The literature on manyserver approximations provides significant simplifications toward the optimal capacity


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1 Publshed onlne ahead of prnt November 13, 2009 Copyrght: INFORMS holds copyrght to ths Artcles n Advance verson, whch s made avalable to nsttutonal subscrbers. The fle may not be posted on any other webste, ncludng the author s ste. Please send any questons regardng ths polcy to Artcles n Advance, pp ssn essn MANUFACTURING & SERVICE OPERATIONS MANAGEMENT Prcng and Dmensonng Competng LargeScale Servce Provders Gad Allon, Ita Gurvch Kellogg School of Management, Northwestern Unversty, Evanston, Illnos nforms do /msom INFORMS The lterature on manyserver approxmatons provdes sgnfcant smplfcatons toward the optmal capacty szng of largescale monopolsts, but falls short of provdng smlar smplfcatons for a compettve settng n whch each frm s decson s affected by ts compettors actons. In ths paper, we ntroduce a framework that combnes manyserver heavytraffc analyss wth the noton of epslonnash equlbrum and apply t to the study of equlbra n a market wth multple largescale servce provders that compete on both prces and response tmes. In an analogy to flud and dffuson approxmatons for queueng systems, we ntroduce the notons of flud game and dffuson game. The proposed framework allows us to provde frstorder and secondorder characterzaton results for the equlbra n these markets. We use our results to provde nsghts nto the prce and servcelevel choces n the market and, n partcular, nto the mpact of market scale on the nterdependence between these two strategc decsons. Key words: competton; games; approxmate equlbrum; asymptotc analyss; heavy traffc; Halfn Whtt regme; servces Hstory: Receved: August 2, 2007; accepted August 24, Publshed onlne n Artcles n Advance. 1. Introducton An mportant attrbute of customer experence n varous servce ndustres s the tme spent on watng for servce. As a result, customers may consder both the prces and the delay guarantees n choosng whch provder to patronze. The purpose of ths paper s to study the equlbra that emerge n markets n whch largescale servce provders compete on both prces and customer delays. We focus on understandng the mpact of market sze on the way n whch dfferent frms make these prcng and servcelevel choces. Toward that end, we analyze a model of competton wth multple largescale servce provders n whch the demand faced by each frm depends on the prces and the servce levels offered by all frms n the market. Quanttatvely, our goal s to characterze the capacty and prcng choces of the frms n the market. Qualtatvely, we wsh to understand how the strategc postonng of the frm depends on ts own characterstcs vsàvs those of ts compettors. To address these ssues, we must frst examne the frm s capacty decson. When servce levels are measured through delays, a decson to mprove the servce level requres an nvestment n ncreased capacty. Hence, n postonng tself n the market, a frm must wegh the benefts of hgh servce levels aganst the assocated capacty costs. The benefts of mproved servce levels are not, however, ndependent of other compettors actons, and hence the task of determnng the tradeoff between effcency and servce qualty s a nontrval one. Ths tradeoff s a nontrval one even for a monopolst. Indeed, when capacty s adjusted by determnng the number of servce representatves rather than by adjustng the servce rates), the problem of optmzng capacty costs versus watngtmerelated costs s a complex optmzaton problem. Although t can often be solved numercally, numercal solutons fal to provde any structural nsghts. An alternatve to exact numercal solutons s the use of approxmatons. Manyserver approxmatons provde a smplfed means to approach ths problem see, e.g., Borst et al. 2004; addtonal references are provded n 2). In ths type of analyss, one consders a sequence of queueng systems wth growng demand and wth capacty that grows accordngly to satsfy 1
2 Allon and Gurvch: Prcng and Dmensonng Competng LargeScale Servce Provders 2 Manufacturng & Servce Operatons Management, Artcles n Advance, pp. 1 21, 2009 INFORMS Copyrght: INFORMS holds copyrght to ths Artcles n Advance verson, whch s made avalable to nsttutonal subscrbers. The fle may not be posted on any other webste, ncludng the author s ste. Please send any questons regardng ths polcy to ths demand). One then dentfes solutons that are asymptotcally optmal as the demand grows. The asymptotcally optmal soluton s nearly optmal for a gven system provded that the demand t faces s large enough. The lterature on manyserver approxmatons not only provdes a tractable way to characterze nearly optmal capacty and prce decsons for monopolsts; t also relates a frm s operatonal regme to the relatve sgnfcance the frm ascrbes to servce levels as opposed to capacty costs see the dscusson of Borst et al n 2). The frm s operatonal regme dctates how the frm should optmally respond to an ncrease n market sze. Some frms should use ther growth to ncrease ther utlzaton and thus ther cost effcency) wthout mprovng ther servce level. These frms are sad to operate n the effcencydrven ED) regme. Ther emphass on effcency results n a stuaton n whch when the market s large) almost all customers experence some delay before beng served. Some frms wll sacrfce effcency for qualty. In response to an ncrease n market sze, these frms wll match an ncrease n utlzaton wth yet a greater mprovement n servce levels. These operate n the qualtydrven QD) regme. In ths regme, almost all customers are served mmedately. An ntermedate regme s the qualty and effcencydrven QED) regme also known as the Halfn Whtt regme after the authors that frst formalzed t see 2) whch corresponds to frms for whch effcency and qualty are of smlar mportance. These frms wll match the ncrease n effcency wth a comparable ncrease n the qualty of servce. In ths regme, a nontrval fracton but not all) of the customers receve servce mmedately, wthout any delay, but, at the same tme, the effcency s very hgh. The regmecharacterzaton results are proved n the lterature for servce provders that are monopolsts n ther respectve markets. For a monopolst, the manyserver approxmatons provde a tractable way to characterze ts optmal capacty choces. The compettve settng s, however, more complex. Not only does the dscrete nature of the capacty choce make the task of dentfyng equlbra and obtanng quanttatve and qualtatve results more arduous, the task s further complcated by the fact that the demand the frm experences s not fxed, nor does t depend solely on the frm s own prcng and servcelevel choces. Rather, the demand depends on the choces made by all frms n the market. It seems plausble, however, that manyserver approxmatons can be embedded wthn a game theoretc analyss to characterze equlbra n these markets. We pursue ths drecton by constructng a formal framework that draws on manyserver approxmatons, as developed for monopolsts, and by applyng t to the study of equlbra n compettve markets. Two fundamental questons are central to the study of equlbra n compettve markets: a) Do Nash equlbra exst n the market exstence)? And b) gven some sort of exstence, s t possble to characterze the set of equlbra to obtan qualtatve nsghts nto market outcomes characterzaton)? Startng wth exstence, we note that the concept of Nash equlbrum may be too restrctve for descrbng servcemarket behavor. It s known that Nash equlbra need not exst even under the most common demand functons, such as Multnomal Logt MNL), and the smplest supply systems, such as the M/M/1 queue see, e.g., Cachon and Harker 2002). Ths nonexstence s often drven by economes of scale, but s further exacerbated by the lumpy nature of the capacty n settngs where the capacty choces are made n a dscrete manner, by adjustng the number of servce representatves. Nonexstence of Nash equlbra does not rule out the possblty to say somethng meanngful about the market outcomes. It s desrable n these cases to fnd a less strngent framework that wll allow for some characterzaton of the market outcomes. The mathematcal framework we propose s desgned to address two concerns: a) n terms of exstence, we want to overcome the restrctve nature of the Nash equlbrum n addressng relatvely general demand functons as well as supply facltes that are more general than the M/M/1 queue, and b) n terms of characterzaton, we want to handle the complex nature of the servce system by combnng approxmatons for the queueng dynamcs wth a game theoretc framework. Our framework stands on three pllars: ) Nash or approxmate) equlbra, ) manyserver approxmatons, and ) market replcaton. The ntroducton of approxmate equlbrum s amed, ntally,
3 Allon and Gurvch: Prcng and Dmensonng Competng LargeScale Servce Provders Manufacturng & Servce Operatons Management, Artcles n Advance, pp. 1 21, 2009 INFORMS 3 Copyrght: INFORMS holds copyrght to ths Artcles n Advance verson, whch s made avalable to nsttutonal subscrbers. The fle may not be posted on any other webste, ncludng the author s ste. Please send any questons regardng ths polcy to to overcome the nonexstence of Nash equlbra. Its eventual benefts, however, go beyond ths ntal objectve when combned wth market replcaton and manyserver approxmatons. We examne the behavor of equlbra not n a sngle market, but rather n a sequence of markets wth ncreasng aggregate demand these are referred to as replcated markets. We emphasze that, when characterzng the equlbrum behavor n these markets, we assume that the set of frms s gven; n other words, we do not consder the possblty of frms extng or enterng the ndustry. Our framework can be thought of as a formalzaton of the use of flud and dffuson models of queueng systems n a compettve settng. In the optmzaton of queueng systems, the orgnal system s often replaced by a determnstc approxmaton a flud model whose analyss sheds lght on frstorder propertes of the underlyng queueng system, such as ts stochastc stablty. In a second step, the orgnal queueng system s replaced by a more refned) stochastc model, whch s often referred to as a dffuson model of the queueng system. The latter s often more tractable than the orgnal queueng system and can be used to dentfy propertes that are asymptotcally correct for the orgnal queueng system. In partcular, the dffuson model s often used to construct nearly optmal solutons for optmzaton problems that are ntractable for the orgnal queueng system. Analogously, our framework constructs approxmate games for the game played among the servce provders. We frst ntroduce a flud game that s obtaned from the orgnal game by dsregardng the stochastc nature of congeston. Buldng on the analyss of the flud game, we then ntroduce a more refned dffuson game. Ths game s obtaned from the orgnal one by replacng each of the servce provders wth ts manyserver dffuson approxmaton. We then relate the equlbra of ths new approxmate game wth the outcomes of the orgnal market. As n manyserver approxmatons, the dea s to show that the equlbra of the dffuson game are, n a sense, asymptotcally correct for the orgnal game. The noton of Nash equlbrum plays a key role n rgorously establshng these approxmatons. The approxmate equlbrum concept provdes a formal way to construct envelopes for the profts of the frms n the market. Although a Nash equlbrum mght not exst and the market mght oscllate, the Nash dentfes a regon wthn whch the profts of all the frms n the market must resde. The ultmate goal of ths paper s, however, to understand market postonng n terms of the actons of the dfferent frms.e., the prces and servce levels that the frms choose). The challenge s, then, to use the envelopes on the proft functons to construct correspondng envelopes n the acton space around the approxmate prce and servcelevel choces. To our knowledge, there are no general results that, gven an Nash equlbrum, dentfy the maxmum that the frms can devate n ther actons wthout causng a devaton n the profts that would compromse the approxmate equlbrum. Such results, whch characterze the maxmal oscllatons of the prces and servce levels around some pont, are thus unque, and are obtaned through the framework that we develop by employng the concepts of replcated markets n conjuncton wth heavytraffc queueng theory. Havng constructed the analyss framework, we use t to provde an analytcal characterzaton of the approxmate equlbra n the market wth multple servce provders. The characterzaton s then used to obtan some nsghts nto the market outcomes. Our nsghts are concerned wth the relatonshp between the prce and servcelevel choces and, n partcular, between the functons n the frm that make these choces marketng and operatons. We dentfy a onesded decouplng phenomenon by whch the frms can be farly close to optmalty by allowng the prcesettng functon to lead, and the operatons functon to follow. The approxmate equlbra n both the flud and dffuson level exhbt a sequental structure: one can frst pretend that the customers n the market are entrely nsenstve to servce levels and solve a smple prce competton game. The real prce and servce level choces, n the approxmate equlbrum, are then a functon of ths nave prce vector but are, otherwse, ndependent of each other. Ths ndependence allows the marketng functon to set the nave prce vector as an ntal estmate for the optmal prce and leave for the operatons functon the task of settng the servce levels and adjustng the prces that are eventually offered to the customers.
