Auctioning a Cake: Truthful Auctions of Heterogeneous Divisible Goods

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1 Auctioning a Cake: Truthfu Auctions of Heterogeneous Divisibe Goods Yonatan Aumann Department of Computer Science Bar-Ian University Ramat-Gan 52900, Israe aumann@cs.biu.ac.i Yair Dombb Department of Computer Science Bar-Ian University Ramat-Gan 52900, Israe yairdombb@gmai.com Avinatan Hassidim Department of Computer Science Bar-Ian University Ramat-Gan 52900, Israe avinatanh@gmai.com ABSTRACT We consider the probem of auctioning a one-dimensiona continuousy-divisibe heterogeneous good (a.k.a. the cake ) among mutipe agents. Appications incude auctioning of time intervas, e.g. auctioning time for usage of a shared device, auctioning TV commercia sots, and more. Different agents may have different vauations for the different possibe intervas, and the goa is to maximize the aggregate utiity. Agents are sef-interested and may misrepresent their true vauation functions, if this benefits them. Thus, we seek auctions that are truthfu. Considering the case that each agent may obtain a singe interva, the chaenge is twofod, as we need to determine both where to sice the interva, and who gets which sice. The associated computationa probem is NP-hard even under very restrictive assumptions. We consider two settings: discrete and continuous. In the discrete setting we are given a sequence of m indivisibe eements (e 1,..., e m), and the auction must aocate each agent a consecutive subsequence of the eements. For this setting we provide a truthfu auctioning mechanism that approximates the optima wefare to within a og m factor. The mechanism works for arbitrary monotone vauations. In the continuous setting we are given a continuous, infinitey divisibe interva, and the auction must aocate each agent a sub-interva. The agents vauations are non-atomic measures on the interva. For this setting we provide a truthfu auctioning mechanism that approximates the optima wefare to within a O(og n) factor (where n is the number of agents). Additionay, we provide a truthfu 2-approximation mechanism for the case that a sices must be of some fixed size. Categories and Subject Descriptors I.2.11 [Distributed Artificia Inteigence]: Mutiagent Systems; G.2.1 [Combinatorics]: Combinatoria agorithms Genera Terms Agorithms, Economics Keywords Cake Cutting; Auctions; Resource Aocation Appears in: Aessio Lomuscio, Pau Scerri, Ana Bazzan, and Michae Huhns (eds.), Proceedings of the 13th Internationa Conference on Autonomous Agents and Mutiagent Systems (AAMAS 2014), May 5-9, 2014, Paris, France. Copyright c 2014, Internationa Foundation for Autonomous Agents and Mutiagent Systems ( A rights reserved. 1. INTRODUCTION Consider a common resource, e.g. high-end video editor, for which severa agents require access. Different agents may attribute different utiities to using the equipment, and this utiity may further vary between different time intervas (nights, mornings, Sundays). When and for how ong shoud each agent get to use the common resource? Simiary, consider a setting where commercias can be paced aongside the screening of some movie. Severa advertisers wish to pace their commercias, and are wiing to pay for doing so. Naturay, different advertisers may have different preferences as to when, and for how ong, they get to air their commercias (possiby depending on the exact content of the movie scenes), thus attributing different vaues to the different possibe time sots. How shoud one decide who gets which time sot and for what price? What is the optima way to determine the time sots? In these cases, as in many others, a natura soution is to auction the common good. Auctioning, if done right, aows for the resource to go to the agent(s) that benefits from it the most, and for the best price. However, in the above exampes the resource in consideration is not a singe item, but rather time : a continuousy divisibe and heterogeneous good. There are many ways to sice-out time (in fact, infinitey many), and each such partition may offer different utiities to the different agents. The question thus arises: How does one auction time? In this work we take the first steps in addressing this fundamenta question. We focus on the traditiona goa of maximizing utiitarian wefare maximizing the tota utiity produced by the auction. Additionay, we require that the auction be truthfu, as agents may misrepresent their vauation functions. We focus throughout on the case where each agent must get a singe, contiguous, time interva. It is interesting to compare this probem to the genera probem of combinatoria auctions [9]. Auctioning time is simiar to combinatoria auctions in that many items are auctioned at once, and participants pace different vaues on bundes of items. However, time aso contains a strong geometric eement, absent in the genera combinatoria setting; with time, the core good is an interva, not a set, and the auction needs to decide how to sice the good among the agents. Thus, on the one hand, there is the additiona chaenge of determining where to pace the cuts. On the other hand, the set of permissibe aocations is restricted to those that aocate a contiguous interva to each agent. We note that the same framework aso appies to auction- 1045

