Pricing Internet Access for Disloyal Users: A Game-Theoretic Analysis

Transcription

1 Prcng Internet Access for Dsloyal Users: A Game-Theoretc Analyss Gergely Bczók, Sándor Kardos and Tuan Anh Trnh Hgh Speed Networks Lab, Dept. of Telecommuncatons & Meda Informatcs Budapest Unversty of Technology and Economcs (bczok, kardos, trnh)@tmt.bme.hu ABSTRACT In ths paper we nvestgate the mpact of customer loyalty on the prce competton between local Internet Servce Provders who sell Internet access to end-users. The man contrbuton of ths paper s threefold. Frst, we develop a repeated game, and show how cooperaton between ISPs resultng n hgher profts can be enforced through a threat strategy n the presence of customer loyalty. Second, we nvestgate the case of a dfferentated customer populaton by ntroducng dual reservaton values, and show how t leads to new, pure strategy Nash equlbra for a wde range of demand functons. Thrd, we develop two novel models for customer loyalty, along wth a smulaton tool that s capable of demonstratng the mpact of the novel models. We argue that our fndngs can brng us closer to the understandng of economc nteractons among ISPs and, at the same tme, can motvate researchers to ncorporate a fnergraned user behavor model nvolvng customer loyalty n ther nvestgatons of such nteractons. Categores and Subect Descrptors C.4 [Performance of Systems]: Modelng technques; J.4 [Socal and Behavoral Scences]: Economcs General Terms Economcs, Theory, Expermentaton 1. INTRODUCTION Advances n networkng technology and affordable servce prces are contnung to make Internet access avalable for bllons of customers. To provde end-to-end network connecton, Internet Servce Provders (ISPs) form a herarchy that spans from local ISPs who sell access to end-users, through regonal ISPs who connect local ISPs to the Internet backbone, to Ter-1 ISPs who form the backbone, and are peerng wth each other. The economc nteractons among servce provders of dfferent levels and endusers have been n the focus of nterest for several years. Furthermore, these nteractons wll contnue to get specal attenton, snce ntatves lke the NSF FIND [1] promote Permsson to make dgtal or hard copes of all or part of ths work for personal or classroom use s granted wthout fee provded that copes are not made or dstrbuted for proft or commercal advantage and that copes bear ths notce and the full ctaton on the frst page. To copy otherwse, to republsh, to post on servers or to redstrbute to lsts, requres pror specfc permsson and/or a fee. NetEcon 08, August 22, 2008, Seattle, Washngton, USA. Copyrght 2008 ACM /08/08...$5.00. economc ncentves as a frst-order concern n future network desgn. Also, decson-makers tryng to work out a plausble soluton for the recently surfaced net neutralty debate would greatly beneft from an n-depth understandng of economc processes nsde the user-isp herarchy. There s broad lterature n the area of modelng nteractons between ISPs wth game-theoretcal means [11] [5] [16]. Whle these papers ntroduce and analyze complex models for the nteracton of ISPs at dfferent levels of the herarchy, they mostly assume a very smple user behavor model when nvestgatng the market for local ISPs: end-users choose the cheapest provder assumng that the qualty of the certan servces s the same. Ths assumpton could be plausble n certan scenaros, but t could be msleadng f there are loyal customer segments present n the market. On the other hand, economsts are well aware of the noton of consumer or brand loyalty, whch s very much exstng n realstc markets. Practcally speakng, a customer s loyal to a brand, when she purchases the product of that brand, even f there are cheaper substtutons on the market. Brand loyalty s rooted n both satsfacton towards a gven brand and customers beng reluctant to try substtute products. There s exstng work dealng wth classfcaton of buyers nto loyalty groups [17], and a recent study develops and emprcally tests a model of antecedents of consumer loyalty towards ISPs [6]. In [12] authors use a game-theoretc framework to prove that f loyalty s an addtonal product of market share and penetraton, customer retenton strateges seem to be consequently more effcent for