Proceedngs of the 4th Hawa Internatonal Conference on System Scences - 27 Second-Best Combnatoral Auctons The Case of the Prcng-Per-Column Mechansm Drk Neumann, Börn Schnzler, Ilka Weber, Chrstof Wenhardt Insttute of Informaton Systems and Management (IISM) Unverstaet Karlsruhe (TH) [neumann, schnzler, weber, wenhardt]@sm.un-karlsruhe.de ABSTRACT One of the man contrbutons of classcal mechansm desgn s the dervaton of the Groves mechansms. The class of Groves mechansms are the only mechansms that are strategy-proof and more mportantly allocatve effcent. The VCG mechansm retans ts propertes for combnatoral allocaton problems. From a computatonal perspectve the VCG has to solve two problems: () the wnner-determnaton (2) the determnaton of the prces. However, both problems are complex (NP-hard), when complementartes are present. The Prcng-Per-Column (PPC) aucton s another approach to solve the combnatoral allocaton problem. In essence, t apples the Vckrey prncple to any possble combnaton of goods and determnes the overall wnnng bds. PPC s computatonally less demandng, however, t can be shown that PPC s not necessarly effcent. Apparently, solvng the tenson between computatonal and game-theoretc propertes s a challengng task n mechansm desgn. Engneerng auctons suggests to lower requrements upon the aucton. In ths paper the evaluaton of the PPC concernng approxmate effcency s presented - n an analytcal and smulatve evaluaton the PPC s compared to the VCG and t s shown that the effcency losses ncurred by the PPC mechansm are very small.. INTRODUCTION Auctons have become an mportant coordnaton mechansm for supportng negotatons n dstrbuted systems lke the Internet. Ther popularty ncreased through a multtude of onlne markets such as ebay, Yahoo!, and Amazon. These platforms lst mllons of tems for sale and attract a multtude of users. They usually make use of tradtonal aucton mechansms such as varants of the Englsh aucton for tradng sngle and multple homogenous goods. However, these aucton mechansms fal when a set of heterogeneous goods are traded smultaneously. One reason for ths les n the dependences between goods n terms of complementartes and substtutes. In such cases, the valuaton for a sngle good s nfluenced by addtonal allocatons of other goods. For nstance, suppose a bdder requres a travel package ncludng a flght, a hotel, and a rental car. Suppose the partcpant has a postve valuaton for the whole bundle. An allocaton of the car wthout the flght s, however, useless for hm. Flght, hotel, and rental car supplement each other and thus are complementartes. The valuaton for the whole bundle s hgher than the sum of the valuatons for the sngle tems. Goods may also be substtutes,.e. they may be replaced by smlar tems. Suppose the bdder s wllng to pay 4 for the hotel at the destnaton. If he gets allocated two rooms smultaneously, hs valuaton for both rooms s stll 4. In ths case, the utlty of gettng allocated two smlar tems s not hgher than the utlty of a sngle good. Aucton mechansms supportng complementartes and substtutes are called combnatoral auctons. Combnatoral auctons are mportant n many real-world problems such as for auctonng spectrum lcenses [2], allocatng arport tme slots [7], or procurng transportng servces [9]. Although combnatoral auctons can be approxmated by multple sngle-tem auctons, ths often results n neffcent outcomes [2]. Effcent mechansms n the presence of complementartes are of the Groves famly. Known as the Vckrey-Clarke- Groves (VCG), the mechansm provdes a domnantstrategy soluton to report preferences truthfully even n the presence of complementartes. The pecularty of the VCG mechansm s that the prce an ndvdual bdder has to pay depends on the bds of other bdders. Despte ts theoretcal soundness and elegance, the VCG ncurs severe drawbacks when complementartes are present: From a computatonal pont of vew, the VCG s hard to solve wth an ncreasng number of goods and partcpants. The wnner determnaton problem n combnatoral auctons s NP-hard, as t s an nstance of the set packng problem (SPP) [3]. In the VCG, ths problem has to be solved N+ tmes, once wth all partcpants and then N tmes more wth each of the N partcpants removed from the allocaton. In addton to ths tractablty ssue, combnatoral auctons are often hampered by the fact that even the preference elctaton problem s hard to solve [3]. For nstance, f three goods are avalable, say A, B and C, the VCG mechansm 53-65/7 $2. 27 IEEE
