Game Study on Collusion in the First-price Sealedbid Reverse Auction

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1 Assocaton for Informaton Systems AIS Electronc Lbrary (AISeL) WHICEB 2013 Proceedngs Wuhan Internatonal Conference on e-busness Summer Game Study on Colluson n the Frst-prce Sealedbd Reerse Aucton Tan Jan School of Economcs and Management, Jangsu Unersty of Scence and Technology, Chna Xu Xnxn School of Economcs and Management, Jangsu Unersty of Scence and Technology, Chna Follow ths and addtonal works at: Recommended Ctaton Jan, Tan and Xnxn, Xu, "Game Study on Colluson n the Frst-prce Sealed-bd Reerse Aucton" (2013). WHICEB 2013 Proceedngs Ths materal s brought to you by the Wuhan Internatonal Conference on e-busness at AIS Electronc Lbrary (AISeL). It has been accepted for ncluson n WHICEB 2013 Proceedngs by an authorzed admnstrator of AIS Electronc Lbrary (AISeL). For more nformaton, please contact elbrary@asnet.org.

2 The Twelfth Wuhan Internatonal Conference on E-Busness Human Behaor and Socal Impacts on E-Busness Track 505 Game Study on Colluson n the Frst-prce Sealed-bd Reerse Aucton Tan Jan 1,Xu Xnxn 1 1 School of Economcs and Management, Jangsu Unersty of Scence and Technology, Chna Abstract:Colluson between supplers n reerse auctons may damage buyers benfts and then lower effcency of resource allocaton. Under frst-prce sealed-bd reerse aucton mechansm, supplers equlbrum bddng strategy and buyers reenue loss was analyzed consderng colluson. Then some general conclusons were gen: supplers bddng strategy s releant to some factors such as aluaton costs, the number of supplers and Cartel; the number of supplers and Carter can lead to buyers reenue loss. Fnally, consderng two dfferent knds of dstrbuton functon, the strateges of mprong buyer s reenue were dscussed. Keywords: reerse aucton; colluson; equlbrum bddng 1. INTRODUCTION Electronc reerse aucton (ERA)s an onlne, real-tme dynamc aucton between a buyng organzaton and a group of pre-qualfed supplers who compete aganst each other to wn the busness to supply goods or serces [1].Notwthstandng ts practcal sgnfcance and adantages such as procurement tme sangs and cost sangs,eras s an emergng research area n recent years.today, ERAs hae become a common method to source a broad range of products and serces. As the electronc reerse aucton hae some adantages the buyers get so much faor from t.reerse aucton theory has gradually enrched,howeer the majorty of the exstng research s focusng on the adantages,rsk, applcaton condtons of reerse aucton and relatons among buyers and supplers,wthout consderng supplers colluson case.colluson among supplers can reduce the aderse effects n reerse aucton procurement practce. The colluson n aucton frstly be researched by some scholars such as Robnson, Graham and Marshall, McAfee and McMllan. Robnson ponted out that the sharng of nformaton led to nstablty of the cartel n the frst prce sealed aucton, whch explaned why the frst prce sealed aucton s frequently used n practce [2]. Graham and Marshall extended the standard model, consderng colluson exstng among the bdders and the auctoneers strategc response to colluson, The man result shows that settng a hgher resere prce by the auctoneer s the optmal strategy [3].McAfee and McMllan characterzed coordnated bddng stateges n two cases:a weak cartel,n whch the bdders cannot make sde-payments;and a strong cartel,n whch the cartel members can exclude new entrants and make transfer payments [4]. The man result shows that when a strong cartel exsted and hghest aluaton exceeded the resere prce, confguraton s ald.then t ponted out that the auctoneer can mproe or hdden the resere prce, nterfere the cartels strategy to preent colluson. Lyk-Jensen accordng to ntroduce the frst prce or second prce pre-aucton knockout mechansm and assume all bdders partcpantng a cartel,t shows that the the colluson s stable, but can not achee the effcency,but effcency can be acheed by allowng the sharng of nformaton and budget balance of the pre-aucton knockout [5].Marshall and Marx dd further reserch on the colluson n the frst and second prce aucton [6].The Jng Hurong and L Chuanzhao compared cartel representate choce and proft dstrbuton n the frst and the second prce sealed pre-aucton knockout n the strong cartel mechansm, got effecte measures to respond to cartels colluson based on the aucton acton strateges [7].Wang Wenju and Yang Yngme combned the colluson between tenders to tenders publcly resered prce strategy,studed the nterests of the allocaton mechansm of the collusonn the second-prce sealed bd auctons,then got the results that reducng the publc

