Equilibrium Prices in Bundle Auctions. Peter R. Wurman and Michael P. Wellman. University of Michigan. Articial Intelligence Laboratory

Transcription

1 Equlbrum Prces n Bundle Auctons Peter R. Wurman and Mchael P. Wellman Unversty of Mchgan Artcal Intellgence Laboratory f pwurman, wellman g@umch.edu Abstract The allocaton of dscrete, complementary resources s a fundamental problem n economcs and of drect nterest to e-commerce applcatons. In ths paper we establsh that compettve equlbrum bundle prces always exst that support the ecent allocaton n dscrete resource allocaton problems wth free dsposal. We beleve that ths s an mportant step n the quest for a mechansm that performs well n the face of complementary preferences. We present a famly of auctons that use ths bundle prcng polcy, and make some ntal observatons on several of ts members, ncludng the new Ascendng k-bundle aucton. Introducton In many potental e-commerce applcatons, agents have complementary preferences for objects n the marketplace. Consder an agent tryng to construct a computer system by purchasng components. Among other thngs, the agent needs to buy a cpu, harddrve, and montor, and may have a choce of several models for each component. The agent's valuaton of a package depends on the combnaton of components that t can accumulate. Ths example s an nstance of the a general allocaton problem characterzed by heterogeneous, dscrete resources and complementartes n agent preferences. Dscrete resource allocaton has been the focus of aucton theory snce Vckrey's semnal work on the second-prce aucton (Vckrey 1961). The allocaton of sngle objects, or multple homogeneous objects, s well understood (Demange, Gale, & Sotomayor 1986; McAfee & McMllan 1987; Mlgrom 1989), however a general soluton to the allocaton of heterogeneous objects has proven elusve. The landscape seems composed of varous mechansms that make derent tradeos among desrable propertes. One branch of research s focused on dentfyng condtons under whch prce equlbra exst. For nstance, t has been shown that prce equlbra exst f a gross substtutablty condton holds (Kelso & Crawford 1982), or f utlty functons satsfy the nocomplementartes condton (Gul & Stacchett 1997). Bkhchandan and Mamer (Bkhchandan & Mamer A B AB Agent Agent Table 1: An example wthout equlbrum prces for ndvdual goods. 1997) demonstrated that equlbra exst the socal value of the optmal allocaton s equal to the value of the correspondng relaxed lnear program. However, t s easy to construct examples n whch prce equlbra fal to exst. A smple example presented by McAfee and McMllan (McAfee & McMllan 1996), reproduced n Table 1, llustrates the problem. One agent values the par of goods at 3 and gets no benet from the goods sngly. The second agent values ether good at 2, but gets no added benet from gettng both goods. The ecent soluton, from a socal perspectve, assgns both goods to agent 1. 1 For any assgnment that ncludes agent 2 recevng one or more goods, both agents can be made better o by a transfer of money from agent 1 to agent 2 n exchange for agent 2's allocaton. In ths example, however, no equlbrum prces exst. In order to exclude the rst agent from the allocaton, the prce of each good ndvdually must be greater than or equal to 2. However, these prces put the cost of the par at 4 above the second agent's valuaton. Another subject of research n ths area s the generalzaton of Vckrey's orgnal results and related results by Clarke (Clarke 1971) and Groves (Groves 1973). Ths mechansm, whch alternately goes by the monker Vckrey-Clarke-Groves mechansm or the Generalzed Vckrey Aucton (GVA), s an ncentve compatble, ndvdually ratonal, ecent, drect revelaton mechansm. The mechansm's attractve propertes come from the fact that an agent pays (receves) the derence n utlty that other agents lose (gan) by ts presence. Thus, the agent's payment s not dependent upon ts own stated preferences, whch removes the ncentve to msrepresent those preferences. 1 Throughout the examples n ths paper we ndcate the socally ecent allocaton n boldface.