2 ing other one dimensiona continuousy divisibe goods, e.g. beach-front and. Foowing the rich iterature on fair and efficient division of an interva (see e.g. [26] for a survey), in the remainder of the paper we refer to the auctioned interva as the cake. Resuts. It was recenty shown [2] that it is computationay hard to find the sociay optima aocation in the setting we consider here, even if the vauations are fuy known, and even when imposing severe restrictions on the vauation functions. Hence, we seek approximations. We consider two setting: discrete and continuous. In the discrete setting we are given a sequence of m indivisibe eements (e 1,..., e m), and the auction must aocate each agent a consecutive subsequence of these eements. For this setting we provide a truthfu auctioning mechanism, which approximates the optima wefare to within a og m factor. This mechanism works for arbitrary monotone vauations. The continuous setting is the one presented above: we are given a continuous, infinitey divisibe interva, and the auction must aocate each agent a sub-interva. The agents vauations are non-atomic measures on the interva (in particuar, they are additive). Note that in this case there are infinitey many possibe sub-intervas, so a major chaenge is to choose the right cuts to consider, whie maintaining truthfuness. We provide a universay-truthfu auctioning mechanism for the continuous setting that approximates the optima wefare to within a O(og n) factor (where n is the number of agents). We aso consider a specia case of the continuous setting, where there is some fixed interva ength such that any aotted interva must be of exacty this ength. This requirement may arise in many rea word settings where the ength of the aotted intervas is determined by externa considerations, e.g. presentations at a conference. The probem remains NP-hard even under this restriction. Here we provide a truthfu mechanism for this case that is guaranteed to output an aocation with wefare at east 1/2 of the optimum. Communication. A major difficuty in combinatoria auctions is the communication requirements. The representation of individua vauations functions may be prohibitivey arge. Our mechanisms pace very ow communication requirements: each agent needs to provide its vauation for ony 2 m intervas, in the discrete setting, and O(n 2 ) intervas in the continuous setting. Reated Work. The use of auctions for aocating goods (typicay discrete and indivisibe) has been mathematicay studied for decades. In the past 15 years, these questions have aso been intensivey studied from a computationa point of view, mainy under the tite of Agorithmic Mechanism Design [23] (see aso [25]). It turns out that the probem of finding an optima aocation is computationay intractabe in many settings. Therefore, many works have considered restricted cases, in which some assumptions are made on the bidders possibe vauations and/or on the set of permissibe aocations [28, 29, 17]. There is a arge body of work concerning auctions of homogeneous divisibe goods [12, 30, 18, 16, 15]. In this setting, agents may have different vauations for different amounts of the good, and the auction needs to determine what amount each agent gets. The resuts from these works, however, do not carry over to our heterogeneous setting, where the vauations are not ony a function of the amount, but aso of the exact ocation of the interva within the good. The probem of dividing a heterogeneous, continuousy divisibe good among different payers is commony referred to as cake cutting. Cake cutting probems have been studied for over haf a century, with the main focus traditionay being on fairness (see [4, 27] for surveys of this iterature). In recent years the cake cutting setting has received increasing attention from the AI and muti-agent community, as a mode for resource aocation among agents, see [26]. Aumann et a. [2] consider the probem of maximizing wefare in cake cutting with contiguous pieces, showing that the probem is NP-hard even for vauations that have a rather simpe structure, and providing a constant-factor approximation agorithm; however, their agorithm does not extend to a truthfu aocation mechanism. Bei et a. [3] study the probem of finding a contiguous pieces partition that maximizes wefare whie maintaining proportionaity. They show that the probem is NP-hard even for piecewise-constant vauation functions, and provide a PTAS for inear functions. Truthfu mechanisms for the cake cutting setting were considered by severa previous works. Chen et a. [6] and Mosse and Tamuz [20] both consider the probem of devising truthfu mechanisms aimed at producing fair aocations, providing poynomia time truthfu mechanisms for various fairness criteria, and various casses of vauation functions. The setting of these works differs from ours in severa ways: their goa is fairness whie ours is aggregate wefare, their mechanism are without payments whie ours may use payments, and they aow non-contiguous pieces whie we focus on contiguous