market leaders. An other study [7] analyzes a duopolstc prce settng game n whch frms have loyal consumer segments, but cannot dstngush them from prce senstve consumers. They demonstrate that consumer loyalty plays an mportant role n establshng the exstence and dentty of a prce leader. The latter two papers provde valuable nsght to the mpact of brand loyalty on certan markets, but also nspre for further nvestgatons. Frst of all, how does customer loyalty affect a dynamc market of Internet access? Second, [7] only consders perfectly nelastc demand and a sngle reservaton prce for the whole customer populaton. Whle these two assumptons may hold n certan scenaros, are they vald f consderng the Internet access market n developng countres or an economcally dfferentated Internet user populaton? Thrd, are there ncentves for cooperatve prcng regardng local Internet Sevce Provders n a market where user loyalty s present? And last, s the smple model, whch s commonly used n game-theoretc frameworks, a good representaton of real-world brand loyalty? Can the real-lfe behavor of customers (such as senstvty to the prce dfference between provders and uncertanty n ther decsons) be ncorporated nto a better user model? We ar- 55

2 Table 1: Payoff matrx for the basc game H M L H (60, 60) (0, 100) (0, 60) M (100, 0) (50, 50) (0, 60) L (60, 0) (60, 0) (30, 30) Table 2: Payoff matrx for the brand loyalty game H M L H (60, 60) (36, 70) (36, 42) M (70, 36) (50, 50) (30, 42) L (42, 36) (42, 30) (30, 30) gue that fndng an answer to these questons can brng us closer to the understandng of economc nteractons among ISPs and, at the same tme, t can motvate researchers to use a fner-graned user behavor model nvolvng customer loyalty n ther nvestgatons. In ths paper we nvestgate the mpact of customer loyalty on the prce competton between local ISPs who sell Internet access to end-users, both qualtatvely and quanttatvely. The man contrbuton of ths paper s threefold. Frst, we develop a repeated game based on the sngle-shot game presented n [7], and show how cooperaton between ISPs resultng n hgher profts can be enforced through a threat strategy n the presence of customer loyalty (Secton 3.1). Second, we nvestgate the case of a dfferentated customer populaton by ntroducng dual reservaton values, and show how t leads to new, pure strategy Nash equlbra. Also, we show how these results hold for a wde range of demand functons (Secton 3.2). Thrd, we develop two novel models for customer loyalty, along wth a smulaton tool that s capable of demonstratng the mpact of the novel loyalty models n prce competton among local ISPs (Secton 4). 2. MOTIVATION For llustratng the effect of brand loyalty consder the followng game [9]. Suppose there are two restaurants sellng pzza n a partcular geographc market. Suppose they each consder three possble prces for pzzas: a hgh prce (H), a medum prce (M) and a low prce (L). The proft per product s known to be $12, $10 and $6 for each frm regardless of the volume of sales. Also let us assume a perfectly nelastc demand functon, D(p) = 10000, so customers buy pzzas wthout regard to ts prce. The game s smlar to the Bertrand game as f the prces of the two frms are dfferent all demand goes to the lower prced frm, and f the prces are equal, frms splt the market evenly. It s easy to see that (p 1,p 2)=(L, L) s the unque Nash equlbrum of the game (see Table 1). Now, we change the game a lttle bt, and ntroduce brand loyalty, such as the frm wth the hgher prce loses some but not all of ts customers to the lower prced compettor. Assume that each frm has a loyal customer base that buys 3000 pzzas, and the frms are competng for the remanng demand of 4000 pzzas. In ths case the unque Nash equlbrum shfts to (p 1,p 2)=(M,M) (see Table 2). It turns out that brand loyalty removes the ncentve to try to undercut the prce of the other frm n order to steal market share. The game above demonstrates qualtatvely how the exstence of brand loyalty can affect the outcome of the prce competton, by changng the equlbrum pont. However, the broad exstng lterature assessng the prcng competton among Internet access provders (local ISPs) does not take brand (or user) loyalty nto consderaton resultng