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1 ExprtMdiatd Sarch Mnal Chhabra Rnsslar Polytchnic Inst. Dpt. of Computr Scinc Troy, NY, USA Sanmay Das Rnsslar Polytchnic Inst. Dpt. of Computr Scinc Troy, NY, USA David Sarn BarIlan Univrsity Dpt. of Computr Scinc Ramat Gan, Isral ABSTRACT Incrasingly in both traditional, and spcially Intrntbasd marktplacs, knowldg is bcoming a tradd commodity. This papr considrs th impact of th prsnc of knowldgbrokrs, or xprts, on sarchbasd markts with noisy signals. For xampl, considr a consumr looking for a usd car on a larg Intrnt marktplac. Sh ss noisy signals of th tru valu of any car sh looks at th advrtismnt for, and can disambiguat this signal by paying for th srvics of an xprt (for xampl, gtting a Carfax rport, or taking th car to a mchanic for an inspction). Both th consumr and th xprt ar rational, slfintrstd agnts. W prsnt a modl for such sarch nvironmnts, and analyz svral aspcts of th modl, making thr main contributions: (1) W driv th consumr s optimal sarch stratgy in nvironmnts with noisy signals, with and without th option of consulting an xprt; (2) W find th optimal stratgy for maximizing th xprt s profit; (3) W study th option of markt dsignrs to subsidiz sarch in a way that improvs ovrall social wlfar. W illustrat our rsults in th contxt of a plausibl distribution of signals and valus. Catgoris and Subjct Dscriptors J.4 [Social and Bhavioral Scincs]: Economics Gnral Trms Algorithms, Economics Kywords Economicallymotivd agnts, Modling th dynamics of MAS 1. INTRODUCTION In many multiagnt systm (MAS) sttings, agnts ngag in onsidd sarch [13, 6]. This is a procss in which an agnt facs a stram of opportunitis that aris squntially, and th procss trminats whn th agnt picks on of thos opportunitis. A classic xampl is a consumr looking to buy a usd car. Sh will typically invstigat cars on at a tim until dciding upon on sh wants. Similar sttings Cit as: ExprtMdiatd Sarch, Mnal Chhabra, Sanmay Das, and David Sarn, Proc. of 1th Int. Conf. on Autonomous Agnts and Multiagnt Systms (AAMAS 211), Tumr, Yolum, Sonnbrg and Ston (ds.), May, 2 6, 211, Taipi, Taiwan, pp. XXXXXX. Copyright c 211, Intrnational Foundation for Autonomous Agnts and Multiagnt Systms ( All rights rsrvd. can b found in jobsarch, hous sarch and othr applications [12, 15]. In modrn lctronic marktplacs, sarch is likly to bcom incrasingly important as, with th prolifration of possibl sllrs of a good, consumrs will turn to artificial agnts, whom w trm sarchrs, to find th bst prics or valus for itms thy ar intrstd in acquiring. Th dcision making complxity in onsidd sarch usually ariss from th fact that thr is a cost incurrd in finding out th tru valu of any opportunity ncountrd. For xampl, thr is a cost to arranging a mting to tst driv a car you ar considring purchasing. Th sarchr thus nds to trad off th potntial bnfit of continuing to sarch and sing a mor valuabl opportunity with th costs incurrd in doing so. Th optimal stopping rul for such sarch problms has bn widly studid, and is oftn a rsrvation stratgy, whr th sarchr should trminat sarch upon ncountring an opportunity which has a valu abov a crtain rsrvation valu or thrshold [18, 13]. Most modls assum that th sarchr obtains th xact tru valu of th opportunitis it ncountrs. Howvr, in many ralistic sttings, sarch is inhrntly noisy and sarchrs may only obtain a noisy signal of th tru valu. For xampl, th drivtrain of a usd car may not b in good condition, vn if th body of th car looks trrific. Th rlaxation of th assumption of prfct valus not only changs th optimal stratgy for a sarchr, it also lads to a nich in th marktplac for nw knowldgbrokrs. Th knowldg brokrs, or xprts, ar srvic providrs whos main rol is to inform consumrs or sarchrs about th valus of opportunitis. An xprt offrs th sarchr th option to obtain a mor prcis stimat of th valu of an opportunity in qustion, in xchang for th paymnt of a f (which covrs th cost of providing th srvic as wll as th profit of th xprt). To continu with th usd car xampl, whn th agnt is intrigud by a particular car and wants to larn mor about it, sh could tak th car to a mchanic who could invstigat th car in mor dtail to mak sur it is not a lmon. Th xprt nd not b a mchanic it could b, for xampl, an indpndnt agncy, lik Carfax, that monitors th rcordd history of transactions, rpairs, claims, tc. on cars. It can also b a rpatd visit of th sarchr to s th car, possibly bringing an xprincd frind for a mor thorough xamination. In all ths cass, mor accurat information is obtaind for an additional cost (ithr montary or quivalnt). In this papr w invstigat optimal sarch and mchanism dsign in nvironmnts whr sarchrs obsrv noisy signals and can obtain (i.., qury th xprt for) th ac