Proceedngs of the 4th Hawa Internatonal Conference on System Scences - 27 requres the defnton of values for {A}, {B}, {C}, {AB}, {AC}, {BC}, and {ABC}. If the number of goods ncreases, the number of feasble sets that need to be valued by the bdders ncreases over-proportonally. To allevate ths preference elctaton problem, recent work suggests teratve aucton formats, e.g. Bundle [4] or Smultaneous Ascendng Aucton wth Package Bddng []. The nformatonal requrements upon teratve mechansms are relatvely mld, consderng the fact that the mechansms need only nformaton about the valuaton for the bundle the bdder values most. Instead of weghng all possble combnatons, one sngle value s suffcent. Besde ths argument, t s often referred to the theoretcal result from sngle tem auctons, where teratve auctons yeld hgher revenue for the seller than one-shot auctons. The lne of argumentaton follows ths ntuton: f valuatons are afflated, teratve mechansms wll be more desrable than one-shot mechansms. Although those arguments are convncng, the concluson should not be that teratve mechansms are always preferable to one-shot mechansms. If, for nstance, the aucton s fully automated by bddng agents, the preference elctaton problem may be tractable for relatvely small number of goods. Snce mmedacy s of concern, teratve auctons are nferor to one-shot auctons. In ths case, t appears to be reasonable to employ one-shot auctons. But here the problem arses: whch one to use? As aforementoned, the VCG mechansm faces the problem of ts NP-hardness when complementartes are nvolved. In ths paper, we wll analyze the Prcng-Per-Column (PPC) mechansm as an alternatve to the VCG [4]. Although t cannot solve the preference elctaton problem, whch s nherently assocated wth one-shot combnatoral auctons, t allevates the computatonal complexty of the prce determnaton problem consderably. It wll be shown that the effcency losses ncurred by the PPC mechansm are very small. The remander of the paper s structured as follows: Frstly, the requrements the aucton should satsfy are lsted. Subsequently, the Prcng-Per-Column aucton s ntroduced as a canddate soluton to those desgn requrements. Then, the aucton s analytcally and numercally evaluated and compared to the VCG. The paper concludes wth a summary and an outlook for future research. 2. Requrements on the Desgn Any desgn task begns wth the elctaton of the requrements the desgn artfact must satsfy. Ths paper addresses a combnatoral aucton format that s sutable for a certan class of domans. For smplcty, t s convenent to focus on one concrete scenaro and derve the requrements upon the aucton format and upon the qualty of the results ths aucton produces. Consder a transportaton scenaro, where several trucks transport freght from one depot to another. Demand wll be conveyed to a central department, whch contracts out the transportaton obs to the subsdary depots. The depots manage ther truck fleet decentralzed and bd on the sngle obs. For nstance, a depot may bd.5 for transportng, kg from Zurch to Prague. Furthermore, transportaton obs are characterzed by complementartes among the obs. Suppose the example from before: the depot bds.5 for a ob from Zurch to Munch, and, n addton.5 from Prague to Warsaw. If the depot s allocated both obs, the depot may rase ts bd from 3. to 4.5. The reason for ths ncreasng value for the bundle stems from the synerges that can be realzed. Instead of returnng from