3 506 The Twelfth Wuhan Internatonal Conference on E-Busness Human Behaor and Socal Impacts on E-Busness Track resere prce can preent colluson [8]. Thomas got the mpact factors on colluson under the repeated aucton.:the number of supplers, the expected cost of supplers, the cost arablty of buyers [9]. Ohash nestgated bdder qualfcaton procedure to mproe the transparency, and got the result that mproed transparency can reduce procurement costs and undermne the stablty of the colluson [10]. Pesendorfer under the assumptons that buyers can sell more than one contract and the supplers aluaton was prate, then t got the colluson that the colluson equlbrum s to conceal ts offer [11]. Based on the aboe studes, ths artcle attempts to determne the strategy of cartel under the condtons of "strong" cartel, a sngle statc nseparable sngle tem only a cartel n the reerse aucton then accordng to the analyss of the supplers bddng strategy as well as the buyers reenue losses.impact factors of bddng strategy and measures to mproe the ncome of buyers through dfferent numercal analyss wll be got. 2. THE DESCRIPTION OF THEQUESTION AND MODEL HYPOTHESIS. Buyers use the electronc reerse aucton as a tool for supportng e-sourcng, Because supplers are symmetrcal and homogeneous,compette bddng wll reduce ther proft and colluson wll ncrease ther proft, whch promotes the cartel formaton. The frst questons faced by the cartel s how to choose cartel representates to partcpate the formal aucton and ratonal allocaton of proft they receed. In the frst prce sealed aucton, all cartel members make a contract: each member quotes a specfed prce, then any one n the cartel who wn the bd should pay to the other members lots of the reenue. But the cartel members may change ther tactcs, they are lkely to reduce the prce, so they can get the bd wth non-proft-dstrbutng whch wll obtan addtonal profts.so a reasonable strategy should be desgned wthn the cartel,the lowest prate aluaton wll wn the bd. In order to make more members not compulsory to jon the cartel, ths mechansm must satsfy the partcpaton constrant condton:if the members does not partcpate n the cartel,hs proft wll be zero,for non-cartel members to partcpate n the cartel can get a certan transfer payments.so t attracts more non-cartel members partcpate n the cartel. In short, an equlbrum strategy s as follows:all the cartel members are ndependently reported ts prate real aluaton, whch s a domnant strategy. In ths paper,we consder the bdders can make sde-payments whch s proposed n McAfee and McMllan [4] n a sngle statc frst prce sealed reerse aucton.equlbrum bd strateges of cartel representates who wll partcpate n the reerse aucton and buyers expected reenue loss wll be studed.for the conenence of research, ths paper make the assumptons as follows: 1) Colluson exsts among supplers n the aucton, the number of supplers s n ( n 3),the cartels number s k( 1 k n). 2) buyer and supplers are rsk neutral and maxmze ther expected reenue n the aucton. 3) the aluaton of the supplers s ( 1,2Λ n) =,whch s prate nformaton and each ( [, ]) s ndependent and dentcally dstrbuted n the cumulate dstrbuton functon ( ) F, ths functon s a contnuous monotoncally ncreasng functon,and the dstrbuton s common knowledge to buyer and supplers. 4) The bddng strategy b of suppler s monotoncally ncreasng.