2 tual valus for a f. Our main contributions ar thrfold: First, w introduc a spcific modl of onsidd sarch with noisy signals and prov that th optimal sarch rul, for a larg class of rallif sttings, is rsrvationvalu basd. Scond, w formally introduc th option of consulting an xprt in such noisy sarch nvironmnts, and driv th optimal stratgy for th sarchr givn a cost of consulting th xprt, as wll as th profitmaximizing pric for th xprt to charg for its srvics. As part of th analysis w prov that undr th standard assumption that highr signals ar good nws (i.., th distribution of th tru valu conditional on a highr signal stochastically dominats th distribution conditional on a lowr signal), th optimal sarch stratgy is charactrizd by a doubl rsrvation valu stratgy, whrin th sarchr rjcts all signals blow a crtain thrshold, rsuming sarch, and accpts all signals abov anothr thrshold, thus trminating sarch, without qurying th xprt; th agnt quris th xprt for all signals that ar btwn th two thrsholds. Finally, w study a markt dsign mchanism (introduction of a subsidy) with th potntial to improv social wlfar in such domains. Ths mchanisms may b implmntd by ithr an lctronic marktplac that mploys th xprt (for xampl a wbsit for usd cars that has a rlationship with a providr of car history rports), or an ntity with rgulatory powr lik th govrnmnt. Along with th gnral thory, w illustrat our rsults in a spcific, plausibl distributional modl, in which th tru valu is always boundd by th signal valu, and th probability monotonically dcays as th discrpancy btwn th two incrass. 2. THE GENERAL MODEL Th standard onsidd sarch problm [13] considrs an agnt or sarchr facing an infinit stram of opportunitis from which sh nds to choos on. Whil th spcific valu v of ach futur opportunity is unknown to th sarchr, sh is acquaintd with th (stationary) probability distribution function from which opportunitis ar drawn, dnotd f v(x). Th sarchr can larn th valu of an opportunity for a cost c s (ithr montary or in trms of rsourcs that nd to b consumd for this purpos) and hr goal is to maximiz th nt bnfit, dfind as th valu of th opportunity vntually pickd minus th ovrall cost incurrd during th sarch. Having no a priori information about any spcific opportunity, th sarchr rviws th opportunitis sh ncountrs squntially and sts hr optimal stopping rul. Th stopping rul spcifis whn to trminat and whn to rsum sarch, basd on th opportunitis ncountrd. Our modl rlaxs th standard assumption that th sarchr rcivs th xact tru valu of an opportunity. Instad, w assum that th sarchr rcivs, at cost c s, a noisy signal s, corrlatd with th tru valu according to a known probability dnsity function f s(s v). In addition, th sarchr may qury and obtain from a third party (th xprt) th tru valu v of an opportunity for which signal s was rcivd, by paying an additional f c. Th goal of th sarchr is to maximiz th total utility rcivd i.., th xpctd valu of th opportunity vntually pickd minus th xpctd cost of sarch and xprt fs paid along th way. Th first qustion that ariss is how to charactriz th optimal stratgy for th sarchr. A scond qustion is how th xprt sts hr srvic f c. In this papr w considr a monopolist providr of xprt srvics. Th sarchr s optimal stratgy is dirctly influncd by c, and thus implicitly dtrmins th xpctd numbr of tims th srvics of th xprt ar rquird, and thus th xprt s rvnu. Th problm can b thought of as a Stacklbrg gam [5] whr th xprt is th first movr, and wants to maximiz hr profits with rspct to th f c sh chargs sarchrs. Th nw sarch modl raiss intrsting nw qustions about markt dsign. Assuming xognity of opportunitis, social wlfar, dnotd W, is a function of th xpctd valu to sarchrs and th xpctd profit of th xprt. (W abstract away from modling th xistnc of sllrs of opportunitis, instad viwing thm as xognous, or ls as bing offrd at som fair pric by th sllr.) W assum that th providr of xprt srvics has alrady prformd th startup work ncssary, and only pays a marginal cost d pr qury, and that social wlfar is additiv. Sinc th procss scals up linarly in th numbr of sarchrs, w can simply considr th intractions involving a singl sarchr and th xprt. Th social wlfar is thn th sum of th xpctd nt bnfit to th sarchr and th xpctd profit of th xprt. It turns out that social wlfar can b significantly affctd (and improvd) if th markt dsignr (or a rgulator lik th govrnmnt) subsidizs quris by compnsating th xprt in ordr to rduc qury costs to th sarchr. W now turn to dvloping th mathmatical machinry to addrss ths problms. 