pieces. Truthfu cake cutting mechanisms without payments and non-contiguous pieces were aso considered by Maya and Nisan [19]. They consider the case of two payers with piecewise uniform vauations and show a tight bound of 0.93 on the utiitarian wefare approximation. Guo and Conitzer [13] and Han et a. [14] consider a simiar setting, but where one needs to divide a set of divisibe goods, each of which is assumed to be homogeneous. Rothkopf et a. [28] consider the discrete setting considered here. However, unike in our setting, they aow bidders to receive mutipe subsequences, and assume that the vaues of such subsequences can be (ineary) added, aowing them to obtain an optima aocation agorithm for this case. Wefare maximization has been aso studied in many other settings of mutiagent resource aocations; for a survey of this iterature see, e.g. [7]. Finay, it is worth noting that, in practice, auctions have been successfuy appied for dividing one-dimensiona goods such as TV airtime [22] and spectrum [8]. 2. PRELIMINARIES AND NOTATION We consider two setting: discrete and continuous. now formay define each. We The Discrete Setting. In the discrete setting there is a sequence of m items, E = (e 1,..., e m) to be divided among a set of n bidders/agents, denoted by the integers [n] = {1,..., n}. Each bidder has a vauation function, v i attributing a non-negative vaue to 1046

3 each possibe subsequence of E. We assume nothing on the vauation functions v i other than being monotone, i.e. that if I and I are subsequences of E, with I I (as sets), then v i(i) v i(i ), for a i. A mechanism aocates nonoverapping subsequences of E to some or a of the n bidders, such that each bidder gets at most one subsequence. Thus, each bidder i is aocated a subsequence I i (which may be empty). Note that we do not require the aocation to aot the entire resource; this assumption is known as free disposa. The Continuous Setting. In the continuous setting, the good at hand is the interva [0, 1], to be divided among the n bidders/agents. Each bidder s vauation function, v i, is a non-atomic measure on [0, 1] (in particuar, v i is additive). The mechanism aocates nonoverapping sub-intervas of [0, 1] (to which we sometimes refer as pots ) to the n bidders, such that each bidder gets at most one subinterva. A usefu consequence of the non-atomicity of the v i s is that we can ignore the endpoints of intervas, as for any a < b and any i [n] it impies that v i([a, b]) = v i((a, b)). We wi therefore consider ony aocations that give bidders open intervas, thus competey avoiding the issue of deciding who gets the boundary point between two adjacent intervas. Socia Wefare. In this work we devise mechanisms aiming to maximize the utiitarian socia wefare; that is, mechanisms that for any set of vauations {v i( )} i [n] try to find an aocation {I i} i [n] with the sum i [n] vi(ii) as high as possibe. However, as we state in Theorem 1 beow, finding an aocation that maximizes the utiitarian wefare is computationay intractabe. We therefore focus on finding mechanisms that approximate the wefare. We say that a mechanism M achieves an α- approximation of the optima wefare if it outputs an aocation {Ii M } i [n] such that for any possibe aocation {I i} i [n] it hods that α v i(ii M ) v i(i i). i [n] i [n] Communication. In genera, a bidder s vauation function is her own private data, unknown to the mechanism. This poses two chaenges for the mechanism designer. The first is finding a way for the bidders to inform the mechanism about their vauations using a reasonabe amount of communication. 1 The second is deaing with the probem that bidders may opt to misreport their vauations in order to increase their gains. The mechanisms presented in this work a operate with very imited communication. More precisey, in the discrete setting, each bidder must ony communicate the vaues of 2m specific subsequences. In continuous setting, each bidder needs to communicate O(n 2 ) such vauations, and O(n) cut points. For continuous setting with a fixed interva size, each bidder must communicate O(1/) vaues. Thus, the amount of communication required by our mechanisms is very modest, and is competey independent of the representation size of the bidders actua vauations. 1 This issue has aso been studied in the case of combinatoria auctions; see [21, 9, 25]. Truthfuness. For deaing with the bidders possibe misrepresentation of their true vauations, our mechanisms make use of payments. Formay, et M be an aocation mechanism that on every vector v = {v 1,..., v n} of vauations produces an aocation giving each bidder i a bunde I i(v) and charges her a price P i(v). We say that the mechanism M is truthfu if for each bidder i, every vector v of vauations, and every aternative vauation function v i( ) of i, it hods that v i(i i(v)) P i(v) v i(i i(v )) P i(v ) where v is the vector of (reported) vauations in which v i is repaced with v i. In other words, we require that truthteing is a dominant strategy for each bidder. 