A Partcpato Icetve Maret Mechasm for Allocatg Heterogeeous Networ Servces Juog-S Lee ad Boleslaw K. Szymas * Noa Research Ceter, Palo Alto, CA 94304 USA * Resselaer Polytechc Isttute, Troy, NY 280, USA Eml: juog-s.lee@oa.com, * szymas@cs.rp.edu Abstract Ths paper studes a aucto based allocato of etwor resources for short-term cotracts for heterogeeous etwor servces. The combatoral wer selecto yelds the optmal resources allocato a sgle-roud aucto for heterogeeous resources. However, the recurrg ature of aucto for etwor servces causes least wealthy bdders to ext the aucto as they persstetly lose uder the tradtoal combatoral wer selecto that focuses oly o reveue maxmzato. Such exts decrease prce competto ad may cause a collapse of the sellg prces ad reveues of etwor servce provders. We troduce ad evaluate a ovel wer selecto strategy for auctog of heterogeeous etwor servces. The proposed strategy prevets collapse of the sellg prces ad the auctoeer reveues, stablzes aucto maret, ad ehaces socal welfare by allowg larger subset of users to become occasoal wers of aucto rouds tha the tradtoal combatoral wer selecto does. I. INTRODUCTION Approprate prcg mechasms ecourage customers to choose servce levels adequate to ther eeds, ad acheve effcet etwor resource allocato Qualty of Servce (QoS eabled etwors. Hece, they are regarded as a effcet soluto to the cogesto cotrol, servce admsso cotrol, far allocato of etwor resources, ad reveue maxmzato. For these reasos, they costtute a essetal compoet of dyamcally adaptable servce maagemet framewors for the sale ad delvery of a wde rage of etwor servces. Several prcg mechasms for sellg multple classes of etwor servces or allocatg shared etwor resources QoS eabled etwors have bee proposed [-7]. I may exstg etwor servce marets, fxed prcg (.e., flat rate prcg or statc tme-dfferetal prcg mechasms have bee used because of ther smplcty. However, the flat rate prcg caot optmze reveue of etwor servce provder because customer s demad dose ot follow a step fucto, but rather gradually shft from o- to off-pea. Accordgly, uder-utlzato of etwor resources arses whe demad s low ad uder-prcg happes whe demad s hgh []. Dyamc prcg mechasms that adapt to cotuously chagg etwor codtos are more effcet maagg etwor resources. Addtoally, such prcg mechasms, prce tself becomes a mportat sgal for etwor maagemet. I such dyamc prcg evromets, however, the servce provder s prce decso ad customer s budget plag are dffcult. A aucto ca mtgate such complextes sce prces auctos emerge a decetralzed way based o the customer s wllgess to pay. Addtoally, auctos are easy to uderstad, ad ca supports automatc egotato process [9]. Geerally, aucto for etwor servces should be regarded as recurrg because etwor servce provders allocate etwor resource repeatedly to satsfy recurret requests of customers []. I a sgle-roud aucto for heterogeeous etwor servces, the combatoral wer selecto of the Geeralzed Vcrey Aucto (GVA mechasm maxmzes reveue of the etwor servce provder by selectg the combato of wers that maxmzes the auctoeer s reveue [8]. However, applyg the tradtoal GVA mechasm to a recurrg aucto for heterogeeous etwor servces results a evtable bdder drop problem that s caused by paradox of cetve compatble mechasm recurrg aucto []. Such dropped bdders decrease the log-term demad for etwor servces, ad, cosequetly, lower the etwor servce provder s reveue. I ths paper, we propose a ew aucto based etwor resource prcg mechasm for QoS eabled heterogeeous etwor servces. Ths mechasm focuses o stablzg reveue of etwor servce provder. The ext secto aalyzes aucto marets for heterogeeous etwor servces ad descrbes the tradtoal GVA mechasm for heterogeeous etwor servces. I secto III, observed challegg problems are dscussed. I Secto IV, the proposed ew aucto based etwor resource allocato mechasm s preseted ad verfed by varous smulato expermets