The Search for Coalition Formation in Costly Environments 1

Transcription

1 The Seach fo Coalto Fomato Costly Evomets 1 Davd Sae 1 ad Sat Kaus 1,2 1 Depatmet of Compute Scece, Ba-Ila Uvesty, Ramat-Ga, Isael {saed, macs.bu.ac.l 2 Isttute fo Advaced Compute Studes Uvesty of Maylad, College Pak, MD Abstact. We study the dyamcs of fomg coaltos of self-teested autoomous buye agets, fo the pupose of obtag a volume dscout. I ou model, agets, epesetg coaltos of vaous szes, may choose to be acuated wth othe agets, hopefully edg up wth a ot coalto stuctue, whch wll mpove the oveall pce. Upo ecouteg potetal pateg oppotutes fo exteded coaltos, the aget eeds to decde whethe to accept o eect them. Each coalto pateshp ecapsulates expected beeft fo the aget; howeve the pocess of fdg a potetal pate s assocated wth a cost. We exploe the chaactestcs of the aget s optmal stateges the eulbum ad develop the euatos fom whch these stateges ca be deved. Effcet algothms ae suggested fo a specfc szetwo vaat of the poblem, ode to demostate how each aget s computato pocess ca be sgfcatly mpoved. These algothms wll be used as a fastuctue fom whch the geeal case algothms ca be extacted. 1. Itoducto The gowg teest autoomous teactg agets has gve se to may ssues coceg coalto fomato. A coalto s a goup of self-teested agets that agee to coodate ad coopeate the pefomace of a specfc task. Though the coalto, the agets as a goup ae able to pefom the task moe effcetly, ad cease the patcpats' beefts [2, 6, 15, 16]. The ma uesto evey coalto fomato applcato s how to deteme the set of agets each specfc aget wll be wllg to fom a coalto wth. Ths s whee each aget s assocated wth a specfc type that captues specal popetes that chaactezes t. We cosde evomets whch a aget s utlty s fully coelated wth ts type. The aget s type s addtve ad thus ca be mpoved by fomg coaltos wth othe agets. Aget types ae odeed accodg to the assocated utlty. The hghe the othe coalto membe s type the hghe the aget s utlty. Each aget ou model epesets a coalto of oe o moe membes. The agets may teact wth each othe to shae fomato egadg the types. Ths fomato s used by each aget ts decso makg pocess of whethe to combe ts cuet coalto wth aothe aget s coalto, theeby fomg a ew coalto of a hghe type. Cosde, fo example, the electoc maketplace whee agets epeset coaltos of buyes teested a poduct. Assume the euested uattes deteme the aget s type, thus the hghe the euested uatty the bette the pce fo the coalto membes. The agets' seach fo coalto oppotutes s costly [10]: at each stage of ts seach a aget has to sped esouces locatg ad teactg wth aothe aget epesetg a coalto of a adom type. I addto, each stage of the seach eflects a coalto coodato cost (commucato wth ts membes), whch s deved fom the umbe of coalto membes. The aget s wllgess to exted the coalto upo ecouteg a ew aget s suffcet. The ew coalto wll be fomed oly f t s mutually accepted by both 1 Ths eseach was suppoted pat by NSF ude gat #IIS

2 agets' pates. A ew aget wll be ceated fo the ew coalto fomed. Ths ew aget wll hadle ad epeset all membes pevously hadled by the agets that fomed the ew coalto. Each membe a gve coalto wll shae the aget s costs elatve to ts type. It wll also shae the utlty of the fal coalto t wll be a membe of, elatve to ts type. Recallg that the types ae addtve ad the aget s utlty ceases wth ts type, the a decso take by a aget to fom a coalto wth aothe aget to mpove ts type, wll eve cause a coflct of teests wth ay of the membes epeseted by ths aget the hghe the cease the aget s type, the moe utlty each membe wll be gag. Theefoe, each aget, ethe ceated fo a ew coalto, o eteg the maketplace as a epesetatve of a sgle coalto membe must cosde two mao uestos: Fst, s t gog to execute ts task mmedately o s t moe beefcal to egage costly seach to exted the coalto? If the latte decso was take the at each stage of the aget s seach pocess, afte evewg the fomato egadg the cuet potetal pate, the aget must make a decso whethe to temate the seach o to cotue. Cotug the seach wll hopefully yeld a hghe type pate to cotue the pocess wth (the aget wll pefe eectg pateshps eflectg sgfcat futue coalto coodato costs). Tematg the seach wll esult fomg a coalto wth the cuet potetal pate f t agees; ths ew coalto wll face the same decsos, as metoed above, all ove aga. We utlze the electoc maketplace evomet [16], as a famewok fo ou aalyss. The model we peset cosdes buyg agets, epesetg oe o moe dffeet buyes, possbly teested extedg the coalto fo buyg a specfc poduct. The beeft fo all patcpats such a coalto s the ablty to obta, as a team, a dscout pce (compaed to the pce each of them would have pad sepaately). Each aget gas ths utlty sepaately, ad the utlty s ot tasfeable (o sde-paymets). Obvously, the lage the uatty a aget s seekg to buy fo the cuet coalto t s epesetg, the geate the beeft fo ts potetal pates of a exteded coalto ad vce vesa. Wheeve a ew coalto s ceated the ew aget wll seek to mmze the oveall costs of such a coalto. The aget s utlty (ethe by puchasg the poduct o by fomg ew coaltos) as well as ts costs wll be splt amog ts epeseted membes, accodg to the pecetage out of the oveall euested uatty. Theefoe ay decso the epesetatve aget wll take s the most beefcal fo all ts epeseted membes. The best pce that ca be obtaed wll be though oe bg coalto whch all agets ae membes. Yet, the toducto of seach costs ad coalto maagemet costs to the model pevets ths type of solutos, ad efoces a geue cost effectveess aalyss fo evaluatg each potetal ew coalto. The same cocepts of coalto fomato though pateshps ae vald ad e-usable othe plausble MIS ad CS elated applcatos. Cosde, fo example, clet-seve evomets, whee dstbuted suboutes ae watg to be pocessed o a cetal seve. Typcally, the seve pocessg tme s a fucto of the uey put. Hee, thee may be a cetve fo a suboute to pate wth othes to ceate a combed uey fo whch the pocessg tme ove the seve s shote tha the aggegated executo of each suboute sepaately. Each suboute s uey chaactestcs ca dcate a type, ad the combed uey possesses smla addtve behavo as descbed above. Adopto of the poposed aalyss ad the suggested algothms fo such applcatos s smple oce we expess the seach ad coalto coodato costs assocated wth each applcato tems of the coalto beeft. Lookg fo a basele fo the coalto though pateshps poblem we addessed AI ad ecoomc lteatue, as wll be descbed secto 2. We cotue secto 3 by pesetg the geeal model fo coalto fomato though pateshps wth seach ad coalto coodato costs. We show the geeal chaactestcs of eulbum ad develop the euatos descbg the agets' optmal stateges. I secto 4 we utlze the basc two-sze

