A Robust Open Ascending-price Multi-unit Auction Protocol against False-name Bids

Transcription

1 A Robust Open Ascendng-prce Mult-unt Aucton Protocol aganst False-name Bds Atsush Iwasak and Makoto Yokoo Kenj Terada NTT Corporaton NTT Corporaton NTT Communcaton Scence Laboratores NTT Informaton Sharng Platform Laboratores 2-4 Hkarda, Seka-cho, Soraku-gun, Mdor-cho, Musashno, Kyoto , JAPAN, Tokyo , Japan yokoo} {wasak, ABSTRACT Ths paper presents a new ascendng-prce mult-unt aucton protocol. As far as the authors are aware, ths s the frst protocol that has an open format, and n whch sncere bddng s an equlbrum strategy, even f the margnal utltes of each agent can ncrease and agents can submt falsename bds. As ever-ncreasng numbers of companes and consumers are tradng on Internet auctons, a new type of cheatng called false-name bds has been notced. Specfcally, there may be some agents wth fcttous names such as multple e-mal addresses. The VCG s not an open format, and truth-tellng s no longer a domnant strategy f agents can submt false-name bds and the margnal utltes of each agent can ncrease. The Iteratve Reducng (IR) protocol wth a sealed-bd format s robust aganst falsename bds, although t requres the auctoneer to carefully pre-determne a reservaton prceforoneunt. Open format protocols, such as the Ausubel aucton, outperform sealed-bd format protocols n terms of the smplcty and prvacy-preservaton. These two advantages are sad to encourage more agents to bd sncerely and to provde the seller wth hgher revenue. We extend the Ausubel aucton to our proposed protocol whch can handle the cases where the margnal utltes of each agent can ncrease. Moreover, t s robust aganst false-name bds and does not requre the auctoneer to set a reservaton prce. Oursmulaton result ndcates that our protocol heren obtans a socal surplus close to Pareto effcent and that t outperforms the IR wth respect to the socal surplus and the seller s revenue. Categores and Subject Descrptors I.2.11 [Artfcal Intellgence]: Dstrbuted Artfcal Intellgence; K.4.4 [Computers and Socety]: Electronc Commerce Permsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. To copy otherwse, to republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. EC 03, June 9 12, 2003, San Dego, Calforna, USA. Copyrght 2003 ACM X/03/ $5.00. General Terms Economcs, Theory 1. INTRODUCTION Ths paper proposes a newly developed ascendng-prce mult-unt aucton protocol. As far as the authors are aware, ths s the frst protocol that has an open format, and n whch sncere bddng s an equlbrum strategy even f the margnal utltes of each agent can ncrease and agents can submt false-name bds. The Internet has recently provded an excellent nfrastructure for executng much cheaper auctons wth lots of sellers and buyers from all over the world. More and more companes and consumers buy and sell varous goods on Internet auctons; therefore, Internet auctons have become a popular part of Electronc Commerce, and a promsng feld for applyng AI technologes [3, 5]. However, the authors have ponted out a new type of cheatng called false-name bds [10, 12, 15]. Specfcally, there may be some agents wth fcttous names, such as multple e-mal addresses who can decrease ther payments usng false-name bds. Because dentfyng each partcpant s vrtually mpossble on the Internet, t s desrable for Internet aucton protocols to be robust aganst false-name bds. The Vckrey-Clarke-Groves (VCG) protocol s a well-known protocol n whch truthtellng s a domnant strategy. However, the authors [10, 12, 15] show that truth-tellng s no longer a domnant strategy n mult-unt or combnatoral auctons f the margnal utltes of each agent can ncrease and agents can submt false-name bds. In other words, an agent can obtan an addtonal payoff by usng false-name bds n that protocol. Furthermore, these papers revealed that no robust aucton protocol exsts aganst false-name bds for mult-unt or combnatoral auctons whose outcomes smultaneously satsfy both Pareto effcency and ndvdual ratonalty. The authors [13, 14] have developed a combnatoral and amult-unt aucton protocol, Leveled Dvson Set (LDS) and Iteratve Reducng (IR) respectvely, whch are robust aganst false-name bds. However, they do have sealed-bd formats. As far as the authors are aware, no aucton protocol exsts that s robust aganst false-name bds and has an open format. Open format auctons [7] such as the Englsh, Dutch and

2 Ausubel types aresadtooutperform sealed-bd format auctons [1]. For example, the VCG aucton, also called the Vckrey aucton, s not prevalent, whle the Englsh aucton s hghly prevalent n the real world, even though the outcomes of these two formats are equvalent n prvate value auctons. Furthermore, Ausubel shows that smplcty and prvacy-preservaton seem to encourage more agents to bd honestly and to provde the seller wth a hgher revenue [1]. Ths smplcty leads agents to understand the Ausubel protocol more easly than the VCG aucton. The prvacypreservaton of the agents values nduces agents to jon auctons snce they need to reveal only parts of ther demand curve. On the other hand, n sealed-bd formats, partcpants may not be very comfortable about truthfully revealng ther entre prvate values, even though dong so s a domnant strategy. In fact, n an expermental study, Kagel and Levne [6] show that subjects tend to bd more aggressvely n a unform prce mult-unt aucton wth an open format than n one wth a sealed-bd format. They also observe that the subjects tend to behave accordng to the equlbrum strategy. The Ausubel aucton protocol [1] s a representatve open ascendng-prce mult-unt aucton. 