descrbed secto V. Fally, the paper cocludes wth a summary of ts cotrbutos ad our future wor gve secto VI. II. GVA FOR HETEROGENEOUS NETWORK SERVICES QoS eabled heterogeeous etwor servces ca be regarded as a aucto maret whch there s oe etwor servce provder actg as a auctoeer ad may customers partcpatg as bdders. The traded etwor servces are varous elastc etwor applcatos, such as real-tme voce ad vdeo applcatos that requre dfferet fxed amout of etwor resources to acheve desred qualty of servce (QoS. Hece, such a maret, bdders request dfferet amout of etwor resources for ther etwor servces. Aucto for such etwor servces s recurrg, because, from the etwor servce provder s pot of vew, a allocato of etwor
resources eeded for a servce s made for a specfc tme oly. Oce the servce s completed, these resources become free ad the etwor servce provder eeds to offer them to the customers aga. From the bdder pot of vew, etwor servces are requested repeatedly for specfc tme tervals. The resources such as badwdth eeded for etwor servces caot be stored warehouse for future sale, ad leavg them uused decreases ther utlzato, so they are pershable []. I cocluso, QoS eabled heterogeeous etwor servce maret requres a recurrg aucto whch bdders (.e., customers request dfferet amout of tme-sestve, pershable etwor resources recurretly for specfc tme tervals. I such a recurrg aucto, bdders demad for etwor resources, bds, ad the umber of avalable etwor resource uts may be dfferet each aucto roud. The combatoral wer selecto s the optmal strategy for etwor resource allocato a sgle roud. It eumerates all possble combatos of bdders, that we wll deote as C, where =,...,2, because 2 combatos of bdders are possble. We wll also deote the set of dexes of all feasble combatos as F. We call combato C feasble f the etwor resources that are requested by bdders C are avalable, that s f: r j R t, ( j C where r j deotes the vector of etwor resource uts requested by bdder j, ad R t deotes the vector of avalable etwor resource uts at tme t. The etwor servce provder selects the combato of bdders C m that maxmzes the sum of products of bds ad requested umbers of etwor resources (.e., maxmzes the resultg reveue amog all feasble combatos of bdders. Hece, the wer selecto the GVA mechasm ca be defed as: max b r (2 C To defe the prce of etwor resources for the wers selected GVA, we deote by x * the vector of wers (f bdder s a wer, x * =, otherwse, x * = 0. Let x * deote the vector of the optmal combato of bdders wthout bdder partcpatg the aucto. The prce of etwor resources for wer s computed by deductg the sum of products of bds ad requested umbers of etwor resources for all other bdders (except bdder x * from such a sum for all bdders x * [8]. Hece, the prce G( b of etwor resources for wer s computed as: G( b = = b r x * = b r x * (3 Based o the prcg rule of Eq. (3, the GVA mechasm s cetve compatble because truthful bd (.e., a bd represetg the true valuato of the resource by the bdder maxmzes the bdder s expected utlty [8]. The followg example llustrates the descrbed combatoral wer selecto ad prcg of the GVA mechasm. Example : There are four bdders B, B2, B3 ad B4, ad oe etwor servce provder. There are three etwor resource uts avalable. B, B2, B3 ad B4 request, 2, ad 2 etwor resource uts for ther desred etwor servces wth bds (.e., true valuatos of $3, $5, $6 ad $8 for a etwor resource utespectvely. I the GVA mechasm, bdder B3 ad B4 are selected as wers. Based o prcg rule of Eq. (3, bdder B3 pays ($9 - $6 / = $3 for each etwor resource ut whle bdder B4 pays ($6 - $6 / 2 = $5 per ut. III. CHALLENGES ARISING IN TRADITIONAL GVA The combatoral wer selecto of GVA has desrable features such as the optmal dyamc prcg ad egotato mechasms for effcet etwor resource allocato. Yet applyg t short-term cotract marets for heterogeeous etwor servces ca cause maret prce collapse