3. OPTIMAL POLICIES In this sction, w analyz th sarchr s optimal sarch stratgy and hr xpctd us of th xprt s srvics, givn th f c st by th xprt. Th analysis builds on th trivial nonnoisy modl and gradually adds th complxitis of signals and having th xprt option. From th sarchr s optimal sarch stratgy w driv th xprt s xpctd bnfits as a function of th f sh sts, nabling maximization of th xprt s rvnu. OnSidd Sarch. Th optimal sarch stratgy for th standard modl, whr th actual valu of an opportunity can b obtaind at cost c s, can b found in th xtnsiv litratur of sarch thory [13, 6]. In this cas, th sarchr follows a rsrvationvalu rul: sh rviws opportunitis squntially (in random ordr) and trminats th sarch onc a valu gratr than a rsrvation valu x is rvald, whr th rsrvation valu x satisfis: c s = y=x (y x )f v(y)dy (1) Intuitivly, x is th valu whr th sarchr is prcisly indiffrnt: th xpctd marginal bnfit from continuing sarch and obtaining th valu of th nxt opportunity xactly quals th cost of obtaining that additional valu. Th rsrvation proprty of th optimal stratgy drivs from th stationarity of th problm rsuming th sarch placs th sarchr at th sam position as at th bginning of th sarch [13]. Consquntly, a sarchr that follows a rsrvation valu stratgy will nvr dcid to accpt an opportunity sh has onc rjctd and th optimal sarch stratgy is th sam whthr or not rcall is prmittd. Th xpctd numbr of sarch itrations is simply th invrs of th succss probability,, sinc this bcoms a Brnoulli 1 1 F v(x )
3 sampling procss, as opportunitis aris indpndntly at ach itration. OnSidd Sarch with Noisy Signals. Bfor bginning th analysis of sarch with noisy signals, w mphasiz that, givn f v(x) and f s(s v), w can also driv th distribution of th signal rcivd from a random opportunity, f s(x), and th distribution of tru valus conditional on signals, f v(v s) (th conditionals ar intrchangabl by Bays law). In many domains, it may b asir to assss/larn f s(s) than f v(v s) as most past xprinc involvs signals, with th actual valu rvald for only a subst of ths signals. Whn th sarchr rcivs a noisy signal rathr than th actual valu of an opportunity, thr is no guarant that th optimal stratgy is rsrvationvalu basd as in th cas whr valus obtaind ar crtain. Indd, th stationarity of th problm still holds, and an opportunity that has bn rjctd will nvr b rcalld. Yt, in th absnc of any rstriction ovr f s(s v), th optimal stratgy is basd on a st S of signalvalu intrvals for which th sarchr trminats th sarch. Th xpctd valu in this cas, dnotd V (S), is givn by: V (S) = c s + Pr(s / S)V (S) + P r(s S)E[v s S] = c s + V (S) f s(s) ds + f s(s)e[v s] ds (2) s / S s S Th fact that th optimal stratgy may not b rsrvationvalu basd in this cas is bcaus thr may b no corrlation btwn th signal and th tru valu of th opportunity. Nvrthlss, in most rallif cass, thr is a natural corrlation btwn signals and tru valus. In particular, a fairly wak and commonly usd rstriction on th conditional distribution of th tru valu givn th signal gos a long way towards allowing us to rcaptur a simpl spac of optimal stratgis. This is th rstriction that highr signal valus ar good nws in th sns that whn s 1 > s 2, th conditional distribution of v givn s 1 firstordr stochastically dominats that of v givn s 2 [19, 14]. Th condition rquirs that givn two signals s 1 and s 2 whr s 1 > s 2, th probability that th actual valu is gratr than any particular valu v is gratr for th cas whr th sarchr rcivs signal s 1. Formally: Dfinition 1. Highr signals ar good nws (HSGN) assumption: If s 1 > s 2, thn, y, F v(y s 1) F v(y s 2). This nabls us to prov th following thorm. Thorm 1. For any probability dnsity function f v(v s) satisfying th HSGN assumption, th optimal sarch stratgy is a rsrvationvalu rul, whr th rsrvation valu, t, satisfis: ( ) c s = E[v s] E[v t ] ds (3) s=t Proof: Th proof is basd on showing that, if according to th optimal sarch stratgy th sarchr should rsum hr sarch givn a signal s, thn sh must ncssarily also do so givn any othr signal s < s. Lt V dnot th xpctd bnfit to th sarchr if rsuming th sarch. Sinc th optimal stratgy givn signal s is to rsum sarch, w know V > E[v s]. Givn th HSGN assumption, y yfv(y s ) dy < yfv(y s) dy holds for y s < s. Thrfor, V > E[v s ], proving that th optimal stratgy is rsrvationvalu. Thn, th xpctd valu of th sarchr whn using rsrvation signal t is givn by: t V (t) = c s + V (t) f s(s) ds + E[v s]f s(s) ds s= s=t = cs + E[v s]fs(s) ds s=t (4) 1 F s(t) whr F s(s) is th cumulativ distribution function of th signal s. Stting th first drivativ according to t of Equation 4 to zro w obtain: V (t ) = E[v t ]. Th scond drivativ for t that satisfis th lattr quality confirms that this is indd a global maximum. Finally, using intgration by parts ovr th drivativ according to t of Equation 4 w obtain Equation 3, and th valu t can b calculatd accordingly. Th social wlfar W is th xpctd gain to th sarchr from following th optimal stratgy, V (t ) = E[v t ]. Th xpctd numbr of sarch itrations is is a Brnoulli sampling procss. 