1 In ths protocol, the auctoneer announces a low prce, and each agent declares her demand. The auctoneer then aggregates the demands and rases the prce untl the market clears. In the process, he allocates unts to agents accordng to a clnchng rule, whch we wll explan n Secton 2. In an aucton wth prvate values, sncere bddng by all agents sanexpostperfect equlbrum. In ths paper, we concentrate on prvate value auctons [8]. In prvate value auctons, all agents know ther own evaluaton values of the tems wth certanty, whch are ndependent of the other agents evaluaton values. We defne an agent s utlty as the dfference between ths prvate value of the obtaned tems and ts payment. Such a utlty s called a quas-lnear utlty [8]. These assumptons are commonly used for makng theoretcal analyss tractable. The Ausubel aucton yelds the same effcent outcome as the VCG aucton, also called the mult-unt Vckrey aucton, f every agent has a dmnshng margnal value n a unt. In general, t s natural to assume that a margnal value tends to dmnsh f the quantty of unts becomes very large. On the other hand, f the quantty of unts s relatvely small, we cannot always assume admnshng margnal value. A typcal example s an all-or-nothng case, where one needs a certan quantty of unts, otherwse one needs nothng (e.g., arplane tckets for a famly trp, or network bandwdth n avdeo-conferenceapplcaton). We extend the Ausubel aucton to our proposed protocol, whch can handle the cases where the margnal utltes of each agent can ncrease. Moreover, the protocol s robust aganst false-name bds and does not requre the auctoneer to set a reservaton prce. We compare the proposed protocol wth the IR protocol. The results show that t performs a socal surplus close to a Pareto-effcent one. Furthermore, t outperforms the IR protocol n terms of the socal surplus and seller s revenue. 1 Bkhchandan and Ostroy [4] and Parkes [9] nvestgate an ascendng-prce and an teratve aucton respectvely. However, they focus not on robust aucton protocols aganst false-name bds, but on aucton protocols that yeld the effcent outcome for the VCG. Ths paper s organzed as follows. Secton 2 ntroduces theausubelaucton protocol. Secton 3 descrbes basc deas and explans the proposed protocol heren va llustratve examples. Secton 4 presents the formal model and analyss. Secton 5 evaluates the proposed protocol and compares t wth the IR protocol. Secton 6 shows dscusses on the proposed protocol, and Secton 7 concludes the paper. 2. AUSUBEL AUCTION PROTOCOL Ths secton ntroduces the Ausubel aucton protocol [1], whch s an open ascendng-prce mult-unt aucton whose outcome s equvalent to the VCG. The auctoneer aggregates the declared demands and allocates the unts of an tem to agents. The prce rses untl the market clears,.e., all unts of the tem are allocated. The allocaton procedure s called a clnchng rule. The allocaton of agent depends not on her declared demand, but on the demand of all other agents. The declared demands of q (p) whcheach agent declares at a prce of p and the quantty of avalable supply of K, determne the resdual supply, whch s defned as Eq. 1. s (p) max 8 < X : 0, K j 9 = q j(p) ;. (1) Theresdual supply s always non-negatve and a non-decreasng functon at a prce p. No unts of an tem are allocated whle the resdual supply of each agent s zero. If a resdual supply facng an agent becomes 1 at a prce, the frst unt of the tem s clnched by. Ifthe resdual supply becomes 2 at one of the followng prces, then the second unt s clnched by at that prce, and so on. Note that the prce for the frst unt s dfferent from the prce for thesecond unt. In examples that follow n ths paper, we wll refer to v = (v 1,...,v k,...,v K )asavaluevector. Here, v k represents agent s evaluaton value for havng total k unts of the auctoned tem. Ausubel defnes sncere bddng as the followng strategy. Defnton 1. The sncere demand of an agent at a prce maxmzes her utlty at that prce. Sncere bddng n the Ausubel aucton s the strategy n whch she always bds her sncere demand, subject to the constrants posed by a monotone actvty rule (Eq. 3) and the quantty/prces of unts she has already clnched. Example 1. Let us assume that there are two agents and two unts of an tem. The evaluaton values of agents are assumed as follows: v 1 = (7, 8), v 2 = (6, 9). For example, whenagent 1 buys two unts of the tem at a total prce of 6, sheobtans the utlty 8 6=2. Let us assume that the aucton begns wth a prce of 1(+ɛ). Here, ɛ denotes a suffcently small quantty. Each agent declares her demand of 1 and 2 f she bds sncerely. The aggregate demand of 3 exceeds the avalable supply of 2, so the aucton must proceed further. However, let us examne ths stuaton carefully from agent 2 s perspectve. The aggregate demand of all agents other than agent 2 s 1, whle2 unts of the tem are avalable. If agent 1

