Approxmatly Maxmzng Effcncy and Rvnu n olyhdral Envronmnts Thành Nguyn Cntr for Appld Mathmatcs Cornll Unvrsty Ithaca, NY, USA. thanh@cs.cornll.du Éva Tardos Computr Scnc Dpartmnt Cornll Unvrsty Ithaca, NY, USA. va@cs.cornll.du ABSTRACT W consdr a rsourc allocaton gam n polyhdral nvronmnts. olyhdral nvronmnts modl a wd rang of problms, ncludng bandwdth sharng, som modls of Adwords auctons and gnral rsourc allocaton. W xtnd th far sharng mchansm for such rsourc allocaton gams. W show that our mchansm smultanously crats approxmatly ffcnt allocatons and approxmatly maxmzs rvnu. W also dvlop a nw approach for analyzng gams of ths typs. At th cor of ths approach s th rlaton btwn th condton for Nash qulbrums of th gam and th dual of a crtan lnar program. Catgors and Subjct Dscrptors F.2 [Analyss of algorthms and problm complxty] Gnral Trms Thory, Algorthm, Economcs. Kywords Gam Thory, Mchansm Dsgn, Effcncy, Rvnu. 1. INTRODUCTION W study a rsourc allocaton gam wth n playrs n polyhdral nvronmnts. Th goal of th gam s to dtrmn a ral valud outcom x 0 for ach playr, whch w thn of as th playr s lvl of actvty or allocaton. Each playr s ntrstd n maxmzng hs valu. Th smplst polyhdral nvronmnt s th sharng of a sngl rsourc, whr x s th amount of th rsourc allocatd Supportd n part by ITR grant CCR-0325453. Supportd n part by ITR grant CCR-0325453, and ONR grant N00014-98-1-0589. rmsson to ma dgtal or hard cops of all or part of ths wor for prsonal or classroom us s grantd wthout f provdd that cops ar not mad or dstrbutd for proft or commrcal advantag and that cops bar ths notc and th full ctaton on th frst pag. To copy othrws, to rpublsh, to post on srvrs or to rdstrbut to lsts, rqurs pror spcfc prmsson and/or a f. EC 07, Jun 13 16, 2007, San Dgo, Calforna, USA. Copyrght 2007 ACM 978-1-59593-653-0/07/0006...$5.00. to playr. If w hav 1 unt of th rsourc, w rqur that th allocaton satsfs x 1. W consdr gams on a gnral class of polyhdrals. Such nvronmnts modl a wd varty of gams, ncludng bandwdth sharng gams [8, 6], som form of Adwords auctons [3, 12], and gnral rsourc allocaton gams, whr fasbl allocatons ar constrand by a nonngatv constrant matrx Ax u, wth ach row of th matrx A corrspondng to a dffrnt rsourc, or just a dffrnt constrant on th allocaton. Bandwdth allocaton s naturally modld va a polyhdral gam: assum ach playr s assocatd wth a path and ach dg has a capacty u. layr s ntrstd n rsrvng capacty x along th path. Th rsourc constrants ar now : x u, corrspondng to th capacty constrant of th dgs. A mor gnral rsourc allocaton gam can hav constrants for dffrnt rsourcs, and ach rsourc corrsponds to a row n a nonngatv constrant matrx Ax u. W wll show that polyhdral nvronmnts can also b usd as a modl of crtan Adwords auctons, whr x s th xpctd numbr of clcs allocatd to bddr. In an aucton mchansm, buyrs submt bds and basd on ths bds th mchansm dcds on th allocaton valus x and th paymnts w, th amount playr has to pay for th allocaton x h rcvs. W assum that ach usr has a concav utlty functon U (x ) for th amount x h s rcvng, and assum that playrs hav lnar and sparabl utlty for mony, so ach playr s ntrstd n maxmzng th total utlty xprssd as U (x ) w. Thr ar two natural, and oftn comptng goals for aucton mchansms: to maxmz socal wlfar and to maxmz th auctonr s rvnu. For a gvn allocaton vctor x w say that th socal wlfar s th total U(x), th sum of all usr s utlts for th allocaton. An altrnat goal for a mchansms s to maxmz th rvnu, whch s th auctonr s ncom. In our contxt, f th mchansm collcts paymnts w 1,..., w n thn th rvnu s w. Our quston s ths: Can w dsgn a smpl mchansm that achvs both goals or at last gvs an approxmat guarant for both objctv functons smultanously? In ths papr w nvstgat ths quston n th contxt of a far sharng mchansm n a gnral class of gams whch w call allocaton gams n polyhdral nvronmnts. In ordr to obtan good bounds on both ffcncy and socal wlfar, w wll nd a fw assumptons, whch w wll xplan mor formally n Scton 4.