2 However, the GVA does have several undesrable propertes. Foremost among these s that, n the twosded market, t s not budget-balanced. Even when we restrct our attenton to the sngle-sded case, the mechansm has very hgh computatonal costs for the auctoneer, and requres complete nformaton revelaton by the agents. We suspect that n many cases t s not necessary for agents to fully dsclose ther utlty functon n order to reach the ecent allocaton. Under some condtons, the payments computed by the GVA can be nterpreted as prces (Leonard 1983). However, n the general case, we run nto two problems. Frst, the GVA computes payments per agent, not prces per object. When a sngle agent s allocated a combnaton of objects, there s no way n whch to determne the relatve prce of each object n the bundle. Moreover, the GVA may not compute anonymous prces, where anonymty mples that every agent has the opportunty to purchase the same object at the same prce. 2 Consder an example wth two buyers and two dentcal unts of a good. Agent 1 values one unt of the good at 5, and two unts at 7. Agent 2 values one and two unts at 5 and 8, respectvely. The ecent allocaton s to gve one unt to each agent. However, the GVA payment for agent 1 s 8? 5 = 3, whereas the payment by agent 2, for a unt of the same good, s 2. Thus, f partcpants care that prces be unform, the GVA wll not be an acceptable mechansm. There are a few notable alternatves to the GVA that attempt to allocate heterogeneous goods under complementartes. The most well-known mechansm s the Smultaneous Ascendng Aucton (SAA) used by the FCC to allocate spectrum rghts (McAfee & McMllan 1996; McMllan 1994). Two other promsng mechansms have been studed n laboratory experments: the Adaptve User Selecton Mechansm (AUSM) ntroduced by Banks, et al. (Banks, Ledyard, & Porter 1989), and a combnaton of the prevous two called the Resource Allocaton Desgn (RAD) aucton (DeMartn et al. 1998). These three mechansms der n many subtle ways. Two dmensons of partcular relevance to ths dscusson are the semantcs of bds and the scope of objects beng prced. SAA accepts bds on ndvdual goods, and sets prces on the ndvdual goods. RAD accepts bds on bundles, but stll computes prces exclusvely for the ndvdual goods. AUSM accepts bds on bundles and makes those bds publc, essentally announcng the prces of the bundles. 3 All three auctons use ascendng rules to assure termnaton, however none of them have been analyzed for ther equlbrum propertes. In partcular, we do 2 The antthess of anonymous prcng s dscrmnatory prcng. 3 Ths s an oversmplcaton. The mechansm announces the prces of the wnnng bundles, and provdes a secondary queue wth whch agents bddng on smaller bundles can coordnate ther bds to dsplace larger bundles. The prces lsted n ths queue mply prces for ther complements wth respect to supersets that have bds. not know whether, gven the nal prces and the nal allocaton, any agent would prefer some other bundle to ts allocaton. In other words, we do not know f compettve equlbra exst for the underlyng allocaton models. In ths paper we present a method for settng prces on bundles that ensures that equlbrum bundle prces always exst that support the ecent allocaton when agents act compettvely. We ntroduce a famly of auctons, the k-bundle auctons, based on ths prce settng scheme. Fnally, we present some observatons about several members of ths famly. Model We consder a benevolent auctoneer wth an objectve to allocate a set of heterogeneous objects n the most socally ecent manner. There are N agents, ndexed by, and M objects, ndexed by j. There are 2 M? 1 derent combnatons of objects (excludng the empty set). Let b 2 f0; 1g M where b j = 1 mples that object j s an element of the bundle b. For two bundles, b and c, we use the superset notaton b c to ndcate that 8j, b j c j. The utlty that an agent derves from a bundle s gven by v (b). We assume the exstence of a currency, denoted m, and that agents have utlty quaslnear n m. U(b) = v (b) + m: Further, we assume free dsposal, whch allows us to assume valuatons ncrease monotoncally as tems are added to a bundle. Formally, b c mples v (b) v (c). A soluton to the allocaton problem s an assgnment of bundles to agents that satses the materal balance constrant. Let x b 2 f0; 1g take the unt value only when b s assgned to. The socally ecent allocaton satses the followng maxmzaton problem: max s:t: b b v (b)x b (1) x b 1; 8 b x b 2 f0; 1g: x b b j 1; 8j The rst constrant mples that each agent receves at most one bundle. The second constrant ensures that no object s allocated more than once. Let be a soluton to ths nteger program. The value of the soluton s smply V ( ) = v (b)x b : b A vector of bundle prces, P, speces a non-negatve real number, p b, for each b. We admt nonlnear prces.