3 bds monotoncally, agent 2 s now mathematcally guaranteed to obtan at least 1 unt of the tem. Thus, agent 2 obtans one unt at the prce of 1. Snce there s stll excess demand, the prce contnues upward. At the prce of 3, the aggregate demand of all agents s 1+1=2, whch s equal to the avalable unts, 2. Here the market has cleared and the aucton ends. Agent 2 has already clnched one unt at the prce of 1. Agent 1 obtans one unt at the prce of 3. Accordngly, agent 1 obtans the utlty of 7 3=4whle agent 2 obtans 6 1=5. The Ausubel aucton yelds the same effcent outcome as the VCG aucton, and sncere bddng by all agents s an ex post perfect equlbrum f every agent has a dmnshng margnal utlty n a unt. However, the Ausubel aucton does not account for such stuatons that the margnal utltes of each agent can ncrease. The next example shows that n such a case, sncere bddng defned above s no longer an equlbrum strategy,.e., an agent can ncrease her utlty by over-declarng her demand. Example 2. Let us assume that there are two agents and two unts of an tem. The evaluaton value of agents s defned as follows. v 1 = (7, 8), v 2 = (0, 10). Suppose that each agent bds sncerely, lke n the Ausubel aucton. Agent 2 clnches her frst unt at the prce of 1. Snce agent 2 s evaluaton value for two unts s all-or-nothng, her demand remans two unts as long as the prce s less than 9 (note that she already clnched the frst unt at the prce of 1). Then, she clnches the second unt at the prce of 7. Agent 1 obtans nothng and her utlty s 0. However, agent 1 can ncrease her utlty by over-declarng her demand. For example, suppose that agent 1 pretends to have a valuaton (7, 12). Because agent 1 keeps her demand untl the prce of 5, agent 2 cannot clnch the frst unt before the prce reaches 5(+ɛ). Atthat prce, agent 2 reduces her demand to 0. Thus,agent 1 can obtan oneuntattheprce of 5 andher utlty becomes 7 5 = 2.Thus,agent 1 receves more utlty by over-declarng than by sncere bddng. 3. BASIC IDEAS AND ILLUSTRATIVE EX- AMPLES In ths secton, we descrbe how the Ausubel acton can be modfed nto the proposed protocol. Then, we show how the protocol works va an llustratve example. The characterstcs of our newly developed protocol are as follows. An agent pays the same unt prce for all unts she obtans n our proposed aucton. To prevent demand reducton les, the proposed protocol enables an agent to choose from multple optons she obtaned. More specfcally, as shown n the prevous example, when usng the Ausubel aucton for the stuatons where the margnal utltes of each agent can ncrease, an agent s demand depends not only on the current prce but also on the quantty/prces of prevous clnched unts. The current demand declared by an agent wll affect the quantty/prces of clnched unts of other agents. Then, these quantty/prces of clnched unts of other agents affect the future demands of these agents. Also, the future demands of these agents affect the future resdual supply of the agent. Therefore, the current demand declared by the agent affects the future resdual supply of the agent. Thsprovdes agent wth an ncentve to over-declare herdemand, as shown n Example 2. Thus, sncere bddng s no longer an equlbrum strategy f the margnal utltes of each agent can ncrease. To elmnate such dependences, n the proposed protocol, an agent pays the same unt prce for all unts she obtaned. By applyng the same unt prce, the demand of the agent depends only on the current prce (not on the quantty/prces of prevous clnched unts). However, f we apply the same unt prce, as n unform prce aucton or M+1 -st prce aucton protocols [2, 7], we need to consder the possblty of a demand reducton le [2]. A demand reducton le means that an agent has an ncentve tounder-declare ts demand so that t can obtan fewer unts wth less prce. In the proposed protocol, each agent does not mmedately obtan unts of an tem, but just receves several optons, each of whch guarantees her to be able to buy a certan quantty of unts at a certan prce. Here, an ordered par, (p, c) representsan opton wth whch an agent can buy c (or fewer) unts at unt prce p. For example, let us assume that an agent has two optons, (5, 1) and (8, 3). In ths case, the agent can buy one unt at the prce of fve, or buy two orthreeunts at the prce of eght. Also, she can choose to buy nothng (and pay nothng). These extensons guarantee sncere bddng to be an equlbrum strategy even f the margnal utltes of each agent can ncrease and agents can submt false-name bds. The proposed protocol heren s an ascendng-prce aucton lke the Ausubel aucton. The auctoneer announces a prce p and the agents respond wth ther demands q (p). The auctoneer then aggregates ther demands and rases the prce untl the market clears. The auctoneer allocates optons represented by (p, c) toagents durng the aucton. Note that an agent s declared demand and resdual supply determne what optons she obtans. We wll llustrate our proposed protocol va the followng example. Example 3 smply llustrates how agents obtan (clnch) ther optons and choose one of gven optons. Example 3. There are fve dentcal tems for aucton, and three agents wth values for the tems gven by v : v 1 = (0, 0, 36, 44, 44), v 2 = (9, 18, 24, 28, 28), v 3 = (12, 12, 12, 12, 12). Suppose that the aucton begns wth the auctoneer announcng a prce of 1(+ɛ). Ifeachagent bds sncerely, 2 that s to say, she honestly bds the quantty of unts that maxmzes her utlty at the current prce, each agent would respond wth demands of 4, 4 and 1, respectvely. The aggregate demand of 9 exceeds the avalable supply of 5, so the aucton must proceed further. Next, when the auctoneer calls a prce of 4, agent 2 declares her demand of 3. Snce the aggregate demand of 8 2 Unlke the Ausubel aucton, each agent need not consder her prevous optons.