Thr ar many dntcal or smlar playrs. Th typ of a playr s dscrbd by th utlty functon U and hs rsourc nds for th dffrnt rsourcs. W wll assum that thr ar at last playrs of ach typ. Our bounds wll mprov as gts largr. Usrs utlts do not dcln too fast, namly U(2x) αu(x) for som α > 1 and all x 0. Both assumptons ar ssntal for guarantng rvnu for th typ of allocaton mchansm usd hr, whr th mchansm s ndpndnt of th usr typs. To s th nd for th frst assumpton, consdr th cas whn thr s only a sngl usr wth hgh valuaton for th rsourc, h wll hav to gt most of th rsourc allocatd (du to dsrng ffcncy), vn f h sgnfcantly msrprsnts hs utlty. And hnc, w cannot xpct to xtract hs valuaton as ncom. Th scond assumpton s also ssntal: To gan hgh rvnu n th prsnc of lmtd dmand for th rsourc, on has to ntroduc a rsrv prc, and possbly lav a larg fracton of th rsourc un-allocatd. Our mchansm shars th rsourc btwn th usrs, and hnc would ncssarly nd up wth low rvnu f th avalabl rsourc xcds th hgh payng ntrst. Our Rsults. In ths papr w adopt Klly s far sharng mchansm [8] for gnral polyhdral nvronmnt, and analyz t for both socal wlfar and rvnu of th outcom. For th cas of sharng a sngl rsourc, whn thr s a sngl constrant, say x 1, th far sharng mchansm assums that ach playr bds th amount w of mony h wants to pay, and th mchansm shard th rsourc proportonal to th paymnts, allocatng x = w /( j wj) to playr. Johar and Tstsls [6] consdr ths mchansm as a gam. Thy show that th mchansm has a unqu Nash qulbrum assumng th utlts U ar concav. Thy also show that th Nash qulbrum s approxmatly ffcnt, showng that ts socal wlfar s at last 3/4 tms th socal wlfar of th most ffcnt allocaton. W xtnd th rsults of Johar and Tstsls [6] to gnral polyhdral nvronmnts and also show that f thr ar multpl dntcal playrs thn ths ffcncy rato tnds to 1 as th numbr of dntcal playr ncrass. Furthr, w nvstgat th rvnu gnratd by ths gam, and show that th gam also approxmatly maxmzs th rvnu of th auctonr, wth th approxmaton rato tndng to 1 f playrs utlts ar lnar and th numbr of dntcal playrs ncrass. In a mor gnral class of utlts satsfyng U(2x) > αu(x) for som constant α > 1, th approxmaton rato of th rvnu wll tnd to α 1. W also dvlop a nw tchnqu for analyzng gams of ths typs. Our tchnqu uss th th smlarty btwn th condton of Nash qulbrums of th gam and th dual of a crtan lnar program. Our man thorm can b clamd mor prcsly as follows: MAIN THEOREM Gvn a constant α > 1, and an ntgr 2, undr th assumpton that ach playr s utlty satsfs U(2x) > αu(x) and for ach playr typ, thr ar at last playrs (dfnd formally n scton 4), th far sharng mchansm (dfnd n scton 3) obtans both approxmatly maxmum ffcncy, and approxmatly maxmum rvnu. Th ffcncy s at last (1 1 ) tms th 4 optmal ffcncy and th rvnu s at last (α 1)(1 1 )(1 1 ) th optmal rvnu. 4 Not that ths bound s vry strong whn utlty s lnar (and so α = 2). For ths cas w hav th rvnu of th mchansm s at last f() = (1 1 )(1 1 ) tms 4 th optmal. Alrady whn thr ar 2 playrs of ach typ (whn = 2) th mchansm achvs 7 tms th optmal 8 ffcncy and almost half of th maxmum rvnu. Th followng numbrs ar th approxmaton ratos of ffcncy and rvnu n th cas of lnar utlty as ncrass. 2 3 4 5 6 7 Rvnu 0.436 0.611 0.703 0.76 0.8 0.82 Effcncy 0.875 0.91 0.937 0.95 0.958 0.964 Rlatd Wor. Most closly rlatd to our wor ar th mchansms of Klly [8] and Johar and Tstsls [6] consdrng th socal wlfar of th far sharng mchansm. W adopt ths far sharng gam for polyhdral nvronmnts. Th gam rqurs all playrs to bd mony on all th rsourcs and allocats rsourcs proportonal to th mony offrd. Johar and Tstsls consdr th ffcncy of th rsultng allocaton. W xtnd thr rsults, and also consdr rvnu. Th far sharng mchansm was motvatd by th nd for a smpl and asy to mplmnt mchansm for th rsourc sharng problm on th Intrnt. Th proposals vary from usng auctons to smpl prcng, but thy shar th basc goal of maxmzng socal wlfar. Th da s to mplmnt a smpl lghtwght mchansm that hlps arrang th socally optmal sharng of rsourcs. Congston prcng, frst proposd by Shnr, Clar, Estrn, and Hrzog [11], has mrgd as a natural way to dcd how to shar bandwdth n a congstd Intrnt. Th far sharng mchansm analyzd n ths papr s a vrson of congston prcng. Whl maxmzng socal wlfar s mportant to p customrs subscrbd to th systm, and to p thm happy, w blv that rvnu should also b consdrd. Onc a mchansm gts mplmntd, th ntwor managrs wll try to ta advantag of th usrs, and am to maxmz ncom, and wll no longr only thn of th mchansm as a way to arrang th bst us of th ntwor by maxmzng socal wlfar. As a rsult, t s mportant that w also undrstand th rvnu gnratng proprts of th proposd mchansms. Th wll-nown VCG mchansm, du to Vcry, Clar and Grovs s maxmzng socal wlfar (s [10]) n vry gnral sttngs. Ths mchansm s truthful, th playrs rport thr utlty functons to th auctonr who thn dcds on th allocaton and paymnts to maxmz socal wlfar. In th contxt of playrs wth complx utlty functons, on dffculty wth th VCG mchansm s that t rqurs usrs to communcat thr whol utlty functon. Rcnt paprs [7, 13, 14] mplmnt th VCG outcom va a smpl mchansm that s analogous to th far sharng mchansm (though t s lss natural). Ths paprs focus on maxmzng socal wlfar, and do not consdr rvnu. Th ssu of maxmzng rvnu n auctons has bn most wdly consdrd n th contxt of Baysan gams, s for xampl [9]. In th contxt of truthful auctons th