2 VCG Payments and MIR Mechanisms. A genera way for achieving truthfuness in aocations is to use the Vickrey Carke Groves (VCG) scheme (see, e.g. [25] Chapter 9). Fix some vector of vauations v = (v 1, v 2,..., v n), and denote by A the optima aocation for these vauations, giving each bidder i a pot A (i). In addition, for every i denote by A i the optima aocation that can be achieved when bidder i gets nothing. A VCG mechanism returns the optima aocation A, and charges each bidder i the price P i = k i v k (A i(k)) k i v k (A (k)). Intuitivey, bidder i is charged for the oss of vaue its presence causes to the other bidders; thus, the bidders individua incentives become aigned with the goba goa of wefare maximization. We note the besides being truthfu, VCG mechanisms (with the above payments) are aso individuay rationa in that the utiity of any bidder is never negative (and so a bidder can ony gain from participating in the auction). A drawback of the VCG scheme is that its truthfuness criticay hinges on that the returned aocation, as we as the aocations used to compute the payments, are a optima. In our setting (and in many other cases), computing optima aocations is infeasibe. However, it is sometimes possibe to get good resuts using the Maxima-in-Range (MIR) approach (see, e.g. [24, 10]), based on the foowing idea. Let A be a sub-range of a possibe aocations such that searching for the optima aocation in A is computationay tractabe; then a VCG mechanism restricted to the sub-range A is truthfu, individuay rationa, and computationay efficient. The price of restricting to the range A is, of course, the degradation in wefare resuting from the fact that the optima aocation in A is typicay inferior to the gobay-optima aocation. However, if this degradation can be bounded (as is the case in our setting), we obtain a truthfu mechanism that approximates the optima wefare. However, in our case, we cannot simpy appy this approach, due to the geometric restrictions of the probem; namey, in the continuous setting the probem is not ony how to aocate a given set of sices, but aso where to sice. Thus, some more work is required before using the MIR technique. 2 Here we aso make the standard assumption that the bidders utiities are quasi-inear; i.e., that being given an interva I i and charged a price P i, bidder i s utiity is v i(i i) P i. 1047

4 3. ALLOCATIONS WITH EQUALLY-SIZED PLOTS We begin with a reativey restrictive case. Namey, we assume that each agent must be given a pot of size (exacty), where is some given parameter. The mechanism we provide for this case gives a 2-approximation. Before going into the detais, however, we state the foowing theorem, showing that even for such restricted aocations and vauations, finding the optima aocation is computationay intractabe. Theorem 1 (Foowing from [2]). Finding an aocation that maximizes the utiitarian wefare is NP-hard, even when given a fixed size such that the a the pieces given by the aocation must be of size exacty. This hods in the discrete setting as we (in which each piece is compsed of the same number of items). The mechanism we provide here finds an aocation that is guaranteed to have at east 1/2 of the optima wefare. Both this mechanism and the one for the more genera case (in which intervas of different sizes may be aocated) use a matching technique. We thus begin with a few definitions that wi be usefu for both agorithms. Partitions. Let < 1, we ca the -partition of [0, 1] the set of intervas { (0, P 0 ) ( ) ( 1 =,, 2,..., ( 1), 1 )} ; in other words, it is a sequence of -ength consecutive subintervas of [0, 1] starting at 0 (hence the superscript). We can simiary define a sequence of such consecutive subintervas beginning at some point 0 < δ < ; we accordingy ca such a sequence a δ-shifted -partition and denote it by { (δ, P δ ) ( ) } = + δ, + δ, 2 + δ,... where the number of intervas in P δ may be either 1 or 1 1, depending on the reation between the sizes δ and 1 1. Partition Graphs. For a partition P we define a compete and weighted bipartite graph G P as foows. The eft set of vertices corresponds to the set [n] of bidders, and the right set of vertices corresponds to the set of intervas in P. For any i [n] and I j P, we create an edge e ij with weight w(e ij) = b i(i j), i.e. the bid of bidder i for interva I j. Ceary, any matching in G P induces an aocation in which each bidder i is aocated the interva to which it is matched (if one exists). We therefore refer to aocations and matchings interchangeaby. The Mechanism. With these notions in hand, we now describe our mechanism for auctioning uniform-size pots. Given the pot size, the mechanism, to which we refer as Mechanism 1, operates as foows: 1. Bidding: Set = 1 1. Create the partitions P 0 and P. For every bidder i, get a bid b i(i) for each interva I P 0 P of the two partitions. 