1 1 F s(t ), sinc this Th Exprt Option. Th introduction of an xprt xtnds th numbr of dcision altrnativs availabl to th sarchr. Whn rciving a noisy signal of th tru valu, sh can choos to (1) rjct th offr without qurying th xprt, paying sarch cost c s to rval th signal for th nxt offr; (2) qury th xprt to obtain th tru valu, paying a cost c, and thn mak a dcision; or (3) accpt th offr without qurying th xprt, rciving th (unknown) tru valu of th offr. In cas (2), thr is an additional dcision to b mad, whthr to rsum sarch or not, aftr th tru valu v is rvald. As in th noxprt cas, a solution for a gnral dnsity function f v(v s) dictats an optimal stratgy of a complx structur. In our cas, th optimal stratgy will hav th form of (S, S, V ), whr: (a) S is a st of signal intrvals for which th sarchr should rsum hr sarch without qurying th xprt; (b) S is a st of signal intrvals for which th sarchr should trminat hr sarch without qurying th xprt (and pick th opportunity associatd with this signal); and (c) for any signal that is not in S or S th sarchr should qury th xprt, and trminat th sarch if th valu obtaind is abov a thrshold V, and rsum othrwis. Th valu V is th xpctd bnfit from rsuming th sarch and is givn by th following modification of Equation 2: V = c s + V f s(s) ds c s S ( V f s(s) V f v(x s) dx+ s {S,S } x=v s {S,S } f s(s) ds+ ) xf v(x s) dx ds + f s(s)e[v s] ds (5) s S Th first lmnt on th right hand sid of th quation applis to th cas of rsuming sarch, in which cas th sarchr continus with an xpctd bnfit V. Th scond lmnt is th xpctd paymnt to th xprt. Th nxt lmnts rlat to th cas whr th sarch is rsumd basd on th valu rcivd from th xprt (in which cas th xpctd rvnu is onc again V ) and whr th sarch is
4 trminatd (with th valu E[v s] obtaind as th rvnu), rspctivly. Finally, th last lmnt applis to th cas whr th sarchr trminats th sarch without qurying th xprt. W can show that undr th HSGN assumption, ach of th sts S and S actually contains a singl intrval of signals, as illustratd in Figur 1. Thorm 2. For f v(y s) satisfying th HSGN assumption (Dfinition 1), th optimal sarch stratgy can b dscribd by th tupl (t l, t u, V ), whr: (a) t l is a signal thrshold blow which th sarch should b rsumd; (b) t u is a signal thrshold abov which th sarch should b trminatd and th currnt opportunity pickd; and (c) th xprt should b qurid givn any signal t l < s < t u and th opportunity should b accptd (and sarch trminatd) if th valu obtaind from th xprt is abov th xpctd valu of rsuming th sarch, V, othrwis sarch should rsum (s Figur 1). Th valus t l, t u and V can b calculatd from solving th st of Equations 68: x=v tl tu V = c s + V f s(s) ds c f s(s) ds+ s= s=t l tu ( V f s(s) V f v(x s) dx ds+ s=t l x= ) xf v(x s) dx ds + f s(s)e[v s] ds (6) s=t u c = c = y=v V (y V )f v(y t l ) dy (7) (V y)f v(y t u) dy (8) Proof: Th proof xtnds th mthodology usd for proving Thorm 1. W first show that if, according to th optimal sarch stratgy th sarchr should rsum hr sarch givn a signal s, thn sh must also do so givn any othr signal s < s. Thn, w show that if, according to th optimal sarch stratgy th sarchr should trminat hr sarch givn a signal s, thn sh must also ncssarily do so givn any othr signal s > s. Again, w us V to dnot th xpctd bnfit to th sarchr if rsuming th sarch. If th optimal stratgy givn signal s is to rsum sarch thn th following two inqualitis should hold, dscribing th supriority of rsuming sarch ovr trminating sarch (Equation 9) and qurying th xprt (Equation 1): V > E[v s] (9) V > c + y=v yf v(y s) dy + V V y= f v(y s) dy (1) Givn th HSGN assumption and sinc s < s, Equation 9 holds also for s, and so dos Equation 1 (which can b formalizd aftr som mathmatical manipulation as: V > c + V + (y V )fv(y s) dy). Th proof for y=v s > s is similar: th xpctd cost of accpting th currnt opportunity can b shown to dominat both rsuming th sarch and qurying th xprt. W omit th dtails bcaus of spac considrations. Th optimal stratgy can thus b dscribd by th tupl (t l, t u, V ) as statd in th Thorm. Thrfor, Equation 5 transforms into Equation 6. Taking th drivativ of Equation 6 w.r.t. t l and quating to zro, w obtain a uniqu t l which maximizs th xpctd bnfit (vrifid by scond drivativ), and similarly for t u. Finally, using intgration by parts ovr th drivativs of Equation 6 w.r.t. t l and t u w obtain Equations 78. Intuitivly, t l is th point at which a sarchr is indiffrnt btwn ithr rsuming th sarch or qurying th xprt and t u is th point at which a sarchr is indiffrnt btwn ithr trminating th sarch or qurying th xprt. Th cost of purchasing th xprt s srvics must qual two diffrnt things: (1) th xpctd savings from rsuming th sarch whn th actual utility from th currnt opportunity (which is not known) turns out to b gratr than what can b gaind from rsuming th sarch (onc it is rvald) (this is th condition for t l ); (2) th xpctd savings from trminating th sarch in thos cass whr th actual utility from th currnt opportunity (onc rvald) is lss than what can b gaind from rsuming th sarch (for t u). Figur 1: Charactrization of th optimal stratgy for noisy sarch with an xprt. Th sarchr quris th xprt if s [t l, t u] and accpts th offr if th worth is gratr than th valu of rsuming th sarch V. Th sarchr rjcts and rsums sarch if s < t l and accpts and trminats sarch if s > t u, both without qurying th xprt. It is notabl that thr is also a rasonabl dgnrat cas whr t l = t u(= t). This happns whn th cost of qurying is so high that it nvr maks sns to ngag th xprt s srvics. In this cas, a dirct indiffrnc constraint xists at th thrshold t, whr accpting