4 Allon and Gurvch: Prcng and Dmensonng Competng LargeScale Servce Provders 4 Manufacturng & Servce Operatons Management, Artcles n Advance, pp. 1 21, 2009 INFORMS Copyrght: INFORMS holds copyrght to ths Artcles n Advance verson, whch s made avalable to nsttutonal subscrbers. The fle may not be posted on any other webste, ncludng the author s ste. Please send any questons regardng ths polcy to The analyss of the dffuson game provdes a refned understandng of the operatonal regme of a frm and the mplcaton of ths regme on the frm s prce choces. We show that both the QED and the ED regmes can emerge n equlbrum, thus, we appear to be the frst to show how these dfferent regmes emerge n a compettve market and, n partcular, how dfferent demand structures lead to the dfferent regmes. We show that, whle the actual choces of servce level and prce depend on the characterstcs of all frms n the market, the operatonal regme of a frm s determned solely by ts own ntrnsc propertes. Consequently, when dfferent frms have dfferent senstvtes, they may operate n dfferent operatonal regmes, and thus poston themselves dfferently n the face of ncreased market sze. We also fnd that the operatonal regme of a frm determnes the degrees of freedom t has n prcng. We show that, compared wth frms that operate n the QED regme, frms operatng n the ED regme have greater freedom n choosng the prces they charge. Ther freedom s reflected by the fact that they have a larger set from whch they can choose ther prces wth hardly any compromse to ther profts. Thus, frms n the ED regme can keep the onesded decouplng n the sense that the marketng functon can pay less attenton to the operatonal sde n determnng the prces. Frms operatng n the QED regme need to pay greater attenton to ther prce choces. For these frms, the decouplng s weaker and a feedback mechansm s requred between the manner n whch the frm operates.e., the operatonal regme), and ts prcng. 2. Lterature Revew Our work bulds on two streams of lterature: a) game theory and ts applcaton to competton analyss, and b) queueng theory and ts applcaton to the study of largescale servce systems. These two streams are not dsjont, and some recent work les at the ntersecton of the two. The lterature on competton n servce ndustres dates back to the late 1970s; see, e.g., Levhar and Lusk 1978). Although t ntally focused on a sngle attrbute prce or servce level or a smple aggregaton of the two) more recent work treats prces and watngtme standards as fully ndependent attrbutes. We follow Allon and Federgruen 2007) n consderng a model wth dfferentated servces.e., a model n whch other servce attrbutes matter along wth the full prce) and n treatng delay and prce as ndependent attrbutes. We refer the reader to Allon and Federgruen 2007) for a systematc dscusson of exstng results n ths context, and to Hassn and Havv 2003) for a general survey of queueng models wth competton. Allon and Federgruen 2007) and others focus on provdng a full analytcal characterzaton of the Nash equlbra that arse n a market n whch the market sze s fxed. In contrast, we focus on understandng the mpact of the market scale on the prces, servce levels, and nterdependences between the two. Furthermore, our framework sgnfcantly expands the famly of models that can be studed. Ths expanson s n two drectons: ) Frst, most of the lterature on competton n servces models the supply sde va M/G/1 queues, whch mples, n turn, that capacty choces are made contnuously by adjustng the servce rates. We, n contrast, allow the servce provder to adjust ts capacty by ncreasng or decreasng the ntegervalued) number of servce representatves, gvng rse to an M/M/N queue, where N s a decson made by the frm. Ths s a common method of capacty management of servce provders and one that renders the Nash equlbrum ntractable for characterzaton. ) Second, n terms of the demand model, our framework allows for sgnfcant generalty n modelng the customers senstvty to servce levels and prces. From the game theoretc perspectve, the noton of Nash equlbra that we use has been used extensvely n the economcs lterature. For the basc defnton we rely on Tjs 1981). Dxon 1987) uses the dea of market replcaton n the context of prce competton. Although our form of replcaton s dfferent Dxon 1987) ncreases the number of frms n the market, whereas we ncrease the aggregate demand volume our analyss s nspred by hs concept. Prevous work n game theory has focused on three types of sequences of games: ) sequences of games n whch the acton space s gettng ncreasngly fner, and although each game has dscrete acton space, the lmtng game has contnuous acton space see, e.g., Whtt 1980); ) sequences of games n whch
5 Allon and Gurvch: Prcng and Dmensonng Competng LargeScale Servce Provders Manufacturng & Servce Operatons Management, Artcles n Advance, pp. 1 21, 2009 INFORMS 5 Copyrght: INFORMS holds copyrght to ths Artcles n Advance verson, whch s made avalable to nsttutonal subscrbers. The fle may not be posted on any other webste, ncludng the author s ste. Please send any questons regardng ths polcy to the number of agents grows Lu et al. 2009); and ) a sequence of replcated markets wth growng market sze see, e.g., Dxon 1987). We use the thrd framework. The applcaton of Nash n the operatons lterature s rare. Lu et al. 2009) use ths concept n a settng where Nash equlbrum does exst but the Nash equlbrum concept stll helps n characterzng the equlbrum n the game when the number of players s large and approaches a contnuum. Dasc 2003) uses ths concept n the context of subgameperfect equlbrum. We appear to be the frst to combne the concepts of Nash, market replcaton, and heavy traffc n the context of operatonal settngs. Ths combnaton allows us to dscuss both stablty and trends n markets of competng servce provders. Wth respect to the relevant queueng lterature, our work bulds on the lterature on manyserver approxmatons of monopolsts, startng wth the semnal work of Halfn and Whtt 1981). Although manyserver approxmatons exsted before, the result of Halfn and Whtt 1981) made such approxmatons relevant for varous applcatons, such as callcenter operatons see the survey paper by Gans et al. 2003) and, more recently, healthcare operatons see, e.g., Jennngs and de Vércourt 2008). Halfn and Whtt 1981) consder a sequence of M/M/N queues and show that, as the demand rate grows, the probablty of delay PW > 0 converges to a number strctly between 0 and 1 f and only f the number of agents grows wth accordng to a squareroot safetystaffng rule,.e., f and only f N = R + R + o R 1) where R= / s the offered load and s a strctly postve constant. In partcular, a servce provder that uses the squareroot safetystaffng rule to determne ts capacty wll utlze ts servers very effcently and, at the same tme, have a nontrval fracton of ts customers enter servce mmedately upon ther arrval. Ths combnaton of hgh effcency and hgh servce level provdes the justfcaton for the name qualtyand effcencydrven) regme. Whereas Halfn and Whtt 1981) dentfed ths regme, Borst et al. 2004) placed manyserver approxmatons wthn a broad economc framework that consders the problem of mnmzng capacty and watngtme costs. They showed how the QED regme emerges as the optmal economcal choce n some cases, but also dentfed condtons under whch other regmes namely, the QD and ED regmes emerge as the optmal choces. A key dea n the framework developed n Borst et al. 2004) s to replace the orgnal optmzaton problem, whch nvolves the ntegervalued number of servers, by a tractable contnuous and convex optmzaton problem. Usng smlar deas, we wll construct a contnuous and tractable game the dffuson game that wll serve as an approxmaton for the orgnal, relatvely ntractable one. Recently, Kumar and Randhawa 2009) extended the work of Borst et al. 2004) to a settng n whch the customers are prce and delay senstve, and consequently the demand s not fxed. Ther work shows how dfferent operatonal regmes emerge dependng on the convexty or concavty) of the delay cost functon. Smlar dependences wll also emerge wthn the compettve settng that we study n ths paper. Addtonal examples of work that provdes staffng and prcng recommendatons for largescale monopolsts facng delay and prcesenstve customers are the papers by Whtt 2003) and Maglaras and Zeev 2003, 2005). All the work mentoned above consders a monopolst wth a demand rate that may depend only on the congeston experenced and the prce charged by ths sngle player. Our paper appears to be the frst to show how the dfferent operatonal regmes emerge n a compettve settng wth multple players and to dentfy the dependences between the operatonal regmes and the prce choces of the varous players n the market Organzaton of ths Paper The rest of ths paper s organzed as follows. We present the model n 3. Secton 4 s concerned wth regme characterzaton. In 5 we ntroduce and characterze the flud game and dscuss ts mplcatons. In 6 we turn to the dffuson game, whch s concerned wth a refned understandng of the frms choces. Conclusons and drectons for future research are dscussed n 7. The ecompanon to ths paper contans numercal examples as well as generalzatons to some of the results n 5 and 6.
