1 Desig of Mlti-it Electoic Exchages thogh Decompositio Pakaj Dayama, Y Naahai Abstact I this pape we exploit the idea of decompositio to match byes ad selles i a electoic exchage fo tadig lage olmes of homogeeos goods, whee the byes ad selles specify magial-deceasig piecewise costat pice ces to capte olme discots Sch exchages ae eleat fo atomated tadig i may e-bsiess applicatios The poblem of detemiig wies ad Vickey pices i sch exchages is kow to hae a wost case complexity eqal to that of as may as NP-had poblems, whee is the mbe of byes ad is the mbe of selles O method poposes the oeall exchage poblem to be soled as two sepaate ad simple poblems: (1) fowad actio ad (2) eese actio, which t ot to be geealized kapsack poblems I the poposed appoach, we fist detemie the qatity of its to be taded betwee the selles ad the byes sig fast heistics deeloped by s Next, we sole a fowad actio ad a eese actio sig flly polyomial time appoximatio schemes aailable i the liteate The poposed appoach has wost case polyomial time complexity ad o expeimetatio shows that the appoach podces good qality soltios to the poblem Note to Pactitioes I the ecet times, electoic maketplaces hae poided a efficiet way fo bsiesses ad cosmes to tade goods ad seices The se of ioatie mechaisms ad algoithms has made it possible to impoe the efficiecy of electoic maketplaces by eablig optimizatio of eees fo the maketplace ad of tilities fo the byes ad selles I this pape, we look at sigle-item, mlti-it electoic exchages These ae electoic maketplaces whee byes sbmit bids ad selles sbmit asks fo mltiple its of a sigle-item We allow byes ad selles to specify olme discots sig sitable fctios Sch exchages ae eleat fo high olme bsiess-to-bsiess tadig of stadad podcts sch as silico wafes, VLSI chips, desktops, telecom eqipmet, commoditized goods, etc The poblem of detemiig wies ad pices i sch exchages is kow to iole solig of may NP-had poblems O pape exploits the familia idea of decompositio, ses cetai algoithms fom the liteate, ad deelops two fast heistics to sole the poblem i a ea optimal way i wost case polyomial time Keywods Mlti-it exchages, magial-deceasig piecewise costat bids, olme discots, fowad actio, eese actio, geealized kapsack poblem Geeal Motos Idia Sciece Lab, Bagaloe, Idia (Wok caied ot as pat of Maste s Dissetatio at Compte Sciece ad Atomatio, Idia Istitte of Sciece, Bagaloe) E-mail:pakajd@csaiisceeti Compte Sciece ad Atomatio, Idia Istitte of Sciece, Bagaloe - 560 012, Idia E-mail: hai@csaiisceeti (coespodig atho) I INTRODUCTION I the ecet time, electoic maketplaces o electoic exchages hae poided a efficiet mechaism fo bsiesses ad cosmes to tade goods ad seices I this pape, we coside sigle-item, mlti-it electoic exchages These ae e-maketplaces whee byes (o byig agets) ad selles (o sellig agets) sbmit bids ad asks fo mltiple its of a sigle-item Sch exchages ae eleat fo high olme B2B (bsiess-to-bsiess) tadig of stadad podcts sch as silico wafes, VLSI chips, desktops, telecom eqipmet, commoditized goods, etc We assme that the bids ad asks sbmitted by the byes ad selles ae magial-deceasig piecewise costat fctios Sch fctios help the byig agets ad sellig agets to specify olme discots (ofte called qatity discots) A bye ca specify a lowe bod o the mbe of its he demads ad a selle ca specify a ppe bod o the mbe of its he ca spply Hee we coside sigle-shot, sealed bid exchages Thee ae two optimizatio poblems ioled with sch exchages 1) Allocatio Poblem (also called tade detemiatio poblem o wie detemiatio poblem): Detemiig how mch will be boght by each byig aget ad how mch will be sold by each sellig aget 2) Picig Poblem: Detemiig the et paymet to be made by each (wiig) byig aget ad the et paymet to be made to each (wiig) sellig aget Ee with may estictios o the stcte of bids ad asks sbmitted by the byes ad selles, the allocatio poblem ts ot to be itactable [1] If, i additio, tth eelatio popeties ae eqied fom the byig ad sellig agets, the comptatio of paymets will iole solig seeal sch itactable poblems [1] I this pape, o appoach to the poblem is to sole it appoximately, sig a simple decompositio idea, ad to come p with a comptatioally efficiet soltio that poides ea optimal soltios A Releat Wok Thee ae excellet sey papes i the aea of actios ad exchages Fo example, the eade is efeed to the books [2], [3], [4] ad the sey aticles [5], [6], [7], [8], [9], [10], [1] Othe popla efeeces ae: [11], [12], [13], [14] Hee, we poide a bief eiew of eleat liteate i the aeas of (1) sigle-item, mlti-it actios ad (2) sigle-item, mlti-it exchages 1) Sigle-Item, Mlti-Uit Actios: O pape ses the eslts i the ecet wok of Kothai, Pakes, ad Si [15] o appoximately stategy-poof ad tactable mlti-it actios