4 number of unts agent 1 s bd resdual supply facng agent 1 1) Agent 1 obtans the frst opton wth whch she can buy one unt at the prce of 4. 3) She s facng the resdual supply of 4 at the prce of 9, but she demands only 3 unts at the same prce. 4) Thus, she obtans the thrd opton wth whch she can buy 3 unts at the prce of 9. 2) She obtans the second opton wth whch she gets one unt at the prce of 6. 5) Fnally, agent 1 chooses the thrd opton, buy 3 unts at the prce of 9, and obtans her utlty of announced unt per prce Fgure 1: Agent 1 s optons under the proposed protocol. See Table 1 for agents 2 and 3. Table 1: Indvdual demand curve, the resdual supply curve and the quantty of unts that each agent obtans as an opton. See also Fgure 1 for the perspectve of agent 1 n detal. p q 1(p) q 2(p) q P 3(p) q(p) s 1(p) s 2(p) s 3(p) c 1(p) c 2(p) c 3(p) stllexceeds the avalable supply of 5, theauctoneer further rases the prce. However, let us examne the stuaton carefully from agent 1 s perspectve. The demands of all agents other than agent 1 total only 4, whle5 unts are avalable for sale. Here, agent 1 clnches an opton wth whch she can obtan 1 unt at the current prce of 4. If each agent s assumed to bdmonotoncally, agent 1 s guaranteed to obtan at least one unt because the demand q (p) declared by each agent s nonncreasng and the resdual supply s (p) facng agent s non-decreasng. At the prce of 6, agent 2 agan drops her demand to 2. Here, the resdual supply facng agent 1 ncreases agan. Therefore, agent 1 clnches her second opton wth whch she can obtan 2 unts at the current prce. Snce there s stll excess demand, the prce contnues upward. At the prce of 8, agent 1 drops her demand to 3. Agent 2 obtans an opton wth whch she can buy one unt at that prce. Fnally, at the prce of 9, agent 2 drops her demand to 0. Snce the avalable supply exceeds the aggregate demand of all bdders,.e., 3+0+1= 4, theaucton ends. Here, agents 1 and 3 smultaneously obtan optons. Agent 1 obtans the thrd opton wth whch she can buy 3 unts at the prce of 9. Although the resdual supply allocates 4 unts to the agent, she demands only 3 unts. Then, she obtans that opton. As such, agent 3 obtans an opton wth whch, at the same prce, she can buy one unt. In summary, at the end of the aucton, agent 1 has three optons: wth whch she can buy one unt at the prce of 4, wth whch she can buy two (or fewer) unts at the prce of 6, and wth whch she can buy three (or fewer) unts at the prce of 9. Agent 2 has one opton: wth whch she can buy one unt at theprce of 8. Fnally, agent 3 too has one opton: wth whch she can buy one unt at the prce of 9. Table 1 summarzes the process descrbed above, and Fg. 1 llustrates the process from the perspectve of agent 1. Each agent chooses one of the gven optons and obtans tems wthn the chosen opton. Thus, f each agent s assumed to choose her opton and obtans the unts to maxmze her own utlty, agent 1 obtans three unts at the prce of 9 and receves her utlty of = 9. Agent 2 chooses to buy one unt at the prce of 8, shethenreceves her utlty of 9 8=1. Agent 3 executes her only opton and receves her utlty of 12 9=3. Example 4 llustrates a case where sncere bddng s stll an equlbrum strategy even f demand reducton occurs. In ths case, agent 2 clnches two optons: (6, 1) and (8, 2). Although she has the opton wth whchshecan buy two unts at the prce of 8, she chooses to buy only one unt at the cheaper prce of 6. As a result, the auctoneer only allocates four out of fve avalable unts for sale. Example 4. There are fve dentcal tems for aucton, and three agents wth values for the tems gven by v : v 1 = (0, 0, 30, 36, 36), v 2 = (11, 20, 27, 27, 27), v 3 = (8, 13, 13, 13, 13). Suppose that the aucton begns wth the auctoneer announcng a prce of 5(+ɛ). Eachagent would respond wth demands of 4, 3 and 1, respectvely. The aggregate demand s stll 8, sothe aucton proceeds further. However, as descrbed n Example 3, agent 1 clnches an opton wth whch she can buy 1 unt at that prce. Next, when the auctoneer calls a prce 6, agent 1 declares her demand of 3. Now agent 2 faces the resdual supply of 1, soagent 2 obtans an opton wth whch she can buy one unt at the prce of 6. Snce the aggregate demand of 7 stll exceeds the avalable supply of 5, theauctoneer further rases the prce. At the prce of 7, agent 2 drops her demand to 2. Agent 1 then clnches an opton wth whch she can obtan 2 unts. Fnally, when the auctoneer calls 8, theaggregate demand s equal to the avalable supply. The aucton ends. Here, each agent s faced wth a resdual supply of 3, 2 and 0, respectvely. Agent 1 clnches an opton wth whch she can buy 3 unts at the prce of 8, whleagent 2 clnches another opton wth whch she can buy 2 unts at the same prce. In summary, at the end of theaucton, agent 1 has three optons: (5, 1), (7, 2) and (8, 3). Agent 2 has two optons: (6, 1) and (8, 2). Fnally,agent 3 has no opton at all. Table 2 and Fg. 2 summarze the process descrbed above. Thus, agent 1 obtans three unts at the prce of 8 and receves her utlty of 6. Agent 2 chooses to buy one unt at the prce of 6, although she can buy two unts at the prce of