3 The Seach fo Coalto Fomato Costly Evomets 3 coalto vaat to suggest algothms fo smplfyg the dstbuted calculato pefomed by each aget. Ths s a mpotat step towads extedg the algothms to hadle the geeal case. We coclude ad peset dectos fo futue wok secto Related Wok Coalto fomato pocesses focus a lot of atteto o mult-aget systems [11, 13]. I ecet yeas, eseach has toduced the coalto fomato pocess also to electoc maket evomets. Recogzg the cetves fo both buyes ad vedos volume dscouts 2, seveal dffeet buye-agets coalto schemes wee poposed [7, 12, 15, 16]. Extesos of the tasacto-oeted coaltos to log-tem oes, wee also suggested [2, 3]. Howeve, most eseach was maly coceed wth the pocedues of egotatg the fomato of coaltos ad dvso of coalto payoffs. Othe elated popula eseach topcs wee fdg the optmal dvso of agets to coaltos though a cetal mechasm ad efocemet methods fo the teacto potocols. The esouces assocated wth aget s seach fo a coalto, ad the fluece of ths facto ove ts decsos, wee ot dscussed ths cotext 3. Most mechasms assumed a aget could sca as may agets as eeded, o smply a cetal vew of the evomet. The evew of ecoomc lteatue eveals that the model of pateshps wth pate seach costs was wdely studed tadtoal maage makets ad ob-seach applcatos, whch evolved fom the aea of seach theoy [8, ad efeeces thee]. These models wee focused o establshg optmal stateges fo the seache, assumg o mutual seach actvtes ad wee classfed ude oe-sded seaches. I a effot to udestad the effect of dual seach actvtes such models, the Two-sded seach eseach followed. Ths oto was exploed wth the eulbum seach famewok [4]. A teestg aalyss though smulato of a two-sded maket vaat was toduced by Geewald ad Kephat [5]. Aothe mpotat eseach aea s the oe called assotatve matchg. Ths aea volves a decetalzed seach of moe tha two heteogeeous aget types. Becke [1] aalyzed a costless matchg maket, whee two dffeet aget types poduce a dffeet utlty whe matched ad othewse o utlty. Becke showed that the uue compettve eulbum has assotatve matchg meag that matched pates ae detcal ( type), fo both the tasfeable ad o-tasfeable utlty cases. Exteded models whch cluded some seach cost elemets fo the o-tasfeable case wee poposed by Smth [14] who modeled seach costs by the dscoutg of the futue flow of gas ad Moga [9] who used addtve explct seach costs. The tasfomato of the suggested cocepts the ecoomcal models, to plausble applcatos ove the teet ad computezed evomets wth seach ad coalto coodato costs s ot tval. All the above ecoomcal models assumed o utlty fo a aget wthout fomg a pateshp, ad most mpotatly, they dd t allow coaltos to exted themselves beyod two agets. We wll efe to such models secto The model We cosde a electoc maketplace wth umeous heteogeeous buyg agets, epesetg dffeet buyes. Each aget s chaactezed by the umbe of buyes t epesets ad the tal teto of buyg a pe-defed uatty of a specfc well 2 The vedo aget was also clamed to possess seveal advatages fo sellg bulk, maly due to deceased advetsemet costs ad dstbuto costs. 3 A excepto ca be foud [10], dscussg settgs whee thee ae too may coalto stuctues to eumeate ad evaluate (due to, fo example, costly o bouded computato ad/o lmted tme). Istead, agets have to select a subset of coalto stuctues o whch to focus the seach.

4 defed ad easly foud poduct. The ecapsulated uatty of a coalto epesetatve aget s a aggegato of all the uattes euested by the buyes ths aget epesets. The poduct pce s a fucto of the puchased uatty - selles offe a dscout fo uatty pce, amg to ecouage puchase of lage uattes a bulk. Deotg the posted pce fo a uatty as P, the pce fucto satsfes the followg: dp < 0, 2 2 d P > d 0 (1) d( P ) > 0, d ( P ) (2) < d d d The fst codto esues the pce pe ut s a mootocally deceasg fucto ( a deceasg ate) of the euested uatty. Ths s maly because selles wsh to ceate the cetve to buy wholesale. The secod codto esues basc ecoomc pcpals of payg oveall moe whe buyg moe. By toducg the above pce fucto, ad assumg that ay uatty ca be suppled, buyg agets have a cetve to fom a coalto wth othe buyg agets ode to ga the pce dscout. Howeve, fdg a pate has ts cost: fo each stage of the seach the pocess duces a specfc seach cost a+b, whee s the umbe of buyes epeseted by the aget. The fxed cost a s elated to the esouces a aget speds o advetsg ts pesece, locatg othe agets ad teactg wth them. The vaable cost b s assocated wth the coalto coodato the aget s eued to mata commucato wth the buyes t epesets thoughout the seach pocess. The agets ae homogeeous the sese that each aget has the same goal (to puchase a specfc poduct at the lowest total cost), howeve they ae heteogeeous the types. We classfy all the agets epesetg buyes, who ae teested buyg a oveall uatty of of the poduct, as agets of type (,). (*,k) wll be used to deote ay aget epesetg k buyes ad (,*) wll deote ay aget euestg a aggegated uatty. The (,) type aget ca ethe egage a seach to exted ts coalto, o buy the euested poduct, ts cuet cofguato wth a oveall cost of P. The agets ae self-teested ad theefoe, gve seveal alteatves, they wll pefe to select the moe beefcal oes. Sce the aget s ot coceed wth a lmted decso hozo ad the teacto wth othe agets does t mply ay ew fomato about the maket stuctue, the ts best seach stategy fo pates s seuetal. Also, spte of the exstece of seach costs, the aget s stategy s statoay a aget of ay specfc type wll ot accept a oppotuty t has eected befoehad. At ay stage of ts seach the aget adomly ecoutes oe othe aget teested the same poduct. At the ecoute, both agets wll eveal the type (oveall euested uatty ad umbe of buyes epeseted by each aget). The, each aget wll make a decso whethe to cotue seachg o to exted ts coalto stuctue by pateg wth the cuet ecouteed aget. I the latte case, the ew coalto wll take effect oly f both agets ae wllg to fom the pateshp. Upo the ceato of a ew coalto, a ew aget wll be ceated, eplacg the two agets epesetg the ot coalto. The seach ad coalto coodato costs accumulated fo each aget wll be mposed o the buyes epeseted by t, elatve to the euested uatty fom the oveall uatty. The ew aget ceated, as ay othe aget the maketplace, ca ethe buy the poduct wth a total cost of ( + ) P, o coduct a seach to futhe exted ts coalto + wth seach ad coodato costs of a + b( + ) - see Fgue 1. We stat by assumg a cotuous flow of ew agets to the maketplace, each epesetg a sgle buye. These agets ae of type (,1) whee s defed ove the teval [, ]. Sce these ae the basc bcks of futue coaltos, the potetal aget types ca be descbed ove the sem-cotuous gd Fgue 2 (possble aget types ae gay).