ssu of rvnu has bn most consdrd for dgtal goods, s for xampl Hartln and Karln [4, 5]. Th fact that proprts of som systms mprov as th numbr of usrs ncrass has bn prvously consdrd n othr sttngs. Edgworth [2] consdrs an xchang conomy, whr usrs com to th mart wth a bast of goods and am to xchang th goods to maxmz thr utlty. H was comparng th concpt of Walrasan compttv qulbrum to th noton of th cor n ths sttng. For an xchang conomy a compttv allocaton s an allocaton rsultng from mart clarng prcs p, whr all playrs sll at prc p and us th rsultng mony to buy thr optmal st of goods. Th cor of th xchang conomy gam s an allocaton of goods whr no subst of usrs can r-contract usng thr ntal allocaton to mprov at last on usr s utlty wthout dcrasng th utlty of any of thm. It s not hard to s that all compttv allocatons ar n th cor of th xchang gam, but n gnral th cor has othr allocatons that ar not supportd by prcs. Edgworth [2] showd that wth two dffrnt playrs f th mart contans many cops of ach playr, th st of cor allocatons convrgs to th compttv allocaton as th numbr of playrs grows. Mor gnrally, th cor n xchang conoms wth many (small) playrs s nown to convrg to th compttv allocatons, s Andrson [1] for a survy. Organzaton of th papr:. In scton 2 w dfn a class of rsourc allocaton gams, and show som xampl of gams n ths class. In scton 3 w dscrb th mchansm n th polyhdral nvronmnt ntroducd n scton 2. Scton 4 dscuss th bound on th rvnu and th ffcncy of ths gam. 2. GAMES WITH OLYHEDRAL DOMAIN Th rsourc allocaton problm dfnd n th ntroducton has n playrs, whr ach playr s to rcv an amount of rsourc x. layr has a concav and monoton utlty functon U (x) for rcvng x amount, such that U (0) = 0. W assum that all playrs hav lnar and sparabl utlty for mony, and f thy hav to pay w for rcvng x amount of th rsourc, thy valu ths at U (x ) w. W call th vctor of amounts x = (x 1,.., x n) 0 an allocaton. Allocatons hav to satsfy crtan constrants, and w say that an allocaton x s fasbl f t s possbl to allocat x to ach playr smultanously. DEFINITION 1. W say that th rsourc allocaton problm s polyhdral f thr xst a non-ngatv matrx A and a vctor u, such that fasbl allocatons ar th nonngatv vctors x 0 satsfyng Ax u Th smplst polyhdral allocaton problm s bandwdth sharng, whr th playrs shar 1 unt of bandwdth, and th constrant for fasblty s that x 1. A mor gnral vrson of ths xampl s th followng bandwdth sharng problm. Exampl 1: Bandwdth Sharng. olyhdral nvronmnts modl bandwdth sharng n a ntwor whr ach usr s sndng traffc along a path and x s th amount of traffc usr can snd along. In ths cas w hav a rsourc constrant assocatd wth ach dg : x u. In mor gnral bandwdth sharng, dffrnt usrs can b usng th rsourcs at dffrnt rats. Th rows of th constrant matrx A corrspond to dffrnt rsourcs, and th coffcnts ndcat th rat at whch th usrs us th rsourc. Exampl 2: Gams wth fnt outcom sts. Anothr ntrstng xampl of gams n polyhdral nvronmnts s th followng. Consdr a gam wth a fnt st of outcoms, ach of whch s xprssd by an allocaton vctor x. Lt {x 1, x 2,.., x N } b th st of possbl outcoms. W consdr a gam by also allowng mxd outcoms, a probablty dstrbuton of th basc outcoms. Choosng btwn th basc allocatons by th probablty dstrbuton p w gt that th vctor of xpctd allocatons to th playrs s j pjxj. Now th st of xpctd allocaton vctors obtand ths way s xactly th convx hull conv(x 1,..., x N ): conv(x 1,..., x N ) = { j p jx j p 0 and j p j = 1}. Ths convx st may not dfn a polyhdral nvronmnt n our sns, as w rqurd that th matrx A has nonngatv coffcnts. In many sttngs, gvn a fasbl allocaton (or xpctd allocaton) x thr s a mthod to rduc th allocaton of a playr. Thrfor w wll also consdr all th vctors y such that y x fasbl allocatons. If ths s th cas, th st of fasbl allocatons s th followng: S = {y 0 x conv(x 1,.., x N ) s.t y x} (1) Ths convx st now satsfs th condton of Dfnton 1, as shown by th followng Thorm, that can b provd usng standard tchnqus n convx gomtry. W provd a proof n th Appndx. THEOREM 1. A st S can b dfnd by (1) for som non-ngatv vctors x 1,.., x N f and only f S s boundd and thr xst a non-ngatv matrx A and a non-ngatv vctor u such that S = {x x 0; Ax u}. Not that ths thorm only clams th xstnc of a matrx A. Fndng such a matrx A algorthmcally can b hardr. In partcular, th sz of th matrx A may b xponntal n both th dmnson and th numbr of vctors