Munch back to the depot after havng completed the frst ob, the second ob can be undertaken, savng the costs ncurred by returnng to the depot. Ths smple scenaro mposes several requrements upon the aucton mechansm and upon the outcome the aucton acheves. Concernng the mechansm, the scenaro depcts a stuaton, where a sngle sded mechansm s necessary that can cope wth combnatoral bds. More precsely, the requrements on the mechansm are as follows: Sngle-sded mechansm: The central entty aggregates demand and contracts them out to the depots. Ths means, only the central entty s the seller (or ssuer) of obs wthout compettors, whle the depots compete aganst each other. Language ncludes combnatoral bds: The depots often demand a combnaton of obs as a bundle to realze synerges. As such, transportaton obs are complementartes,.e. partcpants have superaddtve valuatons for the obs, as the sum of the valuatons for the sngle ob s less than the valuaton for the whole bundle (v(a)+v(b) v(ab)). Suppose a depot bds for the bundle {(Zurch, Munch) and (Prague, Warsaw)}. If one component, e.g. the frst ob, s not allocated to hm, the remanng bundle (consstng of the last ob) has a decreased value for hm snce no synerges can be realzed. In order to avod ths exposure rsk (.e. recevng only a subset of the bundle), the mechansm must allow for bds on. Furthermore, the depots may also want to submt more than one bd on a bundle but many that are excludng each other. In ths case, the obs of the are substtutes. Ths means that the buyer has sub-addtve valuatons (v(a)+v(b) v(ab)) for the obs. For nstance, a depot s wllng to pay a hgh prce for a transportaton ob durng the day 2
Proceedngs of the 4th Hawa Internatonal Conference on System Scences - 27 and a low prce f the ob s done at nght. However, ths transportaton ob can be done only once. As such, the market mechansm must support XOR bds to express substtutes. Clearng and prcng rules that explot the fullrange of the language: Furthermore, clearng and prcng rules have to be desgned that () mpute a desrable allocaton (allocatve effcent) and (2) make usage of all nformaton of the bddng language. One-shot Process: Due to the dynamc nature, the bddng process wll be automatcally conducted by agents. To delmt the tmng of an aucton and to confne the strategc complexty of the bddng agents, the aucton needs to be one-shot. Concernng the outcome of the aucton, the scenaro suggests the followng propertes that the mechansm should satsfy: Allocatve effcency: An allocatve effcent allocaton of obs maxmzes the sum of ndvdual profts. Snce, the depots and the central department belong to one sngle organzaton, the maxmzaton of all profts results n the maxmzaton of the organzaton s proft. Incentve Compatblty: Achevng an allocatve effcent allocaton of the obs requres that all depots truthfully report ther valuatons. The aucton should thus nduce ncentve compatblty,.e. all depots report ther preferences truthfully n equlbrum. In the optmal case, truth-tellng s a domnant strategy, snce the depots have no ncentve to untruthfully report ther preferences n order to ncrease ther ndvdual utlty. Indvdual Ratonalty: Another requrement s that the depots voluntarly on the aucton. Ths n turn requres that the proft the depots derve from partcpaton s greater or equal than before, snce the depots would otherwse decde to opt out. Budget Balance: A mechansm s sad to be strctly budget balanced f the amount of prces sum up to over all depots. In ths case nether are funds removed from the system nor s the system subsdzed from outsde. Strct budget balance s an mportant property snce the resource allocaton can be performed at no costs. In case the mechansm runs a defct, the organzaton has to subsdze the defcary depots. Such a stuaton cannot be sustaned for a longer tme perod [8, 5]. A XOR B (A B) means ether A or B but not both Computatonal tractablty: Computatonal tractablty consders the complexty of computng the outcome of a mechansm from the depots strateges. Wth an ncreasng sze of bds, the allocaton problem can become very demandng. Thus, computatonal constrants may delmt the desgn of the proper aucton mechansm [9, ]. 