5 Aget A (, ) Aget B (, ) Fgue 1 Coalto Fomato The Seach fo Coalto Fomato Costly Evomets 5 Fgue 2 Possble coaltos Notce that at ths stage of the aalyss, the gay aea has o fte boudaes. Ths s smply because we cosde all possble coaltos, egadless of eulbum costats ad seach poftablty. We suggest that the aea epesetg the agets' types egaged seach (of whch we ae teested ths pape) s actually bouded. Itoducg seach ad coalto coodato costs to the model, we ca easly set uppe lmts fo ths aea. Fo each pot the hozotal axs > we ca fd a aget type (,k) fo whch t wll be o-beefcal to egage seach, egadless of the dstbuto of othe aget types cuetly egagg seach. Ths s whee: ( P P ) = a + bk (3) Sce the ght sde expesso of (3) ceases k, the obvously all othe agets of type (,), whee >k, wll pefe ot to egage seach. Theefoe the followg expesso (4) s a uppe lmt fo possble aget types egaged seach eulbum: ( P P a ) (4) ( ) = m, b, A desty fucto ( ) f, s assocated wth each type of agets egaged seach ad euestg a uatty. We assume that agets, whle goat of othe dvdual agets coalto szes ad euested uattes, ae acuated wth the oveall dstbuto of aget types the maket 4. We also assume that ths dstbuto s tme-vaat 5. Cosdeg the populato of agets egaged seach, we pove the followg Theoem (1), fo the aget's stategy eulbum., egagg a seach fo potetal pates, ode to exted ts coalto, wll use a esevato value stategy 6 accodg to a vecto Q, of uatty values (whee Q [ k] epesets the esevato uatty to be, used whe ecouteg a potetal pate of type (*,k)). (b) Q, [ k + 1] Q, [ k]. Theoem 1. (a) I eulbum, a aget of type (, ) Sketch of Poof: Aget C ( +, + ) (a) Assume that all agets ae usg a esevato uatty stategy ad cosde a (, ) type aget whch s wllg to accept aget type (, ) dug the seach. Obvously a,, whee >, wll yeld a bette beeft. Ths ca smply be acheved f the ew aget ceated fo the latte coalto wll mtate the stategy of the oe ceated fo the fst coalto (ad be accepted by all the agets acceptg the fst oe, accodg to the assumpto). At a ceta stage of the seach, alteate coalto wth a aget ( ) M =k+1 =k M =2 =1 aget type ( +, + ) wll evolve to a coalto of type ( ) k, k whch wll pefe to buy the poduct tha to exted the seach. At ths pot, the ogal membes of 4 Usg maket dcatos, spectato agets, etc.. 5 The method fo matag a steady state populato of aget type s mpotat but wll ot be dscussed ths cotext. 6 A esevato value stategy s oe whee the seache follows a esevato-value ule: It accepts all offes geate tha o eual to the esevato value, ad eects all those less tha ths value.

6 coalto ( + ) +, wll be able to puchase the poduct at pce P compaso k to pce P fo ogal membes of the coalto k + ( + ), +. Q k Q k, + 1 smply because the seach costs of the ew coalto, ceated wth, the aget epesetg moe uses, ae hghe. [] (b) [ ] [ ] Notce that Q [ k] s a vecto of a fte sze as the aea of potetal agets to fom a, coalto wth s fte (see euato (4) above). Deotg the expected cost of aget type (, ), whe usg the optmal esevato vecto Q, as V ( ) Q, ad the optmal,, cost of a coalto (, ) as V,, we obta: V, =m{ V ( ) Q, P }.,, (5) Usg euato (5) we wll be able to deteme f a aget wll egage a seach, accodg to ts type. Cosde aget of type (, ) egaged a seach, at ay gve stage of ts seach. Afte evewg the cuet potetal pate s type (, ), t has to make a decso whethe to eect ths pate ad cotue the seach o pate wth ths aget ode to exted the coalto. Cotug the seach the cuet aget cofguato wll esult a expected futue total cost of V ( ) Q. Acceptg the pateshp, wll esult,, a futue cost ( V, ) ( + ) fo aget + + ( ), f the othe aget agees to fom a coalto, o othewse wll efoce the aget to keep seachg wth a expected futue total cost of V ( ) Q. Theefoe, the optmal esevato value s the uatty,, Q [ ] = whee the aget s dffeet to the two optos:, V +, + V ( Q [ ]) V ( Q ) ( Q [ ]) ( Q [ ]),, =,, P <, + P (6), + Resultg : V + Q [ ] V ( Q ) V ( Q [ ]),, +,, =,, =, = 1,2,... (7) Q The tepetato of euato (7) s that fo ay gve aget of type (, ) +, the oveall cost whe usg ts optmal esevato vecto, Q,, euals ts elatve pat (accodg to ts uatty) the oveall cost of a coalto ( + Q [ ], + ), fo ay. Ths, chaactestc wll play a key ole ou futhe aalyss. I ode to toduce the aget s seach euatos, we eed to pove some addtoal chaactestcs of the eulbum. Fst we wll show cosstecy betwee agets decso whethe to egage a seach (Theoem 2). The we wll pove that fo ay two agets egaged a seach, the hghe type aget ( tems of euested uatty) wll use a hghe esevato value fo each (Theoem 3). Fo ths pupose fst we eed to pove Lemma 1. Lemma 1. The mpovemet a aget's utlty whe buyg the poduct a coalto wth ay gve pate, s a ceasg fucto of ts ow oveall euested uatty: d ( P P ), [ ] + dp + (8) = 0 > 0 d d d d The poof fo Lemma 1 as well as moe detaled poofs fo all followg theoems, Lemmas ad algothms ae avalable the full veso of the authos' pape 7. Theoem 2. If ts optmal stategy, a aget of type ( ) seach, the so does ay othe aget of type ( ) >. Sketch of Poof: Cosde agets of types ( ),, whee, ad ( ), chooses to egage a, whee >. Sce all othe agets wll use a esevato value stategy (accodg to Theoem 1) the the latte aget 7 Ca be foud at