x j. Exampl 3: Adwords as gam n polyhdral nvronmnt. An mportant xampl of an aucton allocatng quantts to playrs wth a fnt st of outcoms s th Adwords aucton. Th aucton s for a sngl yword, and th bddrs ar bddng to hav thr bd appar as a sponsord sarch rsult. Thr ar fnt st of outcoms, dpndng on whch bddr gts dsplayd n whch poston. To thn of th Adwords aucton n our framwor, w nd to charactrz th outcom for ach playr wth a sngl valu x. W us th xpctd numbr of clcs th usr rcvs as hs allocaton x. It s rasonabl to modl th bddr s utlty for ths allocaton, as a lnar functon U (x ) = a x, whr a s th xpctd rvnu rcvd for on clc. Tradtonal Adwords auctons us an ordrng (outcom) that dpnds on th bds, and do not us randomzaton [12, 3]. To thn of Adwords auctons as a polyhdral nvronmnt w nd to allow randomzaton n th allocatons of
bddrs to postons. Our modl rqurs that n any fasbl allocaton x, on can dcras th allocaton of a sngl playr. W can mayb do ths by ntroducng a dummy ln on som postons. Our dscrpton of th outcoms assocats a nonngatv vctor of clc-through rats wth ach slcton of bddrs allocatd to th postons. Ths allows us to modl som nds of xtrnalts btwn th bddrs. Th valuaton of say a bddr for bng n poston 2 dpnds on what ad s showng n poston 1. For xampl, N s poston 1 mas th valu of poston 2 lss for a qury of snars compard to havng an unnown brand nam n poston 1. Unfortunatly, our allocaton mchansm wll rly xplctly on th matrx dscrpton Ax u of th fasbl rgon, gvn by Thorm 1. For a gnral st of allocaton vctors x 1,..., x N th rsultng constrants can b rathr complx, and th far sharng mchansm s not a natural aucton mchansm for ths cas. For th cas of Adwords auctons w vw our rsults as a sort of xstnc proof that mchansms that smultanously maxmz rvnu and ffcncy (n an approxmat sns) do xst. W lav t as an opn problm to dcd f on of th smpl and natural Adwords aucton mchansms has ths proprty. 3. THE MECHANISM In th prvous scton w ntroduc th rsourc allocaton gam n a polyhdral nvronmnt, and showd that ths modl capturs a wd rang of problms. W now dscrb th far sharng mchansm for ths class of gams. Th mchansm s an xtnson of th mchansms ntroducd by Klly [8], Johar and Tstsls [6]. Lt E dnot th st of constrants (th rows of A). For smplcty of notaton, w assum that u = 1 for ach E by normalzng ach row. W wll us α to dnot th row of matrx A, whch w wll also call constrant. W now hav th followng dscrpton of th st of fasbl allocatons: α x 1 for all E, x 0 (2) b j α j b, Th mchansm. Whn sharng a sngl rsourc wth constrant x 1 th far sharng [8] mchansm rqurs that ach playr j submts a bd b j, th amount of mony sh wants to pay, and th rsourc s allocatd proportonal to th bds, as x j = b j/ b. W can thn of b as th unt prc p of th good. Th allocaton s drvd from ths unt prc, as usr j gts x j = b j/p amount for th cost w j = b j at ths prc. To xtnd ths mchansm to a sngl constrant wth coffcnts αx 1, w agan rqur that ach playr j submt a bd b j, hr wllngnss to pay, and vw p = b as th unt prc of th good. Rcall that α j s th rat at whch usr j uss rsourc, so usr j nds α jx j allocaton for a valu x j. At th unt prc of p sh gts α jx j = b j/p allocaton, and hnc w nd to st x j = b j/(α jp) = and sh wll hav to pay w j = b j = α jx jp. For nvronmnts wth mor constrants, Johar and Tstsls [6] xtnds th far sharng mchansm by rqurng that usrs submt bds b j sparatly on ach rsourc. As bfor, w can vw th sum of bds p = b as th unt prc of rsourc, and allocat th rsourc at ths prc. Ths allocaton lmts th valu x j for usr j to at most x j = b j/(α jp ). Th da s to as usrs to submt bds b j for ach rsourc, allocat th rsourcs sparatly, ma usr j pay w j = b j, and thn st x j = mn {:α j 0} x j. W nd to xtnd ths mchansm to b abl to dal wth rsourcs that ar undr-utlzd. Som constrants may not b bndng at any soluton, and th far sharng mthod dos not dal wll wth such constrants: usrs wll want to bd arbtrary small amounts as thr s too much of th rsourc. To dal wth such constrants, w allow ach playr to rqust an amount rj wthout any montary bd. For ach rsourc f th prc s 0 (that s p = b = 0) and α r 1 (th rqustd rats can all b satsfd) thn w sttng x j = rj for all j. Th mchansm can b dscrbd formally as follows: THE MECHANISM: Each playr j submt a bd b j and a rqust r j for ach rsourc. For rsourc w us th followng allocaton: If b > 0 thn x j = b j α j ( b ) for j. If b = 0 and α r 1 thn x j = r j for j Els, st x j = 0 for j. For ach playr j, th amount of mony that sh nd to pays s w j = b j and th fnal allocatd x j = mn {:α j 0} x j. Smplfyng assumpton. To smplfy th prsntaton n ths xtndd abstract w wll assum that ach rsourc has at last two ddcatd usrs who only nds rsourc, and who hav nfntsmally small, but lnar utlty. Ths usrs wll guarant that no rsourc s undr-utlzd, but wll not chang thr th optmal allocaton of th Nash qulbrum substantally. Usng