3 That s, for two dsjont bundles b and c, p b[c = p b + p c does not necessarly hold. When adoptng nonlnear prcng, t may be necessary to have some regulator enforce the allocaton. Otherwse, f p b[c > p b + p c, an agent may wsh to purchase b and c separately to assemble the bundle. Smlarly, f p b[c < p b + p c, two agents may have an ncentve to collude n order to purchase the combned set at a dscount, and reallocate the objects later. We requre that the prce of a bundle be at least as great as any of ts subsets, and that the prces be anonymous a sngle prce vector s reported to all agents. A compettve equlbrum s one n whch each agent treats prces as exogenous and demands the objects that maxmze ts surplus. In eect, the agent gnores the mpact of ts actons on prces. Agent s n equlbrum at prces P wth respect to an allocaton f b x b [v (b)? p b ] = max[v (b)? p b ]: (2) b The allocaton and the prce vector P support a compettve equlbrum f (2) holds for all agents. Equlbrum Prces Constructon We start wth the soluton to equaton 2, and wsh to construct equlbrum prces to support the optmal allocaton. The constructon proceeds n two steps. Frst, we compute prces on the bundles that are assgned n, then we compute prces on the rest of the bundles. Let B + be the subset of bundles whch are assgned n, and B? be the unassgned bundles. Let A be the set of all agents, A + the agents whch are assgned nonempty bundles, and A? be the agents that are excluded. For each agent 2 A?, we ntroduce a dummy good whch represents the agent's null allocaton. Let f j 2 A? g. Every agent has a valuaton of zero for every dummy good. Let G B + [, and g 2 G. We have now constructed a pseudo-assgnment problem for whch we already have the soluton. Our goal s to compute prces that support the mappng A! G. To accomplsh ths, we adopt the dual program used by Leonard (Leonard 1983) to compute mnmal prces for the assgnment problem. Frst, we solve LP mn : mn s:t: g p g (3) s + p g v (g); 8; g s ; p g 0; s + g p g = V ( ): The s term represents the surplus acheved by agent. LP mn maxmzes the agent's surplus wthn the range of equlbrum prces that support the ecent allocaton. Note that the ntroducton of the objects n s smply a trck to nclude the agents n A? n the pseudoassgnment problem. In the soluton, s = p = 0 for all 2 A?. LP mn has a complementary problem, LP max, whch computes upper bound prces: mn s:t: s (4) s + p g v (g); 8; g s ; p g 0; s + g p g = V ( ): Clearly the prce vector produced n ndng a soluton to ether of these lnear programs s a compettve equlbrum for the pseudo-assgnment problem the rst constrant requres that an agent cannot gan any more surplus from another bundle than t receves from the one allocated to t. Gven a soluton to ether LP mn or LP max we must now set prces on the bundles n B?. Ths can be done qute smply. For all b 2 B?, p b = max[v (b)? s ]: (5) By constructon, these prces cannot dstract any agent from the allocaton t receves n. Moreover, these prces satsfy the same monotoncty constrant we mposed on utlty functons. Let P be a vector of prces where the p b s are calculated by LP max and (5). Smlarly, P s the prce vector resultng from solvng LP mn and applyng (5). We can now consder a range of equlbrum prces that support the optmal allocaton. Lemma 1 For any k 2 [0; 1], kp + (1? k)p s a compettve equlbrum. Proof: Consder two bundles, b and c, where agent prefers b to c at P. Because P and P both support the same allocaton, must prefer b to c at P. Thus we have v (b)? P b v (c)? P c ; and v (b)? P b v (c)? P c : In both cases, the relaton s nvarant to postve scalar transformatons. Thus, for k 2 [0; 1], and k[v (b)? P b] k[v (c)? P c]; (1? k)[v (b)? P b] (1? k)[v (c)? P c]: Addng the two equatons and smplfyng gves v (b)? kp b? (1? k)p b v (c)? kp c? (1? k)p c :