7 The Seach fo Coalto Fomato Costly Evomets 7 wll be accepted by all agets acceptg the fst. By ceatg a coalto wth ay thd aget ecouteed by these two agets, aget (, ) wll ga moe compaed to the othe type (accodg to Lemma 1). Theefoe, sce both agets seach costs ae detcal, f the fst aget pefeed to seach ove buyg dectly the the secod aget s stategy would be the same. [] A ecessay step towads Theoem 3, whch deals wth the depedecy of the aget s esevato vecto ts uatty s the poof of Lemma 2. Hee we wll pove the cosstecy of the dffeet elemets the esevato vectos of two agets., whee all othe aget types use esevato uattes that cease as a fucto of the euested uattes, the followg holds: If Q [] k Q [ k], fo a specfc k, the, Q [ v] Q [ v], fo ay value, v. Namely, f oe aget s esevato uatty s hghe tha aothe s fo ay coalto sze, the ths wll be the case fo all othe coalto szes. Lemma 2. Fo ay two aget types ( ), ad (, ) Sketch of Poof fo Lemma 2: Recusvely substtute the cost fucto of aget types (, ) ad (, ), whe usg euato (7) wth =k fo the fst teato ad =1 fo each addtoal step. Evetually oe of the agets wll each a coalto fo whch t wll be o-beefcal to egage a seach (accodg to Theoem 2, t wll be aget (, ) ). Ths coalto pce, wll be the expected pce pe ut that aget (, ) wll pay (whe addg the seach costs to calculatos), accodg to (7). The othe aget s expected pce pe ut wll be lowe, sce all othe aget types use esevato uattes that cease as a fucto of the euested uattes. Repeatg the same pocess wth =v wll poduce a cotadctoy esult. [] Theoem 3. A uue Nash eulbum exsts fo the poblem whch each aget egagg a seach uses a esevato value stategy, whee the esevato value Q, [] ceases wth the aget uatty fo evey. Sketch of Poof: Cosde a aget of type (, ). Assumg all othe agets behave accodg to the theoem, we wll pove that the optmal stategy fo ay sgle aget of type (, ) s to act accodg to the theoem. We wll use the otato: [, ] = max{ Q, [ ] }, the maxmal uatty euested to be puchased by ay of the aget types (*,), acceptg aget type (, ). (, ) wll deote the last membe of the vecto Q,. Notce that V ( ) Q ca be wtte as:,, V +, + (9) V ( Q ) = E a + b + ( dualaccept) + V ( Q ) ( ot( dualaccept) ),, 1,, 1 + Hee 1(dualaccept) epesets the dcato of the evet {both agets accept each othe}. If all othe agets act accodg to the theoem, the: ( [ ] ) V,, +, ( ) = V, ( ) (10) 1 dualaccept f, d + = 1 = [ ] + Q, ad: ( [ ], ), (11) V ( ) ( ( )) = ( ) ( ), Q, 1 ot dualaccept V, Q, 1 f, d = 1 = Q, [ ]

8 Substtutg (10) ad (11) (9), we obta: ( [ ], ), V (12) + +, a + b = ( ) ( ) V, Q, f, d = 1 = [ ] + Q, ad accodg to (7): ( [ ], ), [ ] V + + Q = V,,, ( ) (13) a b f, d = [ ] [] 1 = + + Q Q,, Notce that + Q [] > +, fo ay value of wth the teval boudaes., Theefoe, accodg to theoem (2) ethe at least oe of the aget types ( + Q [], + ) ad, ( ) +, + wll egage a seach o both wll puchase the poduct the cuet cofguato. I the latte case, the tegated fucto becomes P Q [ ] P, whch s a ceasg fucto of +, + fo ay value of wth the teval boudaes (accodg to Lemma 1), ad a deceasg fucto of Q, [] (due to the atue of the pce fucto, as eflected (1)). Also, sce the othe aget s esevato uatty s ceasg fucto of, the teval uppe lmt, [, ], becomes a ceasg fucto of. Thus, ceasg must be accompaed wth a cease the teval lowe lmt Q [ ] of at least oe of the aggegated tegals, ode to mata, the eualty. Usg Lemma 2, we coclude that all othe elemets of Q, also cease. [] To coclude the eulbum aalyss, we suggest theoem 4, whch complemets theoem (2). Theoem 4. If ts optmal stategy, a aget of type ( ) seach, the the same holds fo ay othe aget of type ( ) Sketch of Poof: Cosde a aget (, ) fo whch:, chooses ot to egage a, whee <. ( ) [ ],, + + V, + = ( ) (14) a b P f, d = 1 = + Obvously ths aget s dffeet to matag a seach o puchasg the poduct ts cuet coalto stuctue. Notce that sce the lowe boud of the euato s ot flueced by chages the aget s type, the accodg to Lemma 1, the ght had sde of the euato s a ceasg fucto of. Theefoe, thee s a specfc aget type (, ), whee all othe aget types (, ), whee <, wll eve egage a seach. [] Theoems 1-4, fully outle the codtos by whch a aget wll decde to egage a seach ode to exted ts coalto, ad the chaactestcs of ts optmal stategy wth the seach. A mpotat ssue to cosde s the complexty of the euatos fom whch the eulbum stateges ca be extacted. The dffeet chaactestcs of the optmal stateges as descbed ths secto ca ad us ths task. We wll demostate such a pocess usg a vaat of the poblem the followg secto. 4. Specfc Cases I ths secto we dscuss some vaats of the geeal coalto fomato poblem modeled the pevous secto. The fst vaat, whch the coalto coodato costs ae eglgble, s a uppe lmt fo the aget types that wll egage a seach the geeal model. The secod vaat, the two-sze coalto s a good example ad a test bed fo demostatg possble uses of algothms fo dstbuted calculato of the aget s optmal stategy paametes eulbum.