ths assumpton, w can nvr hav b = 0 for any rsourc. Condton for Nash qulbrum. Nxt w analyz th condton for an qulbrum for ths gam. W wll us ths condtons to show that an qulbrum always xsts. Consdr a st of bds b, and a rsultng allocaton x, whr playr gts allocaton x. Whn s ths allocaton at qulbrum? For ach rsourc w us p = b, th sum of th bds, as th unt prc of th rsourc (rcall that w normalzd constrants, so thr s 1 unt of vry rsourc avalabl). Now consdr th optmzaton problm of a playr j assumng bds b for all othr playrs ar st. Th playr j s ntrstd n maxmzng hr utlty at U j(x j) b j. At qulbrum, t must b th cas that x j = x j for all rsourcs that costs mony, or othrws playr j can rduc hr bd b j wthout affctng hr allocaton. So w can thn of th playr s optmzaton problm as dpndnt on on varabl x j, th allocaton sh wll rcv. What bd dos playr j hav to submt for a rsourc to gt allocaton x j = x j? Bds must satsfy th followng condton: If b j > 0 thn: α jx j = b j. b
Assumng all othr bds b ar fxd, w can xprss th bd b j ndd as follows. b j(x j) = α jx j j b. 1 α j xj Not that ths xprsson assums that α jx j < 1, that s, j s not th only usr of th rsourc at qulbrum. It s not hard to s that ths s guarantd by havng at last two ddcatd usrs for ach rsourc. Usr j wll want to choos x j to maxmz hr utlty. For ths nd, t wll usful to xprss th drvatv of th bd whn vwd as a functon of x j. W gt th followng b j (agan assumng α jx j < 1): j b b j(x j) = α j x j (1 α. j xj)2 Substtutng j b = p (1 α jx j) and smplfyng w gt that b x j(x j) = p α j j 1 α. j xj Not that a drvatv of p α j would corrspond to a fxd prc p on rsourc, as ncrasng x j ncrass th us of ths rsourc at th rat of α j. In th allocaton gam, th prc s a functon of th bds, and hs nducs th playrs to shad thr bd for th rsourc by a factor that dpnds on thr shar of th rsourc. Now consdr th optmzaton problm of playr j. Sh wants to maxmz hr utlty U j(x j) b j, whch can now b xprssd as U j(x j) α jx j j b 1 α j xj, as a functon of th sngl varabl x j. Not that ths s a concav functon of x j. Th maxmum occurs at a valu x j, whr th drvatv of ths functon 0, or f th drvatv s ngatv vrywhr, maxmum occurs at x j = 0. Usng th drvatvs w computd abov, w gt th drvatv of usr jth utlty as a functon of hr allocaton x j to b U j(x j) p α j (1 α j xj) Ths drvatv s a strctly dcrasng functon, so w hav th followng Nash condton, whch wll b usd n th analyss of th gam. x s a Nash soluton f and only f: α x 1; x 0 for all E U j(x j) = U j(0) p α j f (1 α xj > 0 and jxj), p α j f x j = 0. Gvn ths qulbrum condtons, w can xtnd th rsult of Johar and Tstsls [6] to prov that th gam always has a Nash qulbrum. THEOREM 2 (Johar-Tstsls). If th utlty functon of ach playr s ncrasng, dffrntabl and concav, thn thr always xsts a Nash qulbrum. Furthr mor (3) f an allocaton x, a bd and a rqust vctor b, r s a Nash qulbrum thn thy satsfy th condton (3). To valuat th outcoms of th gam, w wll compar th socal wlfar and th rvnu wth th optmal socal wlfar, whch can b wrttn as an optmum of th followng a lnar program. n max O T = U (x ) subjct to =1 α x 1; E x 0. 4. REVENUE AND EFFICIENCY Nxt w analyz th rvnu and ffcncy of a Nash qulbrum. W wll us th maxmum socal wlfar OT to masur both ffcncy and rvnu. Not that OT s also an uppr bound on th rvnu. As alrady mntond n th ntroducton, w nd to ma two assumptons to b abl to gt a rasonabl bound on th rvnu. Frst w assum that th playrs utlty functons grow at a rasonably stady rat. Scond, w assum that thr ar at last 2 playrs of ach typ. ASSUMTION(α, ) (4) Th utlty functon U j(x) of all usrs j s non ngatv, ncrasng, dffrntabl, concav, furthr, U satsfs: U (2x) αu (x) for som α > 1. W say that th typ of a playr j s hr utlty functon U j(x) and th rat at whch sh nds th rsourcs, th coffcnts α j for all rsourcs. W assum that thr ar at last playrs of vry typ: that s for vry playr j, thr ar at last ( 1) othr playrs wth th sam typ. Nots. Th class of functons w consdr capturs th lnar functons and contans small dgr polynomals: U j(x) = x ɛ for any 0 ɛ 1. In th contxt of bandwdth sharng, th scond assumpton mans that for ach playr j, thr ar at last 1 othr playrs wth th sam utlty functon and th sam path. W assumd that both th utlty and th rat of us s xactly th sam for all of th usrs. W can rlax ths condtons to rqurng only smlar utlts and smlar rats, by loosng an addtonal factor n th approxmaton bounds for th coffcnt of ths smlarty. Th man rsult of our papr s th followng: THEOREM 3. Undr Assumpton(α, ), th mchansm dfnd n scton 3 approxmatly maxmzs both ffcncy and rvnu. Th loss of ffcncy s boundd by a fracton 1 of and th rvnu s at last (α 1)(1 1 )(1 1 ) tms 4 4 th optmal rvnu. W bound th rvnu n two stps. W compar th rvnu to th socal wlfar obtand at qulbrum: U(x), and thn compar ths valu wth th optmum socal wlfar O T. Th fnal bound s th product of ths two ratos.