5 number of unts agent 2's bd resdual supply facng agent 2 1) Agent 2 obtans the frst opton wth whch she can buy one unt at the prce of 6. 2) She obtans the second opton wth whch she can buy two unts at the prce of 8. 3) She executes her frst opton, buys one unt at the prce of 6, and receves her utlty of 5, though she could buy two unts at the prce of 8. Ths s because she receves a greater utlty by obtanng one unt rather than two unts,.e., demand reducton occurs announced unt per prce Fgure 2: Agent 2 s optons under the proposed protocol. See Table 2 for agents 1 and 3. Table 2: Indvdual demand curve, the resdual supply curve and the quantty of unts that each agent obtans as an opton. See also Fgure 2 for the perspectve of agent 2 n detal. p q 1(p) q 2(p) q P 3(p) q(p) s 1(p) s 2(p) s 3(p) c 1(p) c 2(p) c 3(p) Ths s because she obtans a greater utlty by obtanng one unt rather than two unts. As a result, she receves her utlty of 5. Agent 3 obtans no utlty. 4. FORMATION OF THE MODEL There s an aucton for K avalable unts of an dentcal tem, and N agents. The aucton allocates the P x unts to N agent =1,...,N posed by the constrant =1 x K. Assume that agent has a quas-lnear utlty, U (p,x ), whch s derved from the valuaton for x unts of the tem v x,theobtaned unts x and the payment p. U (p,x )=v x p x. (2) At a round l [0,L], the auctoneer announces a prce, p l (l {0,...,L}) ton agents. Each agent declares her demand q (p l )atthatprce. Notethat, snce the proposed aucton protocol s anascendng-prce aucton, the auctoneer cannot call a lower prce than the prce called before. l, p l 1 <p l. Each agent cannot declare a hgher demand than the demand she declared before. In other words, she cannot ncrease her declared demand. Monotone actvty rule:, p <p, q (p ) q (p P ). (3) We defne an aggregate demand functon, Q l as q(pl ). When an aucton ends at L,.e., the auctoneer calls a prce of p L, Q L K<Q L 1 must be satsfed at that prce. Let us defne the maxmum quantty of unts that an agent can buy at prce p l as c l : n o c l =mn q (p l ),s (p l ), (4) where s (p)represents the resdual supply facng the agent at prce p defned as Eq. 1. Each agent clnches optons represented by the followng ordered par, {(p 0,c 0 ),...,(p l,c l ),...,(p L,c L )}, where (p l,c l )meansthatthe agent can buy x [0,c l ] unts of the tem at prce p l. Fnally, each agent chooses one opton from her clnched optons to maxmze her utlty, exercses the opton l [0,L], and buys x unts 3 at prce p l. x = a (p l ), (5) where a (p l )= max {U (p l,a)} (6) a [0,c l ] and l =arg max l [0,L] {a(pl )}. (7) Fully specfyng the ascendng-prce aucton also requres nformatonal assumptons and strateges because dfferent assumptons and strateges lead to dfferent outcomes. Ausubel focuses on the followng three avalable nformatonal assumptons. Each agent observes: 1) whether the aucton s stll open wth no bd nformaton, 2)the aggregate demand of all agents wth aggregate bd nformaton, or3)allthe bds by all agents wth full bd nformaton. Eachagent s strategy s represented by any functon of a prce, her own valuaton, and observable nformaton to quanttes, posed by the bddng constrants. Ausubel also shows that ex post perfect equlbrum s realzed n hs dynamc aucton wth any nformatonal assumptons (See Ausubel [1] for detal). Now we are ready to ntroduce our theorems. The followng theorem showsthat ths protocol s feasble,.e., the auctoneer never lacks the unts of an tem to allocate. Theorem 1. In the proposed protocol, agent [1,...,N] chooses and executes one opton she has obtaned. Even f all agents obtan maxmum unts wthn ther allocated optons, the auctoneer never lacks the unts of an tem to allocate. Proof. c l allocates the quantty of unts each agent can obtan at a prce p l. The agent can choose to buy from 0 to c l unts at that prce. Here, we are gong to prove Eq. 8. l = {l 1,...,l N }, X [1,N] c l K (8) 3 To smplfy the protocol descrpton, we assume that an agent declares a sngle demand value for the current prce. To ncrease the effcency of the proposed protocol, we can modfy the protocol so that an agent can declare multple demand values f the optmal demand value s not unque. We can choose an arbtrary combnaton of these demands so that Q L becomes equal to K f possble.