n an overly smplfed user model. Ths may lead to mprecse statements regardng equlbrum propertes. But what s the partcular quanttatve mpact of user loyalty on local ISP prcng competton? We apply the smple, statc loyalty model used both n the pzza game and [7] to the scenaro of multple local ISPs competng n prce to attract customers (users) to show how loyalty could ntroduce new equlbra, and how cooperaton between ISPs can be acheved n the presence of loyalty. Later, n Secton 4 we address several shortcomngs of the statc loyalty model and ntroduce two novel models, that enable us to ncorporate more realstc user behavor to prcng competton. We show the mplcaton of these models to ISP prces and profts by means of smulaton. 3. IMPACT OF USER LOYALTY In ths secton we construct games nvolvng local ISPs as players, who compete n prces to attract customers. The loyalty model used n these games s smlar n nature to the one used n Secton 2. There s a fxed loyal user base for each competng servce provder, and there s an addtonal group of potental users, not ted to any frm, seekng the lowest prce on the market. We analyze the outcome of the games and show how cooperaton between two competng local ISPs can be enforced n the presence of user loyalty (see Secton 3.1), and also how a dfferentated user populaton may ntroduce pure strategy Nash equlbra, whch do not exst n a sngle reservaton value scenaro. Note, that proofs of propostons are omtted due to space constrants and can be found at [3]. Before gettng nto the detals, we hereby ustfy our assumptons used n the games throughout ths secton. Flat-rate subscrptons. There are repeatng patterns n the hstory of communcaton technologes, ncludng ordnary mal, the telegraph, the telephone, and the Internet. In partcular, the typcal story for each servce s that qualty rses, prces decrease, and usage ncreases to produce ncreased total revenues. At the same tme, prcng becomes smpler [14]. The schemes that am to provde dfferentated servce levels and sophstcated prcng schemes are unlkely to be wdely adopted. On the other hand, prce and qualty dfferentaton are valuable tools that can provde hgher revenues and ncrease utlzaton effcency of a network, and thus n general ncrease socal welfare. It s also shown that flat-rate prcng wastes resources, requres lght users to subsdze heavy users, and hnders deployment of broadband access [18]. However, t appears that as communcaton servces become less expensve and are used more frequently, those arguments lose out to customers desre for smplcty. A success story of the late 1990s was the -Mode servce n Japan, whch was the frst to offer hgh-speed moble Internet access for a flat rate. It succeeded at a tme, when other moble data servces were falng, partly because of the prcng scheme. The servce s stll popular among users, more than 20 percent of NTT DoCoMo customers n Japan have sgned up for flat rate moble nternet plans [2]. Furthermore, non-flat rate bllng s also resource consumng from a servce provder s vewpont [10]. All of the above, and the fact that most Internet access provders offer flat-rate subscrptons for end-users today, motvates us to assume a flat-rate prcng scheme n our models. Consumer demand for Internet access. The prce elastcty of demand for a partcular demand curve s greatly nfluenced by the degree of necessty or luxury: luxury products tend to have greater elastcty than necesstes. The pro- 56

3 porton of ncome requred to purchase a servce also plays a key role: products requrng a larger porton of the consumer s ncome tend to have greater elastcty [19]. These two observatons suggest that n a developed country, where ncomes are hgh, Internet access s ubqutous and people tend to lean on the Internet by a great degree (n ther work and also durng ther spare tme), almost every household has Internet access, so the demand can be modeled as constant (perfectly nelastc). On the other hand, markets n developng regons are hghly prce senstve, snce people have lower ncomes, and the number of Internet subscrptons would greatly beneft from lower prces. Therefore, the demand for Internet access n such regons can be best modeled as elastc. We use the nelastc model n Secton 3.1 to comply