2 I [15], the athos coside sigle-item, mlti-it actios whee the biddes se magial-deceasig, piecewise costat fctios to bid o homogeeos goods Both fowad actio (sigle selle ad mltiple byes) ad eese actio (sigle bye ad mltiple selles) ae cosideed I the fowad actio, the objectie is to maximize the eee fo the selle ad i the eese actio, the objectie is to miimize the cost fo the bye It is show that the allocatio poblems ae geealizatios of the classical 0/1 kapsack poblem, hece NP-had Comptig VCG (Vickey-Clake- Goes) paymets [1] also is addessed The athos deelop a flly polyomial time appoximatio scheme (FPTAS) fo the geealized kapsack poblem This leads to a FPTAS algoithm fo allocatio i the actio which is appoximately stategy poof ad appoximately efficiet It is also show that VCG paymets fo the actios ca be compted i wost-case time, whee is the ig time to compte a soltio to the allocatio poblem Eso, Ghosh, Kalagaam, ad Ladayi [16] addess the pocemet poblem faced by a bye who wishes to by lage qatities of seeal heteogeeos podcts Spplies sbmit piecewise liea ces fo each of the podcts idicatig the pice as a fctio of the spplied qatity The poblem of miimizig the pchasig cost ts ot to be itactable The athos deelop a flexible colm geeatio based heistic that poides ea-optimal soltios to the bid selectio poblem sig bach ad pice methodology Simila poblems hae bee iestigated i [17], [18] Howee, i all these papes, the isse of picig so as to idce tth eelatio by the agets has ot bee discssed Dag ad Jeigs [19] coside mlti-it actios whee the bids ae piecewise liea ces Maximizig the eee of the actioee is the objectie Algoithms ae poided fo solig the allocatio poblem I the case of mltiit, sigle-item actios, the complexity of the allocatio algoithm!#"$ is whee is the mbe of biddes ad is a ppe bod o the mbe of segmets of the piecewise liea picig fctios The algoithm theefoe has expoetial complexity i the mbe of bids 2) Sigle-Item, Mlti-Uit Exchages: Kalagaam, Daepot, ad Lee [20] coside cotios call doble actios which ae also kow i the liteate as cleaig hoses o call makets I sch a maket, the maketplace collects bids fom byes ad asks fom spplies oe a fixed time peiod ad cleas the maket at the ed of the time peiod A bid specifies qatity ad pice Similaly a ask specifies a qatity ad pice Thee cases ae cosideed: % Ay pat of a bid may be matched with ay pat of ay ask I this case, the allocatio poblem ca be soled i i log liea time % Whe thee ae assigmet costaits, that is, some demads ca oly be assiged to some spplies, the the allocatio poblem ca be soled i polyomial time sig etwok flow algoithms % If the demad is idiisible, that is, a gie demad is costaied to be satisfied by exactly oe ask oly, the allocatio poblem ts ot to be NP-had The aboe eslts ae smmaized i Kalagaam ad Pakes [1] Dailiaas, Saiamesh, Gottemkkala, ad Jhiga [21] coside maketplaces fo badwidth i a etwok seices ecoomy The byes specify bid ces which specify the it pice eqested as a fctio of qatity Similaly the selles specify offe ces that specify the it pice as a fctio of qatity offeed Thee types of objecties ae cosideed: % Pofit maximizatio: maximize pofit o the pice spead betwee the aggegated bids ad aggegated offes % Bye satisfactio: Match the demad of all byes ad fid the best combiatio of selle offes that will maximize the pofit % Miimm liqidity: Match the demads of at least a cetai pecetage of byes while gaateeig some miimm pofit fo the maketplace Both exact ad heistics-based soltios ae exploed fo each of the thee objecties ad a aalysis of the pefomace of the soltios is epoted Sadholm ad Si [22] discss a aiety of allocatio algoithms The athos coside makets whee thee ae mltiple idistigishable its of a item fo sale (o thee ae mltiple its of mltiple items fo sale, bt diffeet items ca be teated idepedetly as belogig to diffeet makets) The bids ae i the fom of spply ces (sellig agets) ad demad ces (byig agets) that specify pice qatity elatioships These ces ae assmed to be piecewise liea The objectie is to maximize the total spls Two diffeet picig schemes ae cosideed: o-discimiatoy (all selles shae the same pice ad all byes shae the same pice) ad discimiatoy ( each selle ad each bye may be associated with a diffeet pice) The athos peset a polyomial time algoithm fo cleaig o-discimiatoy makets ad show that cleaig discimiatoy makets is NP-complete If the spply ad demad ces ae liea, the discimiatoy makets ca also be cleaed i polyomial time B Cotibtios ad Otlie We fid