9 The Seach fo Coalto Fomato Costly Evomets Coodato costs ae eglgble Cosde a maketplace whee the agets ae ot subect to coalto coodato costs, b, ad the oly cost assocated wth the seach s the fxed cost pe seach stage, a. Hee, thee s o sgfcace to the umbe of buyes a potetal pate aget s epesetg, ad the oly elevat decso agumet s the pate s oveall euested uatty. As log as the aget egages seach t wll accept ay aget t ecoutes (the seach cost s a suk cost, ad ay coalto wll cease the coalto uatty). A aget of type (,* ) wll choose to seach, oly f the expected coalto to be eached though the seach wll beeft moe tha buyg the poduct the cuet coalto cofguato. Deotg the oveall cost of aget type (,* ) f t s coductg a seach as V ( seach), ad the cost of puchasg the poduct wthout a seach as V ( buy), we obta: V + V ( seach) = a + E, V ( buy) = P (15) + Remembe that f (, ) s the dstbuto fucto of the aget types egagg a seach, eulbum, theefoe: uppe V V (16) + + E = f ( ) d, + = + I the above euato (16), epesets the hghest uatty fo whch f(,*)>0. Aget uppe (,* ) wll pefe to coduct a seach f the followg euato (17) holds, ad othewse t wll pefe to buy the poduct ts cuet cofguato: uppev + V ( buy) V ( seach) = P a f (, ) d > 0 (17) = + Notce that eulbum all agets of type, whee > uppe, wll ot fom futhe coaltos ad though the tem ( V )/( ) ca be expessed as + + P Aget pas' coaltos I ths secto we cosde a secod vaat of the poblem, coceg coaltos of aget pas. Each coalto ca be see as a beefcal pateshp amog two autoomous agets, of specfc types (,1). The easo fo lmtg the dscusso hee to coaltos of pas s twofold. Fst, the dyamcs of the atom (sze-two) coalto stuctues ceato pocesses s a basc fastuctue fo the aalyss of the lage coaltos. Thus ths ca be a useful test bed fo algothms that ca be exteded late to deal wth the geeal case. Secod, we detfy seveal applcatos whee a aget ca beeft o pefom a task oly by fdg oe pate 8. Cosde, fo example, a clet aget a Kazaa\Gutella-lke fle shag applcato, seachg fo a specfc meda fle A, ad offeg aothe specfc meda fle B. The aget ca fd a pate wth complmetay offegs, statg mmedately wth the dowload ad upload pocesses. Alteatvely, t ca look fo a bette coected aget wth the same offeg, ode to educe the dowload tme. At each stage of the seach, the aget wll have to cosde the tadeoff betwee tme saved (o dowload, by possbly fdg a bette pate) ad tme spet ( seach fo such a pate). I addto, whe cosdeg possble bette pates, the aget should take to accout these agets ow stateges ad asses the wllgess to fom a coalto wth ts type. A totally dffeet applcato ca 8 We do udestad the dffcultes of usg the abstact sze-two model these applcatos. Howeve the aalyss gve, as well as the algothms to follow, ae uue the cotext of coalto fomato fo these applcatos.

10 be foud VoIP etwokg. Hee, we ca utlze the pateshp cocept, fo sevce povdes, lookg fo pates to fom ad-hoc call tematos betwee two destatos. Each sevce povde faces dffeet pates offeg dffeet lk ualtes (tte, packet loss, etc.). Howeve testg a pate s ualty also etals costs. I the cotext of the geeal coalto fomato poblem electoc maketplace, as descbed secto 3, we may cosde ths specfc vaat whe thee mght be techcal mplemetato obstacles fo ceatg a epesetatve aget fo moe tha oe buye. Though, fo a aget that has decded to egage a seach, at each stage of ts seach pocess, afte evewg the fomato egadg the cuet potetal pate, the aget must make a decso whethe to temate the seach o to cotue. Tematg the seach wll esult mmedate puchase of the poduct wth the cuet potetal pate, f t agees, wheeas cotug the seach wll hopefully yeld a bette type pate. Notce that sce the aget s ot coceed wth coalto coodato costs, the oly elevat paamete fo ts decso s the othe aget s euested uatty. Theefoe the aget wll be usg a esevato uatty Q, stead of the esevato vecto Q as the geeal case. Whe a mutual acceptace occus the aget eds ts seach ad puchases the poduct at pce P +. Euato (9) ca ow be expessed as: V Q E k + P 1 dualaccept + V Q ot dualaccept (18) [ ] ( ) = ( ) ( ) ( ( )) 1 + whee V ( Q) V ( [1] ), 1 Q = ad k=a+b. Theoem 5. Fo ths vaat of the poblem a uue Nash eulbum exsts fo the aget types egagg a seach, whch each aget uses a esevato value (esevato uatty) stategy, wth a lowe esevato tha ts euested uatty (deved by ts type), ad ceases wth the aget s type 9. Sketch of Poof: The poof methodology esembles the oe gve fo theoem (3). The euvalet to Euato (11) s: V ( Q ) = E[ k + P ( Q ( )) + V ( Q ) ( < Q < Q )] U (19) Ad usg V ( Q ) = P (the euvalet to euato (13) of the geeal case), we obta: + Q ( ) (20) k = P P f ( )d ( ) + Q = Q + Cocludg that the oly way to mata the eualty, whe ceasg s by ceasg Q. Sce all buyg agets wll accept aget type, the ths aget s type poblem becomes a smple poblem of choosg the best esevato uatty wthout ay estcto (of acceptace by othe agets). Ths leads to Q <, ad so theefoe Q < fo ay. [] As the geeal poblem, f a aget euestg a uatty fds t beefcal to egage a seach, athe tha buy the poduct as a sgle buye, the ay othe aget euestg a uatty > wll fd the seach pocess beefcal as well. Cosde the aget type, whee V = P. Sce fo all agets of type above, V = P Q < P holds, the these agets have o cetve to abado the seach. + Agets of a type lowe tha, wll pefe to leave the maket ad ot to egage a seach at all. The type ca be calculated the cotuous case as follows: ( ) (21) k = P P f ( )d = ( ) + 9 Smla chaactestcs of eulbum as theoem 5 ae descbed [9] fo a model coceg the maxmzato of the aget s utlty whe seachg fo pas.