th offr yilds th sam xpctd valu as continuing sarch, so V = E[v t]. This can b solvd in combination with Equation 4, sinc thr ar now only two rlvant variabls. Expctd numbr of quris: Th sarch stratgy (t l, t u, V ) dfins how many tims th xprt s srvics ar consultd. In ordr to comput th xpctd numbr of quris, w considr four diffrnt typs of transitions in th systm. Lt A b th probability that th sarchr quris th xprt and thn dos not accpt, rsuming sarch, B b th probability that th sarchr rsums sarch without qurying, C b th probability that th sarchr trminats without qurying, and D b th probability that th sarchr quris th xprt and trminats sarch. Thn: A = Pr(t l s t u and v < V ) (11) B = Pr(s < t l ) (12) C = Pr(s > t u) (13) D = Pr(t l s t u and v V ) (14) Lt P j dnot th probability that th sarchr quris th xprt xactly j tims bfor trminating. Th sarchr can trminat sarch aftr xactly j quris in on of two ways: ithr sh maks j 1 quris, thn quris th xprt and chooss to trminat, or sh maks j quris and
5 thn chooss to trminat without qurying th xprt. Factoring in all th possibl ways of intrlaving j quris with an arbitrary numbr of tims that th sarchr chooss to continu sarch without qurying th xprt, w gt: P j = = m= (m + j 1)! m!(j 1)! A j (1 B) j ( D A + C 1 B ) A j 1 B m D + m= (m + j)! A j B m C m!j! Thn th xpctd numbr of quris, whn charging an xprt f c, is givn by j= jpj, yilding η c = E(Numbr of quris c ) = (1 B)D + CA (1 B A) 2 (15) Expctd numbr of opportunitis xamind: Using th sam notation as abov, w s that th probability of trminating th sarch at any itration is C + D, and ths ar indpndnt Brnoulli draws at ach opportunity. Thrfor th xpctation of th numbr of opportunitis xamind is simply η s = 1/(C + D). Expctd profit of th xprt: Lt d dnot th marginal cost of th srvic th xprt is providing. Th xpctd profit of th xprt is thn simply π = E(Profit) = (c d )η c Th xprt can maximiz th abov xprssion with rspct to c (η c dcrass as c incrass) to find th profit maximizing pric to charg sarchrs. Social Wlfar: Th social wlfar is givn by th sum of all partis involvd, thus far just th sarchr and th xprt. Of cours this gnralizs to multipl agnts as wll, sinc ach sarch procss would b indpndnt. W dfin: W = V c + π (16) whr c is th f that maximizs th xprt s profit. 4. MARKET DESIGN Abov, w hav dscribd th basics of sarch in such xprtmdiatd markts. In this sction, w dscrib possibl uss of th thory dscribd abov to improv th dsign of markts in which such sarch taks plac. Th prospct of dsigning or significantly influncing ths markts is not rmot. Considr th dsign of a larg scal Intrnt wbsit lik autotradr.com. Th listings for cars that usrs s ar signals, and thy may b unsur of a car s tru worth. autotradr can partnr with a providr of rports lik Carfax, to mak it asy for usrs to look up a car s worth. In ordr to b gnral, lt us rfr to autotradr as th markt (or in som instancs as th markt dsignr ) and Carfax as th xprt. Th markt wants to attract customrs to it, rathr than to rival markts. Th bst way of doing this is to provid customrs with a high valu shopping xprinc. Th xprt wishs to maximiz its profits. Sinc th markt and th xprt both hav significant powr, it is rasonabl to imagin thm coming up with diffrnt modls of th kinds of rlationships thy may hav. It is notabl that whil w ar thinking about privat markts hr, this ntir discussion is qually rlvant to a big playr lik th govrnmnt as markt dsignr, and indpndnt providrs of xprt srvics. In ordr to provid customrs with th highst valu shopping xprinc, th markt may choos to subsidiz th cost of xprt srvics. A typical problm with subsidization is that it oftn dcrass social wlfar bcaus th tru cost of whatvr is bing subsidizd is hiddn from th consumr, lading to ovrconsumption of th rsourc. In this instanc, howvr, th natural xistnc of many monopolis in xprt srvics, combind with th xistnc of sarch frictions, mak it quit possibl that subsidis will in fact incras social wlfar. W show in Sction 5 that this is in fact th cas for som natural distributions. Th basic framwork of subsidization works as follows. Suppos a monopolist providr of xprt srvics maximizs its profits by stting th qurying cost to c, yilding an xpctd profit π = (c d )η c (this discussion is on a prconsumr basis). Th markt dsignr can stp in and ngotiat a rduction of th f c chargd by th xprt, for th bnfit of th agnts. In rturn for th xprt s agrmnt, th markt dsignr will nd to offr a prconsumr paymnt β to th xprt, which fully compnsats th xprt for th dcrasd rvnu, laving hr total profit th sam. Sinc c < c, η > c ηc (th consumr quris mor oftn bcaus sh has to pay lss). Th compnsation for a rqustd dcras in th xprt s f from c to c is thus β = (c d )η c (c d )η c. Th ovrall wlfar pr agnt in this cas incrass by V c Vc, whr V c and Vc ar th xpctd valu of sarchrs according to Equation 68, whn th xprt uss a f c and c rspctivly, at a cost β to th markt dsignr. Sinc th xprt is fully compnsatd for hr loss du to th dcras in hr f, th chang