6 Assume that Eq. 8 were false, and we then derve Eq. 9. We are then gong to derve a contradcton to show that Eq. 9 does not hold. l = {l 1,...,l N }, K < X [1,N] c l. (9) Let us defne l t as the earlest tme n l 1,...,l N.Here, l t s represented by l t =mn{l 1,...,l N }. Then, Eq. 10 holds from Eq. 4: c lt t =mn{q t(p lt ),s t(p lt )} s t(p lt ). (10) On the other hand, for any agent j, Eq.11holdsfrom Eq. 3 (the monotone actvty rule) and p lt p l j : c l j j =mn{q j(p l j ),s j(p l j )} q j(p l j ) q j(p lt ). (11) Accordngly, from Eqs. 10 and 11, we derve that Eq. 12 holds: P P cl = t cl + c lt t P t q(plt )+s t(p lt ) = P t q(plt )+K P t q(plt ). (12) P Thus, cl K holds. Ths contradcts Eq. 9. As a result, we prove that Eq. 8 s true. An ntutve explanaton of the above proof s as follows. c l s defned as a mnmum value of q l and s (p l ), and s thus constraned by s (p l ). The resdual supply facng an agent s (at most) the avalable supply less an aggregate demand other than the agent. Therefore, the sum of resdual supples of all agents never exceed the avalable supply. Accordngly, Theorem 1 holds. The next lemma proves that an agent cannot decrease her payment usng two dentfers. Lemma 1. If an agent uses two dentfers, and, and obtans q and q unts under the dentfers and, respectvely, then the agent can obtan q + q unts at a payment less than or equal to the sum of payments for q and q unts by usng a sngle dentfer. Proof. Assume that the unt prce for q unts s p and t for q s p.notethatwealso assume that p s not greater than p (p p ). To prove Lemma 1, we prove Eq. 13: q + q s (p ). (13) Here, the rght sde of the equaton represents the number of unts an agent obtans by usng two dentfers: and,and the left sde represents a resdual supply facng the agent at a prce of p by usng a sngle dentfer. In other words, f the Eq. 13 holds, the agent can buy q + q unts at a cheaper prce of p by usng asngledentfer than at aprceofp + p by usng two dentfers. Snce q s (p )andq q (p ) q (p )holdsfrom the defnton of the proposed protocol, q s (p ) = K X j q j(p ) = K X q j(p ) q (p ) j X, K q j(p ) q j, (14) holds. Furthermore, substtutng the defnton of s ( + )(p ) for Eq. 14, we obtan Eq. 15. q s ( + )(p ) q. (15) Then, q + q s ( + )(p ) (16) holds. As a result, we prove Eq. 13. Usng a smlar argument as Lemma 1, we can show that f an agent uses three or more dentfers, she can decrease her payment by usng a sngle dentfer. Thus, n the followng theorems, we can assume that each agent uses only a sngle dentfer. We defne sncere bddng n the proposed protocol as follows. Defnton 2. Sncere bddng s the strategy n whch agent bds her demand that maxmzes her utlty at every prce and after every hstory, subject to the constrants posed by the monotone actvty rule 4 : q sn (p) arg max k {vk p k}. (17) Theorem 2. In the proposed protocol wth prvate values, sncere bddng s a weakly-domnant strategy for every agent wth no bd nformaton. Proof. Let us assume that Theorem 2 does not hold, then we derve a contradcton. Suppose that for a certan combnaton of strateges of other agents, agent uses an nsncere bddng strategy n whch she declares q q sn (p). Then, she obtans x ns q unts at ths prce and her utlty s larger than n the case of sncere bddng. Snce we assume no bd nformaton, nootheragents can realze the strategy of agent because they are nformed only of whether the aucton stll contnues or ends. Therefore, we can assume the declared demands of other agents are the same for the cases of sncere/nsncere bddng. Thus, n the followng, we suppose that s (p) sconstant. Frst, we consder the case where an agent under-declares q <q sn (p) unts at prce p,.e., she can obtan (at least) the same utlty as n the case of nsncere bddng. Ths contradcts the assumpton. Next, we consder the case where an agent over-declares her demand (q <q sn (p)). Then, snce x ns holds, by usng sncere bddng, she can buy x ns her demand (q sn nto the followng two cases. (p) <q ). Ths case can be further dvded 1. In the case where q sn (p) s (p): From Eq. 17, the agent can obtan q sn (p) unts by sncere bddng. By defnton, ths s the optmal quantty of unts that maxmzes agent s utlty,.e., the utlty for obtanng q sn (p) unts s larger than or equal to the utlty for obtanng x ns unts. Ths contradcts the assumpton. 2. In the case where s (p) <q sn (p): From Eq. 4, x ns s (p) holds. Bysncere bddng, she can obtan x ns unts at prce p,.e., she can obtan (at least) the same utlty to the case of nsncere bddng. Ths contradcts the assumpton. 4 If there are multple sncere demands, the demand she bds can be any of these demands. For example, we can choose (p) nf {arg max k {v k p k}}. q sn