wth the assumptons of [7], whle we nvestgate both of them n the games of Secton 3.2. Reservaton prces of customers. Consumer populaton s heterogeneous n the sense that certan groups are wllng to pay dfferent amounts of money for the same servce. In the dream world of ISPs, n whch they were able to perfectly dentfy the reservaton prce of each customer n the market, they could offer ndvdually dfferentated prces, thus squeezng off every cent from the users. Such a perfect dentfcaton of reservaton prces s not lkely n the real world. However, the reservaton prce of exstng customers s generally hgher than that of new customers, because exstng customers tend to exhbt hgher swtchng costs and also hgher brand preference for that product [20]. Furthermore, most of the analytcal lterature on prce dscrmnaton has found that t s optmal to penalze loyals wth hgher prces than swtchers [13] [8]. Whle we do not ntroduce targeted prcng to our models, we stll assume that loyal users nherently tolerate a hgher prce than swtchers, who are only nterested n dscount prces. Ths way, we use dual reservaton values n Secton 3.2 to represent the heterogenety of the user populaton. In Secton 3.1 however, we stck to the assumptons of [7] n order to construct a clear extenson of that model. In all cases, reservaton prces are assumed to be common knowledge. Although the payoff functons of ISPs are pretty smple across ths paper (e.g., margnal cost s set to zero), they are n lne wth flat-rate prcng, consumer demand elastcty and reservaton values dscussed above, and thus they sut our needs. 3.1 Incentve to cooperate Here we present a sngle-shot game of user loyalty whch was ntroduced n [7]. Later, we extend ths game to an nfntely repeated game, and show how a cooperatve maxmum can be enforced, where the long-term proft of ISPs are hgher than that of playng the equlbrum strategy of the stage game n each round. The stage game. Consder a market wth two local ISPs competng n prces for a fxed number of customers. Customers are splt nto three parttons upon ther brand loyalty: the frst group conssts of l 1 customers who are all loyal to ISP 1 n the sense that f ISP 1 s prce p 1 s less than or equal to a reservaton value α, they choose ISP 1 as ther servce provder, otherwse they do not purchase Internet access. The second group conssts of l 2 loyal customers of ISP 2, whle the thrd group contans n swtchers, who buy servce from the cheapest provder, f ts prce s not greater than α. If the provders announce the same prce (p 1 = p 2 <α), then half of the swtchers chooses ISP 1 and the other half chooses ISP 2. The flow of the game s that ISPs announce ther prces smultaneously, then customers make ther choces. Ths game s referred to as G 0. Note, that though values l 1 > 0, l 2 > 0andα>0are common knowledge, group membershp of a gven customer cannot be determned, so there s no prce dscrmnaton possble. Furthermore, for smplcty we assume a constant unt cost of zero for both frms, and that ISP 1 has the larger loyal user base, l 1 >l 2. Gven the above and that p 1 α and p 2 α, ISP 1 s payoff can be expressed as ( (l1 + n)p 1 p 1 <p 2 π 1(p 1,p 2)= (l n)p 1 p 1 = p 2 l 1p 1 p 1 >p 2 (1) It can be shown (see [7] and [13]) that ths game has a unque Nash equlbrum n mxed strateges. In ths case, equlbrum profts are π 1 = l 1α and π 2 = l 2+n l 1 l1α. As t can be +n notced, the equlbrum has shfted compared to the smple Bertrand game wthout consumer loyalty, both partes havng a postve payoff n equlbrum. The repeated game. Now, we extend the prevous model, and show that the nfntely repeated G 0 has a subgame perfect equlbrum, whch can be enforced by a threat strategy, namely the Nash equlbrum strategy of the stage game G 0. In the followng we construct G r as the nfntely repeated extenson of G 0. Payoff s dscounted at step k wth a dscount factor Θ < 1. The game s contnuous at nfnty snce the dscounted payoff n any step s bounded by α(l 1 + n). Ths way we can use the one-step devaton prncple to prove sub-game perfecton of a gven strategy set. Now, f the two provders cooperate and set ther prces equal to the reservaton value α, they wll share swtchers equally, n addton to keepng ther own loyal users. Ths way