in th ovrall social wlfar is V c Vc β. Undr th nw pricing schm c, and givn th subsidy β, th social wlfar is givn by W = V c + π β. In th following sction w illustrat how such a subsidy β can hav a positiv chang ovr th social wlfar. An intrsting spcial cas to considr is whn d =. W can think of this cas as digital srvics, analogous to digital goods lik music MP3s producing an xtra on of ths has zro marginal cost. Similarly, producing an xtra lctronic history of a car, lik a Carfax rport, can b considrd to hav zro marginal cost. In this cas, thr is no socital cost to highr utilization of th xprt s srvics, so subsidy is wlfar improving right up to making th srvic fr. Ths ar th cass whr it could mak sns for th markt dsignr or govrnmnt to tak ovr offring th srvic thmslvs, and making it fr, potntially lvraging th incrasd wlfar of consumrs by attracting mor consumrs to thir markt, or incrasing thir fs. 5. A SPECIFIC EXAMPLE In this sction w illustrat th thortical analysis givn in th formr sction for a particular plausibl distribution of signals and valus. This cas illustrats th gnral structur of th solutions of th modl and dmonstrats how intrvntions by th markt dsignr can incras social wlfar. W considr a cas whr th signal is an uppr bound on th tru valu. Going back to th usd car xampl, sllrs and dalrs offring cars for sal usually mak cosmtic improvmnts to th cars in qustion, and procd to advrtis thm in th most appaling mannr possibl, hiding dfcts using tmporary fixs. Spcifically, w assum signals s ar
6 1.9.8 Signal(s) vs Sarch Cost(c s ) for c =.5 t l t u V(sarchr s utility) vs c s (sarch cost) c =.3 c =.6 c =.12 No Exprt.6 signal >.7 V > c s > c s > (a) Signal thrshold vs c s (b) Variation of agnt utility w.r.t c s and c Figur 2: Effct of c s on th signal thrsholds (t l,t u) and agnt utility V uniformly distributd on [, 1], and th conditional dnsity of tru valus is linar on [, s]. Thus { 1 if < s < 1 for y s f s(s) = f v(y s) = { 2y s 2 othrwis Othrwis W can substitut in ths distributions in Equations 6 through 8 and simplify. From Equation 6: V = c s + V t l c (t u t l ) + V 3 (t u t l ) 3t ut l V = 2t l 3 + V 3 c (from Equation 7) c = V 3t 2 l (V y)f v(y t u) dy = V 3 3t 2 u + 1 t2 l 3 (from Equation 8) W can find fasibl solutions of this systm for diffrnt paramtr valus, as long as th condition t l < t u holds. Othrwis, whn c is high nough that qurying nvr maks sns, a singl thrshold srvs as th optimal stratgy, as in th cas with no xprt. In th lattr cas, w obtain th optimal rsrvation valu to b usd by th sarchr from Equation 3, yilding t = 1 3c s. Th othr thing to not hr is that Equation 8 abov is for th cas whn th support on signal s is unboundd. Whn thr is an uppr limit on s i. s m for som m (as is th cas hr, whr signals ar boundd in [, 1]), onc t u rachs m (w nvr buy without qurying), Equation 8 dos not hold. Now th systm rjcts if th signal is blow t l or quris if it is abov. Figur 2(a) illustrats how th rsrvation valus t l and t u chang as a function of c s for c =.5. Th vrtical axis is th intrval of signals. As can b sn from th graph, for vry small sarch cost (c s) valus, th sarchr nvr trminats sarch without qurying th xprt. 1 Du to th low sarch cost th sarchr is bttr off only qurying th xprt whn a high signal is rcivd. Th xprt option is prfrrd ovr accpting without qurying th xprt for thos high signals, bcaus if a low valu is rcivd from th xprt thn th cost of finding a nw opportunity with a high signal is low. As th sarch cost c s incrass, thr is som 1 Whn sarch costs ar th problm is illdfind. Th first point on th graph shows an xtrmly low, but nonzro sarch cost. In this cas t u = 1 and t l is almost 1, but not xactly, and th xprt is again always qurid. π > Profit of Exprt vs Cost pr qury for c s =.1 d = d =.25 d = Cost pr qury(c ) > Figur 3: Exprt s profit as a function of c and d for c s =.1. bhavior that is not immdiatly intuitiv. Th rsrvation valus t l and t u bcom closr to ach othr until coinciding at c s =.8, at which point th xprt is nvr qurid anymor. Th rason for this is that th ovrall valu of continuing sarch gos down significantly as c s incrass, thrfor th cost of qurying th xprt bcoms a mor significant fraction of th total cost, making it comparativly lss dsirabl. This is a good xampl of th additional complxity of analyzing a systm with an xprt, bcaus in th static sns th cost of consulting th xprt dos not chang, so th fact that th xprt should b consultd lss and lss frquntly is countrintuitiv. Figur 2(b) illustrats th chang in th sarchr s wlfar as a function of th sarch cost, c s, for diffrnt valus of th srvic f, c, chargd by th xprt. As xpctd, th sarchr s wlfar is bttr with th xprt option than without, and th smallr th f chargd by th xprt, th bttr th sarchr s wlfar. Expctd numbr of quris. W can find th xpctd numbr of quris in this cas by using our knowldg of th uniform distribution and th nois distribution in Equations 1115, yilding A = V 2 ( 1 t l 1 t u ); B = t l ; C = (1 t u); D = t u t l V 2 ( 1 t l 1 t u ) which giv th final xprssions: η c = t l t u(t u t l ) t ut 2 l t lv 2 t ut l + t ; 1 uv 2 ηcs = 1 t l V 2 ( 1 t l 1 t u )