11 The Seach fo Coalto Fomato Costly Evomets 11 Oce agets of a specfc type have a cetve to leave the evomet, ew eulbum esevato uattes should be computed fo the emag agets. The ew calculato should be based o the updated pobablty fucto of the emag agets. Ths pocedue should be epeated utl all agets have a cetve to buy though pa coaltos. Cosde euato (20). We have show that the ght had sde of the euato s a ceasg fucto of Q, ad though oce ( ) s kow, Q ca be calculated by settg Q =, ad deceasg ths value utl the ght had sde exceeds the value k. Notce that all the agets wll accept the hghest type aget,, sce a aget s esevato uatty s always smalle tha ts euested uatty. Ths meas that agets of type ca smply calculate the esevato uatty Q by solvg (20) fo ( ) =, egadless of the othe agets esevato uattes. Ths calculato method s vald also fo all agets the peceved teval [ ] Q,, sce all othe agets esevato uattes ae lowe tha Q. Futhemoe, fo each oe of the aget types ths teval, a uue fte seuece of esevato values ca be obtaed. Each membe of such seuece (except fo the fst oe) s the esevato uatty of the fome seuece membe. Each seuece membe (except fo the last oe) s also the hghest type aget wllg to accept agets of the ext seuece membe type. Thus, each seuece membe ca be calculated usg (20), wth the fome seuece membe aget as the tegal uppe boud. Obvously, calculatg all possble seueces wll eveal the esevato values fo all aget types the teval [ ],. Howeve, havg a cotuous age of aget types makes the task mpossble. A computatoal algothm s eeded to fd a esevato uatty of a gve aget of type. Utlzg the uue chaactestcs of the eulbum, alog wth the cocept of the seueces, we suggest a algothm fo boudg the optmal esevato uatty, Q, to a teval of sze ε. Recall that each aget s esevato uatty s smalle tha ts euested uatty, ad ceases as ts type cease. Theefoe Q s bouded by the esevato uattes of two subseuet agets ay seuece. Fo the pupose of boudg Q a teval of sze ε, we should fd two aget types ad, whose esevato uattes dffeece s smalle + tha ε. Ths s doe by chagg the selecto of the two seuece ogatg types the teval [ Q, ]. Usg bay seach ove ths teval wll esue that o each step the teval boudg Q s aowed. The poposed algothm fo computg such a seuece, whch s vald fo ay dstbuto fucto f ( ), ad seach pce k, s specfed below. Algothm 4.1 (FdSeuece( stat, )). (* Iput: (1) A aget type to stat wth ; (2) A aget type stat to wth *) (* Output: a seuece, epeseted by a aay, [], whee [0]= stat, ad [last] *) s the fst seuece membe exceedg type fom below *) 1. [0]:= stat ; 2. [1]:= Q, whee Q solves (20) wth =[0] ad ( ) =[0]; 3. :=1; 4. Whle ( [ ] ) do {++; []:= Q, whee Q solves (20) wth =[-1] ad ( ) =[-2]; } 5. etu []; Usg the above pocedue, the followg scheme ca be used to calculate the esevato uatty of aget type : Algothm 4.1 (CalculateResevato(,ε )). (* Iput: (1) A aget type ; (2) level of pecso ε *) (* Output: the bouded teval fo esevato uatty of type *)

12 1. :=FdSeuece ( uppe, ); 2. If ( uppe [] 1 ) the a. Uppe= Q, whee Q solves (20) wth ad ( ) := ; b. Retu (Uppe, Uppe); 3. = FdSeuece ( lowe uppe [1], uppe [last]); 4. Whle ( uppe [last]- lowe [last]>ε ) do a. :=FdSeuece(( uppe [0]+ lowe [0])/2, uppe [last]); b. f ([last]> ) the uppe =; c. else lowe =; 5. etu( uppe [last]+ lowe [last]); Futhe explaatos ad poofs egadg the coectess of the algothms ae avalable the full veso of the authos' pape. Notce that the absece of the algothm, ay gve aget would have bee eued to compute esevato values fo a cotuous teval of types. The poposed algothm suggests evaluato of the esevato value, a fte umbe of steps, fo ay level of eued pecso. 4.3 Dscete evomet The above algothms ae coceed wth makets chaactezed by eumeated types lke gold, coffee ad chemcals. Howeve fo most poducts the electoc maketplace, the eued uatty ca be expessed oly dscete uts. I the followg paagaphs, we wll demostate how these algothms ca be adusted to hadle such dscete evomets. Ths ca be obtaed by povg that the same eulbum chaactestcs foud fo the cotuous case ae also mataed the dscete evomet. Theoem 6. Cosde a fte set of aget types, each chaactezed by (, g ), whee s the uatty euested by aget type, ad g s the popoto of ths aget type the populato. The thee s a uue Nash eulbum fo the model, whee agets use esevato values as the optmal stategy, ad: (a) A aget s esevato uatty s a ceasg fucto of ts type. (b) A aget s esevato uatty s lowe tha ts euested uatty. (c) Type aget s esevato uatty Q, satsfes: ( ) = Q ( P P ) k = g (22) + Q + Sketch of Poof: Smla poof to the oe gve secto 4.2, esultg euato (22). Usg Lemma 1 (the Lemma coces oly the pce fucto ad ts valdess s ot flueced by the dstbuto of types), we coclude that ode to keep the above euato vald, Q must be a ceasg fucto of. [] As the cotuous case, fom lemma 1 we coclude that the ght had sde of (22) s a ceasg fucto of Q, ad though oce ( ) s kow, Q ca be calculated. Theefoe, a smla algothm to that peseted secto 3.2 fo fdg a esevato uatty of a gve aget of type, ca be appled the dscete case, eplacg euato (20) wth (22). Notce, howeve, that the dscete case thee s a possblty that calculatos based o (22) wll esult a esevato uatty Q, wth o actual aget type assocated to t. Hee, ay esevato uatty take fom the teval [ Q, m( ); Q ] wll esult detcal expected mmal costs. Theefoe, fo each aget we ca defe a esevato type stead of a esevato uatty. The esevato type defes the lowest aget type a aget wll be wllg to accept as a pate. A appopate esevato aget value fo a aget type ca be calculated smply by usg euato (22): fst, fd the uppe dex based o the esevato uattes of hghe types