the followig eseach gaps i the liteate: % The allocatio poblem i the geeal case of mlti-it, sigle item exchages ad actios with magial deceasig, piecewise costat bids is NP-had Polyomial time algoithms hae bee poposed fo the allocatio poblem oly i ey special cases % Most of the papes do ot coside the picig poblem This is a impotat isse becase appopiately compted pices ca idce tthfl bids by all the agets Motiated by these eseach gaps, this pape exploes the followig diectios i the cotext of sigle-item, mlti-it exchages whee the biddes specify magial deceasig piecewise costat pice ces % We se the familia idea of decompositio to sole the allocatio ad picig poblems by solig two sepaate simple poblems: a fowad actio ad a eese actio % We popose two fast heistics to compte the tadig qatity to be sed fo the fowad ad eese actios
L A &(')+*-,###/,021 0 set of byes 34'5)+*-,6//6,8791 :$';*-,/#/#,0 7 set of selles <='>*-,#/6/,87 idex fo byes 0@? idex fo selles &BA : mbe of steps i the bid of bye < mbe of steps i the ask of selle C=';*-,6//#,0@?9DE* FG'>*-,#/6/,H&IAJDK* : idex fo step i the bid of bye < idex fo step i the ask of selle MON &P, 3RQ a lage eogh itege MON & S)UTV1+,H3RQ 0 7 spls with byes ad selles MON T spls whe bye does ot paticipate &P, 3 S)XWY1XQ W spls whe selle does ot paticipate ZU[? C : 1 if bid iteal fo bye is selected \ othewise ] ^ F < 1 if bid iteal fo selle is selected \ othewise _+[? C : mbe of its i iteal allocated to bye `G^ A F < mbe of its i iteal allocated to selle acb d?fe T Vickey discot to bye acg e h+i Vickey spls to selle W TABLE I NOTATION FOR THE ALLOCATION PROBLEM The heistics hae wost case polyomial time complexity ad podce ealy optimal ales of tadig qatity The se of these heistics i a decompositio based appoach has wost case polyomial time complexity wheeas the diect appoach fo solig the allocatio ad picig poblems has a comptatioal complexity eqal to that j /Rkjlk of as may as NP-had poblems whee is the mbe of byes ad is the mbe of selles % Usig appopiate ad extesie meical expeimets, we show the efficacy of the poposed appoach ad the poposed heistics, i tems of qality of soltios podced, comptatioal efficiecy, ad ability to idce tth eelatio by the biddes The pape is ogaized i the followig way Sectio 2 descibes the otatio ad fomlatios that will be sed i the est of the pape Fist, we peset the fomlatio of optimizatio poblems i a mlti-it exchage whee byig agets ad sellig agets sbmit magial deceasig pice fctios We show the fomlatio fo the (1) allocatio poblem ad (2) comptatio of Vickey paymets We also show how the exchage poblems ca be soled sig a simple decompositio appoach iolig a fowad actio ad a eese actio I Sectio 3, we peset two fast heistics to detemie the optimal qatity to be taded, which will be eqied i solig the fowad actio ad eese actio Sectio 4 pesets the eslts of a wide age of expeimets caied ot II ALLOCATION AND PRICING PROBLEMS IN SINGLE-ITEM MULTI-UNIT EXCHANGES The otatio is descibed i Table I The exchage we coside ca be descibed as follows % Thee is a set of byig agets, moqp +sts+s-6j> ts+stsw#j set of sellig agets, lqp, ad a Fig 1 Pice pe it 100 98 95 93 0 10 21 31 46 50 Nmbe of its A bid sbmitted by a byig aget % The byig agets sbmit bids, xyzpxy +sts+s+ x@, espectiely A bid is a list of pais, X w+sts+sw ƒ X ˆ xg, with a ppe bod of O o the qatity, whee Š Š Œ+ŒtŒ X B w ŽŒ+ŒtŒ@ 6 $6 Hee the bidde s alatio i the qatity age 6 is fo each it Note that the bid stcte hee eables the byes to specify qatity o olme discots +s+stsw, % The sellig agets sbmit asks, y p 2 " espectiely H A ask is a list of pais, 6 -ts+s+stt 2 6 9 @ š ˆ R of G o the qatity, whee 6 Œ+ŒtŒœ ŒtŒ+Œ š 3, with a ppe bod Note that the bid stcte hee eables the selles to offe qatity o olme discots % We ca itepet each list of tples as a pice fctio: ž t V VU $6 if, whee ªI X«9+sts+s-6j V if c 9 žj w/ V 2 if 4, whee ± X«9+sts+s- c 9 m ² if 2 A example of a bid sbmitted by a byig aget t is gie i Fige 1 Hee the byig aget bids a pice of pe it fo «9U $! tu«$ qatity i the age, µ pe it fo the age, µ ¹ º$ 6¹ º pe it fo the age, ad µ pe it fo the age Vˆ A example of a ask sbmitted by a sellig aget ¹» is gie i Fige 2 Hee the sellig aget offes a pice of pe it tº6 º fo qatity ½¼ fo the age +º, pe º it fo the age, ad pe it fo the age cˆ A Allocatio Poblem I the exchage descibed aboe, the allocatio poblem is fomlated as follows We choose the spls to the exchage (also called eee to the exchage) as the objectie to maximize The spls is defied as the total paymet eceied fom all the wiig byes mis the total paymet made to all the wiig spplies The mai costait to be satisfied is the total mbe of its sold to the byes shold be less