2. Computing the Aocation: Create the partition graphs G P 0 and G P. Compute maximum weight matchings M0 M in G P 0 and G P (respectivey). and Return the heavier among M0 and M, denoted M. 3. Computing the Payments: For each bidder i, compute the maximum weight matching in G P 0 with i removed, and in G P with i removed; denote the heavier of these matchings by M i. Charge each bidder i a payment P i = w(m i) w(m ) + b i(m (i)). Lemma 1. Mechanism 1 requires ony 2/ bids from each bidder, runs in time poynomia in n + 1/, and is truthfu. Proof. The number of bids is apparent from the description of the mechanism. For the running time, it is easy to see that the main computationa task of Mechanism 1 is the computing a maximum-weight matching in the bipartite graphs G P 0 and G P, each having n + 1/ vertices. Computing such matchings can be done in time O((n+1/) 3 ) [11], and so the tota running time of the mechanism is bounded by O(n (n + 1/) 3 ). For the truthfuness, note that Mechanism 1 in fact uses VCG payments. In addition, the aocation part of the mechanism computes the maximum among a aocations in which the intervas are either a from P 0 or a from P. Therefore, Mechanism 1 is a MIR mechanism with VCG payments, and is thus truthfu. We next show that Mechanism 1 approximates the optima utiitarian wefare we. Lemma 2. The aocation returned by Mechanism 1 approximates the optima wefare by a factor of 2. Proof. Consider the optima aocation A, giving each bidder i an interva A (i). We create two matchings M 0 in G P 0 and M in G P and show that one of them must have weight that is at east haf the tota vaue of A. Thus, the aocation M (which the heaviest among a the matchings in G P 0 and G P ) ceary has at east this vaue. Reca that we denote by = 1 1 the amount of the resource that is eft uncovered in any partition to intervas of size ; ceary <. Let i be a bidder that receives an interva A (i) in the optima aocation; we can write A (i) = ( ) j i + δ i, (j i + 1) + δ i with δ i <. The matching M 0 wi match i to I 0 j i, i.e. to the j i-th interva (from the eft) in P 0, if δ i. If δ i >, M 0 wi match i to I 0 j i +1 (i.e. to the (j i + 1)-th interva from the eft in P 0 ). The matching M wi aways match i to I j i, i.e. the j i-th interva from the eft in P. Bidders that receive nothing in A wi not be matched by either M 0 nor M. We iustrate this with an exampe in Figure

5 {}}{ } {{ } A (i) M (i) M 0 (i) {}}{ } {{ } Figure 1: An exampe of the intervas matched to some i in M 0 and M. The fu ines separate the intervas of P 0 and the broken ones separate the intervas of P. In this exampe, bidder i receives in A an interva of the form (j + δ i, (j + 1) + δ i) with δ i >. It is easy to see that M is indeed a vaid matching: in order for two bidders i and i to be matched to the same interva Ij L, it must be that both A (i ) and A (i ) begin somewhere in the interva [j, (j + 1) ). However, this is ceary impossibe, as A must aocate them both disjoint intervas of ength. To see that M 0 too is a vaid matching, assume that i and i are both matched in M 0 to the same interva Ij 0. Again, it is impossibe that A (i ) and A (i ) begin both within [j, (j + 1) ) or both within [(j 1), j ). Therefore, w..o.g. it is the case that A (i ) = ( ) (j 1) + δ i, j + δ i and A (i ) = ( ) j + δ i, (j + 1) + δ i with δ i > and δ i. However, this impies that the (non-empty) interva (j +δ i, j +δ i ) is common to both A (i ) and A (i ); this is again impossibe, and we concude that M 0 and M are both vaid matchings. We have thus matched every bidder i getting some interva in A to some interva M 0(i) P 0 and some interva M (i) P. It is aso easy to observe that A (i) M 0(i) M (i) for a i; thus, we have that v i(a (i)) v i(m 0(i)) + v i(m (i)) = b i(m 0(i)) + b i(m (i)) where the inequaity foows from the monotonicity and subadditivity assumptions, and the equaity from truthfuness. Summing over a bidders, we get that w(m 0) + w(m ) v(a ), which impies that at east in one of the graphs G P 0 and G P there exists a matching of weight v(a )/2. Combining Lemma 1 and Lemma 2, we get: Theorem 2. Mechanism 1 is an efficient and truthfu 2- approximation mechanism for the probem of auctioning a continuous resource with fixed pot size. Note that the agorithm aso works for sub-additive vauation functions, as the proof of Lemma 2 uses ony subadditivity. 4. THE DISCRETE SETTING We now consider the discrete setting, in which a sequence of m indivisibe items E = (e 1,..., e m) is auctioned, and each bidder can get a at most a singe sub-sequence of E. For this case we show a mechanism that approximates the optima wefare without the need to assume anything except monotonicity. To make the presentation sighty simper, assume that the number of items is m = 2 r for some integer r; as we show ater, this is without oss of generaity. 1. Bidding: For every t = 0,..., r create the 2 t -partition (without any shifting), in which each interva in the partition is a consecutive sequence of 2 t items. We denote the t-th partition by P t. For every bidder i, get a bid b i(i) for each interva I t Pt in these partitions. 2. Computing the Aocation: For every t = 0,..., r create the partition graph G Pt. Compute a maximum weight matching Mt in the graph G Pt. Among a matchings M t, return the heaviest one, denoted M. 3. Computing the Payments: For each bidder i consider a the graphs G Pt with i removed; among a the matchings in these graphs, find the heaviest one, and denote it by M i. Charge each bidder i a payment P i = w(m i) w(m ) + b i(m (i)). Lemma 3. Mechanism 2 requires at most 2m bids from each bidder, runs in time poynomia in n+r, and is truthfu. Proof. The number of bids per bidder is r t=0 m 2 t = m r t=0 1 2 t < 2m. The arguments proving the rest of the emma are indentica to the ones proving Lemma 1. Lemma 4. The aocation returned by Mechanism 2 approximates the optima wefare by a factor of r + 1. Proof. Let A be an optima aocation (giving each agent i an interva A (i)) and et I = t Pt be the set of a the intervas in a of the partitions P t. We show a one-to-one correspondence f( ) between the set of agents receiving non-empty pots in A and the intervas of I such that v i(a (i)) v i(f(i)). Since f( ) is one-to-one, restricting it to the intervas of any singe P t yieds a matching in 1049