3. VCG and PPC Mechansms For modelng the aucton mechansms for the transportaton scenaro, a wdely used prvate value model s employed, where depots have ncomplete nformaton about the preferences of the other depots [c.f. 8]. There are N depots and defnes the set of all possble types for depot. Ths type for depot specfes the preferences of and also s nformaton about other depots. A mechansm M s defned as the avalable bds and the rules how to resolve them; that s: mechansm M s a par (M, h(m),t (M), t 2 (M),..., t N (M) ), where M = M M 2 M N and h:m A and t :M. The term M refers to the message space of, whereas h denotes the allocaton functon that computes who gets what and t denotes the payments of each depot. For one-shot (or so-called drect) mechansms the message space smplfes to M =. In such a one-shot mechansm, the strategy of depot s ˆ. The reported type ˆ can equal the true type, but can also be another type. Gven a ont strategy ˆ, 2,..., N, the outcome generated by ˆ s denoted by h(ˆ ) and t (ˆ ). Depots are assumed to be rsk neutral and have quas lnear utlty functons. That s, utlty functon of depot s u ((h,t ), ) = v (h, ) + t. 3. The VCG mechansm One of the most promnent mechansms n mechansm theory are the so-called VCG (Vckrey-Clarke-Groves), pvotal, or Clarke mechansms. What make the VCG mechansm powerful n mechansm desgn are the nce propertes assocated wth t, whch wll be presented n secton 4. The allocaton rule h of a VCG mechansm s specfed by the maxmzaton of the reported valuatons. Generally, ths s denoted by h * ( ˆ) arg max v ( h, ˆ ). hx The transfer rule (n ths case of a pvotal mechansm) for depot amounts to * ˆ ˆ * t ( ˆ) ( ( ), ) (, ˆ v h v h ). 3
Proceedngs of the 4th Hawa Internatonal Conference on System Scences - 27 The nterpretaton of the transfers s nstructve [8]: f depot s presence does not make a dfference n the maxmzng problem (vz. agent s not part of the optmal allocaton), the payments are zero. Otherwse s presence s pvotal, as the socal welfare,.e. the sum of all agents, s affected by the partcpaton of depot. The payments exactly reflect the loss n valuaton of the other depots, whch s ncurred by the partcpaton of depot. The VCG mechansm ncorporates the margnal mpact on the other valuatons by the announcement of ˆ nto the payment functon nternalzng ths external effect. At the bottom-lne the ndvdual depot s thus forced to consder also socal welfare when makng hs choce. The VCG mechansm can also be appled to combnatoral allocaton problems. The mechansm can then be formalzed as follows: Let G be a set of sngle tems and s G be a bundle whch can be allocated to the partcpants. An effcent allocaton can be computed as follows: S* arg max v ( s ) (3) S { s... s } I s.t. s s,,. (4) The obectve (3) maxmzes the total utlty of the allocaton. The frst constrant (4) ensures, that no tem s allocated to more than one partcpant. Let V * denote the total value of the allocaton ncludng all partcpant and let * ( V ) denote the value of the allocaton wthout partcpant. The payment rule for a partcpant can be calculated as the dfference between the reported valuaton and ts mpact on the allocaton: * * pvick, ( s k ) v ( s k ) ( V ( V ) ). (5) Example : The VCG mechansm Consder there are four depots A, B, C and D competng for three transportaton obs G, H, and T. The valuatons for the obs are gven by Table. The depots would report these valuatons truthfully. Jobs Depot Table : Valuaton Matrx {G} {H} {T} {G,H} {H,T} {G,T} {G,H,T} A 3 4 65-2 -3 B 5-3 -3 4 8 8 C - 7 6 4-2 45 D 5 4 35 4 6-3 5 The VCG mechansm would allocate obs {G, T} to depot B and {H} to depot C, snce ths maxmzes the sum of all ndvdual valuatons. The payments would be for depot A and D, as they do not enter the allocaton. Depot B would have to pay ts valuatons mnus the dfference of the sum of all valuatons and the sum of all valuatons n the allocaton except depot B. The optmal allocaton wthout depot B would be {G,H} to depot A and {T} to depot C accrung a total valuaton of 6 + 65 = 25. Hence, depot B faces a payment of 8 (5-25) = 55. Analogously, depot C s subect to a payment of 4. 