To ma th proof asy to follow, w frst consdr th smpl cas of th gam, whr thr s only on constrant on th fasbl allocaton st. W xtnd th rsult to th gnral cas n Scton 4.2 usng lnar programmng dualty. 4.1 roof for a smpl cas Consdr th smplst vrson of th gam, whr w hav only on constrant for th fasbl allocatons st: Sfrag x rplacmnts 1. Ths s th basc bandwdth sharng problm consdrd also by Johar and Tstsls [6]. Johar and Tstsls show that th socal wlfar at Nash s at last 3/4 tms th optmum. W mprov thr bound from 3/4 to (1 1 4 ) usng our assumpton. To bound th rvnu, w show a bound of (α 1)(1 1 ) on th rato btwn rvnu and socal wlfar at an qulbrum. W start wth th scond bound. THEOREM 4. Wth Assumpton(α, ), n th gam whos fasbl allocaton st s x 0 such that x 1, th rato btwn rvnu and socal wlfar at th Nash qulbrum s at last (α 1)(1 1 ). roof. Lt x b a Nash qulbrum. Accordng to condton (3) from th prvous scton, w gt: Ethr x = 0 or U (x ) = p, 1 x whr w us agan th notaton that p = j bj. W can wrt th bd b of playr as b = px. Thus, n both cass w hav: b = px = U (x )(1 x )x Thrfor th rvnu of th gam, whch s th sum of all th bds, s: b = U (x )(1 x )x. W nd to compar ths rvnu wth U(x), th socal wlfar at th Nash qulbrum. Th rato s U (x )(1 x )x U U(x) max (x )(1 x )x. U (x ) W bound ths rato by boundng th ndvdual ratos on th rght hand sd. Now, consdr two dntcal playrs and j, wth dntcal utlty functons U (x) = U j(x). W clam that dntcal playrs gt qual shar of th rsourc at a Nash qulbrum. To s why, rcall that by (3) w thr hav x = x j = 0 (f U (0) = U j(0) < p) or x and x j ar both th unqu soluton to p = U (x)(1 x). Hnc for vry, thr wll b at last 1 othr playrs gttng th sam allocaton x, and hnc x 1 (1 x ) 1 1. Consdr now U (x )x U (x. Th utlty functon U s concav functon, and hnc th slop of U (x ) s gratr than ) th slop of th ln connctng th ponts (x, U (x )) and (2x, U (2x )). (S Fgur 1). Thus w hav th followng nqualty: U (x ) U(2x) U(x) 2x x. Du to Assumpton(α, ), w hav U (2x ) αu (x ), thus: U (x ) (α 1) U(x) x U (x )x U (x ) (α 1) U(2x) U(x) U (x)x x U (x) 2x Fgur 1: Th utlty functon Thrfor for vry playr, th followng nqualty holds: U (x )(1 x )x U (x ) (1 1 )(α 1) Ths nqualts mply that th rato btwn th total rvnu and th socal wlfar of ths gam s also at last (1 1 )(α 1). Nxt w bound th rato btwn th socal wlfar of th Nash qulbrum and th optmal socal wlfar OT. Johar Tstsls [6] gv a 3/4 bound on ths rato. Usng Assumpton(α, ), w can mprov ths bound to (1 1 4 ). THEOREM 5. Undr Assumpton(α, ) th far-sharng mchansm for th smpl rsourc sharng problm x 1 obtans a soluton wth th socal wlfar at last (1 1 ) 4 tms th optmal. roof. W wll show th rsult for th spcal cas of pur lnar utlty functons U (x) = a x for vry playrs. Johar Tstsls [6] showd that th worst socal wlfar rato occurs n th cas of pur lnar utlts wth a = U (x ), whr x s th allocaton of playr at th Nash qulbrum. For compltnss w nclud a stch of th proof n th Appndx. Th maxmum socal wlfar s th optmum of ax whr x 1. Thus t s qual to O T = max a. Lt s assum that a 1 = max a. From th Nash condton: U (x ) = a = p f x > 0 (1 x ) on obtans: f x > 0 thn a > p and a 1(1 x 1) = p, and thus f x > 0 thn a > a 1(1 x 1). By Assumpton(α, ), n th orgnal gam thr ar at last playrs who hav th sam utlty functon as playr 1 and hnc thy gt th sam allocaton x 1 (as th valu at qulbrum s unqu). Ths playrs provd a total utlty that s at last a 1x 1 and all othr playrs fll out th bandwdth of 1, so thy hav total shar of 1 x 1 and hav utlty coffcnts a a 1(1 x 1). Ths gvs us a total utlty of at last a x a 1x 1 + a 1(1 x 1)(1 x 1).