because of the recurrg ature of etwor resource allocatos ad pershable ature of etwor resources. I a recurret aucto, bdders lear from prevous aucto results, ad try to adapt to the maret codtos. Hece, the combatoral wer selecto s optmal oly for a sgle roud of aucto as t focuses solely o reveue maxmzato (.e., t oly cosders curret bds durg resource allocato. As the result, t ca cause starvato for the etwor resources amog the least wealthy bdders (.e., bdders wth low true valuatos of resources the recurrg aucto because the true valuatos of bdders defe ther maxmum bds. Sce the wealth of bdders s dstrbuted uevely amog the bdders, such wer selecto strategy wll reward oly the bdders wth the hghest wealth (.e., the hghest true valuatos. The cetve compatblty of GVA mechasm maes t easy for a bdder who persstetly lost the prevous aucto rouds to coclude that hs true valuato s ot large eough to ever become a wer oce the wers of the aucto starts to repeat (each wer of a aucto roud may sp several subsequet rouds sce her demad for servces may be satsfed for a whle. Whe ths happes, there s o cetve for persstet losers of the aucto to partcpate the future aucto rouds. Cosequetly, they drop from the aucto ad fd other ways to satsfy ther demad for the desred etwor servces. Such a drop the umber of bdders decreases the prce competto the aucto ad may ultmately result reveue collapse of the etwor servce provder. I short, GVA mechasm, the bdders who do ot w for several rouds may drop out of the aucto after they coclude that t s mpossble for them to w ths aucto maret. I the log ru, oly bdders who belog to oe of the optmal combatos of subsequet aucto rouds are left the aucto maret. The, for at least oe
aucto roud, the left ad rght terms of Eq. (3 wll become equal, that s * * b r x = b r x for wth x * = (4 = = Hece, based o the prcg of GVA gve by Eq. (3, the resultg prce of etwor resource for each selected wer becomes zero. Wth all partcpats becomg wers sooer or later, the aucto s o loger cetve compatble, because lowerg the bd does ot mae the bdder elgble for wg, but merely delays hs wg roud. Hece, the bdders may decrease ther bds below ther true valuatos, thereby trggerg collapse of reveues for the etwor servce provder. We call ths pheomeo a paradox of the cetve compatble mechasm recurrg aucto because by achevg the goal of motvatg the bdders to bd ther true valuatos, the mechasm, whe appled to a recurrg aucto, vtes the maret collapse after oly several rouds of aucto. Our prevous research demostrated ad evaluated the reveue collapse caused by the bdder drop problem o-cetve compatble recurrg aucto mechasms [-2]. The followg example llustrates t the cetve compatble recurrg GVA aucto. Let s cosder Example of secto II aga. Example 2: Bdders B3 ad B4 are selected as wers ad pay $3 ad $5 for a etwor resource utespectvely. If those bdders partcpate every aucto roud, bdders B ad B2 have o cetve to partcpate the aucto ad drop out of t. Whe bdders B3 ad B4 are selected as wers after such a drop, ther paymets become $6 $6 = $0 for bdder B3 ad ($6 $6 / 2 = $0 for bdder B4. Therefore, the reveue of etwor servce provder collapses to zero. More terestg case arses f all bdders sp a roud after each w. The, bdders B3 ad B4 are wg odd umbered aucto rouds, payg $3 ad $5 for a etwor resource ut, respectvely, whle bdders B ad B2 w eve umbered aucto rouds, but pay $0 for allocated resources. All bdders w regardless of the value of ther bds, so they may decrease ther bds. For example, bdders B3 ad B4 may woder f payg $3 or $5 per ut of resource s worth avodg watg a aucto roud for the w. As the result, bds wll reflect the true valuato of oe-roud delay of access to resources stead of the true valuato of ever accessg the resources ad the reveue of the servce provder wll be accordgly lowereflectg ths dfferece true valuatos. IV. NETWORK SERVICE ALLOCATION USING