6 G Pt, which we denote by M f t. The emma then foows, as w(m ) 1 r r + 1 w(mt ) t=0 1 r r + 1 w(m f t ) = 1 r + 1 v i(f(i)) t=0 i 1 r + 1 v i(a (i)) = 1 r + 1 v(a ). i It thus remains to show a correspondence f( ) with the above properties. We construct f to map each agent i (getting a non-empty pot in A ) to the smaest interva in I that contains A (i). It is easy to observe that for each nonempty A (i), the choice of f(i) is unique; furthermore, since f(i) A (i), our monotonicity assumption guarantees that v i(f(i)) v i(a (i)). To see that f( ) is one-to-one, assume that there exists some interva I I and two agents k k such that f(k) = f(k ) = I. This interva ceary cannot beong to P 0, in which each interva contains exacty one item. Thus, I P t+1 for some t 0; et us expicity refer to the endpoints (i.e. indices of the first and ast item) of I, and write I = [a, c]. We know that I aso contains the two intervas [a, b] and [b+1, c] (where the ength of [a, b] and [b+1, c] is haf of the ength of [a, c]), which are both in P t. By the definition of f, neither of A (k) or A (k ) is contained in [a, b] or in [b + 1, c], but both are contained in [a, c]. However, this impies that both A (k) and A (k ) must contain an item from [a, b] as we as an item from [b + 1, c], and thus both must contain the items b and b + 1, which is impossibe as they cannot intersect. Consider now the case in which m is not an integer power of two. In this case, we set r = og m, and set the ast partition P r to simpy contain a singe interva of a the items 1,..., m. To make sure we cover a the items in the other partitions as we, we add to each partition P t that does not cover a the items the ast interva of the partition P t 1. 3 It is easy to see that this maintains that f is oneto-one (as any interva in P t+1 is the union of at most two intervas in P t), and thus Lemma 4 hods for this case as we. Combining Lemma 3 and Lemma 4, we obtain: Theorem 3. Mechanism 2 is an efficient and truthfu ( og m + 1)-approximation mechanism for the probem of auctioning a discrete resource. 5. ALLOCATING A CONTINUOUS CAKE In this section we treat the cassic case, where bidders vauation functions are non-atomic measures on the interva [0, 1], and there is no imitation on the size of each subinterva a bidder can get. The mechanism works in two stages. In the first stage it chooses n/2 bidders at random, and uses them to spit the interva [0, 1] to at most O(n 2 ) segments. In the second stage, it invokes Mechanism 2 to divide the cake between the bidders who did not participate in the first stage: Mechanism 3 3 To eiminate ambiguity in the definition of f, we simpy define it to map each agent i to the smaest interva in the partition with the minima t containing A (i). 1. Creating sub intervas: Choose n/2 bidders at random. Denote this set by S. For every bidder i S, ask i to divide [0, 1] to intervas of equa worth to that bidder. Generate a partition J by taking the union of a boundary points reported by the bidders of S. 2. Computing the Aocation and payments: Treat every interva in J as a singe indivisibe item, and invoke Mechanism 2 on the bidders in [n] \ S and on the items in J. Note that bidders in S ony hep define a partition, but do not receive any cake (nor pay any money). Lemma 5. Mechanism 3 requires at most 2 bids from each bidder, runs in poynomia time, and is universay truthfu. Proof. The bidders in [n] \ S see a mechanism having at most n 2 fixed intervas (or discrete items), and they get sequences of them. Hence, for these bidders, the emma foows directy from Lemma 3. For the bidders in S, the mechanism is truthfu since they do not get any utiity no matter what they do, and thus they have no incentive to misreport. 4 The bounds on the communication and computationa compexity foow straightforwardy from Lemma 3. Note that bidders in S are asked questions of the form given a point a, what is the smaest point b such that v i(a, b) = v i([0,1])? ; such queries are known as cut queries in the cake-cutting iterature. If we restrict ourseves to mechanisms with with evauation queries ony (as in Mechanisms 1 and 2), it is easy to show that the best possibe approximation ratio possibe for this setting is n, which can be obtained by giving the entire cake to a singe bidder. Lemma 6. The aocation returned by Mechanism 3 approximates the optima wefare by a factor of O(og n) in expectation. Proof. Consider the (random) discrete instance created by the mechanism, in which the items of J are aocated to bidders from [n]\s. Denote by A J the optima aocation for this (discrete) instance, and by v(a J) its wefare. We show that v(a J) is expected to be a constant fraction of v(a ), i.e. that E [ v(a J) ] v(a )/α for some constant α. Pugging this into Lemma 4 yieds the desired resut. The remainder of the proof is therefore dedicated to estabishing the above inequaity. 