Oligopolistic Pricing with Online Search

Transcription

1 Oligoolistic Pricing with Online Search Lizhen Xu, Jianqing Chen, and Andrew Whinston Lizhen Xu is a Ph.D. candidate in the Deartment of Information, Risk, and Oerations Management, Red McCombs School of Business, University of Texas at Austin. He holds an M.S. in economics from the University of Texas at Austin and a B.S. in engineering in comuter science from Tsinghua University, Beijing, China. His current research focuses on economic issues and marketing ersectives in online advertising and risk management in rocurement auctions. His aers have been ublished or are forthcoming in academic journals such as Journal of Marketing Research, Journal of Management Information Systems, and Decision Analysis. He received the Best Paer award for the Fifteenth Conference on Information Systems and Technology in 00. Ji a n q i n g Ch e n is an assistant rofessor of information systems at Haskayne School of Business at the University of Calgary. He received his Ph.D. from McCombs School of Business at the University of Texas at Austin in 008. His general research interests are in electronic commerce, economics of information systems, and suly chain management. His aers have been ublished or acceted for ublication in Decision Analysis, Decision Suort Systems, Economics Letters, Information Systems Research, Journal of Marketing, and Production and Oerations Management. Andrew Whinston is the Hugh Cullen Chair Professor in the Deartment of Information, Risk, and Oerations Management, Red McCombs School of Business, University of Texas at Austin. He is also the director at the Center for Research in Electronic Commerce and editor-in-chief of Decision Suort Systems. His recent aers have aeared in Information Systems Research, Journal of Management Information Systems, Management Science, Marketing Science, Journal of Marketing, and the Journal of Economic Theory. He has ublished over 00 aers in the major economic and management journals and has authored 7 books. In 005, he received the Leo Award from the Association for Information Systems for his long-term research contribution to the information systems field. In 009, he was named the Distinguished Fellow by the INFORMS Information Systems Society in recognition of his outstanding intellectual contributions to the information systems disciline. Ab s t r ac t: In this aer, we set u a game-theoretic model to examine oligoolistic rice cometition, considering two features of online search: the existence of a common search ordering and shoers who have nonositive search cost. We find that in equilibrium firms set their rices robabilistically rather than deterministically, and different firms follow different rice distributions. The equilibrium ricing attern exhibits an interesting local-cometition feature in which direct rice cometition occurs only between firms adjacent to each other. Further, we incororate consumers search strategies into the model so that both search order and stoing rules are Jianqing Chen is the corresonding author. Journal of Management Information Systems / Winter 00, Vol. 7, No., M.E. Share, Inc. 074 / 0 $ DOI 0.75/MIS

2 Xu, Chen, and Whinston determined rationally by consumers. We show that similar atterns may continue to hold in the fully rational framework when consumers have higher insection costs for inferior ositions. Ke y w o r d s a n d h r a s e s: local cometition, oligoolistic cometition, online search, rice disersion, ricing. Th e In t e r n e t h a s g r e at ly i m r o v e d t h e e f f i c i e n c y of sharing information. On the one hand, the Internet greatly reduces the hysical cost of accessing roduct information; on the other hand, the raidly develoed e commerce alications, esecially online sonsored advertising, brings merchants selling similar roducts together and facilitates consumers searching. Nevertheless, the rediction of the law of one rice has not been realized in the Internet era. Price disersion has been well documented and discussed in the information systems literature [5, 7, 7]. This work draws uon consumer online search behavior and studies the oligoolistic equilibrium ricing to offer a theoretical exlanation of online rice disersion. One classical view in economic theory attributes rice disersion to the heterogeneity in consumers search behavior [, 8]. These works tyically study rice cometition in traditional offline search markets, and their model settings may not fit the online search very well. Comared to the traditional offline search, two features are more salient in the online environment: first, because of considerable differences in the location or the visibility of business ositions (e.g., the hyerlinks), there often exists a common ordering in consumer search. Second, consumers search costs are highly diversified; in articular, there exists a substantial ortion of shoers who have nonositive search costs. Differences in location and the way of resenting on a Web age, or different virtual ositions in the hyerlinking network on the Internet, cause firms links to differ greatly in terms of their visibility or rominence level. For examle, search engines commonly dislay a limited number of remium-sonsored links in a highlighted area with a large size and a bright color on the to of a search result age; they also dislay a list of regular-sonsored links in the right column of a Web age with no highlighting, among which the to ositions are commonly believed to be suerior to the lower ositions. When dislay sace is limited (e.g., on mobile devices), sometimes only one or two ads are dislayed at a time, and users have to switch to another age to view additional ads. Because of human eye movement atterns and information-rocessing habits, consumers usually ay different levels of attention to different ositions. Exerimental studies have shown that consumers ay more attention to content with colors and a large size [], and comared to aer media, consumers are more likely to focus on ads near the heading of electronic lists [0]. Online statistics also show a significant decrease in traffic from the to-sonsored link downward and much fewer visits after the first age []. As a result, most consumers browse links following a common order: they insect the most rominent ositions first, and then some sto while others continue to insect the less rominent ones. In fact, there are also examles in the hysical world that resemble