PI-GVA Applyg a recurret aucto to short-term cotract-based heterogeeous etwor servces requres resolvg bdder drop problem to prevet maret collapse. To acheve ths goal, we troduce the Partcpato Icetve Geeralzed Vcrey Aucto (PI-GVA wth oe tme sealed bddg. Our addtoal desg goals clude preservg cetve compatblty ad low egotato cost of the GVA mechasm. A. Networ Resource Allocato Strategy To prevet bdder drop problem from arsg, the PI-GVA mechasm rewards bdders for partcpato a roud of recurret aucto through the wg score S r ( b [2] that s used, stead of a bd, for selectg wers. The bddg score of bdder roud r s defed as: b =, (5 S r ( b p w where b deotes the average bd of bdder utl roud r, p,r represets the cumulatve umber of rouds whch the bdder partcpated up to ad cludg roud r, w,r stads for the cumulatve umber of ws utl roud r, ad s a costat that cotrols the effect of bd values o the wg score. Sce ( b p / represets the expected umber of ws, the wg score S r ( b measures the dfferece betwee the expected ad real umbers of ws at roud r for bdder. The hgher the wg score s, the more below bdder s expectatos wgs are ad therefore the hgher the probablty of hm droppg out of the aucto s. I each aucto roud, the PI-GVA mechasm selects as wers the combato of bdders for whch the sum of wg scores of all selected bdders s the largest. Hece, the etwor resource allocato strategy (.e., the wers selecto strategy usg PI-GVA s b max ( b = max ( p w (6 F F C C I each aucto roud, the feasble etwor resource allocato s restrcted by the resource costrat defed by Iequalty ( ad the wg score costrat defed as follows: b ( b > for C (7 The wg score costrat requres that the wg score of each wer s hgher tha zero; a bdder eeds to ear eough expected ws before the real w ca happe. By Eq. (5, the partcpato of a loser the last aucto roud s rewarded drectly by creasg her wg score for the curret ad future aucto rouds. The creased wg score mproves the bdder s chaces to w the future aucto rouds. Therefore, the PI-GVA mechasm cotrols bdder drop problem by ecouragg bdders partcpato future aucto rouds. Sce the cumulatve average bd b s used the Eq. (5, decreasg a bd durg a aucto roud decreases the expected umber of ws, eve f the bdder partcpates every aucto roud, thereby decreasg hs wg score. Hece, the PI-GVA mechasm ecourages the bdders to eter ever hgher or at least the same bds durg recurrg aucto.
The coeffcet cotrols effect of the average bd o the wg score. If s creased, the effect of bd value s dmshed, mprovg the chace of wg by the bdders wth lower bds. Reversely, f s decreased, the effect of bd value s creased ad the chace of wg wth lower bds s decreased. Therefore, the optmal strategy for decdg the value of coeffcet depeds o etwor servce provder s goals. I ths paper, based o varous expermetato results, we set equal to the rato of tal average bds of all bdders to the umber of wers (wth ths value of, the sum of all wg scores remas costat f all bdders partcpate all rouds wth uchaged bds; ths selecto maes t also easy to satsfy the wg score costrat defed by Iequalty (7. B. Prcg Networ Resources To descrbe the prcg rule of the proposed PI-GVA mechasm, we deote by x the vector of wers defed by the PI-GVA wer selecto strategy expressed Eq. (6 (f bdder s a wer, x =, otherwse, x = 0, whle x deotes the vector of such wers whe bdder s removed from the aucto roud. The prcg rule of the PI- GVA mechasm s a modfcato of the tradtoal prcg rule of GVA. The prce s establshed a two-step procedure. The frst step s to compute the paymet wg score I ( b of the selected wer wth the average bd b, accordg to the followg equato: (8 I ( b = ( b r x ( b r x = Hece, the paymet wg score of wer s computed by deductg the sum of products of wg scores ad requested umbers of resources of all wers, except bdder, x from such sum for all wers x. From the paymet wg score of the selected