Ë m É É É Í m m m µ 4 immediately implies that the aboe allocatio poblem is itactable Fig 2 Pice pe it 40 38 37 0 5 16 36 50 Nmbe of its A ask sbmitted by a sellig aget tha the total mbe of its poced fom the selles The otatio is descibed i Table I Maximize sbject to À ² Á 8À 6Á X ÃÂÃÂÃÂà š½ tàå - ÃÂÃÂà9 6 Ç (1) 6Á - ÃÂÃÂའtàå Á X ÃÂÃÂÃÂÃ Ç È É (2) ÊÌË=Í ªI +sts+sw6j š È (3) ± +sts+st m ÊOËÍ Ð m (4) 6Á X ÃÂÃÂÃÂà š Ê Í (5) Á X ÃÂÃÂà@ y (6) /! ÊÌËÍ ªŠ +sts+sw6j (7) 6 ÊÌËÍ $t ÑÒ ÓUÔ ± +sts+sw m (8) ªŠ +sts+sw6j (9) ± +sts+sw m ² (10) Ô+Õ (11) Costait gaatees that the mbe of its sold will ot exceed «the mbe of its poced Costait assigs if É 8¹» Costaits ad efoce the exchage mechaism to choose items fom jst oe bid i- 68º wt ¼V- teal fo each bye ad selle Costaits -t 6 ese that ad lie i the age of the ª»Ö8 iteal fo Ö8 Ë bye ad the Ö8 selle espectiely The classical 0/1 kapsack poblem, which is a well kow NP-had poblem, is a special case of this poblem ad this B Picig Poblem The picig poblem ioles detemiig the actal paymets to be made by the wiig byes to the exchage ad the actal paymets to be made to the wiig selles by the exchage VCG (Vickey-Clake-Goes) paymets ae those that ese that the bids ad asks fom the byes ad selles eflect the te ales [1] Maket mechaisms that follow VCG paymets ae ofte called stategy poof mechaisms VCG paymets fo each of the wiig agets ca be detemied as follows Fist, sole the allocatio poblem by etaiig the aget i the poblem ad detemie the total spls geeated Next, emoe that aget fom the scee, sole the allocatio poblem, ad detemie the total spls (with the aget emoed) The decease i the spls de to the absece of the aget is gie as Vickey discot if the aget is a byig aget ad is gie as Vickey spls if the aget is a sellig aget It is easy to see that we eed to sole p to 8j B itactable poblems, oe fo each wiig bye ad wiig selle, to detemie VCG paymets C The Decompositio Appoach We decompose the poblem of a sigle-item, mlti-it exchage ito two atal, sepaate poblems: fowad actio ad eese actio The appoach ioles the followig steps: 1) Detemiig the tadig qatity, ØÚÙ, that is, the qatity of its that will be exchaged betwee the byes ad the selles 2) Solig the sepaate poblems: (a) Reese Actio: Based o the bids sbmitted by the sellig agets, poce a tadig qatity Ø of the goods so as to maximize the total ale of the sellig agets (b) Fowad Actio: Based o the bids sbmitted by the byig agets, sell the Ø goods to the byig agets, so as to maximize the total ale fo the byig agets We descibe the eese actio poblem The fowad actio poblem ca be fomlated o simila lies 1) Reese Actio: The fomlatio hee is doe o the lies of [15] Miimize sbject to tàå tàå Á - ÃÂÃÂà@ Á X ÃÂÃÂà@ š Á - ÃÂÃÂÃÂÃ Ù Ø È Ê ±Û Ê ±Û ts+s+s+ m ts+s+s+ m
Í Ë Ë Ù Ê ÜÍ + ÑÒ»Ó6Ô Ô+Õ Ê ±Û +sts+sw m Hee the fist costait eses that the total mbe of its poced is geate tha o eqal to the tadig qatity Ø This fomlatio is the same as that of a geealized kapsack poblem [15] Kothai, Pakes, ad Si [15] hae poposed a! time 2-appoximatio algoithm fo the geealized kapsack poblem aisig i eese actio ad also hae peseted a flly polyomial time appoximatio scheme based o this 2-appoximatio III HEURISTICS FOR DETERMINING TRADING QUANTITY The decompositio appoach podces a high qality soltio oly if we se the optimal tadig qatity Detemiig the tadig qatity to be sed by the decompositio method is ths a citical poblem We addess this poblem i this sectio by poposig two heistics to compte a almost optimal tadig qatity A Heistic 1 Based o the bids ad asks sbmitted, it is easy to detemie a lowe bod (Ý ) ad a ppe bod ( ) o the tadig qatity betwee which the optimal qatity will lie Oce this age is detemied, fo diffeet tadig qatities i this age, o idea is to se a geedy method to detemie the allocatio to the selles ad byes, ad detemie the spls We choose the qatity that maximizes this spls Fist we detemie: miimm demad, Þ " 8À maximm demad, Þ 8À miimm spply, " tàå maximm spply, tàå 2 @ Coside ƒ " Þ " Hee, thee cases ae possible 1) " Þ " Þ 2) " Þ ƒ " Þ 3) " Þ ƒ " Þ wß Fo case (1) ad case (2), Ýl " X wß case(3), Ýà " X Þ by the byes, á the tples sbmitted by the selles, alatio of the byes ã ad fo Now, sot the tples sbmitted i descedig ode of it pice ad sot i ascedig ode of it pice Fo diffeet tade qatities, we compte total ad total alatio of the selles as discssed i the algoithm below We sca thogh the soted list ad detemie a feasible allocatio The tadig qatity Ø is chose as a ale betwee ã Ý ad sch that ad is maximm the