6 Allon and Gurvch: Prcng and Dmensonng Competng LargeScale Servce Provders 6 Manufacturng & Servce Operatons Management, Artcles n Advance, pp. 1 21, 2009 INFORMS Copyrght: INFORMS holds copyrght to ths Artcles n Advance verson, whch s made avalable to nsttutonal subscrbers. The fle may not be posted on any other webste, ncludng the author s ste. Please send any questons regardng ths polcy to Our approach n presentng the results s to state them formally wthn the paper, accompaned by varous examples for llustraton. Most of the detaled proofs are relegated to the onlne appendx that can be downloaded from edu/faculty/gurvch/personal/. 3. The Model We consder a market wth a set I = 1 I of competng servce frms, each operatng as an M/M/N faclty and servng arrvng customers n a frstcome, frstserved manner. Frm postons tself n the market by selectng a prce p and a delay guarantee T. We restrct our attenton to servcelevel guarantees that are gven n terms of the customers delay rather than ther whole sojourn tme n the system. Havng chosen the delay target T, the servce provder guarantees that the followng servcelevel constrant wll be satsfed: W >T 2) where W s the steadystate delay wth the th provder, and 0 << 1 s the satsfacton probablty. Ths form of servcelevel constrant s consstent wth the ndustry practce that commonly uses = 02 correspondng to 80% of the servce requests beng answered wthn target; see, e.g., Anton 2001). 1 In ths paper, we study a model of competton where both the prces and the servce levels are set smultaneously. Our results contnue to hold f the strateges are chosen sequentally prce frst or servce level frst). Servce rates are assumed to be fxed and equal to for frm, and the capactes are adjusted through the choce of the number of agents or servce representatves), denoted by the ntegervalued decson varable N. We assume that there s an upper bound T >0 on the acceptable servce levels. For example, n call centers, t s clear that a watng tme of more than a day s unacceptable. Frm chooses T 0 T and needs to adjust ts capacty, N, so as to guarantee that the servcelevel constrant s satsfed for the chosen target. We let = I =1 0 T. Gven the target T and the demand rate, the requred capacty for frm s gven by N = mnn + PW N > T 1 Our results are easly extended to the case where s allowed to vary among dfferent frms. where W N s the steadystate delay n an M/M/N queue wth arrval rate, servce rate, and N servers. 2 We wrte N = R + ê T 3) where R = / s the offered load gven the demand faced by frm, and ê T = N R s the excess capacty requred to satsfy the servcelevel target. Naturally, we defne ê T = 0 whenever = 0, but we note that ê must be postve whenever > 0 to guarantee stablty. The two terms n 3) represent the two components of the requred capacty: The offered load s the volumebased capacty, namely, t s the base capacty ensurng that the servce process s stable. The second component ensures that the desred servce levels are acheved and s referred to as the servcebased capacty. Frm ncurs a cost c per customer served and a cost per agent, per unt of tme. Ths corresponds to the cost of capacty beng lnear n the number of agents. 3 The prce p s chosen from a compact nterval p mn p max. Because frm wll always select a prce p, whch results n a nonnegatve gross proft margn p c /, we may assume, wthout loss of generalty, that p mn The upper bound, p max = c + I 4), s allowed to obtan any value n p mn. We set = p mn p max and = I =1. In full generalty, the demand rates are specfed as general functons of all prces and delay guarantees,.e., p T, where p = p 1 p I and T = T 1 T I. Assumpton 1 Regularty Assumptons on the Demand Functons for Dfferentated Servces). For each I, the functon + s strctly postve, contnuous, and dfferentable n all arguments, and strctly decreasng n p and T. Frm s longrun average proft s then gven by p T = p T p c N 2 N can be calculated by teratvely usng the ErlangC formula. Freeware calculators can be found, for example, at technon.ac.l/serveng/4callcenters/downloads.htm or koole/ccmath/erlangc. 3 See 7 for a dscusson of more general capactycost models.
7 Allon and Gurvch: Prcng and Dmensonng Competng LargeScale Servce Provders Manufacturng & Servce Operatons Management, Artcles n Advance, pp. 1 21, 2009 INFORMS 7 Copyrght: INFORMS holds copyrght to ths Artcles n Advance verson, whch s made avalable to nsttutonal subscrbers. The fle may not be posted on any other webste, ncludng the author s ste. Please send any questons regardng ths polcy to whch, usng 3), s rewrtten as follows: p T = p T p c ) ê T 5) The assumpton of largescale servce systems s ntroduced by consderng a famly of markets ndexed by a marketscale multpler 0 so that the demand grows wth the marketscale multpler n a natural way. Specfcally, we let p T = p T 6) be the demand facng frm n the th market. The proft functons n the th market are then gven by p T = p T p c ) ê T I 7) For future reference, we make the followng formal defnton. Defnton 1 The Market Game). The th market game s the Iplayer game wth proft functons, I and strategy space. As s the case n heavytraffc analyss, the key dea of our market replcaton procedure s to embed the real market wth fxed market sze) nto a sequence of markets wth growng demand. Lookng at the sequence of markets, we are nterested n understandng how the stablty of the market and the market outcomes change wth the ncrease n market sze. Followng conventonal notaton, we let pt =p 1 T 1 p 1 T 1 p +1 T +1 p I T I We denote by T p T and p p T, respectvely, the delay and prce components of frm s best response to p T n the th market game. The exstence of a best response for any actons p T follows from the contnuty of the demand functons and the compactness of the strategy space. When the best response s not unque, we arbtrarly choose one best response. The way ths best response s chosen wll be mmateral for our results. As dscussed n the ntroducton, the market game s ntractable for drect Nash equlbra analyss. Ths s a consequence of the complexty of the expressons for the servcebased capacty, the dscreteness of ths capacty, and the concavty of the capacty cost functon. 4 Instead, we take an ndrect approach that explots the benefts of largescale asymptotc analyss wthn an Nashequlbrum framework Nash Equlbra The noton of Nash equlbrum s adopted from Tjs 1981). Rather than defnng t n general terms, we provde the defnton as t apples to our settng. To ths end, gven p T and p T 0 T, we let p T p T = p 1 T 1 p 1 T 1 p T p 1 T 1 p I T I Defnton 2 Nash Equlbrum for the th Market Game). Fx 0. Let = 1 I be a postve vector. We say that x, s an Nash equlbrum of the th market game f, for each I and any x 0 T, x x x + Nash equlbrum s a specal case of Nash n whch = 0. The generalzaton from Nash to Nash allows us to construct an envelope around the market outcomes, and thus obtan key nsghts about the market behavor even n cases n whch Nash equlbra do not exst. The ablty to construct such envelopes s useful also n cases n whch Nash equlbra do exst but are dffcult to characterze. In these cases, f s small enough, the characterzaton of the Nash equlbra can shed lght on the Nash equlbrum. We wll be formally constructng such envelopes as well as analyzng the gaps between the Nash and Nash equlbra whenever the latter exst NotatonalConventons. For two sequences of postve vectors a 0 and b 0 wth elements n d +, we say that a = Ob f lm sup a /b < for = 1d. We say that a = ob f lm sup a /b = 0 for = 1d. Fnally, for d = 1, we say that a b f a = Ob 4 It s possble to construct contnuous versons of the servcebased capacty see, e.g., 4 of Borst et al. 2004). Ths, however, would stll leave the market game ntractable for exact analyss.