4 The mechanism as described provides ony that truthteing is a weaky dominant strategy for bidders in S. This is standard in auction theory (see e.g. [1, 5] for a discussion on strict and weak dominance). By sighty modifying the mechanism we can get, in addition, that for any vauation function of any singe bidder, there exist vauations for the other bidders for which truthteing is stricty dominant. This can be achieved by initiay not teing the bidders if they are in S or not, and requiring a bidders to provide their divisions in stage (1) of the mechanism. Then, in stage (2), the vauations provided by the bidders not in S must agree with their previousy reported vauations. 1050

7 Let us ca a bidder i happy if v i(a (i)) v i([0,1]), i.e. if in the optima aocation she gets at east 1 from her vaue for the entire cake. Denote by H = { i : v i(a (u)) } vi([0, 1]). the set of happy bidders. An easy fact it that the happy bidders must contribute at east haf of the wefare in the optima aocation, i.e. i H v i(a (i)) v(a ) 2. (1) To see this, assume that this is not the case; then v(a ) = v i(a (i)) + v i(a (i)) i H < v(a ) 2 v(a ) 2 + i [n]\h + i [n]\h i [n]\h v i(a (i)) v i([0, 1]) which is equivaent to 1 n i [n]\h vi([0, 1]) > v(a ). Since [n]\h has at most n bidders, at east one of them must have v i([0, 1]) > v(a ), and thus giving her the entire cake yieds wefare stricty greater than the optima soution, which is impossibe. Let us now order the happy bidders by the order of their pots in A (from eft to right), and write H = {i 1,..., i H }; i.e. i 1 is the happy bidder that gets the eftmost piece, i 2 is the happy bidder getting the second-eftmost piece, etc. (We ignore a non-happy bidders that may get a piece paced between the pieces of happy bidders.) We further ca a happy bidder i k good if i k [n] \ S and i k 1, i k+1 S, i.e. if she is not in the set S but both her neighbors are. (If k = 1 or k = H we treat it as if the missing neighbor is indeed in S.) We denote the set of good bidders by G. We can now show that in the discretized instance created by the mechanism, there is an aocation A J giving each good bidder i k G a piece worth at east the vaue of her piece in the optima division v ik (A (i k )). To see that, consider the piece A (i k 1 ) of her eft happy neighbor. Since i k is good, it must be that i k 1 S. Since i k 1 is a happy bidder, by definition we have that v ik 1 (A (i k 1 )) vi k 1 ([0, 1]). Thus, when i k 1 is asked to divide the cake into pieces of, at east one of the boundaries must fa within A (i k 1 ); this boundary can be the eftmost boundary of A J(i k ). We can symmetricay show how to set the rightmost border of A J(i k ) to somewhere within A (i k+1 ). vaue v i ([0,1]) k 1 Figure 2 iustrates this with an exampe. Thus, we have obtained that v ik (A J(i k )) v ik (A (i k )). Combining the above, the wefare of the optima division in the discretized instance can be ower-bounded by v(a J) v(a J) v ik (A (i k )) (2) i k G Note, however, that this vaue depends on the set G which is generated randomy. Ceary, for any singe i k H, we have } {{ } A (i k 1 ) A J (i k) {}}{ }{{}}{{} A (i k ) A (i k+1 ) Figure 2: An exampe of a piece A J(i k ). The crossedout portions are pieces which A gives to non-happy bidders, which we ignore. The dotted ines mark the division of the cake into pieces of vaue 1 for bidder i k 1, and the dotted ines mark the same division for bidder i k+1. If i k is good, i.e. if i k [n] \ S and i k 1, i k+1 S, then in the discrete instance it is possibe to give i k the interva A J(i k ), which contains A (i k ). that Pr [ i k G ] 1. Thus, by the definition of expectation 8 we can concude that E [ v(a J) ] [ ] E v ik (A (i k )) i k G = v ik (A (i k )) Pr [ i k G ] i k H 1 8 v ik (A (i k )) i k H v(a ) where the first inequaity foows from (2), and the ast one from (1). Combining Lemma 5 and Lemma 6, we obtain: Theorem 4. Mechanism 3 is an efficient and universay truthfu O(og n)-approximation mechanism for the probem of auctioning a continuous resource. 6. CONCLUSION AND OPEN PROBLEMS We studied the probem of auctioning contiguous subintervas of a one-dimensiona good. The chaenges in this setting are two fod: to determine where to cut the cake, and which agent gets what piece. The setting is characterized by an inherent geometric structure, absent in combinatoria auctions. As the probem of finding an optima aocation of such goods is computationay hard (even without incentives), we provide approximation mechanisms. There are many natura extensions to this work, most of which are open probem for future research, such as: Variations on the Setting. In the continuous case we assume that the bidders vauations are additive. This is used in the first stage of the mechanism which determines the division into sub-intervas but it is not necessary for the second. Can this assumption be weakened? 1051