Hnc th rato btwn socal wlfar at th Nash qulbrum x and th optmum on s: ax a1x1 + a1(1 x1)(1 x1) = 1 x 1 + x 2 1. a 1 a 1 Ths xprsson s mnmzd whn x 1 = 1/(2) whn th rato s 1 1/(4) as clamd. 4.2 roof of th man thorm Now w prov th thorm n th gnral sttng, whch s a nontrval xtnson of th prvous rsult. To do ths w nd to xtnd th rsult to handl multpl rsourcs, and dffrnt rats at whch playrs us ths rsourcs. Our man tool s lnar programmng dualty. Frst w show that undr Assumpton(α, ), th rato btwn rvnu and socal wlfar at Nash qulbrum s boundd by (α 1)(1 1 ). Consdr a Nash qulbrum x. Rcall that th qulbrum s not now to b unqu n th gnral cas. Howvr, playrs of dntcal typ must gt dntcal allocaton. LEMMA 6. If two playrs and j hav th sam typ, thn n any Nash qulbrum, thy gt th sam allocaton. roof. By th Nash qulbrum condtons (3) both x and x j ar 0 f U (0) = U j(0) < α p and othrws both ar th unqu solutons quaton of Nash n (3). Th man obsrvaton that allows us to us lnar programmng s that th condton for a Nash qulbrum (3) s closly rlatd to th dual of a crtan lnar program rlatd to th Optmum socal wlfar (4). Rcall from th proof of Thorm 4 that bcaus of th assumpton about th utlty functons, w hav: U (x )x (α 1)U (x ). W wll consdr th followng prmal-dual programs, whr a = U (x ): RIMAL : s.t: DUAL : s.t: max n a x =1 α x 1; E and x 0. mn ỹ α ỹ a and ỹ 0. LEMMA 7 (Wa dualty). If x and ỹ, ar fasbl solutons of th prmal and th dual programs abov, thn ỹ a x THEOREM 8. Undr Assumpton(α, ), th rato btwn rvnu and socal wlfar at Nash s boundd by (α 1)(1 1 ). roof. W want to show that th prcs p of th rsourcs form a dual soluton f w scal thm by a factor (1 1 ). Th dual objctv s p whch s xactly th rvnu. All fasbl dual solutons hav valu that uppr bounds th prmal objctv. W wll gt th clamd bound by consdrng th scalng ndd to ma p dual fasbl, and th rlaton of U (x ) to th lnar objctv U (x )x. In a Nash qulbrum dntcal playrs rcv th sam allocaton (by Lmma 6). Bcaus of Assumpton(α, ), for vry thr ar at last 1 othr playrs wth th sam rsourc constrant gttng th sam valu x. Thrfor: 1 = j α jx j α x, and so 1 α x 1 1. Hnc, from th Nash condton (3), on obtans: U (x ) α p 1 α x Now, lt a = U (x ) and y = p 1 1 α p 1 1,. (5). Th nqualty (5) shows that th vctor y s a fasbl soluton of th DUAL program. Th Nash allocaton x s a fasbl allocaton, and thrfor, a fasbl soluton for th RIMAL program. By Lmma 7 w hav: p 1 1 = y a x = U (x )x Now w can bound th rvnu as follows: p (1 1 ) U (x )x (α 1)(1 1 ) U (x ), whr th last nqualty follows from: U (x )x (α 1)U (x ), as xpland abov th lnar program. Nxt w consdr ffcncy. Hr w us th mprovd bound of Thorm 5 for th smpl cas, and agan us lnar programmng dualty to bound th optmal valu. THEOREM 9. Undr Assumpton(α, ) th rato btwn socal wlfar at Nash and th optmal on s at last 1 1 4. roof. W wll gt th bound from Thorm 5 and lnar programmng dualty by consdrng sparat gams for ach rsourc, along th lns of th proof of Johar and Tstsls [6]. Consdr a Nash qulbrum x. As bfor w can assum wthout loss of gnralty that th utlty functon s lnar U (x) = a x for all playrs, as was shown by [6] (s th proof n th Appndx for compltnss). W us a = U (x ). Wth ths assumpton, socal wlfar at Nash s U (x )x, and th optmal socal wlfar s th optmal valu O T of th RIMAL program whr a = U (x ). Consdr th Nash condton (3): U j(x j) = U j(0) p α j, f (1 α xj > 0 and jxj) p α j f x j = 0. W wll consdr a sparat gam for ach rsourc. In th gam corrspondng to rsourc playr s ntrstd n gttng an allocaton z wth th constrant j z j 1, and a lnar utlty functon v z, whr v p =. If (1 α x ) w st z = α x thn by (3) th allocaton vctor z forms a fasbl allocaton at qulbrum, wth total utlty v z. W want to apply Thorm 5 for ach rsourc. To b abl to do ths, w nd to s that th nw gam also satsfs Assumpton(α, ). To s why, not that playrs of dntcal typ wll also hav dntcal v valus (as dntcal playrs gt th sam allocaton) and hnc rman of dntcal typ n th nw gam.