8 Allon and Gurvch: Prcng and Dmensonng Competng LargeScale Servce Provders 8 Manufacturng & Servce Operatons Management, Artcles n Advance, pp. 1 21, 2009 INFORMS Copyrght: INFORMS holds copyrght to ths Artcles n Advance verson, whch s made avalable to nsttutonal subscrbers. The fle may not be posted on any other webste, ncludng the author s ste. Please send any questons regardng ths polcy to but a ob. For a vector x d, we let x = d k=1 x k. When appled to a vector x d, the absolute value operaton should be nterpreted componentwse,.e., x = x 1 x d. Smlarly, for x d +, x = x1 x d. The notaton stands for convergence as unless explctly stated otherwse. We use 0 to represent the 0 vector n d, and the dmenson of the vector wll always be clear from the context. Fnally, we use the abbrevated notaton nstead of p T whenever the arguments p and T are clear from the context. 4. Regme Characterzaton In ths secton, we dscuss the optmal operatonal regmes of the frms n the market QED, ED, or QD) and relate ths regme choce to the underlyng demand models. As dscussed n the ntroducton, a frm s operatonal regme s the outcome of the frm tradng off ts capacty cost and the servce level t provdes. For a monopolst, ths tradeoff s solely a functon of the frm s own scale economes. In an olgopolstc settng, however, the value of a servce level for a gven frm depends on ts compettors decsons, thus makng the tradeoff more subtle. The outcome of the regme analyss wll be a mappng from frm s demand structure to a quantfer r. Ths quantfer characterzes the order of magntude of the optmal servcelevel choce for frm. Some frms wll have r = 1/, and we wll show that, for these frms, t s optmal to use a servcebased capacty that s of the order of. Consequently, these frms wll operate n equlbrum) n the QED regme. Other frms wll have r, whch s sgnfcantly larger than 1/. These frms wll optmally use a servcebased capacty of a magntude o and wll operate n the ED regme. To motvate our results, note that, gven a prce vector p and a servcelevel choce T by ts compettors, frm s best servcelevel choce s gven by T arg max p x p T x 0 T = arg max p x p T x 0 T p c ) ê x Equvalently, T arg max p x p T x 0T p p 0 p T c ) ê x 8) The order of magntude of T s determned, then, by optmally balancng the loss of market share due to customer delays whch we nformally refer to as the delay cost and s gven by p x p T p 0 p T aganst the servcebased capacty cost ê x. 5 To dentfy ths order of magntude, Lemma 1 provdes us wth estmates on the order of magntude of the servcebased capacty, and Assumpton 2 allows us to control the delay cost. Lemma 1. Fx a sequence p T 0 such that p T for all 0. Then, { 1 ê T mn } T Assumpton 2 Behavor Around T = 0). For each I there exsts > 0 such that lm sup x 0 and lm nf x 0 sup p T 0 p T x < p T x 0 T I 1 nf p T 0 p T x > 0 p T 0 T I 1 Pluggng nto 8) Assumpton 2 and Lemma 1, we have nformally) that [ x mn T arg max x 0T Hence, we should have that x T arg mn x + 1 x 1/ x T 1 x )] A smple calculaton then yelds that T r where { r 1 = max 1 } 9) 1/1+ 5 Here, the loss of market share parallels the role of the delay cost n the monopolst settng of Borst et al. 2004).
9 Allon and Gurvch: Prcng and Dmensonng Competng LargeScale Servce Provders Manufacturng & Servce Operatons Management, Artcles n Advance, pp. 1 21, 2009 INFORMS 9 Copyrght: INFORMS holds copyrght to ths Artcles n Advance verson, whch s made avalable to nsttutonal subscrbers. The fle may not be posted on any other webste, ncludng the author s ste. Please send any questons regardng ths polcy to so that r = 1/ for all 1 and r = 1/1+ otherwse. Assumpton 2 requres that, n the vcnty of T = 0, the demand volume of a frm decreases proportonally to some power of ts delay guarantee T. The power may be dfferent for dfferent frms. Ths assumpton s satsfed by most known demand models, but may not be satsfed n general; see Examples 1 3 at the end of ths secton. The quantfer r, whch depends on Assumpton 2 through 9), plays an mportant role n determnng the operatonal regme of a frm. Our nformal dscusson above suggests that r provdes an orderofmagntude estmate for the servcelevel choce of frm, and by Lemma 1, 1/r then provdes an estmate of the servcebased capacty for that frm. Ths s formally stated and proved n the followng theorem. Theorem 1 Regme Characterzaton). Suppose that Assumpton 2 holds. Let p T 0 be a sequence such that p T for all 0. Then, and T p T r I > 1 10) T p T = Or I = 1 11) T p T = or I < 1 12) Furthermore, QED regme: f 1/r / and N, then ê T lm sup PW > 0< 1 and lm nf PW > 0> 0 ED regme: If 1/r / = o, and = N 13) = o, then ê T = lm PW > 0 = 1 14) Here, s the demand faced by frm when the other frms play p T and frm plays the best response T p T p p T. The random varable W s the steadystate delay at frm under ths best response. Interestngly, Theorem 1 mples that even f the market oscllates between dfferent ponts n, the operatonal regme of a frm remans unchanged. Moreover, t shows that whereas the actual choce of servce level by a frm depends on the characterstcs of all frms n the market, ts operatonal regme ED, QD, or QED depends only on ts own ntrnsc propertes as reflected n the quantfer r. Remark 1 The QD Regme). The QD regme, n whch the probablty of delay, PW > 0, approaches 0 as, does not emerge n our settng. Ths s a consequence of the structure of the servcelevel constrants that we use n our model. Specfcally, when a frm s servce level s defned va PW >T for that s strctly postve and exogenously gven, t can not do better than settng T = 0. In ths case, Proposton 1 n Halfn and Whtt 1981) tells us that, to have PW > 0 for 0 1, t suffces to use the squareroot safetystaffng rule and, n partcular, to use a servcebased capacty that s proportonal to the square root of the demand. Hence, t cannot be optmal for a frm to operate n the QD regme because ths requres the servcebased capacty to be orders of magntude greater than. The framework that we provde n ths paper can, however, be appled to alternatve settngs n whch the QD regme may emerge n equlbrum. We expect, for example, that f servce levels are defned va guarantees of the form EW T, the QD regme wll emerge as a possble outcome. Indeed, under some demand models t may be optmal for some frms to guarantee an average delay T such that T = o1/. These frms wll operate n the QD regme; see 9 of Borst et al. 2004). We conclude ths secton by pontng out some wdely used demand models that satsfy Assumpton 2. The multnomal logt and the Cobb Douglas models are two such examples. Example 1 The MNL Demand Model). Let p T = e a T b p v 0 + j e a j T j b j p j I 15) where v 0 > 0 s a constant and a T = a k T, for postve constants a, k, and. Then, t s easly verfed that n the defnton of a T plays the role of the exponent n Assumpton 2. It s mportant to note that n a market n whch the demand experenced by each frm s characterzed by the multnomal logt model, some frms may be operatng n the ED regme and others may be operatng n the QED regme, dependng on the senstvty of the attracton values of each frm to ts own servce level.
10 Allon and Gurvch: Prcng and Dmensonng Competng LargeScale Servce Provders 10 Manufacturng & Servce Operatons Management, Artcles n Advance, pp. 1 21, 2009 INFORMS Copyrght: INFORMS holds copyrght to ths Artcles n Advance verson, whch s made avalable to nsttutonal subscrbers. The fle may not be posted on any other webste, ncludng the author s ste. Please send any questons regardng ths polcy to Henceforth, whenever we menton the MNL demand model we are referrng to the one n Example 1. Example 2 Demand Models wth Taylor Expanson Around T = 0). A large famly of models for whch Assumpton 2 s satsfed are those n whch the functon p T has a Taylor seres expanson around T = 0. In these cases, expandng around T = 0, we can wrte p T x = p T 0 + k l l T p T l=1 T =0 x l + ox k Assumpton 2 s then satsfed wth an exponent that corresponds to the frst nonzero dervatve wth respect to T at the pont T = 0, provded that the dervatve s unformly bounded away from 0. Formally, we wll have = k where k s such that k k T p T a b and T =0 l l T p T = 0 l < k T =0 16) for some 0 < a < b < and for any vectors p and T 0 T I 1. Of course, Assumpton 2 holds also n many examples n whch such a Taylor expanson does not exst; see Example 1 above. In that example, a Taylor expanson need not exst when a T = a T for a nonnteger exponent. The followng example llustrates a demand model for whch all parameter values lead to the exponent = 1 and, consequently, to the QED regme. Example 3 The Cobb Douglas Demand Model). Fx I and assume that p T = v p T v 0 + j v j p j T j wth v p T = c T / T T a b p for strctly postve constants a, b, and c. Here, t s easly verfed that Equaton 16) holds wth k = 1 so that we always have = 1 n Assumpton 2. Consequently, n a market n whch all the frms face a Cobb Douglas demand model, only the QED regme emerges as the equlbrum choce. Table 1 Three Dfferent Games Game Strategy space Payoff functon Market p T as n 7) Flud P p = p 0 p c Dffuson p T = p T p c ) ) f T 4.1. Roadmap for Rest of ths Paper In 5 and 6 we ntroduce the flud game and the dffuson game, respectvely. The flud game provdes a frstorder characterzaton of the market outcome. It also serves as an essental buldng block n the ntroducton and characterzaton of the more refned) dffuson game. These two approxmate games dffer from each other, and from the market game of Defnton 1, both n terms of the payoff functons and n terms of the strategy spaces. These dfferences are outlned n Table 1. The exact dervaton of the payoff functons as well the defnton of the functon f n Table 1 appear n 5 and 6; see Defntons 3 and 4. In passng, t s mportant to note that the strategy space of the flud game s only the prce doman, and no servcebased capacty cost appears n the payoff functon. It s also mportant that, n the dffuson game, the servcebased capacty cost for frm s replaced by an expresson that depends only on the servcelevel choce of frm. Ths stands n contrast to the market game n whch the servcebased capacty s ê p T T, and hence depends on the complete vector p T. For each of the above approxmate games we wll characterze the equlbra: a sngle Nash equlbrum p 0 for the flud game, and a sequence of Nash equlbra p T 0 for the sequence of dffuson games. We wll show that, at dfferent levels of precson, these equlbra approxmate the outcomes of the orgnal market game. The precson of the approxmaton wll be a functon of the quantfers r dentfed earler n ths secton. Startng the market n one of the approxmate equlbra, we wll characterze the maxmum proftable devaton that a frm wll devate n ts payoffs and actons servce level and prce.