7 Thus, Theorem 2 holds snce we derve a contradcton assumng the negaton s true. An ntutve explanaton of the above proof s as follows. Frst, snce an agent can choose to obtan a smaller number of unts than her opton, gettng an opton wth a large number of unts never hurts. Therefore, under-declarng her demand s totally useless. On the other hand, obtanng an opton that s larger than the optmal demand s useless f the agent can obtan an opton wth the optmal demand. Therefore, over-declarng her demand s also useless. As aresult,sncere bddng s a weakly-domnant strategy for every agent wth no bd nformaton. Sncere bddng s not always a domnant strategy wth full or aggregate bd nformaton, snce there mght be an agent who has astrategythatchanges ts acton by respondng to the obtaned bd nformaton. As shown n [1], agent j may take a (werd) strategy nwhchshemantans that q j(p l )=q j(p 0 )aslongasq (p l )=q (p 0 ), whle she reduces her demand to 0,.e., q j(p l+1 )=0,rght after agent reduces her demand,.e., q (p l ) <q (p 0 ). Here, agent may mprove her utlty by reducng her demand mmedately, snce agent j never decreases her demand unless agent reduces her demand frst. It s mpossble for an agent to take such a strategy respondng wth others bds wth no bd nformaton because she observes only a prce that an auctoneer calls. On the other hand, t s possble for her to take such a strategy wth aggregate or full bd nformaton because she observes both acurrent prce and others bds. However, sncere bddng by all agents s an ex post perfect equlbrum, even wth aggregated or full bd nformaton case. Theorem 3. In the proposed protocol wth prvate values, sncere bddng by all agents s an ex post perfect equlbrum wth any nformatonal assumpton. Proof. The same argument for provng Theorem 2 can be appled to any strategy of other agents, as long as the strategy does not react to agent s acton. Clearly, the sncere bddng strategy satsfes ths property,.e., the demand of an agent s determned only by the current prce and her evaluaton value, not bythedeclared demands of other agents. Thus, sncere bddng by all agents s an ex post perfect equlbrum. 5. EVALUATION Ths secton compares the obtaned socal surplus and seller s revenue of the proposed protocol heren wth that of the IR protocol whch s an exstng non-trval protocol that can handle mult-unt auctons and s robust aganst false-name bds. Ths protocol requres the auctoneer to pre-defne a reservaton prce for each unt. We evaluate these two protocols n the dentcal settng used n [14], then represent the evaluaton value of each agent as a sngle step functon,.e., all-or-nothng. Agent s values are generated as follows. Frst, a bnomal dstrbuton B(m, p) derves the x unts of an tem that she desres to obtan. Specfcally, the probablty wth whch she desres to obtan the x unts s gven by p x (1 p) m x m!. (x!(m x )!) Then, v,whch she s wllng to pay for the x unts s drawn randomly n [0,x ]. Snce we assume that the evaluaton rato of socal surplus Proposed protocol (0.947) IR (max 0.672) reservaton prce Fgure 3: Comparson of socal surplus (sngle step) average revenue Proposed protocol (4.20) VCG (4.13) IR (max 3.70) reservaton prce Fgure 4: step) Comparson of seller s revenue (sngle value of an agent s all-or-nothng, she s wllng to pay v for x (or more) unts whle she does not want to pay anythng for unts smaller than x. We perform 100 smulaton sets where there are ten unts of an tem and ten agents for each aucton,.e., m = 10, n = 10 and p = 0.2. Fgure 3 llustrates the average rato of the obtaned socal surplus to the Pareto effcent socal surplus by varyng the reservaton prce. We can see that the rato of the proposed protocol reaches around 95%, whch s far superor to that of the IR protocol (70% at maxmum), even wth the most approprate reservaton prce. Fgure 4 shows the average of the obtaned seller s revenue by varyng the reservaton prce. We can see that the average revenue of the proposed protocol heren s around 4.2, whch s far superor to that of IR protocol (3.70 at maxmum), even wth the most approprate reservaton prce, and better than n the VCG (around 4.13), n whch we assume that no false-name bds exst and all agents truthfully declare ther evaluaton values.