3.2 The PPC mechansm The PPC mechansm [4] dstngushes tself from the VCG mechansm by the payment functon. Thus, the allocaton rule h of PPC mechansm s analogous to the VCG mechansm h *( ˆ) arg max v ( h, ˆ ) (3) hx The transfer rule of the PPC mechansm for depot s denoted by the valuaton that would occur f depot s not present keepng the dstrbuton of constant. Ths can be easly explaned by referrng to Table. If t s assumed that the depots reveal ther valuatons truthfully, depot B and C would stll receve the allocaton of {G,T} and {H}, respectvely. The prce depot B has to pay s accordng to the PPC scheme, the hghest valuaton for {G,T} f B would be present. Accordngly, the columns of the allocaton are fxed and the second prce s used. Hence, B has to pay the second hghest prce of the column, whch amounts to 45 reported by depot C. Ths ntutve transfer rule s specfed by: t ˆ arg max v ( h, ˆ ) (4) k* for all wnnng bds assumed that there are at least two bds on the column and otherwse. Furthermore, only bds that are lower or equal the wnnng bd n a column are consdered. Note that k* refers to the column ndex, whch s part of the wnnng allocaton. 4
Proceedngs of the 4th Hawa Internatonal Conference on System Scences - 27 In essence, both the VCG and the PPC mechansm dffer n ther payment rule defnton and the prces assocated wth t. Snce the payments depots have to pay have an nfluence on ther bddng behavor, the results are lkely to be dfferent. 4. Evaluaton The evaluaton of the mechansms wll be twofold. In the frst part, an analytcal approach wll be conducted. The analytcal approach, nonetheless, s not suffcent enough to shed lght nto the queston, whch mechansm s ultmately to be preferred. Hence, a numercal smulaton wll explore those questons, for whch the analytcal approach s slent. 4. Analytcal Evaluaton The theory of mechansm desgn provdes a theoretcal toolbox for desgnng nsttutons wth a partcular emphass on ncentves []. The problem of desgnng a mechansm,.e. game form, s to mplement a mechansm ( M, y), such that the equlbrum outcome satsfes a partcular socal choce functon. A socal choce functon f() s denoted as a par (h(),t (), t 2 (),..., t N () ). That s, gven any preference profle of the depots, the socal choce functon chooses one allocaton of obs and correspondng payments. In essence, the task of the mechansm desgner (n our example, the managers of the transportaton frm) would be deally to choose the allocaton that maxmzes valuatons of the socety (.e. all depots of the frm). Such a socal choce functon, whch satsfes harg max N h H v h, s denoted to be effcent. It can be shown that t s not suffcent to maxmze the sum of reported preferences ˆ to mplement an effcent socal choce functon. The depots wll attempt to manpulate ther reports such that the can extract more beneft from the mechansm. Payments are needed n order to ncentvze depots to report ther valuatons truthfully at all tme, regardless of what the other depots bd. Ths refers to strategy proofness, whch denotes that truth-tellng s a domnant strategy: For all N, the ndvdual beneft from truth-tellng s at least as hgh as from lyng u, h, t u,., Mechansm desgn has found out that one class of mechansms, the so-called Groves mechansms, s strategy proof. A Groves mechansm s defned by the followng () allocaton and () payment rule. () hargmax v h, h H N () t ˆ x v h, where functon x : ndependent of s. Theorem : Groves mechansms (Groves 73): Groves mechansms are strategy proof. The proof that Groves mechansms are strategy proof and, hence, lead to an effcent allocaton of the