4 A B C AB AC BC ABC Agent Agent Agent A B AB Agent Agent Agent Table 2: An example wth three agents. AB C 3 Agent Agent Agent Table 3: The pseudo-assgnment problem. 2. The k parameter here s analogous to the parameter used n the k-double aucton (Satterthwate & Wllams 1989). P and P represent the range of prces for whch supply equals demand. Reducng any prce n P would create excess demand for some of the objects. There s a slghtly weaker analogy for the upper bound sde. We cannot rase the prce of any b 2 B + wthout dsequlbratng supply and demand. However, n some cases we can ncrease the prces of bundles n B? wthout adverse aects. Example Consder the example n Table 2. The ecent allocaton s to assgn AB to agent 2 and C to agent 1. Agent 3 gets nothng. Ths allocaton has a socal welfare of 13. In order to construct equlbrum prces, we ntroduce the dummy good 3. We then solve the lnear programs LP mn and LP max for the pseudo-assgnment problem n Table 3. The soluton to LP mn for ths problem s p AB = 7, p C = 1, and of course p 3 = 0. Ths leaves s 1 = 4, s 2 = 1, and s 3 = 0. We leave the computaton of the soluton to LP max as an exercse for the reader. The nal upper and lower equlbrum prce vectors are gven n Table 4. Dscusson Unfortunately, by expandng the prcng space we admt necent equlbra. For example, the prce vector h2; 2; 2:5 supports a compettve equlbrum for both the problem n Table 1 and the problem n Table 5. However, n the latter case the allocaton t supports s necent agent 1 gets both goods. For ths example, the prce vector h2; 2; 3 supports the ecent A B C AB AC BC ABC P P Table 4: Equlbrum prces. Table 5: An example wth multple equlbrum bundle prce vectors. allocaton. 4 Clearly, one of the challenges n developng an ecent allocaton mechansm wll be avodng these degenerate equlbra. Recently, Bkhchandan and Ostroy (Bkhchandan & Ostroy 1998) (B&O) thoroughly analyzed the equlbrum propertes of a partcular package assgnment model. They demonstrated that dscrmnatory, nonlnear prcng s requred to ensure that compettve equlbra exst n the sngle-seller case. However, ther allocaton model ders from the one presented heren n a crucal respect. In B&O's model, the seller must desre to sell the packages n exactly the bundles that buyers demand. In our model, we have gnored the seller and assumed that all of the goods are avalable for reallocaton by the auctoneer. More mportantly, we assume that the auctoneer s nterested n maxmzng ecency, and not revenue. It s also approprate to queston whether the compettve assumpton holds for such small problems. We argue that n ths complex doman, compettve equlbrum analyss s a good startng pont for analyss. A more thorough treatment of agent ncentves and strateges wll need to be done n the context of specc aucton mechansms based on the k-bundle prces. It s worth notng that some recent algorthms for computng the optmal allocaton from bundle bds have performed surprsngly well when the bds are sparse and exhbt complementartes (Fujshma, Leyton-Brown, & Shoham 1999; Sandholm 1999). The Famly of k-bundle Auctons Although we have presented the prce settng polcy under the assumpton of an omnscent medator, the algorthms can be used drectly on bds. Thus, we can combne the k-bundle prce polcy wth the full set of aucton parameters (Wurman, Wellman, & Walsh 1998; Wurman, Walsh, & Wellman 1999). Ths gves potentally hundreds of new aucton types to explore. We have not completed an n-depth analyss of any member of ths aucton famly, but have a few prelmnary observatons. Throughout ths secton we assume that bds are of the form r (b), where r (b) 2 R +, but s not necessarly equal to v (b). An agent can bd on any or all bundles. There s an mportant subtlety n the nterpretaton of 4 We present the upperbound prce vector that s computed by our algorthm, but n ths problem, the ecent soluton actually has an equlbrum that s supported by prces on ndvdual goods.