wer defed Eq. (8 ad the wg score fucto defed Eq. (5, the paymet G ( b of wer for a etwor resource ut s computed as: G(b = I (b b (9 r S(b Ths prcg rule guaratees that the paymet of each wer s lower tha hs bd. = C. Optmal Strateges of Bdders The bdder s optmal strategy PI-GVA mechasm ca be defed from two perspectves: the bd value ad the partcpato level. I aucto for heterogeeous etwor servces, each bdder tres to choose the bd that maxmzes the utlty fucto U(b defed as: U(b = (t G(b q(b, (0 where t deotes the true valuato of a resource ut by bdder, G(b s the prce pad for each etwor resource ut ad q(b stads for the probablty of wg aucto roud r. Based o the prcg rule defed by Eq. (9, the utlty fucto of Eq. (0 ca be rewrtte as: I( b b U ( b = ( t G( b q( b = ( t q( b r ( b As show the Appedx, creasg the bd creases the proft factor (t G(b of the above utlty fucto. Moreover, creasg the bd ether creases or matas the probablty of wg q(b. Therefore, bddg the true valuato maxmzes each bdder s expected utlty fucto U(b because the true valuato s the upper boud of the bd of each bdder. I terms of partcpato, the proft factor (t G(b s a creasg fucto whle the probablty of wg q(b s a o-decreasg fucto of the umber of rouds whch the bdder partcpated. Hece, creasg bdder s partcpato aucto rouds creases the expected utlty of ths bdder. Therefore, the domat bdder s strategy uder PI-GVA mechasm s to partcpate as may rouds as eeded to w ad to bd each of those rouds the true valuato of the etwor servce ut. The detaled proof of the optmalty of ths strategy s gve the Appedx. V. EXPERIMENTATION To evaluate the proposed PI-GVA mechasm, we performed smulato-based expermets. I those expermets, we compared the followg two aucto mechasms for short-term cotract maret for heterogeeous etwor servces: ( the Tradtoal Geeralzed Vcrey Aucto (TGVA descrbed secto II as GVA; (2 the Partcpato Icetve Geeralzed Vcrey Aucto (PI-GVA troduced secto IV. A. Expermetato Setup There are 5 bdders who are also customers ad a auctoeer who s also the etwor servce provder. I each aucto roud, 0 etwor resource uts are traded for short-term use by customers requestg etwor servces. Each bdder requests dfferet umber of etwor resource uts to satsfy her desred qualty of servce for heterogeeous etwor servces. Hece, each bdder requests the desred umber of uts of etwor resources ad bds how much he s wllg to pay for each requested ut of etwor resource. The desred umber of uts s uformly dstrbuted betwee [, 3]. Ths umber remas costats over the etre smulato. I addto to the perceved trsc value of the traded etwor resources, the wealth of each bdder lmts her wllgess to pay ad mpacts her true valuato of the uts of etwor resource. For smplcty, we cosder oly the dstrbuto of the customer s true valuatos here. We use three stadard dstrbutos of true valuatos wth mea 5: the expoetal dstrbuto, the uform dstrbuto over the rage [, 9], ad the Gaussa dstrbuto. Hece, wthout loss of geeralty, we ca assume that the average prce of etwor resource ut s 5. Based o
Fg. The average sellg prce of etwor resource ut these true value dstrbutos, each bdder bds her true valuato uder each aucto mechasm to maxmze her expected utlty the recurrg aucto sce the two aucto mechasms beg compared are cetve compatble. To model bdder drop behavor, we troduce a cocept of the Tolerace to Cosecutve Losses, abbrevated as TCL. It deotes the maxmum umber of cosecutve losses that a customer ca tolerate before extg a aucto [7]. Ths oto captures the bdders lmted tolerace to loses before they ext the aucto cocludg that ther true valuatos prevet them from ever becomg a wer ths aucto. Hece, f the cosecutve losses exceed the bdder s TCL, the bdder exts the aucto. We assume, that the bdders who exted, ever retur aga to the curret aucto maret. TCL of each customer s uformly dstrbuted over the rage of [2, 6]. Based o these settgs, each