13 The Seach fo Coalto Fomato Costly Evomets 13 (statg wth the hghest type aget ad backwad). The, check the ght had value whe settg Q = ; =, 1,..., utl the calculated tem exceeds the value k. Ths wll eue oveall (-+1) esevato type calculatos. Notce that oce the esevato type cocept has bee adopted, a aget s esevato type s a weakly ceasg fucto of the aget s type (seveal agets may have the same esevato aget). I addto, the aget s esevato type ca actually be ts ow type (ad ot ecessaly a lowe type). Fo the pupose of calculatg Q, we ca utlze the ew acceptace types cocept, suggestg a moe effcet heustc. The algothm woks o the aay agets[] whee each membe, aget[], holds, addto to the euested uatty, ad type s pobablty felds, also potes to the hghest aget type that accepts type, ad the lowest type accepted by aget. The algothm uses two fuctos: - FdUppe(,tal) - used to assocate a gve aget of type wth the hghest aget type that wll accept t. Ths value wll be used as ( ) the calculato of Q (22). The fucto s talzed wth a hghe o eual aget type, tal, whch s kow to accept type. - FdSeuece(, stat ) A smla fucto as the cotuous case. Ths tme, howeve, each membe s the lowest aget type accepted by the fome seuece membe. The last membe the aay s the fst seuece membe smalle tha type. Algothm 4.2 (FdUppe(,tal)). Algothm 4.2 (FdSeuece( stat, )). (* Iput: (1) Aget type fo whch the calculato s euested *) (* Iput: (1) A aget type to stat wth *) stat (* (2) A tal aget type to stat wth *) (* (2) A aget type to wth *) (* Output: Updatg agets[].uppe wth appopate value *) (* Output: aay, [], epesetg a seuece, whee [0]= *) (*, ad [last]= fst aget type smalle tha *) stat 1. If (agets[].uppe==ull) (* belogg to such a seuece *) the { 2. =tal; 1. [0]:= stat ; FdUppe([0],[0]); 3. Whle (<) do { 2. [1]:= CalculateResevato([0]); 4. =+1; 3. :=1; 5. If(CalculateResevato()>) 4. Whle (( [ ] ) ad ([]!=[-1])) do{ agets[].uppe=-1; a. ++; } b. FdUppe([-1],[-2]); 6. If (=) the c. []:= Q, whee Q solves (22) wth agets[].uppe:=; =[-1] ad ( ) =[-2];} } 5. etu []; Algothm 4.2 (CalculateResevato()). (* Iput: (1) A aget type fo whch the esevato type should be calculated *) (* Output: Retus aget type 's exact esevato type *) 1. Base cases: If s hghest o lowest type aget, o uppe dex was aleady calculated the etu(calculate Q by Solvg (22)); 2. Heustc: f -1 type s kow the FdUppe(,agets[-1].uppe) ad etu(calculate Q by Solvg (22)); 3. Use fdseuece(,) to set uppe ad lowe as the two tal boudg aays as secto Fst Segmet: Set Agets[].uppe= fo evey uppe [1] ; 5. Repeat a. s fst segmet: If ( lowe [ 0] < ) the set agets[].uppe= lowe [] ad etu(calculate Q by Solvg (22)); b. Both seueces collde: If lowe [ ] = uppe[ ] fo ay legth( lowe ), the:. Shote both aays so that uppe wll stat wth ts membe, ad lowe wth ts membe +1;

14 . Fo ay [] k [ ], FdUppe( k [ ] lowe < lowe, ); uppe c. Adacet membes of seueces: If ( uppe [ ] lowe [ ]) = 1 the:. fo ay lowe[ + 1] < k < uppe[ + 1] Set agets[k].uppe= lowe [] ;. Shote both aays so uppe ad lowe wll stat wth the +1 membe; d. s ew fst segmet: If (( lowe [0]) ) the. FdUppe(, uppe [ 0] );. Retu (Calculate Q by Solvg (22)); e. = fdseuece(( uppe [] 0 + lowe[0])/ 2), uppe[ last]; f. Update boudg seueces: If ([last]=) the. FdUppe([last],[last-1]);. Retu (Calculate Q by Solvg (22)); g. Else f ([last]>) the uppe = ; h. Else lowe = ; The fucto CalculateResevato etus the lowest aget type a aget wll accept ts eulbum optmal stategy. I the wost case, whee each aget type s esevato uatty euals ts ow uatty, the algothm wll calculate a esevato uatty fo all agets wth a type hghe tha (sce we ae cosdeg dscete types ad Q ca be easly calculated usg (22)). Usg smulatos, we tested the poposed algothm pefomaces whe ceatg adom aget types wth adom pobabltes. Fo each smulato we set a adom seach cost, ad used a typcal pce fucto 10 (accodg to (1) ad (2)). Aget type dstbuto was ceated adomly fo each smulato, ad the dscete types (uattes) wee adomly daw fo each smulato fom the teval (1,10). We summaze the esults of the smulatos Fgue 3. Fgue 3 descbes the aveage 11 extet of usage of euato (22), whch s the most esouce tesve calculato module ay algothm wth ths poblem s cotext. The hozotal axs epesets the umbe of aget types used. The vetcal axs epesets the pecetage of aget types fo whch we had to calculate a esevato uatty, out of the total umbe of agets wth a hghe type tha the eued oe. I the absece of a heustc, a aget s foced to calculate all esevato uattes fo types geate tha ts ow (epeseted as the oe huded pecet le the gaph), thus the tme eued by ou poposed heustc s sgfcatly lowe tha ay staghtfowad algothm. Notce that the algothms pefomace mpoves as the evomet (umbe of aget types) ceases. Pecetage of eued fome types esevato 100% 80% 60% Oveall cost No Seach 2 o SC 2 wth SC 40% 20% Numbe of Aget types 0% Usg Algothm Regula Calculato Fgue 3 Algothms pefomace N o SC Reuested uatty No seach 2 - wt h SC 2 - o SC N - o SC Fgue 4 Utlty Compaso 10 We used a pce fucto of type a-b*l(c*x). 11 Each gaph pot s the aveage of 5000 adom smulatos fo calculatg a esevato aget fo a type outsde the fst (tval) segmet.