4 The Twelfth Wuhan Internatonal Conference on E-Busness Human Behaor and Socal Impacts on E-Busness Track EQUILIBRIUM BIDDING STRATEGY OF THE CARTEL IN THE REVERSE AUCTION. In the frst prce sealed reerse aucton,assumng that supplers who has the lowest quotaton wll wn the subject and get the payment.the supplers expected reenue functon s U = ( b - ) Pr( b b, j ) In whch Pr ( ) s the lowest probablty n all bdb,obously, ths probablty s That s When the cartel members U (, )) = ( b - ) ))] bddng strategy: cartel representates drectly quoted prce of j [1 F ( ))], k equal to the number of the total number of supplers n, the cartels optmal,the other members quoted prce whch s hgher than. If there s no colluson, supplers quoted ther true costs, then the cartel representates should transfer payment to the unon When the cartel members k u =,then the proceeds wll be dstrbuted n the unon. s less than the number of total number of supplers n, cartel representate consder other cartel members acton.the followng proposton wll be got: Proposton Consderng the cartel exsts, Influence factors n equlbrum prce of supplers n the frst prce sealed reerse aucton : prate aluaton of supplers ( of supplers ( n ), the cartel numbers( k ). ), supplers bddng strategy ( F ( ) ), the number Proof:In the reerse aucton, for non-cartel members, he wn the bd when hs bd s the lowest bd the expected reenue and payment s dfferernt regardless of the exstng of the cartel,. Actually the number of partcpatng n the reerse aucton s n - k + 1.When a supplers prate cost s,hs bddng strategy s b,so hs expected reenue s U (, )) = ( b - )[1 F ( ))] n k The frst order condtons optmzaton s U (, )) = 0 b That s U (, )) -1 = ))] + ( n - k)( b - ) ))] [ ))] b = 0 That s ))] + ( n - k)( b - )[ ))] = 0 Supplers wll bd ther true alue when meetng Bayesan Nash equlbrum,that s b ( ) = Assume that the supplers aluaton,hs bd s, the boundary condtons b ) = (.Aboe equaton

5 508 The Twelfth Wuhan Internatonal Conference on E-Busness Human Behaor and Socal Impacts on E-Busness Track satsfes the frst-order lnear dfferental equatons and boundary condtons,supplers equlbrum bddng functon s ] du b ( ) = (1) + )] Proposton3.1 shows that: 1) Supplers equlbrum bddng functon b () s nterrelated wth supplers bddng strategy, the number of supplers n, the the cartel scale k than ; 2) In order to guarantee the reenue,supplers,and hs equlbrum bddng s hgher set reasonable ealuaton on the project, and the hgher the ealuaton s, the hgher equlbrum bddng strategy s; 3) In order to get more proft,supplers would form cartel to obtan addtonal ncome, obously cartels scale wll affect supplers equlbrum bddng strategy; 4) the more supplers to partcpate n the aucton,the lower of the bddng strategy s,then supplers wll get less profts, the buyer wll get hgher proft. Therefore, the buyer would consder the number of supplers on ther proft, he wll attract more supplers to partcpate n the bddng. aucton. Supplers equlbrum bddng functon has the followng propertes: Property 3.1.1,Supplers equlbrum bddng mproes consderng the exstence of cartel n the reerse Proof:Known from lterature[8],non-colluse condtons, n the standard frst prce sealed aucton supplersequlbrum bddng functon s: (1) subtracted (2),that s ] ( ) = + du b (2) )] ) [ ] ) ) k + 1 [ ) Because u >, 0 < < 1, > 0, s the monotone ncreasng functon We can get That s(1)>(2) ) F ( > ), > 0. ) That s supplers equlbrum bddng mproes consderng the exstence of cartel n the reerse aucton. Property :Supplers equlbrum bddng mproes consderng the mprong scale of cartel when fxng the same number of supplers n the reerse aucton. Proof:Fxed the number of supplers n,when the cartel scale ncreases, from k to k + 1 Proof equlbrum bddng s b -1 ] ( ) = -1 ] du du + (3) )] The analyss process s the same as aboe,we can obtan (1) <(3)..The

6 The Twelfth Wuhan Internatonal Conference on E-Busness Human Behaor and Socal Impacts on E-Busness Track 509 That s supplers equlbrum bddng b () mproes wth the ncreasng scale of cartel k. Property 3.1.3:Supplers equlbrum bddng reduced wth the ncreasng number of supplers when fxed the same scale of cartel n the reerse aucton,and then t wll closer to the cost aluaton. Proof:By (1), when the number of supplers ncreased from n to + 1 functon of the equlbrum bddng s It s easy to obtan (1)> (4). n+ k ] du b ( ) = + (4) n+ k )] n and fx the cartel scale k, the The equlbrum bddng b () s reduced when the number of supplers n ncrease, but t s hgher than the cost aluaton. From the analyss aboe, we can get that fxng the cartel scale, the more the supplers, the lower the cartel the lower the bddng strategy s,the greater the probablty of wnnng the bd. When the the supplers bd s lower and closer to the true cost,hs transfer to the other members of the cartel s smaller.therefore, n order to achee greater gans, Cartel representates wll make efforts to expand the scale of the cartel, whch ndrectly harms the nterests of buyer. But n a certan extent more supplers wll mproe the competton among supplers, may ncrease the buyers ncome.therefore, the number of supplers, the cartel scale wll affect the buyers ncome. 4. BUYER EXPECT REVENUE ANALYSIS WHEN CARTEL EXISTS. Defne the loss of buyers expected reenue R = R 1 R2.In whch R 1 s the buyer expected reenue wth non-colluson, R2 s the buyer s expected reenue when cartel exsts. The optmal aucton mechansm can make buyer to maxmze hs expected reenue. Under the condton of non-colluson, the buyers expected payment s R1 = n [ F ( ) + )][1 )] Just as bddng payment equalence theorem n the lterature [13]. That s n whch Rley [12] consdered that the buyers expected payment s n tmes of the supplers expected payment. n U (, b) F ( ) d = n d ( b - )[1 )] U (, b) = ( b - )[1 )] When the colluson exsts. Essentally, the number of suppler s n - k + 1,that s The the buyers expected reenue loss s R2 = ( n - k + 1) [ F ( ) + )][1 )] F ( ) d R = R1 R2 = ( k -1) [ F ( ) + )][1 )] d. d