Now by Thorm 5, th socal wlfar of ach gam at Nash s at last (1 1 ) tms th optmal on: 4 v z (1 1 4 ) max {v } = (1 1 4 )v, (6) whr w us v = max {v }. Summng th lft hand sds ovr all rsourcs and substtutng th valus for z w gt: v z = v α x = x v α. Now, v α = p α (1 α, and du to condton (3) of x ) Nash qulbrum, ths valu s qual to U (x ) unlss x = 0. Thrfor: x v α = U (x )x. Combnng ths wth th bound (6) w gt: U (x )x (1 1 4 ) W now us th wa dualty Lmma 7 to prov that v s at last th Optmal socal wlfar, and ths wll stablsh th thorm. Bcaus v = max v, w hav: α v α v = p α (1 α x) U (x ). Th last nqualty s du to th condton of Nash qulbrums. Ths nqualty shows that v s a fasbl soluton of th DUAL program whr a = U (x ). Thrfor v s at last th optmum of th RIMAL program O T, whch s th optmal socal wlfar of th gam. Combnng th bound on rvnu n Thorm 8 and th bound on ffcncy abov, Thorm 3 s provd. 5. REFERENCES [1] R.M. Andrson, Th Cor n rfctly Compttv Economs, Chaptr 14 n Robrt J. Aumann and Srgu Hart (dtors), Handboo of Gam Thory wth Economc Applcatons, volum I, 1992, 413-457. Amstrdam: North-Holland ublshng Company [2] F.Y. Edgworth, Mathmatcal sychcs, London: Kgan aul, 1881. [3] B. Edlman, M. Ostrovsy, and M. Schwartz. Intrnt advrtsng and th gnralzd scond prc aucton. NBER Worng apr, 11765, Novmbr 2005. [4] A. Fat, A. Goldbrg, J. Hartln, and A. Karln. Compttv Gnralzd Auctons, STOC 2002 [5] J. Hartln and A. Karln. roft Maxmzaton n Mchansm Dsgn, n Algorthmc Gam Thory, (N. Nsan, T. Roughgardn, E. Tardos, V. Vazran, ds), Cambrdg Unvrsty rss, to appar. [6] R. Johar and J. N. Tstsls. Effcncy loss n a ntwor rsourc allocaton gam. Mathmatcs of Opratons Rsarch,29(3):407 435,2004. [7] R. Johar, J.N. Tstsls. Communcaton rqurmnts of VCG-l mchansms n convx nvronmnts. In prparaton (2006). An arlr vrson appard n th Allrton Confrnc (2005). v [8] F.. Klly. Chargng and rat control for lastc traffc. Europan Transactons on Tlcommuncatons, 8:33 37, 1997. [9] R. Myrson. Optmal Aucton Dsgn. Mathmatcs of Oppratons Rsarch, 6:58 73, 1983. [10] N. Nsan. Introducton to Mchansm Dsgn (for Computr Scntsts), n Algorthmc Gam Thory, (N. Nsan, T. Roughgardn, E. Tardos, V. Vazran, ds), Cambrdg Unvrsty rss, to appar. [11] S. Shnr, D. Clar, D. Estrn, and S. Hrzog, rcng n computr ntwors: rshapng th rsarch agnda, Tlcommuncatons olcy, vol. 20, no. 3, pp. 183-201, 1996. [12] H. Varan. oston auctons. To appar n Intrnatonal Journal of Industral Organzaton. [13] S. Yang and B. Haj. An ffcnt mchansm for allocaton of dvsbl good and ts applcaton to ntwor rcourc allocaton. 2004. Submttd. [14] S. Yang and B. Haj, VCG-Klly mchansms for allocaton of dvsbl goods: Adaptng VCG mchansms to on-dmnsonal sgnals, 40th Annual Confrnc on Informaton Scncs and Systms, March, 2006. AENDI roof of Thorm 1. Gvn a st S = {x x 0; Ax u}, lt x 1,.., x N b th xtrm ponts of S. Bcaus S s boundd w hav S = conv(x 1,.., x N ). Howvr, A s nonngatv, thrfor x S mpls y S 0 y x. Thus: S = {y 0 x conv(x 1,.., x N )s.t y x} W now prov th othr drcton. Gvn S dfnd as abov, w wll show that for vry y S, thr xsts a non-ngatv vctor a such that a T x 1 < a T y x S. Onc w hav that, wth th obsrvaton that th st of th xtrm ponts of S s fnt, (snc S s dfnd by a fnt st of ponts x 1,.., x N ), w can conclud that thr xsts a non-ngatv matrx A and non-ngatv vctor u such that S = {x x 0; Ax u}. Now, w show that thr xsts such an non-ngatv vctor a for vry y S. Bcaus of th dfnton of S: f x S thn any non-ngatv vctor lss or qual to x s also n S, w hav: y S mpls y + z S z 0. Both S and {y + z z 0} ar convx, furthrmor S s compact, thr xsts a vctor a such that a T x 1 < a T (y + z); x S and z 0. But bcaus a T (y + z) > 1 z 0, a has to b a non-ngatv vctor. By ths w fnshd th proof. LEMMA 10. [6] Gvn an nstanc of th gam wth th concav utlty U, and lt x b soluton satsfyng th Nash condton. Consdr th gam whr th utlty U s rplacd by th functon W (z) := U (x )z. Th allocaton x stll satsfs th Nash condton n th nw gam and th rato btwn socal wlfar at Nash and th optmal dos not ncras. roof. W frst modfy th utlty functon U to th lnar functon V wth th slop U (x ) such that V (x ) =
U (x ). That s V (z) = U (x )(z x ) + U (x )). S fgur 2. Bcaus th drvatv of th nw utlty functon at x V U (x ) Sfrag rplacmnts U W x Fgur 2: Nw utlty functons dos not chang, thrfor x stll satsfs th Nash condton of th nw gam. Furthrmor, th socal wlfar of th soluton x stays th sam. On th othr hand, bcaus U s concav, thus V (z) U (z) z, thrfor th nw optmal socal wlfar can only ncras. As a rsult, n th modfd gam, th rato btwn Nash and Optmal socal wlfar dos not ncras. Nxt w consdr nw utlts W obtand by shftng V to th orgn. That s W (z) = U (x )z. Th dffrnc btwn V and W s a constant. Lt c b th sum of ths dffrncs ovr all. If th N and O ar rspctvly th Nash and th optmal socal wlfar of th gam wth utlty V, thn th Nash and optmal socal wlfar of th gam wth utlty functons W ar N c and O c, rspctvly. Snc w now N O and 0 c mn{n, C}, w hav: N O N c O c, whch shows that th rato also dcrass. By ths w fnshd th proof.