11 Allon and Gurvch: Prcng and Dmensonng Competng LargeScale Servce Provders Manufacturng & Servce Operatons Management, Artcles n Advance, pp. 1 21, 2009 INFORMS 11 Copyrght: INFORMS holds copyrght to ths Artcles n Advance verson, whch s made avalable to nsttutonal subscrbers. The fle may not be posted on any other webste, ncludng the author s ste. Please send any questons regardng ths polcy to 5. A Flud Game In ths secton we ntroduce the flud game. Ths a smplfed game n whch only the frstorder mpact of the frms actons s modeled. That s acheved by replacng the servce facltes the M/M/N queues) by ther flud approxmatons. We wll establsh that the flud game does ndeed provde a frstorder approxmaton to the orgnal game n that the market prces wll always le wthn some small neghborhood of the fludgame equlbrum, and the servce levels wll be close, n a sense, to T = Defnton and Characterzaton Defnton 3 The Flud Game). The flud game s the Iplayer game wth proft functons P = p 0 p c ) I and strategy space. Note that the flud game has the orgnal unscaled) demand functons I. In the flud game, the players compete only on prces ths s a pureprce competton game n whch the strategy space,, of each player s a compact subset of + so that there exst numerous suffcent condtons for the exstence and unqueness of equlbra. Exstence of equlbra s guaranteed under the assumpton that P s contnuous and quasconcave wth respect to p see 2.3 of Cachon and Netessne 2004). Ths suffcent condton s satsfed, for example, for attracton models such as the multnomal logt demand model or the Cobb Douglas demand model n Examples 1 and 3, respectvely. We wll assume that there s a unque equlbrum and later dscuss some concrete examples n whch ths assumpton ndeed holds). We formally state ths requrement n the followng assumpton. Assumpton 3 Exstence and Unqueness of Equlbrum for the Flud Game). The flud game has a unque Nash equlbrum p = p 1 p I. For the rest of ths paper, whenever Assumpton 3 holds, we use the notaton p when referrng to the unque equlbrum of the flud game The Qualty of the Approxmaton Wth the restrctons n Assumptons 1 and 3 we show now that the flud game serves as a frstorder approxmaton for the orgnal market game. Combned, Theorems 2 and 3 show that approxmate equlbra exst, and all such equlbra are concentrated n a small neghborhood of p 0 wth p as n the Nash equlbrum of the flud game. Theorem 2 Exstence of Approxmate Equlbra). Suppose that Assumptons 1 and 3 hold, and let 0 be a sequence of vectors n I + such that, for all I, 0 and lm nf > 0 17) 0 Then, there exsts a sequence T 0 such that T, T 0 for all I and, for each, the vector p T = p1 T 1 p I T I s an Nash equlbrum for the th market game. Moreover, T can be chosen so that T1 = T 2 = = T I Theorem 3 FrstOrder Characterzaton). Suppose that Assumptons 1 and 3 hold, and let p T 0 and 0 be such that p T s an Nash equlbrum for the th market game and such that 17) holds. Then, as, T 0 and p p I 18) Theorems 2 and 3 are drven by economes of scale. Lemma 1 shows that the servcebased capacty ê grows at a lower rate than the volumebased capacty, even for small delay guarantees. Consequently, for any sequence p T 0 wth p T, ê T = o and, n turn, the proft functons satsfy the followng property: p T = p T p c ) + o Due to the relatvely low cost of the servcebased capacty, frms wll choose to provde relatvely hgh servce levels correspondng to small values of T ). Accordngly, we expect that a game wth proft functons P p = p0 p c ) = p0 p c ) I 19)
12 Allon and Gurvch: Prcng and Dmensonng Competng LargeScale Servce Provders 12 Manufacturng & Servce Operatons Management, Artcles n Advance, pp. 1 21, 2009 INFORMS Copyrght: INFORMS holds copyrght to ths Artcles n Advance verson, whch s made avalable to nsttutonal subscrbers. The fle may not be posted on any other webste, ncludng the author s ste. Please send any questons regardng ths polcy to and strategy space wll provde a frstorder approxmaton for the th market game. Dvson by the common scalar yelds the flud game n Defnton 3. In a sense, Theorems 2 and 3 show that all sequences of approxmate equlbra and n partcular, a sequence of Nash equlbra, f such exst) must converge to p 0. Our ultmate goal n ths secton s to characterze the convergence rate of T to 0 and of p to p, and relate the convergence rate to the bounds on the proft functons. Before movng toward that goal, we dscuss some practcal mplcatons of the flud game. Remark 2 Interpretng as the Level of Suboptmalty). One may nterpret as the level of suboptmalty for a frm f t chooses to prce accordng to the flud game prce equlbrum p. To llustrate ths pont, consder the specal case n whch the market conssts of a sngle frm a monopolst. The mplcaton of Theorem 2 for ths specal case s that the monopolst cannot ncrease ts proft by more than by devatng,.e., that p T p T + 20) for any sequence of prces and servce levels p T 0 as long as T 0. Here, s a sequence such that / 0 and lm nf 0 / > 0. In partcular, let p T be the true optmal decson for ths monopolst when the market scale s. Assumng such a soluton exsts, Equaton 20) then mples that p T p T 0 as Hence, p T satsfes the standard noton of fludscale asymptotc optmalty. In turn, our results wth respect to the flud game are the game theoretc verson of the fludscale asymptotc optmalty for monopolsts. In the same sprt, our equlbrum results for the dffuson game n 6 are a generalzaton of the dffusonlevel asymptotc optmalty results for monopolsts. Remark 3 ServceLevel Dfferentaton). A fundamental mplcaton of Theorem 2 s that the market s n an approxmate equlbra f all frms set ther prces accordng to p and choose a common but very good) servce level. Although there mght be many plausble explanatons for the use of ndustry standards, Theorem 2 provdes one such explanaton n that t shows that followng ndustry standards s not an rratonal choce for frms competng on servce levels and prces. In partcular, frms need not sgnfcantly dfferentate themselves n terms of servce level. Ths result can also be nterpreted as a onesded decouplng result between prces and servce levels at least at the frst order). The companes may set ther prces accordng to p. Once the prces are fxed, a frm can explot ts largescale effcency and, n partcular, the relatve low cost of the servcebased capacty to match the servce level of the compettor wthout movng sgnfcantly away from the equlbrum. Theorems 2 and 3 requre only Assumptons 1 and 3. Most standard demand models, however, satsfy also Assumpton 2, whch, when mposed, allows us to mprove on the convergence results n these theorems. To ths end, recall that r { = max 1 1 } 1/1+ where I are as n Assumpton 2. Theorem 4 Dstance from the Flud Game). Suppose that Assumptons 1 3 hold. Then, there exsts a sequence = O1/r 1 1/r I such that, for each 0, p 0 s an Nash equlbrum for the th market game. Moreover, and T p 0 r I > 1 21) T p 0 = Or I = 1 22) T p 0 = or I < 1 23) Theorem 4 characterzes the best servcelevel responses to the fludgame equlbrum p 0. The frst part of the theorem strengthens our results n Theorems 2 and 3 by characterzng the growth rate of as a functon of r. Theorem 4 does not yet provde bounds on the prce devatons. For that, we wll need to mpose addtonal restrctons on the demand models n consderaton. We now gradually ntroduce the concepts and condtons that are requred for that