7 510 The Twelfth Wuhan Internatonal Conference on E-Busness Human Behaor and Socal Impacts on E-Busness Track 5. NUMERICAL ANALYSIS. In order to get more ntute explanaton of nfluence factors on supplers equlbrum bddng strategy b and buyers expected reenue R, such as the number of supplers ( n ), the prate cost aluaton ( ) and the cartel scale ( k ). Let = 0, = 1, [0,1],take the followng two knds of aluaton dstrbuton functon,and t s monotoncally ncreasng alue nteral[0,1].all meet the assumptons. Case 1: The suppler aluaton unform dstrbuton and the aluaton of the dstrbuton functon s: F ( ) = The supplersequlbrum bddng functon s The buyers expected reenue loss s 1- b1 = + n - k + 1 2( k 1) R = n( n + 1) Case 2: Suppler aluaton obey polynomal dstrbuton and the aluaton of the dstrbuton functon s: The supplersequlbrum bddng functon s The buyers expected reenue loss s G( ) = b2 = + 2( n - k) + 1 ( k 1)(4n 1) R = n(2n + 1)(2n 1) Respectely let n = 3, k = 1, 2 ; n = 10, k = 1, 2,8 ; n = 15, k = 1, 2,8,14,15 ; = 0.2, 0.5, 0.9. By calculatng the equlbrum bddng functon and the buyers expected reenue loss s as shown n Table 1 and Table 2 below. Table 1.Bddng Strateges of Supplers between Colluson and Non-colluson * * b = + (1 ) /( n k 1) b = + (1 ) /( 2( n k ) 1) n = 3 n = 10 n = 15 n = 3 n = 10 n = 15 k = k = k =

8 The Twelfth Wuhan Internatonal Conference on E-Busness Human Behaor and Socal Impacts on E-Busness Track k = k = Table 2. Expected Reenue Loss Analyss of Buyers between Colluson and Non-colluson R = 2 ( k 1)/ n( n + 1) R = 2 ( k 1)( 4 n 1) / n ( 2 n + 1)( 2 n 1) n = 3 n = 10 n = 15 n = 3 n = 10 n = 15 k = k = k = k = k = (1) When ) s the same, the number of supplers s the same, then the supplers prate cost aluaton s dfferent, there s no colluson exsts, the hgher supplers prate cost aluaton s, the hgher ther equlbrum bddng strategy s. Expected reenue of the suppler (e balanced bd mnus cost aluaton) Wth the rse of the prate cost aluaton s gettng smaller and smaller, for example, when n = 15, respectely, by ,0.0333, down to When the colluson exsts, the more the number of colluson s, the hgher the equlbrum bd s, when n = 15 = 0. 5,the bd s ,0.5625, The supplers expected reenue s ncreasngly hgh, buyers loss s also ncreasng correspondngly. (2) When ) s the same,the number of supplers s dfferent, the supplers of prate cost aluaton s the same, there s no colluson, the more the number of supplers s,the lower equlbrum bd s,for example, when = 0. 5, the bd s , down to When the colluson exsts, the more the number of colluson, that s the greater cartel scale s, the hgher the balanced bd, ncreased from to Buyers loss s ncreasng correspondngly. (3) When ) s the same and there s no colluson, under the deal condtons, buyers expected reenue loss s 0.When all supplers partcpate n the colluson, the equlbrum bd s the hghest alue of , as was set before.the maxmum loss of the buyers s So the buyer must take measures to reduce ts loss of reenue. (4)For the two dfferent equlbrum bddng functon F ( ),the buyers expected reenue loss and the supplers equlbrum bddng functon are dfferent but the aboe conclusons are also ft, that s not rreleant to F ( ).In order to facltate the study,ndependent unform dstrbuton s also assumed n the reerse aucton. 6. CONCLUSIONS Supplers bd domnant strategy n the cartel s all cartel members ndependently report ther prate real