smulato s executed for 2000 aucto rouds. At the ed, we measure for each mechasm the reveue of etwor servce provder ad the bdder retag ablty as measured by the umber of remag actve bdders. To estmate reveues of etwor servce provder, we measure the average sellg prce of etwor resource ut at the ed of every 50 rouds over the etre recurrg aucto. To compute the bdder retag ablty, we measure the percetage of users dropped each compared mechasm at the ed of the recurrg aucto. B. Expermetato Results As show the Fg., all three dstrbutos of bdders wealth (.e., the dstrbutos of true valuatos, TGVA caot prevet collapse of the sellg prce for a etwor resource ut as a result of bdders droppg out of the aucto. The combatoral wer selecto strategy of TGVA focuses o bds oly so t excludes the lower true valuato bdders from ever becomg wers. Cosequetly, those bdders drop out of the aucto after TCL rouds, decreasg prce competto. Fally, oly bdders who at least occasoally become wers rema the aucto ad the resultg sellg prces drop for those bdders to zero based o the prcg rule of TGVA. Ths pheomeo llustrates the paradox of cetve compatble mechasm recurrg aucto. Therefore TGVA, the bdder drop s the sole cause of reveue collapse the recurrg aucto. I cotrast to TGVA, the combatoral wer selecto based o wg score of Eq. (5 the proposed PI-GVA mechasm allocates etwor resources effcetly ad yet prevets bdders drop out of the aucto, thereby matag prce competto ad stablzg the sellg prce of etwor resource uts at about $5. Table. Percetage of dropped users Uform Expoetal Gaussa TGVA 73.33% 73.33% 66.67% PI-GVA 40.00% 46.67% 40.00% I terms of actve bdder retag level, Table shows that the proposed PI-GVA mechasm retas over 50% of the tally actve user throughout the etre recurrg aucto. Oly the bdders wth very low true valuatos are preveted from ever wg by the wer selecto based o wg score of Eq. (5. Yet, TGVA, about 70% of bdders drop out of the aucto. Hece, the proposed PI-GVA mechasm acheves hgher socal welfare because more users beeft from the etwor resource allocatos. VI. CONCLUSIONS AND FUTURE WORKS The combatoral wer selecto of the GVA mechasm allocates resources to bdders optmally for a sgle roud of a aucto. However, the recurrg ature of aucto for etwor resource allocato may cause bdder drop resultg from the paradox of cetve compatble mechasm recurrg aucto. Therefore, desgg aucto based dyamc prcg mechasms for heterogeeous etwor servces, the addtoal pheomeo, bdder droppg, should be cosdered. We troduced the PI-GVA mechasm that provdes a ovel combatoral wer selecto strategy. The proposed strategy prevets bdder drop by provdg cetve for bdder s partcpato each aucto roud whch results eough bdders remag the aucto to eep the prces stable the maret. I our future wor, we wll attempt to desg more effcet bdder drop cotrol algorthms ad wll exted our study to the case whch dropped bdders retur to the aucto. APPENDIX Lemma : The proft factor (t G (b of the utlty fucto of Eq. (0 s a creasg fucto of bd value b. Proof: The proft factor (t G(b ca be rewrtte as (t (I (b b r S(b. Note that the paymet wg
score I (b of bdder does ot depedet o bd b (see Eq. (9 ad the requred umber of etwor resource uts r ad true valuato t ca be regarded as costats. Hece, the proft factor ca be flueced oly by a fracto ( b ( b. Hece, to prove Lemma, we eed to show that creasg the bd decreases ths fracto. Ideed, we have: b ( b α w (A- = + > ( b + α ( b + α b ( b + α ( p, r w where bd cremet α > 0. Iequalty (A- s true because b, α,, ad w,r are always larger tha zero, ad the curret wg score S r ( b s also larger tha zero thas to the wg score costra of Iequalty (7. Therefore, creasg the bd creases the proft factor (t G(b. Lemma 2: The probablty of wg q(b the utlty fucto of bdder s a o-decreasg fucto of bd b. Proof: I aucto roud r, the wg score coeffcet, the cumulatve umber of ws w,r, ad