichael. Their friendship and encouragement has been beyond words. Last, but by no means the least, I would like to thank my family for their love and encouragement.. 15.

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(192) 3. Swedish summary. Kombinatoriska auktioner (CA) En auktion a¨ r en o¨ ppen f¨ors¨aljning av saker, vilken f¨oljer en upps¨attning regler som s¨ager hur vinnaren best¨ams och hur mycket han skall betala. Reglerna kan dessutom ge begr¨ansningar p˚a till˚atna bud. Auktioner anv¨ands i m˚anga transaktioner. M˚anga varor fr˚an snittblommor, fisk och tobak till antikviteter och fastigheter s¨aljs genom auktioner. De viktigaste sk¨alen f¨or att h˚alla auktioner a¨ r: att ge information om k¨oparnas v¨arderingar, om s¨aljaren anv¨ander ombud, att f¨orhindra ohederliga uppg¨orelser mellan ombudet och k¨oparen, och snabbhet. N˚agra av de vanligast anv¨anda auktionerna a¨ r: den o¨ ppna stigande pris-auktionen, eller Engelsk auktion, den o¨ ppna fallande pris-auktionen, eller Holl¨andsk auktion, sluten f¨orstapris-auktion, och sluten andrapris-auktion. En nackdel med dessa mekanismer a¨ r att k¨opare intresserade av att k¨opa flera enheter inte fullt ut kan ge uttryck f¨or sina preferenser. I kombinatoriska auktioner kan k¨opare ge bud inte bara f¨or individuella varor utan ocks˚a f¨or kombinationer av varor, kallade paket. Genom att l˚ata k¨opare bjuda p˚a paket ger kombinatoriska auktioner k¨oparna mer frihet att uttrycka sig, vilket ofta o¨ kar b˚ade effektiviteten och int¨akten samtidigt som det reducerar risken f¨or k¨oparna. Denna inneboende flexibilitet g¨or kombinatoriska auktioner anv¨andbara i en m¨angd resursallokeringsproblem. N¨ar s¨aljaren v¨al har f˚att alla bud m˚aste han best¨amma den m¨angd av disjunkta bud som maximerar hans totala int¨akter. Detta problem a¨ r k¨ant som vinnarproblemet (WDP). Rothkopf, Pekeˇc, och Harstad [15] p˚apekade att WDP a¨ r ekvivalent med ett packningsproblem p˚a en viktad hypergraf, och d¨arf¨or NP-fullst¨andigt. Vidare har H˚astad [5] visat att, om inte varje problem i NP kan l¨osas i probabilistisk polynomiell tid, s˚a a¨ r f¨or varje ε > 0, storleken av den st¨orsta klicken i en graf med n noder sv˚ar att approximera i polynomiell tid inom en faktor n1−ε . Det f¨oljer att det inte finns n˚agon algoritm i polynomiell tid som kan garantera en approximativ l¨osning till WDP inom en faktor n1−ε fr˚an en optimal l¨osning, d¨ar n a¨ r antalet givna bud. Som reaktion p˚a denna nedsl˚aende observation har f¨oljande alternativa metoder f¨oreslagits f¨or konstruerande av kombinatoriska auktioner. • att l¨agga restriktioner p˚a till˚atna kombinatoriska bud s˚a att WDP kan l¨osas i polynomiell tid. • att approximera l¨osningen till WDP 17.

(193) • att a¨ nd˚a anv¨anda en optimal algoritm - en som hittar en bevisligen optimal allokering men utan garanti att den blir klar p˚a polynomiell tid • att undvika sv˚arigheten att finna en l¨osning p˚a WDP genom att l¨agga ber¨akningsb¨ordan p˚a k¨oparna, l¨amnande s¨aljaren med ett ber¨akningsm¨assigt enkelt problem [8]. Avhandlingen best˚ar av fyra artiklar, som kan ses som tv˚a delar. Den f¨orsta delen a¨ r av ett mer till¨ampat slag och inneh˚aller tv˚a artiklar. I den f¨orsta artikeln studerar vi uppf¨orandet hos en viss teknik f¨or cachning i en optimal algoritm f¨or en kombinatorsk algoritm med multipla enheter. I den andra artikeln studerar vi uppf¨orandet av en kombinatorisk andraprisauktion j¨amf¨ort med simultana andraprisauktioner av varorna var f¨or sig. V˚ar analys g¨ors i samma situation som tidigare studerats av Krishna and Rosenthal [10]. V˚ar f¨orsta resultat a¨ r i mots¨attning till Krishna and Rosenthal. Vi visar att om synergiparametern a¨ r liten, s˚a ger den kombinatoriska auktionen h¨ogre f¨orv¨antad int¨akt a¨ n simultana auktioner. Vi visar ocks˚a att, i s˚adana situationer, strategin f¨or den globala budgivaren i denna modell inte a¨ r optimal (dominant). Vi j¨amf¨or ocks˚a de kombinatoriska och simultana auktionerna under antagandet att budgivarna a¨ r riskoben¨agna. S˚adana budgivare a¨ r mer k¨ansliga f¨or f¨orlust (vinnarens f¨orbannelse) och de tenderar att bjuda mindre aggressivt, vilket leder till l¨agre int¨akter. Vi ger numeriska resultat som visar vilken auktionsform som ger st¨orst int¨akter f¨or s¨aljaren beroende p˚a v¨ardena p˚a synergiparametern och riskoben¨agenhetsparametern. Den andra delen best˚ar av de a˚ terst˚aende tv˚a artiklarna. Den a¨ r mer teoretisk men vi har f¨ors¨okt att presentera den i termer av kombinatoriska auktioner. Vi analyserar h¨ar det asymptotiska beteendet hos en girig algoritm f¨or ett problem inspirerat av kombinatoriska auktioner. Mer precist studerar vi ett specialfall d¨ar varje bud omfattar exakt tre olika varor, och vi anv¨ander en Poissonprocess f¨or att modellera en auktion med ett slumpm¨assigt (Poissonf¨ordelat) antal bud. Detta speciella fall a¨ r l˚angt ifr˚an en fullfj¨adrad kombinatorisk auktion, men dess WDP a¨ r ekvivalent med en maximal packning av tripler p˚a en viktad hypergraf, och d¨arf¨or NP-fullst¨andig. Den giriga algoritmen approximerar detta specialfall inom en faktor 3. I den tredje artikeln a¨ r v˚art m˚al att hitta den asymptotiska f¨orv¨antade storleken av den partiella allokeringen som den giriga algoritmen producerar, och dess motsvarande f¨orv¨antade totala int¨akt, f¨or en given f¨ordelning av budens priser. I den fj¨arde artikeln visar vi att, efter l¨amplig normalisering, b˚ade antalet bud accepterade av den giriga algoritmen och antalet avgivna bud konvergerar i f¨ordelning mot en kontinuerlig tv˚adimensionell Gaussisk process n¨ar antalet varor i auktionen, s¨ag M , g˚ar mot o¨andligheten. Vi studerar ocks˚a fallet med ett givet (deterministiskt) antal bud som a¨ r proportionellt mot M , och visar att antalet accepterade bud, a˚ terigen efter l¨amplig normalisering, konvergerar mot en normalf¨ordelning n¨ar M → ∞. 18.

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