9 512 The Twelfth Wuhan Internatonal Conference on E-Busness Human Behaor and Socal Impacts on E-Busness Track alue n the frst prce sealed reerse aucton.the factors that affect the supplers equlbrum bddng strategy s as follows: the supplers cost aluaton, the supplers dstrbuton functon of the the aluaton,the number of supplers, the cartel scale.the supplersequlbrum bddng mproed when the colluson exsts and the more the number of the cartel, the hgher the supplersequlbrum bddng;the hgher the number of all supplers, the lower the supplers equlbrum bddng,and t s closer to hs cost aluaton.colluson lead to buyers reenue loss, and the loss of reenue s releant to and the number of all supplers and the scale of the cartel. Therefore the buyer may take measures to control the sze of the cartel and ncrease the number of supplers to mproe hs reenue. Ths artcle s n deal condtons :the "strong" cartel mechansm, n sngle statc nseparable sngle tem reerse aucton only consderng a cartel exsts.the equlbrum bddng strateges s studed under the colluse and non-colluse condtons and buyers loss of reenue consderng the exstence of only a cartel and then numercal analyss s showed. Only two dfferent dstrbuton functons s consdered n ths artcle, conclusons merely reflects the functon of the mpact of changes,other factors and the presence of multple cartels,and the unersalty of the formula needs do further research;how to extend the model to mult-tems and dynamc reerse aucton need further research. Compared wth the tradtonal aucton, n reerse aucton the buyer s facng large-scale tem and hghly compette,and more complcated process. Wth the growng popularty of e-commerce, network prodes more nformaton to the supplers at the same tme ts conenent to colluson among supplers. how to effectely reduce the occurrence of colluson s also one of the focus of future research n the network enronment. REFERENCES Brd R B, Stewart W E. (1960).Lghtfoot E N. Transport Phenomena. New York: John Wley & Sons Inc, 75 [1] Tan Jan, Tang Xaowo.( 2009). Reew of research on onlne reerse aucton[j]. Management Reew, 21(12): (n Chnese) [2]Robnson M S. (1985).Colluson and the choce of aucton[j]. The RAND Journal of Economcs, 16(1): [3]Graham D A, Marshall R C.(1987).Colluse bdder behaor at sngle-object second prce and englsh auctons [J]. Journal of Poltcal Economy, 95(6): [4]McAfee R P, McMllan J. (1992) Bddng rngs [J]. The Amercan Economc Reew, 82(3): [5]Lyk-Jensen P. Some suggestons on how to cheat the auctoneer: Colluson n auctons when sgnals are afflated [Z]. Workng Paper, Unersty of Copenhagen, [6]Marshall R C, Max L M. (2007).Bdder Colluson[J]. Journal of Economc Theory, 133(1): [7]Gou Hurong,L Chuanzhao.( 2008).Two cartel colluson mechansm n aucton[j].journal of Industral Engneerng, /Engneerng Management, 22(3): (n Chnese) [8]Wang Wenju,Yang Yngme.( 2011).Game analyss of colluson among bdders and optmal publc resere prce[j].research on Economcs and Management, (10):77-85.(n Chnese) [9]Thomas, C.J.( 2005).Usng Resere prces to Deter Colluson n procurement Competton[J].The Journal of Industral Economcs, 53(3): [10]Ohash, H.( 2009). Effects of transparency n procurement practces on goernment expendture: A case study of muncpal publc works[j]. Reew of Industral Organzaton, 34(3): [11]Pesendorfer, M.( 2000).A Study of Colluson n Frst-Prce Auctons[J]. Reew of Economc Studes, 67(3): [12]Rley J G, Samuelson W F. (1981). Optmal Auctons[J].Amercan Economc Reew, 71(3): [13]Yang Yngme,Wang Wenju.(2007).Game analyss of bddng payment equalence and optmal resere prce[j].research on Economcs and Management, (11):54-58.(n Chnese)