8 6. DISCUSSIONS Our smulaton results ndcate that the proposed protocol outperforms the IR protocol, and that t also performs as well as the VCG n terms of the socal surplus and the seller s revenue. If we assume that no false-name bds exst, the socal surplus obtaned by the VCG s optmal. Therefore, the socal surplus obtaned by our protocol s always nferor to (or at best, equal to) that obtaned by the VCG. On the other hand, the seller s revenue obtaned by ths protocol may be less than or more than that obtaned by the VCG. The IR protocol requres the auctoneer to set a reservaton prce. As we notced n Secton 1, t was dffcult for hm to set the approprate reservaton prce, thus t holds an advantage over the IR n that the proposed protocol does not requre the auctoneer to set a reservaton prce. Our protocol cannot guarantee to obtan a Pareto effcent allocaton. In the VCG, we need to solve a combnatoral optmzaton problem to fnd a Pareto effcent allocaton. Our protocol does not perform such an optmzaton, so t s natural that our protocol does not always acheve a Pareto effcent allocaton. Moreover, as shown n [10, 15], t s mpossble to desgn a protocol that acheves a Pareto effcent allocaton n a domnant strategy equlbrum when false-name bds are possble. The smulaton results suggest, though, that an allocaton that s very close to a Pareto effcent allocaton s realzed n the protocol. Furthermore, n our protocol, the requred computatonal costs for an auctoneer s much lower than the cost of performng the VCG. Although a dynamc programmng technque can be used for fndng the optmal allocaton [11], t s necessary to repeatedly solve an optmzaton problem to determne the payment of each wnner. 7. CONCLUSIONS Ths paper develops a new ascendng-prce mult-unt aucton protocol. Ths protocol has an open format, and n whch sncere bddng s an equlbrum strategy, even f the margnal utltes of each agent can ncrease and agents can submt false-name bds. To the best of the authors knowledge, ths s thefrst non-trval protocol that has these characterstcs. We show that sncere bddng by all agents s an ex post perfect equlbrum. In addton, t s a weaklydomnant strategy wth no bd nformaton. The smulaton result shows that t outperforms the IR protocol n terms of the socal surplus and the revenue. Moreover, the obtaned socal surplus s very close to Pareto effcent. In future works, we would lke to analyze the protocol under stuatons where agents have nterdependent values and extend t to heterogeneous tems, whle to nvestgate t further by conductng subject experments. [2] L. M. Ausubel and P. Cramton. Demand Reducton and Ineffcency n Mult-Unt Auctons [3] M. Babaoff and N. Nsan. Concurrent Auctons Across the Supply Chan. In Proceedngs of the 3rd ACM Conference on Electronc Commerce, pages 1 10, [4] S. Bkhchandan and J. M. Ostroy. Ascendng Prce Vckrey Auctons [5] W.Conen and T.Sandholm. Preference elctaon n combnatoral auctons. In Proceedngs of the 3rd ACM Conference on Electronc Commerce, pages , [6] J. Kagel and D. Levn. Behavor n Mult-Unt Demand Auctons: Experments wth Unform Prce and Dynamc Vckrey Auctons. Econometrca, 69(2): , [7] V. Krshna. Aucton Theory. Academc Press, [8] A. Mas-Colell, M. D. Whnston, and J. R. Green. Mcroeconomc Theory. Oxford Unversty Press, [9] D. C. Parkes. An Iteratve Generalzed Vckrey Aucton: Strategy-Proofness wthout Complete Revelaton. In Proceedngs of AAAI Sprng Symposum on Game Theoretc and Decson Theoretc Agents, pages 78 87, March [10] Y. Sakura, M. Yokoo, and S. Matsubara. A Lmtaton of the Generalzed Vckrey Aucton. [11] S. van Hoesel and R. Müller. Optmzaton n Electronc Markets: Examples from Combnatoral Auctons. Netnomcs, 3:23 33, [12] M. Yokoo, Y. Sakura, and S. Matsubara. The Effect of False-name Declaratons n Mechansm Desgn: Towards Collectve Decson Makng on the Internet. In Proceedngs of the Twenteth Internatonal Conference on Dstrbuted Computng Systems (ICDCS-2000), pages , [13] M. Yokoo, Y. Sakura, and S. Matsubara. Robust Combnatoral Aucton Protocol aganst False-name Bds. Artfcal Intellgence, 130(2): , [14] M. Yokoo, Y. Sakura, and S. Matsubara. Robust Mult-unt Aucton Protocol aganst False-name Bds. In Proceedngs of 17th Internatonal Jont Conference on Artfcal Intellgence (IJCAI-2001), pages , [15] M. Yokoo, Y. Sakura, and S. Matsubara. The Effect of False-name Bds n Combnatoral Auctons: New Fraud n Internet Auctons. Games and Economc Behavor, forthcomng, Acknowledgments We would lke to thank Shgeo Matsubara, Yuko Sakura and anonymous revewers for helpful dscussons and comments. 8. REFERENCES [1] L. M. Ausubel. An Effcent Ascendng-Bd Aucton for Multple Objects