diffeece betwee The followig descibes o algoithm fo detemiig the tadig qatity Ø Algoithm: Heistic-1 fo Detemiig Tadig Qatity 1) Sot all pais fom the byes i descedig ode of it pice ad all pais fom the selles i ascedig ode of it pice 2) Vay the qatity to be taded, fom Ý to 3) Compte total alatio as follows: % Set jp»ävå Ë 5 of the byes fo qatity, fo all bids x, Ë +s+stsw#j Iitialize the emaiig qatity Ë to be sold, ; the qatity allocated to bye, Ø ; % Sca the pais i soted ode Let the selected pai be á % jp»ävå Ë if ad Ø 6 ç è àè whee, 6 is the diffeece i the alatio of bye fo! ad his alatio fo Ø its k Ø $6 ; Ø $6 go to scaig step; % jp»ävå Ë if ad Ø 6 ç è àè whee, is the diffeece i the alatio of bye fo Ø its ad his alatio fo Ø its et else go to scaig step; ad % jp»ävå Ë if 6! ç è àè Ë Ì6 whee, is the alatio of bye fo its jp»ävå Ë ; Ø 6 ; k Ø go to scaig step; % jp»ävå Ë if ad @ 6 ç è àè whee, is Ë the alatio of bye fo et its ã 4) Compte total alatio of the selles fo qatity as follows: % jp»ävå Ë Set, fo all asks Ë, ts+stsw# Iitialize the emaiig qatity to be poced, ; ã ; the qatity allocated to bye Ë, Ø % Sca the pais i soted ode Let the selected pai be % jp»ävå Ë if go to scaig step; % jp»ävå Ë if ad H 6! ã è ã éè Ë H R6 whee, is the alatio of selle fo its jp»ävå Ë ; Ø @6 ; k Ø go to scaig step; % jp»ävå Ë if @ ad K Gê 6 ã è ã éè whee, ã is Ë the alatio of selle fo et 5) if ã jû ë!/í-îfë! Øq jû ë! íwîïëc ; pdate 6) if, its
Ý ô ú ú p p ª ± ÿ ÿ ô ª ± ê 6 go to step 2; Oce the iitial sotig is doe, jñ # the algoithm takes ig time, whee ð b Spply ce B Heistic 2 Hee we come p with a faste heistic fo detemiig the tadig qatity based o the cocept of detemiig the call maket pice-qatity pai Fist, we discss a algoithm fo cleaig the call makets [23] We will modify it fo detemiig the tadig qatity to be sed A call maket is a sealed-bid, oe-shot exchage which ca be descibed as follows: % Thee is a set of byig agets, moqp +sts+s-6j>, ad a set of sellig agets, lqp ts+stsw#j % The byig agets sbmit bids, xòópx ts+sts+ 8 U x, whee espectiely Ë A bid x is of fom, x bye is willig to accept p to its at it pice % The sellig agets sbmit asks, yðp 2 ts+s+st ", espectiely A ask @ is of fom, @@ H U whee selle is willig to sell p to it at it pice Ù I a call maket, all tades clea at a maket-cleaig pice A algoithm fo cleaig a call maket fom [23] is descibed below Algoithm: Call Maket Cleaig Algoithm % Sot the bids i deceasig ode of it pice Let the soted ode be ô @Ù;ô Ù Let the soted ode be s+s+s Ù;ôÌ % Sot the asks i ascedig ode ofs+sts it pice " % At the by side, the qatity of item aailable at pice is whee, õ À ã Ë/ö ô $ø á#ùú ä!û Ë/ü % Similaly at the sell side, the qatity of item aailable at pice is whee, ý tàå Þ Ë/ö $ á#ùú ä!û Ë/ü % Plot a gaph betwee the pice ad cmlatie qatity of item aailable both fo selles ad byes (see Fige 3) % The itesectio poit gies the optimal tadig qatity Ø Hee the qatity Ø ã Þ will maximize the spls ad the Notice that maket cleaig pice þ will be 9ÿ þ -ÿ Fig 3 pais Uit Pice a Cmlatie Qatity Q Demad ce Spply ad demad ces fo detemiig maket pice-qatity o exchage (sigle-item, mlti-it exchage) is a aiatio of the aboe call maket i the followig ways 1) Fo byes, each bid coespods to a age ie # 6 wu x p 6 Ì -X s+sts p # ĤX sts+s whee š s+s+s ad X w6 2) Each bye sbmits XOR bids of the type: 3) Fo selles, each ask coespods to a age ie @@ p X 2 -U s+sts p 2 X 2 ˆ 6 9 sts+s whee 2 s+sts ad P 4) Each selle sbmits XOR asks of the type: We ow popose o heistic fo detemiig tadig qatity fo o exchage based o the call maket cleaig algoithm Algoithm: Heistic-2 fo Detemiig Tadig Qatity 1) Sot H the bid pices of byes ts+sts+u U -+sts+sw +sts+stu i descedig ode +sts+s+ ts+sts+ ô Let the soted ode be ôo is the mbe of tems i 2) At the by side, the maximm qatity of item aailable at pice ô is: whee, ã õ 8 $ Ó $ÔtÕ ²Ñ#Ô 8À ã 6 @ ô 3) Sot the ask pices of selles i ascedig ode Let the soted ode be 8 ts+stsw6 +sts+swu is the mbe of tems i 4) At the sell side, the maximm qatity of item aailable at pice is: whee, Þ ýj tàå Ó$Ô+Õ²Ñ#Ô Þ ˆ ˆ