13 Allon and Gurvch: Prcng and Dmensonng Competng LargeScale Servce Provders Manufacturng & Servce Operatons Management, Artcles n Advance, pp. 1 21, 2009 INFORMS 13 Copyrght: INFORMS holds copyrght to ths Artcles n Advance verson, whch s made avalable to nsttutonal subscrbers. The fle may not be posted on any other webste, ncludng the author s ste. Please send any questons regardng ths polcy to purpose. To ths end, gven p, let p be a best response of player n the flud game) to prces p of the compettors. If the best response s not unque, we arbtrarly but consstently) choose one. We put p = 1 p 1 I p I. By Assumpton 3, the vector p s the unque soluton to p p = 0. One expects that f p s a pont n whch no frm can sgnfcantly mprove ts profts by devatng from p, then p should be close to the unque equlbrum p. Combned, Lemmas 2 and 3 dentfy condtons under whch ths ntuton s vald. Lemma 2. Suppose that the followng two condtons hold: C1) For each I and T, the demand functon p T s twce contnuously dfferentable n p. C2) There exsts > 0 such that for all p and I, 2 P 2 p 6 p Then, there exsts > 0 such that f p and 0 I satsfy they also satsfy P p p P p I 24) p p M 25) for some constant M>0 that s ndependent of p. Lemma 2 mples that, under the proper condtons on the demand model, f p s a prce vector such that no frm can ncrease ts proft sgnfcantly n the flud game) by unlaterally devatng from p, then p should be close to a correspondng best response vector p. Under some condtons, one can take one step further and show that f a vector p s close to ts best response vector p, then t must be close to the unque equlbrum of the flud game, p. 7 In Lemma 3 we show 6 Ths can be mposed drectly on the demand functon by requrng that / 2 p p 0p c / + 2/p p 0<. 7 Note that the queston of whether p that s close to p must be close to p s essentally a queston about the soluton to the set of possbly nonlnear) equatons p = p. Indeed, puttng F p = p p, what we want s that, f the set of equatons F p = 0 has a unque soluton p, then any p that satsfes F p wll be close to p n a way that s, to some extent, proportonal to. Condtons on the functon F that guarantee the valdty of such statements appear, for example, n the lterature on convergence of algorthms for the soluton of nonlnear equatons; see, e.g., Gould et al. 2002) and the references theren. that one such condton s the dagonal domnance condton, whch requres the followng: C3) There exsts C<1 such that p p C k k I p I Example 4 at the end of ths secton shows how ths condton s verfed for the case of the multnomal logt demand model by means of the mplct functon theorem. Condton C3) s well known as a suffcent condton for unqueness of the equlbrum p for the flud game see, e.g., Theorem 5 n Cachon and Netessne 2004), but n our context, t gves us more than that. Whenever Condton C3) holds, we say that the flud game s lnearly contnuous. Ths term s motvated by the followng lemma. Lemma 3 Lnear Contnuty of Flud Game). Suppose that C1) C3) hold. Then, for all I + small enough, p p mples that p p 1 1 C B 1 for the nvertble matrx B, wth B = 0 and B j = 1 for all j. Consequently, by Lemma 2, there exst constants M > 0 such that P p p P p I 26) for 0 I mples that p p M 1 C B 1 27) Lemma 3 complements Lemma 2 to show that, under the dagonal domnance condton, f p s such that no frm n the flud game) can ncrease ts profts sgnfcantly by unlaterally devatng from t, then p must be close to p. In other words, under condtons C1) C3), bounds n payoff space as n Equaton 26)) mply bounds n acton space as n Equaton 27)). 8 Havng ntroduced Condtons C1) C3), we can now extend the bounds n Theorem 4 to nclude bounds on devatons n both servce level and prce. The matrx B that s used n the statement of the theorem s as n Lemma 3. 8 In the ecompanon, we provde a framework for the contnuty of the flud game that replaces Condton C3) wth a more general condton.
14 Allon and Gurvch: Prcng and Dmensonng Competng LargeScale Servce Provders 14 Manufacturng & Servce Operatons Management, Artcles n Advance, pp. 1 21, 2009 INFORMS Copyrght: INFORMS holds copyrght to ths Artcles n Advance verson, whch s made avalable to nsttutonal subscrbers. The fle may not be posted on any other webste, ncludng the author s ste. Please send any questons regardng ths polcy to Theorem 5 Dstance from the Flud Game). Suppose that Assumptons 1 3 hold. Then, there exsts a sequence = O1/r 1 1/r I such that, for each 0, p 0 s an Nash equlbrum for the th market game, and Equatons 21) 23) hold. If, n addton, C1) C3) hold, then p p 0 p = OB 1 I 28) wth = 1/r + r. Theorem 5 uses the lnear contnuty of the flud game to relate the market game to the flud game. It adds to Theorem 4 by establshng the prce bounds n 28). Condtons C1) C3) are crucal n the proof of ths theorem. To hnt nto how these condton are used n that proof, note that the bounds on the servce levels n 21) 23) allow us to show that the best response prces p p 0 consttute an r Nash equlbrum for the flud game. Namely, that p p 0 p p 0 r for all I. Havng establshed ths property, Condtons C1) C3) and Lemma 3 can be used to obtan the prce bounds n 28). Evdently then, Equaton 28) bulds heavly on Condton C3) n obtanng the prce bounds from the servcelevel bounds. Fortunately, varous demand models satsfy C3). We conclude ths secton wth one such example. Example 4 The MNL Model). Consder the demand model n 15). The correspondng flud game has demand functons P p = p 0 = and payoff functons P e a 0 b p v 0 + j I e a j 0 b j p j a 0 b p e p = v 0 + j I e a j 0 b j p j I p c ) Gven p, the best response for frm satsfes the equaton 1 P p p p c ) = 1 29) b In partcular, because there exsts > 0 such that P p < 1 for all p, we have that there exsts such that p >c + / + for all p. Usng the mplct functon theorem and dfferentatng 29) wth respect to p j for j, we get P p p p p c ) j = ) p p P j p p p p c ) ) 1 P p p 30) where /p j P p p and /p P p p are the partal dervatves wth respect to p j and p, respectvely, at the pont p = p p. Pluggng 29) nto 30) as well as the dervatves of P p wth respect to p and p j, we have that p p = b j P p p P j p p j I j j I b 1 P p p The rghthand sde s strctly less than 1 provded that j I b j P j p < 1 Ip 31) b Ths condton s the domnant dagonal condton for the MNL model. Hence, Condton C3) holds for the MNL model provded that 31) holds. By dfferentatng P p twce, one can also verfy that C1) and C2) hold for the MNL model. These example conclude the analyss of the flud game and we turn to the dffuson game. 6. A Dffuson Game The dffuson game that we ntroduce n ths secton s more refned than the flud game and can be nterpreted as a secondorder approxmaton for the market game. The sequence of dffuson games s constructed by replacng the servce facltes the M/M/N queues) by ther correspondng dffuson approxmatons. We wll then show that the dffuson game provdes an equlbrum characterzaton that s asymptotcally correct n dffuson scale thus parallelng the dffusonscale asymptotc optmalty results for monopolsts.
15 Allon and Gurvch: Prcng and Dmensonng Competng LargeScale Servce Provders Manufacturng & Servce Operatons Management, Artcles n Advance, pp. 1 21, 2009 INFORMS 15 Copyrght: INFORMS holds copyrght to ths Artcles n Advance verson, whch s made avalable to nsttutonal subscrbers. The fle may not be posted on any other webste, ncludng the author s ste. Please send any questons regardng ths polcy to 6.1. Defnton and Characterzaton In constructng the dffuson game, we frst use Lemma 4 below to replace the dscontnuous servcebased capacty functon ê wth a contnuous functon. The lemma reles on Borst et al. 2004) n approxmatng the delay dstrbuton by an expresson that uses the asymptotc verson, P, for the probablty of delay as dentfed n Halfn and Whtt 1981). Hereafter, we put R p T = p T / for p T. Lemma 4 M/M/N Lemma). Fx a sequence p T 0 such that, for each, p T. Then, for all I, ê T = R p T T + o R p T T ) R p T ) ) R p T where, gven y 0, y s the unque soluton to Here, Pxe xy = [ Px = 1 + xzx ] 1 zx where z and Z are, respectvely, the standard normal densty functon and ts cumulatve dstrbuton functon. Furthermore, the functon s a contnuously dfferentable and convex decreasng functon on 0. Fgure 1 The Functon for = y The functon that s defned n Lemma 4 s depcted n Fgure 1. Usng Lemma 4, we can wrte p T = p T p c ) R p T T ) R p T ) ) + o R p T T R p T 32) The expressons n 32) are stll complcated for equlbrum analyss due to the dependence of the R term on the entre vector p T. Theorem 3 shows, however, that a sequence p T 0 of approxmate Nash equlbra must satsfy p T p 0 and, by the assumed contnuty of the demand functons n Assumpton 1, that R p T R p 0 0 as 33) R p T These observatons motvate the ntroducton of the dffuson game as an approxmaton for th market game. Defnton 4 The Dffuson Game). Fx 0. The th dffuson game has I players, proft functons p T = p T p c ) R p 0T ) R p 0 I and strategy space. In defnng the new proft functons, we have replaced the servcebased capacty, ê, by a smpler term that depends on the prce equlbrum of the flud game, p, but s otherwse ndependent of the actual prce vector p and of the servce levels, T, of the compettors. Moreover, by Lemma 4, ths term s convex and contnuous n T. Ths relatve smplcty renders the dffuson game tractable for Nash equlbrum analyss. For example, t suffces to requre that, for each I, the demand functon p T s jontly concave n the decson p T of frm. For future reference, we assgn ths condton a number: C4) For each I and each p T, the demand functon p T s jontly concave n p T. 9 9 If the demand functons are twce contnuously dfferentable, the concavty of p T n p T and the monotoncty n Assumpton 1 combned mply the concavty of the functon p T p c /.