ther payoffs (π coop ) would be hgher than n the equlbrum case (π eq ), snce π coop 1 =(l n)α >π eq 1 = l 1α, and π coop 2 =(l n)α >π eq 2 = l 2+n l 1 l1α f n>l1 2l2. +n In the cooperatve case the ont proft of the two ISPs s the maxmum achevable (n + l 1 + l 2)α. Ths cooperaton s hghly benefcal for both partes. If somehow one ISP tres to grab the whole free market n a sngle step k, the other ISP can counteract from step k + 1 by chargng the Nash equlbrum prce from G 0 further on, whch results n a decreased payoff for the trator. We show that ths Nash reverson assures sub-game perfecton for the followng strategy profle under the stated condtons. Proposton 1. The strategy profle Cooperate untl the other player devates and then play accordng to the equlbrum n G 0 s a sub-game perfect Nash equlbrum for the repeated game G r,fn>l 1 2l 2 and Θ > 1 + l 1 n+l 1 l n+l n Ths means that both the ISP wth the smaller and the ISP wth the larger loyal user base have an ncentve to cooperate n order to maxmze ther proft on the long run. Whle explct cooperaton may be llegal, ths ncentve may lead to dscussons between servce provders. Note, that a two- ISP settng may seem artfcal, t s certanly not, e.g., a large fracton of Internet users n the US can only choose between the local cable and phone company. 3.2 Dfferentated reservaton prces Here we construct and analyze sngle-shot games modelng the prce competton between local ISPs fghtng for customers wth dfferent reservaton prces. Frst, we deal 57

4 D( p) a 1 a 2 0 p (a) Inelastc demand D( p) a 1 0 p a 2 (b) Elastc demand Fgure 1: Demand functons for G 1 and G 2 wth the case of nelastc demand, and later, we ntroduce elastc demand. Inelastc demand. We need to change the sngle-shot game G 0 a lttle bt to reflect the dualty n reservaton prces. We construct G 1 by ntroducng α 1, the reservaton value for swtchers, and α 2, the reservaton value for loyal users (α 1 <α 2), nstead of the sngle reservaton value α. Thus the demand functon for G 1 s the followng: 8 < n + P N P D(p) = N : =1 l =1 l 0 p α1 α1 <p α2 (2) 0 p>α 2 where p s the prce charged to users and N s the number of competng ISPs. The demand functon can be seen n Fgure 1(a). From that, we can defne the payoff functon Π (p )ofisp, whch has a form of 8 < p `l + n p m =mn p α 1 Π (p) = p l mn p <p α 2 (3) : 0 p >α 2 where m s the number of ISPs chargng the same mnmum prce, therefore sharng swtchers equally. Proposton 2. Consder G 1 wth two players (N =2). Let us defne A = nα 1 and B =(α 2 α 1)l for =1, (p 1,p 2)=(α 2,α 2) s a pure strategy Nash equlbrum, f A<B 1 and A<B 2; 2. (p 1,p 2)=(α 2,α 1) s a pure strategy Nash equlbrum, f A<B 1 and A>B 2; 3. (p 1,p 2)=(α 1,α 2) s a pure strategy Nash equlbrum, f A>B 1 and A<B 2; 4. There s no pure strategy Nash equlbrum f A>B 1 and A>B 2. Elastc demand. A common model used for elastc demand s a lnear demand functon [19]. We construct the game G 2, by substtutng the demand functon n G 1 wth the one n Fgure 1(b): 8 < (α 1 p)+ P N =1 l 0 p α1 P D(p) = N : =1 l α1 <p α2 (4) 0 p>α 2 From that, we can defne the payoff functon Π (p )ofisp, whch has a form of 8 >< p `l + α 1 p p m =mn p α 1 p Π (p) = l mn p <p α 2 or (5) >: α 1 <p α 2 0 p >α 2 where m s the number of ISPs chargng the same mnmum prce. The equlbrum propertes of G 2 are as follows. Proposton 3. Consder G 2 wth two players (N =2). Let p max = argmax p [0,α1 ] `l p + α 1 p m. Let us defne A =(α 1 p max)p max and B =(α 2 p max)l for =1, (p 1,p 2)=(α 2,α 2) s a pure strategy Nash equlbrum, f A<B 1 and A<B 2; 2. (p 1,p 2)=(α 2,p max) s a pure strategy Nash equlbrum, f A<B 1 and A>B 2; 3. (p 1,p 2)=(p max,α 2) s a pure strategy Nash equlbrum, f A>B 1 and A<B 2; 4. There s no pure strategy Nash equlbrum f A>B 1 and A>B 2. The logc behnd Propostons 2 and 3 s the followng. If an ISP has a large loyal user-base, and the ISP can charge them a prce hgh enough, t does not have to deal wth dsloyal users, snce the dfference between the proft at prce α 2 wth loyal users only, s larger than the proft at any prce below the α 1 threshold wth both loyal and all dsloyal users. So f the loyal user populaton has a hgh enough reservaton value (α 2), ther provder can mlk them, and t s not nterested n undercuttng other ISPs to grab swtchers. Note, that we can generalze Proposton 3 to multple servce provders and any reasonable demand functon D(p) =f(p)+ P N =1 l. Please consult [3] for detals. 