15 The Seach fo Coalto Fomato Costly Evomets 15 Fally, we peset Fgue 4, a example of the aget s utlty gaph, the cotext of a seach (Recall that the lowe the oveall cost, the bette the aget s utlty). Fou sceaos wee aalyzed usg dect calculato ad smulato, whee the aget types wee daw adomly fom the teval [1,10]. Seach cost was ~0.5 cets, ad poduct cost 0.2+2/x- 1/x^2, whee x s the eued uatty. I the fst sceao (epeseted by the most uppe cuve, maked as o seach the leged), agets dd ot egage a seach ad puchased the poduct wth a cost coelated to the eued uatty. Secodly, we used the smulato to obta the aget s costs whe coductg a seach a costly evomet (epeseted by the mddle uppe cuve, maked as 2 wth Seach Costs the leged). The esevato uattes wee calculated fo each aget accodg to the poposed algothms. The, a uppe lmt fo the seach beefts was set, by calculatg the aget s cost whe seachg fo pates a o-costly evomet (epeseted by the mddle lowe cuve, maked as 2 o Seach Costs the leged). Ths s ot the case of oe uted coalto but athe a soluto whee each aget foms a coalto wth aothe aget of ts ow type 12 (a kow esult fo the two-sze costless model see [1]). A theoetcal -sze coalto wth o seach Seach Cost costs s epeseted by the lowe cuve (maked as 7.00 N o Seach Costs). Hee all agets fom 6.50 togethe oe bg coalto, ad buy the poduct 6.00 wth a pce P. Ths cuve ca be see as a 5.50 lowe boud fo cost of coaltos the -sze 5.00 sceao wth seach costs. Aget 4.50 Type I Fgue 4, all agets have a cetve to coduct a seach. The case whee aget s cost whe buyg dectly the poduct s hghe tha whe coductg a seach, s well suppoted by the 3.69 model, ad ths aget type wll abado the seach eulbum. Ths was fully aalyzed secto Oveall Cost 3. Theefoe eulbum, all agets egage Fgue 5 Aget type ad seach cost effects seach wll beeft moe tha the cost of buyg dectly the poduct Notce that fgue 4 the dffeece betwee the esults obtaed fom t he costly seach ad those belogg to the theoetcal costless evomet ae sgfcatly small. Howeve, ths s well coelated wth seach cost. A example of the effect the seach cost has o the peceved utlty fo a gve aget type s gve fgue 5. As the oveall seach costs cease, two cotast effects occu: the aget becomes moe attactve to hghe type agets sce the beeft of beg selectve deceases, ad the aget s ow paymets fo seach ceases. These two factos ae the easo fo the local maxmum fgue Coclusos I ths pape we aalyzed the model ad eulbum optmal stateges fo agets egaged costly seaches fo potetal coalto pates. The cetve fo coopeato s As f all buyes get togethe oe place ad fd pas that belogs to the coe.

16 a volume dscout ad the seach s chaactezed wth a fxed cost fo locatg pates ad vaable costs fo sub-coalto coodato towads the fal coalto stuctues. Udestadg the chaactestcs of aget stateges eulbum s the ma bck stoe fo ay algothm ad heustc to be used fo solvg the geeal case. We have show a compehesve aalyss of two algothms fo solvg specfc vaats of the poblem whee the coalto sze s estcted to pa pateshps. We do see these algothms as a basc fastuctue fo suggestg futhe algothms fo the geeal -sze case. Ths s maly because of the smla eulbum stuctue (the cotuum of agets egaged a seach) ad the specal chaactestcs of the aget s stateges as we poved ths pape (esevato vectos\values cease type). The poposed tools fo calculatg the eulbum stateges of agets whe seachg pa coaltos ca be used seveal plausble applcatos ad evomets (Gutella\Kazaa, VoIP sevce povdes, etc.). We pla to exted the eseach towads completg the algothms to be used the geeal case ad smulatos that ca descbe the evoluto of steady-state aget types' dstbutos. Though we have focused o the o-tasfeable utlty case, we see geat mpotace udestadg the chages such models whe the agets ca egotate ove the suplus of the pateshp. Refeeces 1. G. S. Becke. Theoy of maage: Pat I, Joual of Poltcal Ecoomy, 81(4): , S. Beba, J. Vassleva. Log-tem Coaltos fo the Electoc Maketplace, B. Spece (ed.) Poceedgs of the E-Commece Applcatos Wokshop, Caada AI Cofeece, Ottawa, C. H. Books, E.H. Dufee ad A. Amstog. A Itoducto to Cogegatg MultAget Systems. I Poc. of ICMAS, Bosto, K. Budett ad R. Wght. Two-sded seach wth otasfeable utlty, Revew of Ecoomc Dyamcs, 1: , A. R. Geewald ad J. O. Kephat. Shopbots ad pcebots. I Poc. of IJCAI, K. Lema, O. Shehoy. Coalto Fomato fo Lage Scale Electoc Makets. Poc. of ICMAS C. L ad K. P. Sycaa. Algothm fo combatoal coalto fomato ad payoff dvso a electoc maketplace. AAMAS 2002: J. McMlla ad M. Rothschld. Seach, Ch.27 of Hadbook of Game Theoy wth Ecoomc Applcatos, Vol. 2, , ed. Robet J. Auma ad Segu Hat, Amstedam, P. Moga. A Model of Seach, Coodato, ad Maket Segmetato, mmeo, SUNY Buffalo, T. Sadholm, K. Laso, M. Adesso, O.Shehoy, F. Tohme. Coalto stuctue geeato wth wost-case guaatees. Atfcal Itellgece. 111: , T. Sadholm ad V. Lesse. Coaltos amog computatoally bouded agets. Atfcal Itellgece, 94:99-137, S. Se ad P.S. Dutta. Seachg fo optmal coalto stuctues. I Poceedgs of ICMAS, O. Shehoy ad S. Kaus. Methods fo task allocato va aget coalto fomato. Atfcal Itellgece, 101: , L. Smth. The maage model wth seach fctos. Joual of Poltcal Ecoomy, 2003 (To appea). 15. N. Tsvetovat, K. P. Sycaa, Y. Che ad J. Yg. Custome Coaltos Electoc Makets, AMEC, J. Yamamoto ad K. Sycaa. A Stable ad Effcet Buye Coalto Fomato Scheme fo E- maketplaces. Poceedgs of Autoomous Agets, 2001.