16 Allon and Gurvch: Prcng and Dmensonng Competng LargeScale Servce Provders 16 Manufacturng & Servce Operatons Management, Artcles n Advance, pp. 1 21, 2009 INFORMS Copyrght: INFORMS holds copyrght to ths Artcles n Advance verson, whch s made avalable to nsttutonal subscrbers. The fle may not be posted on any other webste, ncludng the author s ste. Please send any questons regardng ths polcy to 6.2. The Qualty of the Approxmaton We now analyze the qualty of the dffuson game as an approxmaton for the market game. To that end, we need to strengthen Assumpton 2 by replacng the lm sup and lm nf there wth actual convergence: C5) There exsts 0 < T and a contnuous functon f 0 T I 1 such that lm p T x p T 0 f x 0 x p T for every p T 0 I 1. Condton C5) holds for varous demand models; see Example 5 below as well as the lneardemand model n A1 of the onlne appendx. Wth C5) we have the result n the followng theorem. In the statement of the theorem, the matrx B s as n Lemma 3 t s the I I matrx gven by B = 0 = 1I and B j = 0 for all j. As before, r s gven by r { = max 1 1 } 1/1+ Theorem 6 Dstance from the Dffuson Game). Suppose that Assumptons 1 and 3 hold n addton to Condtons C1) C5), and let p T 0 be such that, for each, p T s a Nash equlbrum for the th dffuson game. Then, there exsts a sequence = o1/r 1 1/r I such that p T s an Nash equlbrum for the th market game. Moreover, where = r. T p T = T + or and p p T = p + ob 1 To provde some ntuton nto the role of Condton C5) n Theorem 6, assume that a Nash equlbrum p T does exst for the th market game and recall that, n that case, T arg max p x p T x 0 T p p 0 p T c ) ê x From 4 we know that T wll be close to 0; hence, we may heurstcally replace the watng cost p x p T p x p T by an approxmaton of ts behavor around 0. Assumpton 2 only guarantees that the behavor wll be proportonal to x. Although ths s suffcent to obtan the relatvely crude results of Theorem 4, t s not suffcent for the fner characterzaton n Theorem 6 above. For ths result, we need to dentfy the functonal form of the behavor around T = 0, and such a functonal form s provded by Condton C5). Theorem 6 mples that a Nash equlbra, p T, of the dffuson game provdes a refned approxmaton for the real market outcomes, thus mmckng the role of the dffuson approxmaton n the context of monopolsts. More precsely, by usng the dffuson game to determne the frms decsons, the compromse n profts s neglgble wth respect to the cost of the servcebased capacty, whch s, n turn, proportonal to 1/r. Ths s ndeed remnscent of the noton of asymptotc optmalty used n the context of monopolsts as explaned n the followng remark. Remark 4 The Sze of and DffusonLevel Asymptotc Optmalty). Consder the specal case n whch the set I conssts of a sngle frm a monopolst. Let ths be frm 1. An equlbrum of the dffuson game s then a maxmzer of 1 p 1T 1, where 1 s the proft functon n Defnton 4. Pck p 1 T 1 arg max 1 p T p T.e., p1 T 1 s a maxmzer of the dffusongame proft. Theorem 6 mples for ths settng that, for any sequence p 1 T 1 0 of prces and servce levels, lm nf 1 p 1 T 1 1 p 1 T 1 0 1/r1 where 1 s the proft functon n the th market game; see Defnton 1. In other words, the optmalty gap for ths monopolst, f t chooses to use the outcome of the dffuson game, s of the order of o1/r1. If the monopolst has r1 = 1/, then the optmalty gap s o, whch corresponds to the prevalent noton of asymptotc optmalty n the Halfn Whtt or QED) regme. Thus, the Nash equlbra result n Theorem 6 reduces, n the monopolst settng, to asymptotc optmalty n the sense of Borst et al. 2004). Theorem 6 does not only provde bounds on the optmalty gap wth respect to profts, but also wth
17 Allon and Gurvch: Prcng and Dmensonng Competng LargeScale Servce Provders Manufacturng & Servce Operatons Management, Artcles n Advance, pp. 1 21, 2009 INFORMS 17 Copyrght: INFORMS holds copyrght to ths Artcles n Advance verson, whch s made avalable to nsttutonal subscrbers. The fle may not be posted on any other webste, ncludng the author s ste. Please send any questons regardng ths polcy to respect to the prce and servcelevel decsons. Specfcally, the theorem shows that, wth p 1 T 1 beng an optmal soluton for the monopolst when the market scale s, the sequence p 1 T 1 0 must satsfy T 1 = T 1 + or 1 and p 1 = p 1 + o ) B 1 1 where the vector p1 T 1 s the optmal soluton to the dffuson game. It turns out that, under the condtons of Theorem 6, we can be more precse about the servcelevel characterzaton. The followng lemma shows that the servcelevel choce under the dffusongame equlbrum can be characterzed n closed form up to an error of sze or. Here, the functon P s as n Lemma 4. Lemma 5. Suppose that Assumptons 1 and 3 hold n addton to Condton C5). Let p T 0 be such that, for each, p T p T p 0 as. Then, T p T r where = 0 f < 1 and = arg max f p 0 0 p 0 as I p c ) ) 1/ +1 I f 1. Here, s the unque soluton x to Pxe x p 0/ = whenever = 1, and t s the unque soluton to e x p 0/ 1/ +1 = when > 1. Lemma 5 allows us to go one step beyond Theorem 6 and replace the requrement of exstence of a Nash equlbrum for the dffuson game Condton C4) wth a condton on a much smpler perturbed flud game. To ths end, let the flud game on T be the Iplayer game wth proft functons T P p = p T p c ) I and strategy space. The flud game on T = 0 s the flud game from Defnton 3. Theorem 7 below provdes a characterzaton of the Nash equlbrum n terms of the Nash equlbrum of the flud game on = 1 r 1 I r I wth 1 I as n Lemma 5. Note that, n contrast to Theorem 6, here we do not mpose Condton C4). Instead, we assume unqueness of equlbrum for the flud game on. The matrx B n the statement of the theorem s as n Lemma 3. Theorem 7. Suppose that Assumptons 1 and 3 hold n addton to Condtons C1) C3) and C5). Assume that, for all large enough, the flud game on = 1 r 1 I r I has a unque Nash equlbrum p. Then, there exsts a sequence = o1/r 1 1/r I such that p s an Nash equlbrum for the th market game. Moreover, where T p = + or and p p = p + ob 1 = r 34) Example 5 Back to the multnomal logt Demand). By Example 4, C1) C3) all hold for the MNL demand model provded that j I b j P j p b < 1 Ip 35) It remans to show that C5) holds and that a unque equlbrum exsts for the flud game on. Frst, we clam that C5) holds wth f p T = k v p 01 p T j v j p j T j + v p 0 The smple but detaled argument s gven n the onlne appendx. The proof can be useful as a gudelne toward the verfcaton of C5) for other demand models. Provded the flud game on T = 0 satsfes Condton C3), the flud game on T for T n a suffcently small neghborhood of 0) wll satsfy Condton C3) by vrtue of the contnuty of the best response functons and ther dervatves. Condton C3), n turn, guarantees the unqueness of equlbra for that game. Because 0 as, we have that the MNL demand model satsfes the condtons of Theorem 7. Remark 5 Herarchcal Decouplng). Combned, Lemma 5 and Theorem 7 justfy referrng to demand models that satsfy C5) as demand models that admt a herarchcal decouplng. Indeed, Lemma 5
18 Allon and Gurvch: Prcng and Dmensonng Competng LargeScale Servce Provders 18 Manufacturng & Servce Operatons Management, Artcles n Advance, pp. 1 21, 2009 INFORMS Copyrght: INFORMS holds copyrght to ths Artcles n Advance verson, whch s made avalable to nsttutonal subscrbers. The fle may not be posted on any other webste, ncludng the author s ste. Please send any questons regardng ths polcy to shows that servcelevel choces depend on the actons of a frm s compettors mostly through ther prces and not ther servce levels). Moreover, they depend on these prces only through ther fludgame equlbrum p. Practcally, ths suggests that servcelevel and prce choces can be made n a sequental manner rather than jontly. The frms wll frst choose ther prces based on the flud game,.e., dsregardng servcelevel consderatons. Based on the these prces the frms wll make ther servcelevel choces. Although the frms mght choose to adjust ther prces at a later stage n response to the actons of the competton, they wll not need to revst ther servcelevel choces. These can reman fxed wthout any sgnfcant compromse to the frms profts. Remark 6 Global Stablty). The results stated n ths secton focus on the exstence and, to some extent, unqueness of approxmate equlbra. Accordngly, n the sprt of equlbrum analyss, the focus s on unlateral devatons. In A3 of the ecompanon, we strengthen these results by provng a global stablty result. We show that, for any startng pont p T, the market converges to a neghborhood of the dffuson game equlbrum, and ths neghborhood s exactly the one characterzed n Theorem 7. Remark 7 Acton Predctons and Freedom n Prcng). Note that Theorem 7 s not an fandonlyf result. It states that a sequence of Nash equlbra must satsfy the acton bounds n 34). It does not say, however, than any sequence that satsfes 34) s a sequence of Nash equlbra wth = or 1 r I. The proof of Theorem 7 reveals, however, that any sequence p T such that T = or and p p = o ) r s an Nash equlbrum for = o1/r 1 1/r I. There remans a gap then between the necessary gap of ob 1 r that s stated n Theorem 7 and the suffcent bound of or. Our asymptotc stablty result n A3 of the e companon closes ths gap by showng that, provded that the flud game s globally stable, these bounds can be tghtened. In that case, we show that p p = p + o 36) wth = r Hence, we have that p consttutes an Nash equlbrum for = o1/r1 1/r I f and only f p p = o r. These strengthened prce bounds underscore the mplcaton of the operatonal regme of a frm on the degree of freedom t has n choosng ts prce. For llustraton, consder a market wth two frms I = 1 2 such that 1 = 1 and 2 = 2. In that case, by 9), r1 = 1/ and r2 = 1/1/3. Theorem 1 states then that frm 1 operates optmally n the QED regme, whereas frm 2 operates optmally n the ED regme. By 36), we then have that frm 1, the QED frm, cannot devate n prce from p by more than o1/ 1/4 wthout compromsng ts profts. Ths stands n contrast to frm 2, the ED frm, that can afford devatng from p by any quantty that s o1/ 1/3. Thus, a QED frm has to be more precse n settng ts prce than an ED frm. Ths result s drven by the relatve nvestment n servcebased capacty requred by QED and ED frms. For example, assume that the two frms consdered above decrease ther respectve prces n a manner that leads to dentcal ncrease n demand for both frms. To mantan the same tme guarantee, the QED frm wll have to ncrease ts capacty by an order of the square root of the ncrease n demand. The ED frm, n contrast, wll be requred to make an ordersofmagntude smaller nvestment. Consequently, an ED frm can make bgger changes n prces because ther mpact on ts proftablty through the requred nvestment n capacty) wll be smaller. We conclude ths secton wth a bref summary of our results thus far. In 5 we ntroduced and analyzed the flud game. Intally, under very mld assumptons on the demand functon, we showed that the fludgame Nash equlbrum provdes an o approxmaton for frms proft and an o1 approxmaton for frms actons. By ntroducng addtonal assumptons on the demand model, we were able to strengthen to bounds and dentfy how these scale as the market sze ncreases. In 6.1 we then ntroduced the more refned dffuson game. The Nash equlbra of the dffuson game provde an asymptotcally correct estmate for the market outcome where the asymptotc correctness s analogous to dffusonscale asymptotc optmalty for a monopolst. These results together wth the approprate assumptons, condtons, and theorems) are summarzed n Table 2.