4. MODELING USER LOYALTY When the payoff functons of ISPs are known, game theory can be used to fnd the optmal strategy to maxmze proft. But what f you do not know the payoff n advance? In realty, an ISP can hardly ever know t exactly. In the smplest model, the proft of an ISP s the product of the demand for ts servce and the access prce. The bggest problem here s modelng the demand, because t depends on many factors, not only on the gven access prce. It also depends on the prce of the compettors, the qualty of the gven servce, and even on human factors. Unfortunately, t s very dffcult to ncorporate all these factors n a closed form demand functon, and hence, t s very dffcult to analyze them wth game-theoretc tools. Instead, we ntroduce two extended user loyalty models, whch capture mportant aspects of the nature of demand, and assess the mpacts of the models n a smulator. We argue that although ntutvely, user loyalty has a crucal effect on demand and proft of ISPs, ths subect has not been addressed properly so far n the lterature. Through smulatons, we show how an ISP can set ts prce to reach maxmum proft n the presence of user loyalty, and also, exactly how much an ISP can beneft from a loyal user base. 4.1 Two Models Whle the games n Secton 3 have appled a very smple user loyalty model, such a model may not capture the real world characterstcs of a prcng competton. The am of ths secton s to present two novel approaches for modelng the overall loyalty of a user populaton. Both models are constructed for usng n a repeated prce settng scenaro, where servce provders repeatedly (but smultaneously) set ther access prces, e.g., monthly, tryng to attract customers. The frst model ncorporates the prce dfference among ISPs, whle the second model ntroduces uncertanty n human decsons to user loyalty. Dealng wth prce dfference the determnstc model. One logcal mprovement n loyalty modelng s to determne the amount of swtchers (the change n the demand) n a sngle step based on the relatve prce dfference 58

5 between ther current ISP and other ISPs. The ustfcaton of ths method s that swtchng provders comes together wth some cost to the user (e.g., termnatng ts current contract, leasng a new access devce, etc.), so t s only worth t f the prce dfference s large enough. Snce we focus on the prce competton among ISPs where users are not players n a game-theoretc sense, such a factor can only be ntroduced on a per ISP bass. We acheve ths by calculatng the number of swtchers proportonal to the prce dfference between ther current provder and the mnmum-prced provder(s). Furthermore, because of the tme and admnstraton demand on an ISP for termnatng the contract of a huge user populaton (and also on the newly selected ISP for contractng the same amount), there s a hard constrant on the number of swtchers at a sngle step. To model ths constrant, we have ntroduced a threshold to lmt the number of mgratng users. Based on the above, for a gven servce provder ISP,the number of users t loses to or gans from other provders n round k s defned as!! ΔU (k) = U (k 1) mn p(k) p (k) max mn,l, L, mn p (k) + p (k) where U (k 1) s the number of users assocated wth ISP n round k 1, p (k) s the access prce charged by ISP n round k, andl [0, 1] represents the admnstraton constrant of ISP, and t s a smulaton parameter. Note that the number of users n the system (across all ISPs) s modeled as constant, and s normalzed to 1. Dealng wth human uncertanty the stochastc model. In an attempt to cope wth uncertanty n human decsons, we reach back to the concept of ndvdual loyalty. We descrbe a user s ndvdual loyalty by a random varable X wth a cumulatve densty functon of F (x). Snce ISPs have a large number of users, we then apply the well-known Central Lmt Theorem to ndvdual loyalty varables to get the loyalty varable of the whole user base of an ISP. Let X 1,...,X n be dentcally dstrbuted, ndependent random varables wth E(X )=μ and Var(X )=σ for 1 n, where n s the number of users. S n = X X n s the