obs can found at [6]. Theorem 2: VCG and Groves mechansms: The VCG mechansm s a specal case of the Groves mechansm It s trval to show that the VCG mechansm s part of the group of the Groves mechansms wth an x max v h,. hh Theorem 3: Groves unqueness (Green and Laffont 77): The class of Groves mechansms are the only mechansms, whch are effcent and strategy proof. The proof can be found at [5]. Groves mechansms lead to strong economc propertes. However, from a computatonal pont of vew, Groves mechansms may be ntractable [5]. The PPC mechansm has been ntroduced to allevate the computatonal problem. Nonetheless, the propertes of the PPC have not been analyzed. From theory, t s known that only Groves mechansms mplement an effcent socal choce functon n domnant strateges. As such, t s suffcent to show whether the PPC mechansm belongs to the class of Groves mechansms. Theorem 5: PPC Ineffcency: The PPC mechansm s not strategy proof. To show theorem 5, all that s necessary s to prove that the PPC mechansm s not a Groves mechansm. Snce Groves and PPC mechansms have an dentcal allocaton rule, the attenton can be restrcted to the payments rule. In essence, t needs to be demonstrated that the PPC payment functon cannot be represented as a specal Groves payment. The ntuton that ths cannot be made stems from the fact that the ndvdual Groves-payment does not depend on that ndvdual s reported valuatons. In the PPC the payments, on the contrary, depend on the ndvdual reports, snce the reports can determne the dvson of the goods nto ts consttuent (e.g. the columns) and hence the prce. For example a depot that s not part of the optmal allocaton, n Example the bdder A, can enter the allocaton by overstatng hs valuaton for {G, H} by 4 addng up to 5 (note that A s true valuaton for ths bundle s 65). In ths case, A s now part of the allocaton, whle C receves {T}. A s payment amounts to the second hghest prce on the bundle {G, H}, whch s 4. By ths overstatement, depot A can realze a gan of 65 4 = 25, whch s postve. The full proof can be found n the Appendx. 5
Proceedngs of the 4th Hawa Internatonal Conference on System Scences - 27 Theorem 5 s n combnaton wth theorem and 3 strct n ts message: the PPC mechansm cannot mplement an effcent socal choce functon n domnant strateges. Manpulatng the reports can tremendously bas the allocaton. Nonetheless, theorem 5 does not reveal any nformaton about how the depots would manpulate ther valuatons, f t all. In our fxed example, t s ratonal for depot A to overstate hs valuaton, knowng the reports of all other depots. The PPC mechansm, however, s sealed n nature. Hence, the depots have no clear nformaton about the other reports at all. Overstatng the own valuaton s dangerous, as the depots run the rsk of sufferng a loss. For nstance, f C s true valuaton for bundle {G,H} s 7 nstead of 4. If depot A manpulate hs bd to 5, he would receve {G,H}, whch s worth 65 for hm, whle he pays 7, resultng n a loss. In essence, other depots takng part n the mechansm are allevated the manpulaton behavor. It s the clam of ths paper that depots reveal ther valuatons approxmated truthfully, as the (game-theoretc) optmal strategy s too complex to determne. Referrng to Smon [2], humans (.e. depots n ths context) wll lkely adopt very smple heurstcs. The smplest heurstc would be to report the valuaton truthfully. A numercal smulaton wll be used to evaluate ths clam. 4.2 Numercal Evaluaton In the PPC, depots (as bdders) may have an ncentve to msrepresent ther valuatons n order to gan hgher utlty. Ths potental gan wll be measured by performng a stochastc smulaton. The VCG and the PPC are mplemented n a Java based smulaton envronment 2. For solvng the wnner determnaton problem, CPLEX 9.3 s used. As reports, buyers submt a set of XOR concatenated bds on all possble bundle combnatons. For nstance, n a scenaro wth three dfferent obs (A, B, C), buyers bd on the {A}, {B}, {C}, {AB}, {BC}, and {ABC}. The