the umber of rouds whch bdder partcpated p,r ca be regarded as costats. Hece, creasg bd b creases the wg score. The PI- GVA mechasm ras bdders the decreasg order of ther wg scores gve by Eq. (5. Hece, creasg wg score of a bdder wll ether mprove or mata the ra of ths bdder. Sce the PI-GVA combatoral wer selecto strategy of Eq. (6 maxmzes the sum of wg scores of wers, the followg three cases are possble terms of probablty of wg whe a bdder crease hs bd. Case ( Ra of bdder s mproved ad hs probablty of wg s creased (e.g., the bdder ow outras aother bdder the optmal combato wth the same resource request, so the former replaces the latter as the wer. Case (2 Ra of bdder s mproved but there s o chage hs probablty of wg (e.g., the bdder was elgble to be a part of optmal combato because of resource costrats, so hs mproved ra does ot gve hm ay advatage. Case (3 Ra of bdder s uchaged, ad there s o chage hs probablty of wg. I all of three cases, the probablty of wg caot decrease wheever a bdder creases hs bd. Addtoally, the creased wg score wll carry over to future aucto rouds. Hece eve f the creased wg score does ot chage probablty of wg by the bdder the curret aucto roud, t stll may crease the probablty of wg the log ru. Based o Lemmas ad 2, we coclude that by creasg hs bd, bdder creases hs expected utltyu(b. Addtoally, each bdder s bd s lmted by hs true valuato. Hece, bddg each bdder s true valuato maxmzes the bdder s expected utlty the proposed PI-GVA mechasm. I short, PI-GVA s a cetve compatble mechasm. From the partcpato pot of vew, creasg the umber of rouds whch bdder partcpates decreases the fracto (b S(b of the proft factor (t G(b. Hece, t creases the resultg proft factor. Addtoally, creasg the umber of rouds whch bdder partcpates creases the wg score thas to partcpato cetve defed Eq. (5. As show Lemma 2 proof, creasg the wg score creases or matas the probablty of wg by the bdder. Hece, the probablty q(b of wg by bdder s odecreasg fucto of umber of rouds p, whch bdder r partcpates. Therefore, creasg umber of rouds whch the bdder partcpates creases hs expected utlty U(b. I cocluso, the bdder s optmal strategy the PI-GVA mechasm s to bd hs true valuato ad to partcpate as may aucto rouds as possble. ACKNOWLEDGMENT Ths research s cotug through partcpato of B.K. Szymas the Iteratoal Techology Allace sposored by the U.S. Army Research Laboratory ad the U.K. Mstry of Defece. REFERENCES [] S. Ju ad V. Prav, Smart Pay Access Cotrol va Icetve Algmet, IEEE J. Selected Areas Commucatos, 24(5:05-060, Jue 2006. [2] W. X ad S. Heg, Prcg Networ Resource for Adaptve Applcatos, IEEE/ACM Trasactos o Networg, 4(3:506-59, Jue 2006. [3] H. Ara, T. Eva ad W. Tom, A etwor prcg game for selfsh traffc, Proc. 24 th Aual ACM Symp. Prcples of Dstrbuted Computg, pp. 284-29, May 2005. [4] S. Shaotta ad R. at, Ecoomcs of etwor prcg wth multple ISPs, Proc. IEEE INFOCOM 2005, Mam, FL, pp. 84-94, March 2005. [5] C. Courcoubets ad R. Weber, Prcg Commucato Networs Ecoomcs, Techology ad Modellg. Wley, 2003. [6] A. Surea ad P.R. Wurma, Applyg the geeralzed Vcrey aucto to prcg relable multcasts, Proc. It. Worshop Iteret Chargg ad QoS Techologes (ICQT, pp. 283 292, Oct. 2002. [7] L.A. DaSlva, Prcg of QoS-Eabled Networs: A Survey, IEEE Commucatos Surveys & Tutorals, 3(2, 2000. [8] M. Bchler, The Future of e-marets: Multdmesoal Maret Mechasm. Cambrdge Uversty Press, 200. [9] R. McAfee ad P.J. McMlla, Aucto ad Bddg, J. Ecoomc Lterature, 25: 699 738, 997. [0] J.G. Rley ad W.F. Samuelso, Optmal Aucto, The Amerca Ecoomc Revew, 7(3:38 392, Ju. 98. [] J. S. Lee ad B. K. Szymas, A Novel Aucto Mechasm for Sellg Tme-Sestve E-Servces, Proc. 7 th It. IEEE Cof. E-Commerce Techology (CEC, Much, pp. 75 82, 2005. [2] J.-S. Lee ad B.K. Szymas, "Auctos as a Dyamc Prcg Mechasm for e-servces", Servce Eterprse Itegrato, Cheg Hsu (ed., Kluwer, New Yor, pp. 3-56, 2006.