sum of those random varables. Then for a large n, E(S n)= nμ and Var(S n)= nσ. Furthermore, lm n F (s n)= s Φ n nμ nσ. If the thrd central moment E((X ) 3 )exsts and s fnte, then the speed of convergence s at least on the 1 order of n (see Berry-Essen theorem [15]). We defne the loyalty of a user populaton, S, asthesum of random varables representng ndvdual loyalty, denoted by S n above. Snce X 1,...,X n have to be d for the theorem to hold, we make the assumpton that ndvdual users loyalty do not affect each other, rather t s a congental qualty. Also the user populaton under observaton should be relatvely bg, as we need n to be a farly large number for the convergence to take effect. On the other hand, we can model an ndvdual user s loyalty wth any proper probablty dstrbuton. We also keep the admnstraton constrant L and the dependence on the prce dfference between the respectve ISP and other ISPs. The number of swtchers for an ISP at step k s calculated as followng: ΔU (k) =max mn S (k),l, L, where S (k) q N U (k 1) μ, U (k 1) and σ s a smulaton parameter. «σ and μ = mn p (k) p (k) mn p (k) +p (k) 4.2 Smulaton Results We have developed a smulator to study the mpact of the novel loyalty models. Smulatons analyze the behavor of competng local ISPs. Each ISP has some end users, a share of the market. We suppose that the overall demand functon for ther servces s constant: no users enter or leave the market. Ths model s relevant to a saturated market, where everybody can afford to have Internet connectvty, and nternet connectvty s a must (see Secton 3 for detals). We suppose that the user market s nfntely dvdable among the fxed number of ISPs. The total end user market s normalzed to 1, e.g., f there are 2 ISPs wth equal market share, then both of them have a market share of 0.5. ISPs compete for customers by settng ther access prces n each round. The prce scheme used s flat rate and we assume a homogeneous user populaton (sngle, common reservaton prce). The lowest prce an ISP can set for a round s 0, whle the hghest prce s 100, whch corresponds to the reservaton value common to all users. Before the frst round, ntal market shares are set. In each round, end users may mgrate from ther respectve provder wth regard to the appled loyalty model. Swtchers choose the cheapest ISP. For smplcty we assume that f there are two mnmally-prced ISPs, half of the mgratng users ons one and the other half ons the other one. In each round the ISP s try to maxmze ther nstant proft. Each of the ISP s uses the same smple and greedy strategy. They suppose that the prces offered last by ther competton stay the same for the next round. Wth ths n mnd, they calculate ther proected market share change and proft by probng all possble prces they can set, and fnally, they choose the prce that would maxmze ther proft n the next round and then play t. The results presented n ths secton are only ntended to flash some nterestng ssues concernng the mpact of user loyalty on the prcng competton of ISPs n dfferent scenaros. For a more comprehensve analyss on smulatons for local ISP competton please refer to [3]. Intal market shares. The frst nterestng results have been produced usng the stochastc loyalty model. The ntal market share of ISP 1 was set to 1.0 (total market), and the market shares of all other ISPs was set to 0. Fgure 2(a) show the prces and market shares (also correspondng to nstant profts) of 3 competng ISPs n tme. What can be seen s that lower market share ISPs start grabbng the market from ISP 1 by settng lower prces. Ths makes the hgher share ISP lower ts prces as well, untl a state close to equal market shares s reached. The proft chart teaches us exactly what s logcal ntutvely. If you have a large loyal user base, you make the hghest proft by settng hgh prces. Small companes make the hghest proft by settng dscount prces, but can only dream of the profts of the large players. The larger your loyal user base s, the more nstant proft you make. Determnstc vs. stochastc loyalty model. Now let us take a look at the game, when the ntal market shares are equal, but let us use dfferent loyalty models. If we compare the profts of 3 ISPs wth dfferent loyalty models, t can be notced that n the determnstc model all the ISPs get the same proft, whle n the other case, stochastcty results n slghtly dfferent profts for dfferent provders (see Fgure 2(b)). The determnstc curve has a perodc shape. The explanaton behnd ths s that t s worth gradually lowerng your access prce to undercut others and steal market share. Of course there s a certan prce under whch the respectve ISP would get less money by further undercuttng than by mlkng ts currently assocated customers. When ths 59