ä ü Ù Ù õ 5) Obsee that iceases with decease i ô This is becase each of the bids sbmitted by the byes is magially deceasig piecewise costat alatio fctio So, as the pice deceases the cmlatie qatity of item aailable ý iceases Similaly, iceases with icease i 6) A gaph of the total cmlatie qatity of item ad pice both fo selles ad byes is simila to the oe show i ä Fige 3 7) Iitialize ü ; Let õ õ be ã the ý total alatio ÿ of the byes fo qatity ad ý be the total ÿ jp /ë/íwîïëc alatio of the selles fo qatity is sed to stoe maximm spls Pefom the followig steps ) ä while ( ü ad % if ô cÿ õ ý ÿ ad if õ ã ý ÿ jû ë!/í-îfë! Ø õ jû /ë/íwîïëc ; pdate % õ if Ù ý ÿ ü % õ ý ÿ if ä The aboe algoithm gies the tadig qatity Ø Bt it may ot be the optimal qatity becase we do ot coside the lowe bod of each age of the bids ad the asks The jû ig jçf =j time#gf # of the algoithm ca be easily see to be IV EXPERIMENTAL RESULTS I this sectio, we peset eslts of o meical expeimets to show the pefomace of the poposed decompositio appoach ad the poposed heistics A Expeimetal Setp We sed a ILOG CPLEX sole package o a 3 GHz Xeo see with 2 GB RAM to compte exact soltios We efe to this as the diect soltio appoach We sed the same see fo implemetig o heistics, o decompositio appoach, ad the FPTAS algoithms fo fowad actio ad eese actio The bids ad asks eqied fo the meical expeimets wee geeated to be as epesetatie as possible The bids ad asks ae magial deceasig piecewise costat alatio fctios We codcted expeimetatio with fo sets of data These sets of data diffe with espect to the age of ales fo choosig the lowe bod ad ppe bod o the mbe of its fo each bid ad ask Table II gies these ages fo the fo sets of data I all the expeimets, we cosideed 10 selles ad 10 byes Also, we assmed the maximm mbe of steps i a bid o a ask j to be 10 Fo each bye, we geeated the mbe of steps ( 9 ) i the bid thogh a discete ifom adomts+s+st+ aiate i the age 1 to, we chose the Ë Ë 10 Fo each bye, say bye ( miimm ad maximm mbe of its ( ƒ ad ) i his bids sig ifom adom aiates i the appopiate age Fo each step (ª X ), we geeated the pice pe it ( ) adomly i the age $s +! sch that Û 7 w Û Œ+ŒtŒ We followed a simila method fo each of the 10 selles The expeimets o fo diffeet data sets wee codcted 20 times sig idepedet samples We compted the aeage of the soltio ales fo 2, 3,, 20 eplicatios ad fod that afte 20 eplicatios, the aeages of the soltio ales emaied iaiat The eslts epoted ae ths aeaged oe the 20 expeimets codcted fo each data set Expt No Lowe Bod Rage Uppe Bod Rage 1 (10,60) (80,280) 2 (25,70) (100,350) 3 (50,150) (200,600) 4 (100,200) (350,850) TABLE II LOWER BOUND AND UPPER BOUND RANGES FOR BIDS AND ASKS B Pefomace of the Decompositio Appoach with Optimal Tadig Qatities O fist expeimet is to iestigate how effectiely the decompositio idea woks Fo this, we fist soled the allocatio poblem to optimality sig a diect soltio appoach (that is, withot sig decompositio) ad obtaied the optimal ale of the total spls (call it ) ad the ale of optimm qatity taded (call it Ø ) Usig the ale Ø i o decompositio appoach ad the FPTAS algoithms fo fowad actio ad eese actio poblems, we the obtaied the total spls (call it ) Table III compaes the ales of the total spls obtaied sig the diect soltio appoach ad the decompositio appoach The table clealy shows that the allocatio detemied thogh the decompositio appoach is ey ealy optimal Note that this expeimet ses the optimal tadig qatity i the decompositio appoach ad hece shows how well the FPTAS algoithms i the decompositio appoach appoximate the total spls Expt No M M d 1 1326 30552 30515 2 1647 38766 38766 3 2794 63452 63452 4 4199 96434 96426 TABLE III COMPARISON OF OPTIMAL SOLUTION WITH THE SOLUTION OBTAINED BY DECOMPOSITION APPROACH USING OPTIMAL TRADING QUANTITY C Compaiso of the Heistics Hee, we se the heistics peseted i Sectio IV to detemie the tadig qatity ad se this tadig qatity fo solig the fowad actio ad eese actio poblems sig the FPTAS algoithms Table IV fist compaes the tadig qatities obtaied sig the two heistics, Ø ad Ø, with the optimal tadig qatity Ø (compted sig a diect soltio appoach) The it compaes the total spls
8 ales obtaied sig the decompositio appoach with that compted sig a diect soltio appoach ( ) is the spls ale obtaied sig the decompositio appoach employig the tadig qatity ØŠ (Ø ) is the spls ale obtaied sig a diect soltio appoach thogh a ILOG CPLEX sole I the table, we hae omitted the factioal compoet of the spls ales (by tcatig the ales to the eaest itege) M M M! Expt No "! 