8 Lower Bounds. The ony negative resut we are aware of is that no FPTAS exists for the probem uness P=NP [2], and this resut hods even without requiring truthfuness. Can a stronger ower bounds on the approximabiity of wefare by truthfu mechanisms be proven? Mutipe Pieces per Bidder. We have considered ony aocations in which each bidder can receive at most one contiguous piece of the resource. A natura reaxation of this requirement is aowing bidders to get mutipe such pieces. If the number of pieces per bidder is a constant, or given as a parameter, it is not hard to see that the aocation probem remains NPcompete. How we can truthfu mechanisms approximate the wefare in that case? What can be achieved when dropping the contiguity requirement atogether (whie sti having a minimum aowed piece size)? 2-Dimensiona Resources. Whie in many cases it is reasonabe to mode the resource as having ony one-dimension, other resources are inherenty mutidimensiona, e.g. and. Deaing with such resources seems to require different toos and techniques, and poses interesting chaenges for a mechanism designer. 7. REFERENCES [1] D. Abreu and H. Matsushima. Virtua impementation in iterativey undominated strategies: compete information. Econometrica: Journa of the Econometric Society, pages , [2] Y. Aumann, Y. Dombb, and A. Hassidim. Computing sociay-efficient cake divisions. In AAMAS, pages , [3] X. Bei, N. Chen, X. Hua, B. Tao, and E. Yang. Optima proportiona cake cutting with connected pieces. In AAAI, [4] S. J. Brams and A. D. Tayor. Fair Division: From cake cutting to dispute resoution. Cambridge University Press, New York, NY, USA, [5] J. Chen, A. Hassidim, and S. Micai. Robust perfect revenue from perfecty informed payers. In ICS, pages , [6] Y. Chen, J. Lai, D. C. Parkes, and A. D. Procaccia. Truth, justice, and cake cutting. In AAAI, [7] Y. Chevaeyre, P. E. Dunne, U. Endriss, J. Lang, M. Lemaitre, N. Maudet, J. Padget, S. Pheps, J. A. Rodriguez-Aguiar, and P. Sousa. Issues in mutiagent resource aocation. Informatica (Sovenia), 30(1):3 31, [8] P. Cramton. The fcc spectrum auctions: An eary assessment. Journa of Economics & Management Strategy, 6(3): , [9] P. Cramton, Y. Shoham, and R. Steinberg. Combinatoria auctions. MIT Press, [10] S. Dobzinski and N. Nisan. Limitations of vcg-based mechanisms. In STOC, pages , [11] J. Edmonds and R. M. Karp. Theoretica improvements in agorithmic efficiency for network fow probems. J. ACM, 19(2): , Apr [12] R. J. Gibbens and F. P. Key. Resource pricing and the evoution of congestion contro. Automatica, 35(12): , [13] M. Guo and V. Conitzer. Strategy-proof aocation of mutipe items between two agents without payments or priors. In AAMAS, pages , [14] L. Han, C. Su, L. Tang, and H. Zhang. On strategy-proof aocation without payments or priors. In WINE, voume 7090 of Lecture Notes in Computer Science, pages , [15] R. Johari and J. N. Tsitsikis. Efficiency of scaar-parameterized mechanisms. Operations Research, 57: , Juy-August [16] I. Kremer and K. G. Nyborg. Divisibe-good auctions: The roe of aocation rues. RAND Journa of Economics, pages , [17] B. Lehmann, D. J. Lehmann, and N. Nisan. Combinatoria auctions with decreasing margina utiities. In ACM Conference on Eectronic Commerce, pages ACM, [18] R. T. Maheswaran and T. Basar. Nash equiibrium and decentraized negotiation in auctioning divisibe resources. Group Decision and Negotiation, 13:2003, [19] A. Maya and N. Nisan. Incentive compatibe two payer cake cutting. In WINE, voume 7695 of Lecture Notes in Computer Science, pages Springer, [20] E. Mosse and O. Tamuz. Truthfu fair division. In Agorithmic Game Theory, pages Springer, [21] N. Nisan. Bidding and aocation in combinatoria auctions. In ACM Conference on Eectronic Commerce, pages 1 12, [22] N. Nisan, J. Bayer, D. Chandra, T. Franji, R. Gardner, Y. Matias, N. Rhodes, M. Setzer, D. Tom, H. Varian, and D. Zigmond. Googe s auction for tv ads. In Automata, Languages and Programming, voume 5556 of Lecture Notes in Computer Science, pages Springer Berin Heideberg, [23] N. Nisan and A. Ronen. Agorithmic mechanism design (extended abstract). In STOC, pages , [24] N. Nisan and A. Ronen. Computationay feasibe vcg mechanisms. In ACM Conference on Eectronic Commerce, pages , [25] N. Nisan, T. Roughgarden, E. Tardos, and V. V. Vazirani. Agorithmic Game Theory. Cambridge University Press, New York, NY, USA, [26] A. D. Procaccia. Cake cutting: not just chid s pay. Communications of the ACM, 56(7):78 87, [27] J. Robertson and W. Webb. Cake-cutting agorithms: Be fair if you can. A K Peters, Ltd., Natick, MA, USA, [28] M. H. Rothkopf, A. Pekeč, and R. M. Harstad. Computationay manageabe combinationa auctions. Management Science, 44(8): , [29] M. Tennenhotz. Some tractabe combinatoria auctions. In AAAI/IAAI, pages , [30] S. Yang and B. Hajek. Vcg-key mechanisms for aocation of divisibe goods: Adapting vcg mechanisms to one-dimensiona signas. Seected Areas in Communications, IEEE Journa on, 25(6): ,