6 Threshold=0.1 Threshold=0.2 Threshold=0.3 Threshold= Total profts (a) Prces and user bases stochastc model (b) Profts n tme Fgure 2: Smulaton results Round (c) Overall cumulated profts of 2 ISPs value s reached, the ISPs set the prce sgnfcantly hgher. Note, that all ISPs thnk the same (because of the smple strategy appled, see above), so ther prces (and profts and market shares) wll be exactly the same across tme (that s why the curves overlap). On the other hand, when user base changes are random, ISPs do not follow exactly the same lnes of thought, so they get dfferent payoffs. It s worth mentonng that profts fluctuate around the even shares. Level of loyalty n the user populaton. Another mpact of loyalty affects the overall proft of ISPs. Here, we dscount profts n tme wth a dscount factor of In Fgure 2(c) the overall sum proft of 2 ISPs can be seen n tme. Dfferent lnes denote dfferent levels of loyalty: the hgher the threshold, the more users can swtch provders at a sngle step (see Secton 4.1). Results show that the hgher the level of loyalty (.e., the lower the threshold), the hgher overall sum proft can be acheved over tme, mplyng that ISPs are nterested n users beng loyal to them, snce stronger loyalty results n hgher overall profts. Dscusson. Our smulaton results are consstent wth the fndngs of recent emprcal surveys on loyalty n the wred and wreless ISP market [4]. Frst, they pont out the exstence of a truly loyal customer segment ( 38%) whch tolerates hgher prces, and s lkely to pay for new servces and looks for a long-term busness relatonshp. On the other hand, there are hgh rsk customers ( 30%)who are wllng to swtch provders at the earlest opportunty, and are drven by both lower prces and (congental) behavor. Second, whle 78% of the customers are satsfed wth the servce they get, only the above-mentoned 38% are truly loyal, hence there s more to loyalty than beng satsfed (behavoral patterns). Thrd, they show that ISPs wth the most loyal customers ( loyalty leaders ) can expect sgnfcantly larger revenues, faster growth and hgher stock prce performance than ther compettors. We beleve that these results ustfy the mportance of user loyalty modelng wth regard to prcng Internet access. 5. CONCLUSION In ths paper we have studed multple facets of the mpact of customer loyalty on the prce competton of geographcally co-located ISPs. We frst showed how cooperaton between ISPs resultng n hgher profts can be enforced n the presence of customer loyalty. Second, we nvestgated the case of a dfferentated customer populaton by ntroducng dual reservaton values, and show how t leads to new, pure strategy Nash equlbra for a wde range of demand functons. Thrd, we developed two novel models for customer loyalty, along wth a smulaton tool that s capable of demonstratng the mpact of the novel models, and showed the numercal effect of user loyalty on the prce competton through these more realstc models. Of course, there s remanng work to be done: we would lke to ncorporate our novel loyalty models nto a game-theoretc framework, furthermore, we would lke to nvestgate the case of targeted prcng on the Internet access market. 6. REFERENCES [1] NSF Future Internet Network Desgn Intatve. [2] NTT DoCoMo. [3] Socoeconomcs Group Webste, Department of Telecommuncatons and Meda Informatcs, Budapest Unversty of Technology and Economcs. [4] The Walker Loyalty Report for Communcatons Servces. Techncal report. [5] X.-R. Cao, H.-X. Shen, R. Mlto, and P. Wrth. Internet Prcng wth a Game Theoretcal Approach: Concepts and Examples. IEEE/ACM ToN, 10(2): , [6] J.-S. Chou. The Antecedents of Consumers Loyalty toward Internet Servce Provders. 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