1 1349 1314 1326 302 299 305 2 1646 1630 1647 387 386 387 3 2900 2840 2794 631 629 634 4 4057 4032 4199 958 957 964 TABLE IV COMPARISON OF THE HEURISTICS The table clealy shows that the ales of qatity to be taded obtaied sig heistic 1 ad heistic 2 ae qite close to the optimal qatity ad also the spls geeated is qite close to the optimal oe Heistic 1 seems to poide bette estimates compaed to heistic 2 as show by the table This is becase heistic 1 does a exhastie seach o a set of shot listed cadidate ales wheeas heistic 2 may ot always podce the optimal ale (see Sectio IIIB) Howee, o expeimetatio j (ot epoted hee) fo lage ales of ad has show that the tadig qatities estimated by the two heistics ae almost the same I tems of ig time, howee, heistic 2 is mch faste tha heistic 1 (see Sectio IVE fo a discssio o this) The spls ales podced by the decompositio appoach with the help of heistics ae qite close to the optimal spls ales, which shows the efficacy of the heistics D Degee of Stategy Poofess of the Decompositio Appoach ) ad ), espectiely, fo wiig byes ad selles compted by solig the poblem to optimality Table V compaes the total Vickey discot ( 2Þ total Vickey spls ( sig a diect soltio appoach with the ales ( GÞ ad ) whe the poblem is soled sig o decompositio appoach I the decompositio appoach, we sed heistic 1 to detemie the tadig qatity The table clealy shows that the total Vickey discot ad total Vickey spls ales obtaied by the decompositio appoach ae qite close to those obtaied whe the poblem is soled optimally To iestigate this at a moe detailed leel, we compted the idiidal Vickey discots (YÞ ad BÞ ) ad Vickey splses ( ad ) fo the 10 byes ad 10 selles Table VI shows these eslts fo data set 1 I the tables, we hae omitted the factioal compoet of the spls ales (by tcatig the ales to the eaest itege) The eslts i this table also sggest that, i most of the cases, ee at the leel of idiidal byes ad selles, the Vickey discots ad Vickey splses obtaied ae close to the VCG ales as compted by the diect soltio appoach This shows that o appoach is appoximately stategy poof M d M d The eslts peseted i Tables V ad VI fo the case of decompositio appoach se heistic 1 Simila eslts ae obtaied if heistic 2 is sed istead Expt No #%$"&' #%$ #%$'& #($ 1 212 101 207 94 2 298 121 302 133 3 423 294 433 302 4 626 414 629 412 TABLE V COMPARISON OF TOTAL VICKREY DISCOUNT AND TOTAL VICKREY SURPLUS OBTAINED BY DECOMPOSITION APPROACH WITH THOSE OF THE EXACT SOLUTION d M M $ $ d Bye $'& $'& Selle 1 150 162 1 17 08 2 492 496 2 104 123 3 220 228 3 126 116 4 244 234 4 110 90 5 288 275 5 086 0 6 92 74 6 149 147 7 77 56 7 340 332 8 179 178 8 152 152 9 212 198 9 123 134 10 169 166 10 76 69 TABLE VI COMPARISON OF INDIVIDUAL VICKREY DISCOUNTS AND VICKREY E Comptatioal Saigs SURPLUSES FOR EXPT 1 Note that the cleaace of a sigle-item, j4ü mlti-it exchage by the diect method will iole solig NP-had poblems j i the wost case, whee is the mbe of byes ad is the mbe of selles By sig the decompositio appoach, this complexity is edced to that of solig the followig thee poblems: 1) Detemiig a tadig qatity, fo which we hae poided two heistics Befoe applyig these heistics, we fist sot the bids fom jû byesjçf =j ad selles, #Gf # which has wost case ig time Heistic 1 has wost case ig time # of Ý, whee > j ad ad Ý ae as descibed i Sectio jû 41jçf =j ad Heistic #G6 2 has wost case ig time of 2) Reese actio sig a FPTAS algoithm 3) Fowad actio sig a FPTAS algoithm Sice all the aboe steps hae polyomial time complexity, the decompositio appoach will lead to sigificat saigs i ig time, compaed to the diect method Table VII compaes the soltio time of the decompositio j appoach with that of the exact appoach Hee, is the mbe of byes ad is the mbe of selles paticipatig i the exchage ad deote the comptatio time i secods of the diect appoach (sig ILOG CPLEX sole) ad the decompositio appoach, espectiely ad deote the
9 total spls obtaied sig the diect appoach ad the j decompositio appoach, espectiely Fo each choice of ad, the expeimet was codcted 20 times ad the comptatio time epoted is a aeage oe these 20 eplicatios To make the poblem iteestig fom a comptatioal iewpoit, we itodced a additioal bsiess costait i this expeimet, amely, that o sigle bye is to be allocated geate tha 50 pecet of the total qatity taded We sed heistic 2 to compte the tadig qatity i the decompositio appoach The *** ety i the table idicates that the ILOG CPLEX sole was able to sole the istace ee i 3600 secods (1 ho comptig time) The table clealy shows the temedos speedps achieed by the decompositio appoach fo lage poblem istaces Also, the spls ales compted by the decompositio appoach ae qite close to the optimal ales (wheee the optimal ales cold be compted) Notice the o-mootoicity i the seqece 6, j 6, 5, 8, 6 This is a ted obseed fo small ales of j ad Mootoicity is obseed fo highe ales of ad I j fact, The comptatio time stats isig shaply oly afte # d M M d t The eslts peseted i Table VII fo the case of decompositio appoach se heistic 2 Simila eslts ae obtaied if heistic 1 is sed istead m #) 700 700 72 6 127e04 127e04 800 800 78 6 146e04 144e04 900 900 132 5 163e04 163e04 1000 1000 