7 Th monopolist xprt s optimal stratgy. Using th abov drivations, it is now asy to calculat numrically th valu of c that maximizs π, trading off dcrasing numbr of quris η and incrasing rvnu pr qury c. Figur 3 shows xampls of th graph of xpctd profit for th xprt as a function of th xprt s f, c, for diffrnt valus of d, th marginal cost to th xprt of producing an xtra xprt rport, for c s =.1. Subsidizing th xprt. As discussd abov, th markt dsignr or th govrnmnt can guarant th rduction of th xprt s charg from c to c, kping π constant, by paying a prconsumr subsidy β to th xprt. Figur 4 shows th improvmnt in social wlfar, dnotd δ(w ), as a function of th subsidy paid to th xprt, β, for various d valus (whr c s =.1). From th graphs, w do indd find that subsidization can lad to substantial incrass in social wlfar, vn whn thr is a significant marginal cost of producing an xprt rport. Whil this could b from a rduction in sarch and qury costs or an incras in th xpctd valu of th opportunity finally takn, th data in Tabl 1 indicats that th lattr xplanation is th dominant factor in this cas. It is also worth noting that social wlfar is maximizd at th point whr th sarchr pays xactly d pr qury, thus fully intrnalizing th cost to th xprt of producing th xtra rport. If th sarchr had to pay lss, it would lad to infficint ovrconsumption of xprt srvics, whras if sh had to pay mor, th xpctd dclin in th valu sh rcivs from participating in th sarch procss would outwigh th savings to th markt dsignr or govrnmnt from having to pay lss subsidy. 6. RELATED WORK Th autonomous agnts litratur has oftn considrd th problm of sarch which incurs a cost [2, 9, 1]. Th undrlying foundation for such analysis is sarch thory [4, 15], and in particular, its onsidd branch which considrs an individual squntially rviwing diffrnt opportunitis from which sh can choos only on. Th sarch incurs a cost and th individual is intrstd in minimizing xpctd cost or maximizing xpctd utility ([13, 7, 16], and rfrncs thrin). To th bst of our knowldg, non of th onsidd sarch litratur in ithr sarch thory or multiagnt systms, has considrd th rsulting markt dynamics whn obsrvations ar not accurat and mor accurat information can b purchasd from a slfintrstd agnt. Rlaxation of th prfct signals assumption is typically found in modls of twosidd sarch [1], including marriag or dating markts [3] and markts with intrviwing [11]. Th litratur has not to this point focusd on th dcision problms facd by slfintrstd knowldg brokrs, or how thir prsnc affcts th markt. In trms of markt intrvntions, th twosidd sarch litratur has considrd th impact of sarch frictions on labor markts (th 21 Nobl Priz in Economics was awardd for this work [15, 4]). On classic rgulatory intrvntion in ths modls is th introduction of a minimum wag, which can b shown to b wlfar incrasing in many contxts [8], but w ar unawar on any work on subsidizing providrs of xprt knowldg, as w discuss hr. 7. DISCUSSION AND CONCLUSIONS Th powr of modling markts using sarch thory is wll stablishd in th litratur on conomics and social scinc [12; 1, intr alia]. It has ld to brakthroughs in undrstanding many domains, ranging from basic bilatral trad [17] to labor markts [15]. Whil knowldg has always bn an conomically valuabl commodity, its rol continus to grow in th Intrnt ag. Th ubiquity of lctronic rcords and communications mans thr is an incrasing rol for knowldg brokrs in today s marktplacs. For xampl, it is now fasibl for agncis lik Carfax to collct th availabl rcords of vry rcordd accidnt, insuranc claim, oil chang, inspction, and so on for vry car. Th prsnc of such knowldg brokrs ncssitat that w tak thm into account in modling th sarch procss of consumrs. This papr taks th first stp in this dirction. W introduc a sarch modl in which agnts rciv noisy signals of th tru valu of an objct, and can pay an xprt to rval mor information. W show that, for a natural and gnral class of distributions, th sarchrs optimal stratgy is a doubl rsrvation stratgy, whr sh maintains two thrsholds, an uppr and a lowr on. Whn sh rcivs a signal blow hr lowr thrshold, sh rjcts it immdiatly. Similarly, whn sh rcivs a signal abov hr uppr thrshold, sh accpts it immdiatly. Only whn th signal is btwn th thrsholds dos sh consult th xprt, dtrmining whthr to continu sarching or accpt th offr basd on th information rvald by th xprt. In such modls, thr is scop for an authority lik a markt dsignr or rgulator to improv social wlfar by subsidizing th cost of qurying th xprt. Th bnfit to th sarchr of having lss friction in th procss could potntially mor than offst th cost to th authority. Th downsid would b that if th authority providd too much subsidy, this could lad to infficint ovrconsumption of costly (to produc) xprt srvics. By solving th modl for a natural combination of th