valuatons for each of the bundle bds are drawn from a unform dstrbuton. Each scenaro s repeated 5 tmes wth dfferent ntalzatons; the results are averaged. Only smple msrepresentatons by % are consdered, where % of the buyers ncrease ther reported values by %. Instead of observng only symmetrc Nash-equlbra as n the analyss by [6], where partcpants ether msrepresent ther valuaton prce by or %, the rato of msrepresentng buyers to the total number of partcpants s also vared. A rato of =5%, for nstance, denotes that 5% of the buyers msrepresent ther valuatons by %, whle the other 5% report truthfully.4 By explorng the ont strategy space (.e. varyng the share of msrepresentatve and truthful depots as well as the percentage of msrepresentaton) the average utlty gan of manpulatng depot can be measured. Ths analyss provdes nformaton on whether or not the total utlty of depots can be mproved through manpulaton. The utlty, whch depots can gan by manpulaton, only depends on the prces depots have to pay and s thus calculated as: UG pˆ ( s ) p ( s ) s S Iˆ s S I, where Î s the set of buyers who are part of the allocaton n the treatment wth manpulaton partcpants and I s the set of successful buyers n the truthful treatment. For a better comparablty of the results, the utlty gan UG s further specfed as the percentage of the truthful scenaro. In the frst settng wth 5 buyers and 5 dfferent obs (.e. 3 dfferent ), the utlty gan of manpulatng depots are shown n Fgure and Fgure 2. Fgure : Utlty gan usng PPC, buyers, 5 obs, 3 Manpulaton factor,4,6,5,4 Fgure 2: Utlty gan usng VCG, buyers, 5 obs, 3,3,2, 5-5 - Utlty gan -5-2 -25-3 Partcpants manpulatng 2 See http://www.w.un-karlsruhe.de/case for detals. 3 CPLEX s a mathematcal optmzaton engne for solvng lnear programs (http://www.log.com/ ). 4 Ths restrcton s beng made, as the results above 5 % manpulaton suggest a tremendous decrease n ndvdual utlty and are thus left out. 6
Proceedngs of the 4th Hawa Internatonal Conference on System Scences - 27-2 -4-6 -8 Utlty gan n - -2-3 -4 Utlty gan -5 Manpulaton factor,4,6,5,4 In both cases usng the VCG and the PPC buyers mostly do worse by overbddng,.e. revealng a hgher value than ther true valuaton. The VCG penalzes all manpulaton attempts by a utlty loss. In the PPC, utlty can be ganed only when few depots manpulate. The hghest utlty gan (9.5%) s acheved when % of the buyers bd 4% of ther valuaton. Ths smulaton result suggests that the PPC mechansm compared to the VCG results n nearly equal overall utlty and thus may have accurate ncentve propertes. Wth a varyng number of buyers (to 5 and 3), nearly the same results are observed. In a second settng, the number of 3 dfferent obs and thus 7 dfferent were used. Fgure 2 shows the result of the smulaton. In both cases, no utlty can be ganed by overbddng. As seen n the frst scenaro, the VCG agan penalzes manpulaton more than the PPC. The smulaton shows that the PPC nearly acheves the same propertes than the VCG. Fgure 3: Utlty gan usng PPC, buyers, 3 obs, 7,3,2, - -2 Partcpants manpulatng -5 - Utlty gan -5 Manpulaton,4 factor,6,5,,2,4,3 Partcpants manpulatng The frst two scenaros suggest that the PPC works farly well wth an average number of obs (scenaro ) and s nearly comparable to the VCG wth a very low number of obs (scenaro 2). Obvously, a hgher number of obs lead to a lower ncentve compatblty. Hence, the number of obs s ncreased n a thrd scenaro to emphasze ths effect. The thrd scenaro comprses 7 dfferent obs and as such 27 dfferent bundle combnatons. Fgure 3 and Fgure 6 depct the results for the PPC (left) and VCG (rght). The ncreasng ncentve compatblty effect s emphaszed. If a small number of partcpants overbd, they can acheve a hgher utlty gan than n the frst scenaro. Nevertheless, the hghest utlty s only X% f Y% of the partcpants overbd ther valuaton by Z%. In summary, the smulatons have shown that t s reasonable to beleve that partcpants wll not strongly devate from ther truth valuatons. Although partcpants average utlty gan can be mproved through manpulatons f the number of obs s hgh enough, the partcpants ncreasngly also rsk not beng executed n the aucton. Ths rsk actually ncreases the more partcpants use manpulaton. The smulaton result suggests that the PPC has accurate ncentve propertes resultng n farly mld allocatve effcency losses. -6-7 -8-2 Manpulaton factor,4,5,4-25,,3,2 Partcpants manpulatng M Fgure 4: Utlty gan usng VCG, buyers, 3 obs, 7 7