19 Allon and Gurvch: Prcng and Dmensonng Competng LargeScale Servce Provders Manufacturng & Servce Operatons Management, Artcles n Advance, pp. 1 21, 2009 INFORMS 19 Copyrght: INFORMS holds copyrght to ths Artcles n Advance verson, whch s made avalable to nsttutonal subscrbers. The fle may not be posted on any other webste, ncludng the author s ste. Please send any questons regardng ths polcy to Table 2 Summary of Results Qualty of approx. Qualty of approx. Game Equlbrum payoff space acton space Assumptons and condtons Theorems Flud p 0 = o o1 Assumptons 1 and 3 3 Flud p 0 = O1/r Or for servce Assumptons 1 3 4, 5 OB 1 for prce Condtons C1) C3) Dffuson p T = o1/r or for servce Assumptons 1 3 6, 7 7. Dscusson In ths paper, we study markets wth multple largescale servce provders. To do so, we develop a novel framework that combnes the notons of Nash equlbrum, market replcaton, and heavy traffc to study market equlbra. The Nash framework allows us to go beyond the scope of models of competton for whch Nash equlbrum exsts and use relatvely general demand and capacty models. The noton of market replcaton allows us to dscuss trends n the market outcomes along sequences of markets. For example, t allows us to characterze the mpact of the market scale on the nterdependence between the prcng and servcelevel decsons. Combned wth the noton of heavy traffc, whch s well studed for monopolsts, ths framework allows us to characterze equlbrum behavor and obtan nsghts for markets n whch a Nash equlbrum does not necessarly exst. The framework developed n ths paper can be appled to other compettve settngs n whch congeston and queueng play mportant roles. The framework s especally relevant n settngs that satsfy two condtons: a) a Nash equlbrum does not exst or s ntractable for characterzaton, and b) there are avalable approxmatons for the underlyng queueng systems that can be used to construct a tractable approxmate game. In our settng, the dffuson game whch s based on manyserver heavytraffc approxmatons plays the role of ths approxmate game, but ths need not be the case. Indeed, one can apply the same approach to markets wth sngleserver supplers n whch the servce rate, rather than the number of servers, s the capacty decson varable. In these cases, we expect that the socalled conventonal heavytraffc approxmatons n whch the number of servers s kept fxed and the ob 1 for prce Condtons C1) C5) load approaches one would play a key role n supplyng the approxmatons that would replace each of the supplers n the constructon of the dffuson game. In these sngleserver settngs our approach can be used to study demand models n whch the customers are senstve to the whole sojourn tme rather than solely to the watng tme n queue. To llustrate ths latter clam, we consder a market wth two frms such that frm faces the followng logt demand: a f 1 f 2 = m e bf v 0 + a 1 e b 1f 1 + a2 e b 2f 2 Here, m plays the role of the market sze. Also, f s the full prce charged by frm, that s, f = p + s, where s s the average sojourn tme of customers served by frm. Hence, rather than treatng prce and servce level as ndependent attrbutes, ths game wll consder only ther lnear combnaton. Both frms operate through a sngle server faclty so that frm adjusts ts servce rate rather than the number of servers. The cost of capacty s then c for c > 0. m f 1f 2 = f c 2 c = 1 2 where c s the cost of a unt of capacty. Ths model s very smlar apart from the exstence of an outsde opton) to the one consdered n Cachon and Harker 2002). We use the parameters c 1 = c 2 = 375 and a = b = 1, for = 1 2, as n the example of Cachon and Harker 2002, Fgure 3). As n Cachon and Harker 2002), an equlbrum does not exst when m = 1. For larger values of m, however, the market seems to have a Nash equlbrum. Stll, ths Nash equlbrum need not be unque. The flud game allows us, however, to obtan a frstorder approxmaton of
20 Allon and Gurvch: Prcng and Dmensonng Competng LargeScale Servce Provders 20 Manufacturng & Servce Operatons Management, Artcles n Advance, pp. 1 21, 2009 INFORMS Copyrght: INFORMS holds copyrght to ths Artcles n Advance verson, whch s made avalable to nsttutonal subscrbers. The fle may not be posted on any other webste, ncludng the author s ste. Please send any questons regardng ths polcy to the fullprce equlbrum. Followng our approach n ths paper, we frst defne a flud game by removng the servcebased capacty cost 2 c. The mth flud game s the game wth proft functons m f = f c m f 1f 2. Ths flud game does have the equlbrum f = Moreover, t can be easly verfed that the dagonal domnance condton n Equaton 31) holds for the above parameters, so that ths equlbrum s the unque equlbrum of the flud game. We numercally compute equlbra for each value of m for the orgnal game wth proft functons m m f. We plot the equlbrum full prces, f1 f 2 m, on the graph n Fgure 2. Because the equlbra are all symmetrc, wth f1 m = f2 m, each such equlbrum s descrbed by a sngle pont on the dashed lne. The sold lne n Fgure 2 corresponds to the fludgame equlbrum f. Note that the y axs covers only an nterval of sze 03 so that the convergence of f m toward f s very quck. For m = 100, the dstance s less than 0.25, whch s less than 4%. The gap for the last pont n the graph s less than 0.4%. In ths model, n whch the servce provders are modeled as M/M/1 queues and the competton s only on full prce, the dffuson game s dentcal to the orgnal game, and hence an addtonal step s not requred. Of course, f the arrvals were not Posson and servce tmes were not nonexponental, the dffuson game and the orgnal game would Fgure 2 Max full prce A SngleServer Model Twofrm, sngleserver example Marketscale multpler 10 4 no longer be dentcal. Rather, G/G/1 approxmatons would be used to construct a dffuson game that provdes approxmatons for the complex orgnal game. Another example of a settng n whch our framework s readly applcable s the settng wth segmented markets as consdered n Allon and Federgruen 2009), n whch each servce provder serves multple customer classes. In ths model, we expect that the avalable approxmatons for multclass queueng systems would be used n the constructon of the dffuson game. Yet another model that seems amenable to analyss through our framework provded that the market under consderaton s large) s one n whch the competton s ncorporated wth learnng. These are markets n whch the demand characterstcs as well as the prce and servce level actons of all the players n the market are not necessarly observable. Largescale approxmatons have been used recently n the context of learnng and prcng n revenue management see, e.g., Besbes and Zeev 2009), and t seems that these can be combned wthn our framework to characterze the equlbra n these very realstc, but hghly ntractable, settngs. Fnally, n ths paper we consdered only the case of lnear capacty costs. We made ths choce so as not to dstract attenton from the man deas n the proposed framework. Gven our framework, the ablty to address more general cost functons, as n Borst et al. 2004), would follow from the ablty to do so n a monopolst settng. Hence, the fact that such cost functons are treated n the lterature on monopolsts suggests that these could also be treated n the compettve settng. Electronc Companon An electronc companon to ths paper s avalable on the Manufacturng & Servce Operatons Management webste Acknowledgments The authors are grateful to the revewers and the assocate edtor for ther numerous comments, whch lead to sgnfcant mprovements over the orgnal manuscrpt. They also thank Robert Shumsky for helpful dscussons n the early stages of ths work.
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