distribution of signals and th conditional distribution of th tru valu givn th signal, w can analyz such qustions mor spcifically. W show that in our xampl, subsidis can in fact b wlfarnhancing, and, in fact, social wlfar is maximizd whn th sarchr has to pay xactly th marginal production cost of xprt srvics. Both th modl and our rsults ar significant for dsignrs of markts in which consumrs will sarch and th nd for xprt srvics will aris naturally (lik an Intrnt marktplac for usd cars), bcaus by nhancing social wlfar, th markt dsignr can tak markt shar away from comptitors, or prhaps charg highr commissions, bcaus it is offring a bttr marktplac for consumrs. Thr ar svral dirctions for futur rsarch, from both th xprt s prspctiv and th marktdsignr s. An intrsting problm for th monopolist xprt is th optimal pricing of bundls of quris, whr agnts must purchas a bundl, instad of individual rports. Mor ralistic modling of startup costs and hnc th avrag supply curv of xprt srvics (instad of th marginal cost considrd hr) may also xplain a richr rang of bhaviors. Th xistnc of th xprt has ramifications byond onsidd sarch, our focus in this papr. For xampl, in twosidd sarch markts lik labor markts, thr may b diffrnt typs of xprts: thos who conduct background chcks, for xampl, or providrs who run indpndnt tsting srvics to vt potntial mploys. What ar thir incntivs, and
8 Without Subsidy With Subsidy d c η c c c sη s Worth V W subsidy c η c c c sη s Worth V W Tabl 1: Th diffrnt componnts of social wlfar with and without subsidy for c s =.1. Worth is th xpctd valu of th opportunity vntually pickd. Initially th dcras in qury cost contributs mor to th incras in social wlfar, but as d incrass, this contribution bcoms lss significant. Not that th first two columns in th cas without subsidy ar similar bcaus th profitmaximizing c is th sam and th sarchr s cost dpnds only on valu of slctd c, not d..25 Incras in Social wlfar vs Subsidy for c s =.1 and d = 6 x 1 3 Incras in Social wlfar vs Subsidy for c s =.1 and d =.25.1 Incras in Social wlfar vs Subsidy for c s =.1 and d = Incras in social wlfar >.15.1 Incras in social wlfar > Incras in social wlfar > Subsidy(β) > Subsidy(β) > Subsidy(β) > for d = for d =.25 for d =.5 Figur 4: Incras in social wlfar vs subsidy. Whn thr is no marginal cost (d = ), it is bttr for th markt dsignr to mak th srvic availabl for fr but whn thr is som marginal cost involvd, thn incras in social wlfar is a concav pakd at marginal cost. how do ths affct twosidd sarch markts? From th markt dsignr s point of viw, nw altrnativs to subsidization as a mans for improving social wlfar can b xplord,.g., inducing comptition, or provision of xprt srvics by th markt dsignr hrslf (.g., a govrnmnt taks ovr th rol of providing xprt srvics). 8. ACKNOWLEDGMENTS W ar gratful to th USIsral BSF for supporting this rsarch undr Grant Das also acknowldgs support from an NSF CAREER award (952918), and Sarn is partially supportd by ISF grants 141/9. 9. REFERENCES [1] K. Burdtt and R. Wright. Twosidd sarch with nontransfrabl utility. Rviw of Economic Dynamics, 1:22 245, [2] S. Choi and J. Liu. Optimal timconstraind trading stratgis for autonomous agnts. In Proc. MAMA, 2. [3] S. Das and E. Kamnica. Twosidd bandits and th dating markt. In Proc. IJCAI, 25. [4] P. Diamond. Aggrgat dmand managmnt in sarch quilibrium. Th Journal of Political Economy, 9(5): , [5] D. Fudnbrg and J. Tirol. Gam Thory. MIT Prss, [6] J. Gilbrt and F. Mostllr. Rcognizing th maximum of a squnc. Journal of th Amrican Statistical Association, 61:35 73, [7] A. GrosfldNir, D. Sarn, and I. Spiglr. Modling th sarch for th last costly opportunity. Europan Journal of Oprational Rsarch, 197(2): , 29. [8] L. Ho. Wag subsidis as a labour markt policy tool. Policy Scincs, 33(1):89 1, 2. [9] J. Kphart and A. Grnwald. Shopbot conomics. JAAMAS, 5(3): , 22. [1] J. O. Kphart, J. E. Hanson, and A. R. Grnwald. Dynamic pricing by softwar agnts. Computr Ntworks, 32: , 2. [11] R. L and M. Schwarz. Intrviwing in twosidd matching markts. NBER Working Papr, 29. [12] S. Lippman and J. McCall. Th conomics of job sarch: A survy. Economic Inquiry, 14: , [13] J. McMillan and M. Rothschild. Sarch. In R. Aumann and S. Hart, ditors, Handbook of Gam Thory with Economic Applications, pags [14] P. Milgrom. Good nws and bad nws: Rprsntation thorms and applications. Th Bll Journal of Economics, pags , [15] D. Mortnsn and C. Pissarids. Nw dvlopmnts in modls of sarch in th labor markt. Handbook of labor conomics, 3: , [16] M. Rothschild. Sarching for th lowst pric whn th distribution of prics is unknown. Journal of Political Economy, 82: , [17] A. Rubinstin and A. Wolinsky. Middlmn. Th Quartrly Journal of Economics, 12(3): , [18] M. Witzman. Optimal sarch for th bst altrnativ. Economtrica: Journal of th Economtric Socity, 47(3): , [19] R. Wright. Job sarch and cyclical unmploymnt. Th Journal of Political Economy, 94(1):38 55, 1986.
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