Proceedngs of the 4th Hawa Internatonal Conference on System Scences - 27 Fgure 5: Utlty gan usng PPC, buyers, 7 obs, 27 Fgure 6: Utlty gan usng VCG, buyers, 7 obs, 27,4,4,6,6,5,5,4,4,3,3,2 5. Concluson Combnatoral mechansm desgn s partcularly dffcult, as there s a tenson between economc characterstcs, on the one hand, and computatonal propertes, on the other hand. From an economc pont of vew, the VCG mechansm s the only drect mechansm that acheves () an effcent allocaton of goods, () voluntary partcpaton of the bdders and () ncentvecompatblty. Nonetheless, the VCG mechansm lacks tractablty, as the wnner determnaton problem (.e. allocaton rule), and also the N+ payment computatons, are NP hard. In addton to those problems, the VCG mechansm needs the preference values for all possble combnatons of the goods from each bdder to retan ts desrable economc propertes. Ths so-called preference elctaton problem s also NP-hard for any partcpatng bdder. Hence, state-of-the-art mechansm desgn has turned ts attenton from drect, one-shot auctons to teratve auctons. In teratve auctons, the bdders need,2,, 5-5 - -5-2 -25-3 -2-4 -6-8 - -2-4 only to bd on the bundle that provdes them wth the hghest utlty. In ths paper, t s argued that t can make sense to apply drect mechansms n favor of teratve mechansms. The PPC mechansm s a very smple mechansm, whch eases the payment computatons. As classcal mechansm desgn suggests, the PPC mechansm s not strategy proof any more. Nonetheless, classcal mechansm desgn cannot state how the bdders wll behave when exposed to ths mechansm. In our numercal smulaton, we show that devatng from the true valuatons does not mprove the ndvdual utlty f the number of competng bdders s suffcently hgh wth respect to the avalable goods. Ths property s lost, f the number of avalable goods s ncreased. Ths leads to the concluson that when the sze of the aucton s very large, strategzng does not pay off. Competton drves the bdders to reveal ther true valuatons. As the number of bdders ncrease relatvely to the number of goods, the results of PPC mechansm converge to the ones of the VCG mechansm. In that case, the PPC mechansm can be seen as a knd of second-best mechansm. For decson support systems that gude the bdders n formng ther bds, ths has the mplcaton that the emphass s shfted from devsng bddng strateges towards preference elctaton. To fully explot the merts of the PPC mechansm, several problems need to be overcome. Frstly, the preference elctaton problem remans complex, as all valuatons need to be reported. Hence, the mechansm needs bddng support, when the number of goods becomes large. Even though bddng support can be reduced to extractng the true valuatons, ths can become cumbersome when the valuatons are not exactly known. In ths case, bddng support may need to apply some heurstcs to approxmate the valuatons. The effects on the mechansm result when estmaton errors are present are wdely unknown. Secondly, teratve mechansms yeld hgher revenues when valuatons are afflated. Htherto, the ssue of afflaton has been left out of the analyss. Thrdly, experments wth few goods are needed to verfy the conectures made n ths paper. Appendx Proof of Theorem 5 (PPC Ineffcency) All we have to show s that the payment functon of the PPC cannot be represented as the payment functon of a Groves mechansm. Based upon the aforementoned PPC Groves defntons, that s t t. 8
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