167 8 181e04 181e04 1100 1100 224 6 201e04 200e04 1200 1200 *** 9 *** 116e04 1300 1300 *** 21 *** 234e04 1400 1400 *** 27 *** 262e04 1500 1500 *** 30 *** 267e04 5000 5000 *** 164 *** 926e04 TABLE VII COMPARISON OF COMPUTATION TIME (IN SECONDS) OF DECOMPOSITION APPROACH (USING HEURISTIC 2) WITH THAT OF EXACT APPROACH V SUMMARY AND FUTURE WORK I this pape, we hae sed a simple, atal method of decomposig a mlti-it, sigle-item exchage poblem ito fowad actio ad eese actio poblems We hae peseted two heistics fo detemiig the qatity to be taded which is eqied fo solig the fowad actio ad eese actio poblems idepedetly We hae sed kow flly polyomial time appoximate algoithms fo solig these idiidal poblems O specific cotibtios i this pape ae as follows: % Establishig that the decompositio appoach is a attactie appoach to clea sigle-item, mlti-it exchages with meical expeimetatio % Polyomial time heistics fo detemiig tadig qatity to be sed i the decompositio appoach Thee is plety of scope fo fthe wok i seeal diectios (1) We hae looked at sigle-item exchages hee The ext immediate poblem wold be to look at mlti-it combiatoial exchages sig the decompositio based appoach (2) The stategy poofess popeties of the mechaism whe the decompositio appoach is sed eeds to be fomally iestigated (3) Fomal eo bods o the ale of the tadig qatity whe heistic 1 ad heistic 2 ae sed eed to be iestigated (4) Fomal eo bods o the ale of the objectie fctio whe the decompositio appoach i cojctio with the heistics is sed also eed iestigatio REFERENCES [1] J Kalagaam ad D Pakes, Actios, biddig, ad exchage desig, i Hadbook of Spply Chai Aalysis i the E-Bsiess Ea, D Simchi- Lei, D W, ad Z She, Eds Klwe Academic Pblishes, 2005 [2] V Kisha, Actio Theoy Academic Pess, 2002 [3] P Milgom, Pttig Actio Theoy to Wok Cambidge Uiesity Pess, 2004 [4] P Klempee, Actios: Theoy ad Pactice Olie Book, wwwpalklempeeog, 2004 [5] R McAfee ad J McMilla, Actios ad biddig, Joal of Ecoomic Liteate, ol 25, pp 699 738, 1987 [6] P Milogom, Actios ad biddig: a pime, Joal of Ecoomic Pespecties, ol 3, o 3, pp 3 22, 1989 [7] J Kagel, Actios: A sey of expeimetal eseach, i The Hadbook of Expeimetal Ecoomics, J Kagel ad A Roth, Eds Piceto Uiesity Pess, Piceto, 1995, pp 501 587 [8] E Wolfstette, Actios: A itodctio, Ecoomic Seys, ol 10, pp 367 421, 1996 [9] P Klempee, Actio theoy: a gide to the liteate, Joal of Ecoomic Seys, pp 227 286, 1999 [10] M Heschlag ad R Zwick, Iteet actios-a popla ad pofessioal liteate eiew, Electoic Commece, ol 1, o 2, pp 161 186, 2000 [11] S Matthews, A techical pime o actio theoy, Nothweste Uiesity, Tech Rep, 1995 [12] M Hhs ad J M Vidal, Olie actios, IEEE Iteet Comptig, ol 3, o 3, pp 103 105, 1999 [13] Y Tg, R Gopal, ad A Whisto, Mltiple olie actios, IEEE Compte, ol 36, o 2, pp 100 102, 2003 [14] J Hatley, M lae, ad Y Hog, A exploatio of the adoptio of e-actios i spply maagemet, IEEE Tasactios o Egieeig Maagemet, ol 51, o 2, pp 153 161, 2004 [15] A Kothai, D Pakes, ad S Si, Appoximately stategy poof ad tactable mlti-it actios, i Poceedigs of ACM Cofeece o Electoic Commece (EC-03), 2003 [16] M Eso, S Ghosh, J Kalagaam, ad L Ladayi, Bid ealatio i pocemet actios with piece-wise liea spply ces, IBM Reseach, Yoktow Heights, NJ, USA, Reseach Repot RC 22219, 2001 [17] A Daepot ad J Kalagaam, Pice egotiatios fo diect pocemet, IBM Reseach, Yoktow Heights, NJ, USA, Reseach Repot RC 22078, 2001 [18] G Hohe, J Rich, E Ng, G Reid, A Daepot, J R Kalagaam, S Lee, ad C A, Combiatoial ad qatity discot pocemet actios poide beefits to mas, icopoated ad to its spplies, Itefaces, ol 33, o 1, pp 23 35, 2003 [19] V Dag ad N Jeigs, Optimal cleaig algoithms fo mlti-it sigle-item ad mlti-it combiatoial actios with demad-spply fctio biddig, i Poceedigs of the Fifth Iteatioal Cofeece o Electoic Commece, Pittsbgh, USA, 2003, pp 25 30 [20] J Kalagaam, A Daepot, ad H Lee, Comptatioal aspects of cleaig cotios call doble actios with assigmet costaits ad idiisible demad, IBM Reseach, Yoktow Heights, NJ, USA, Reseach Repot RC 21660 (97613), 2000 [21] A Dailiaas, J Saiamesh, V Gottemkkala, ad A Jhiga, Pofitdie matchig i e-maketplaces: Tadig composable commodities, IBM Reseach, Yoktow Heights, NJ, USA, Reseach Repot, 1999 [22] T Sadholm ad S Si, Optimal cleaig of spply-demad ces, i Poceedigs of the 13th Aal Iteatioal Symposim o Algoithms ad Comptatio (ISAAC), Vacoe, Caada, 2002 [23] M Sattethwaite ad S Williams, The Bayesia theoy of the k- doble actio, i The Doble Actio Maket: Istittios, Theoies ad Eidece, Sata Fe Istitte Stdies i the Scieces of Complexity Peses Pblishig, Cambidge, MA, 1993, pp 99 123