5 bds by the auctoneer. When an agent does not bd on all bundles, the bds on the other superset bundles must be nferred. For example, f an agent bd 2 on A, and has not bd on AB, the mechansm must assumes that t \values" AB at 2. Two problems plague auctons that allocate complementary preferences. The rst s the exposure problem: an agent desres a bundle but can only purchase elements of the bundle ndvdually, and n order to avod gettng stuck wth only a subset of ts desres, does not express ts true valuatons. The second s the free rder problem: one agent s wnnng a bundle and two agents value the components of the bundle at a greater combned value, but do not to rase ther bds n the hopes that the other agent wll shoulder more of the burden of dsplacng the wnner. We expect these two problems to be arse to varyng degrees n the analyss of partcular k-bundle auctons. Sealed-Bd k-bundle Auctons The sealed-bd k-bundle aucton accepts bds for all bundles. Then, at a speced tme, t computes the prces and allocaton from the bds usng the above method. Frst, we consder the sealed-bd bundle aucton where k = 0. From the followng example, t s clear that the ncentve compatblty results of the assgnment problem (Leonard 1983) do not hold for our generalzed allocaton problem. Consder agan the example n Table 5. Assume that agents 2 and 3 reveal ther true valuatons. Can agent 1 be better o by lyng about ts valuaton? The answer s yes. By reportng ts true valuaton, agent 1 s allocated nothng and receves a surplus of zero. Suppose, nstead, that agent 1 reports r 1 (AB) = 5. The mechansm wll calculate the optmal allocaton to be the one n whch agent 1 receves AB, and a supportng prce vector P = h2; 2; 2. Thus, by lyng, agent 1 receves a surplus of 1. 5 Now we turn out attenton to the sealed-bd bundle aucton where k = 1. Ths prcng strategy would dscourage manpulatons lke the one above, because agent 1 would be forced to pay 5. However, lke the standard sealed-bd rst-prce aucton, t suers from potental ecency losses due to strategc behavor. Ths can be seen n the example n Table 1 n whch agent 1 has an ncentve to bd 2 +. If the agent's nformaton s mperfect, t rsks underbddng ts opponents and losng the good. Ascendng k-bundle Aucton We now ntroduce a mechansm that performs qute well n our early tests. We call t the Ascendng k- Bundle (AKB) Aucton. In partcular, we focus on the stuaton where k = 1. The aucton accepts bds on bundles, but requres that an agent's new bd, r 0(b), beat max(p b ; r (b)) by some xed. The aucton clears 5 In fact, ths holds for any r 1(AB) 4. the market and closes after no new bds are receved wthn a speced perod. After each bd, the aucton calculates a new prce vector and announces t as the prce quote. Unlke n RAD, an agent can compare ts bd to the prce quote and tell whether t s wnnng. Because of the manner n whch the prces are determned, the agent knows t s wnnng the good that maxmzes ts revealed surplus, r (b)?p b. However, f more than one bundle provdes t wth the maxmal revealed surplus, the agent may not be able to tell from among these whch one t s wnnng. Ths dculty could be addressed by postng the tentatve wnnng combnatons (though not the wnner's dentty). Because agents can bd on bundles n AKB, they do not suer from the exposure problem n whch they are reluctant to express ther true valuaton for a bundle when they need to purchase the elements ndvdually. However, there s stll an ncentve for an agent to free rde. It may be necessary to adopt actvty rules lke those n SAA and RAD to encourage partcpaton. We have not yet performed large scale expermentaton wth AKB, nor emprcally compared t wth RAD or AUSM. However n our early experments wth myopc, best response agents, the results are promsng. Concluson The allocaton of dscrete, heterogeneous resources when agents have complementartes n preferences s a very general problem that s lkely to arse n many e-commerce applcatons. In ths paper we have establshed that compettve equlbrum bundle prces always exst to support the ecent allocaton. We accomplsh ths by solvng a pseudo-assgnment problem for the bundles that are assgned n the optmal allocaton, and then constructng prces on the rest of the bundles usng a smple rule. We beleve that ths s an mportant step n the quest for a mechansm that performs well n the face of complementary preferences. We ntend to contnue our nvestgaton of the famly of k-bundle auctons n general, and the Ascendng k- Bundle aucton n partcular. Acknowledgments Ths research s funded by the DARPA nformaton survvablty project and the IBM Insttute for Advanced Commerce. Helpful comments were provded by Enno Stacchett, Je MacKe-Mason, Wllam Walsh, Wolfram Conen, Davd Parkes, and the partcpants of the Santa Fe Insttute 1997 Graduate Workshop n Computatonal Economcs. References Banks, J. S.; Ledyard, J. O.; and Porter, D. P Allocatng uncertan and unresponsve resources: An expermental approach. Rand Journal of Economcs 20:1{22.

6 Bkhchandan, S., and Mamer, J. W Compettve equlbrum n an economy wth ndvsbltes. Journal of Economc Theory 74:385{413. Bkhchandan, S., and Ostroy, J. M The package assgnment model. Techncal report, Unversty of Calforna at Los Angeles. Clarke, E. H Multpart prcng of publc goods. Publc Choce 11:17{33. Demange, G.; Gale, D.; and Sotomayor, M Mult-tem auctons. Journal of Poltcal Economy 94:863{72. DeMartn, C.; Kwasnca, A. M.; Ledyard, J. O.; and Porter, D A new and mproved desgn for multobject teratve auctons. Techncal report, Calforna Insttute of Technology. Fujshma, Y.; Leyton-Brown, K.; and Shoham, Y Tamng the computatonal complexty of combnatoral auctons: Optmal and approxmate approaches. In Sxteenth Internatonal Jont Conference on Artcal Intellgence (IJCAI-99). Groves, T Incentves n teams. Econometrca 41:617{631. Gul, F., and Stacchett, E Walrasan equlbrum wthout complementartes. Techncal report, Prnceton Unversty and Unversty of Mchgan. Kelso, A. S., and Crawford, V. P Job matchng, coalton formaton, and gross substtutes. Econometrca 50:1483{1504. Leonard, H. B Elctaton of honest preferences for the assgnment of ndvduals to postons. Journal of Poltcal Economy 91(3):461{479. McAfee, R. P., and McMllan, J Auctons and bddng. Journal of Economc Lterature 25:699{738. McAfee, R. P., and McMllan, J Analyzng the arwaves aucton. Journal of Economc Perspectves 10(1):159{175. McMllan, J Sellng spectrum rghts. Journal of Economc Perspectves 8(3):145{162. Mlgrom, P Auctons and bddng: A prmer. Journal of Economc Perspectves 3(3):3{22. Sandholm, T An algorthm for optmal wnner determnaton n combnatoral auctons. In Sxteenth Internatonal Jont Conference on Artcal Intellgence (IJCAI-99). Satterthwate, M. A., and Wllams, S. R Blateral trade wth the sealed bd k-double aucton: Exstence and ecency. Journal of Economc Theory 48:107{33. Vckrey, W Counterspeculaton, auctons, and compettve sealed tenders. Journal of Fnance 16:8{ 37. Wurman, P. R.; Walsh, W. E.; and Wellman, M. P A parameterzaton of the aucton desgn space. Submtted for publcaton. Wurman, P. R.; Wellman, M. P.; and Walsh, W. E The Mchgan Internet AuctonBot: A congurable aucton server for human and software agents. In Second Internatonal Conference on Autonomous Agents, 301{8.