3 Oligoolistic ricing with online search such ordering: most consumers first look at shelf slots at the eye level in a suermarket [9] or first visit the storefronts near the main entrance of a shoing mall, and then some of them continue to the floor-level shelf slots or the corner storefronts. In reality such a common ordering could simly be the direct result of human habits and not necessarily a strategic decision after sohisticated calculation; nevertheless, we can also exlain such ordering in a fully rational framework. As we show, the ordered search can be derived as an equilibrium outcome originating from differences in the insection costs for different ositions. In articular, the inferior ositions incur higher insection costs than the suerior ones. A higher insection cost can be interreted as the sychological resistance to overcoming the information-rocessing habits, or the extra effort in locating a less-rominent link or switching Web ages. Given such differences, consumers choice to follow a certain order is a rational search strategy in equilibrium. The second feature of online search behavior owes to the convenience brought by the Internet, which reduces consumers search costs from driving to the store to making several clicks of the mouse. In addition, it has also been shown that some consumers derive hedonic utility from shoing online [6]. As a result, a few consumers actually have a nonositive (zero or even negative) net search cost. We call them shoers. However, not everyone shoing online has such time luxury. The convenience of electronic commerce attracts eole with time constraints whose only goal is to find a roduct within the shortest amount of time. Thus, these consumers have ositive search costs. They do not conduct an exhaustive search and sto searching at certain stages, usually after samling only a few sites (e.g., [] emirically shows that online customers tend to search very few sites on average). To study the equilibrium ricing attern, we set u a game-theoretic model caturing the two features of the online search. We consider oligoolistic cometition in which multile firms comete for consumers in a roduct market. Firms are differentiated in the rominence level of their ositions, which are reflected in the ranks in consumers resumed search sequence. Consumers are differentiated in their search behavior. In articular, a few consumers have nonositive search costs and conduct a thorough search. We eliminate heterogeneity among firms and consumers in all other dimensions excet firms osition and consumers search behavior to show that the driving forces of the equilibrium ricing attern are the two distinctive features of the search behavior. We find an interesting equilibrium ricing attern when first taking consumers search strategies as exogenous. The equilibrium exhibits the feature of local cometition, in which firms comete directly with their neighbors along consumers search order only. We show that in equilibrium all firms mix their rices over different suorts. Overlas occur only in the suorts of two firms adjacent to each other. Hence, there is no direct cometition between any two firms that are not next to each other. We further show that behind the local-cometition feature lies a global mutual deendence across all firms. We then endogenize consumers search strategies so that consumers make fully rational decisions on their search order and stoing rules. We show that similar ricing atterns may continue to hold in equilibrium when consumers insection costs for different ositions are different.

4 4 Xu, Chen, and Whinston The main contribution of this work to the economic theory on search and ricing lies in that we study the asymmetric mixed-strategy equilibrium ricing in oligoolistic cometition. To the best of our knowledge, the local-cometition attern revealed in this study is absent in the revious literature. The investigation on endogenizing the consumer search enriches the studies that exlore asymmetric equilibrium ricing with otimal search strategies. Diamond [8] raises the famous aradox that when consumers have ositive search costs, an endogenous search model leads to a trivial equilibrium in which all firms charge the monooly rice and consumers do not search. Varian [0] suggests that when there exist consumers who are informed of all firms rices, the equilibrium outcome may involve mixed-strategy ricing, which leads to rice disersion. Stahl [8] studies a random search model in which consumers randomly ick a firm to insect and all firms are symmetric. That article considers the existence of a ortion of shoers who have zero search cost and derive a symmetric equilibrium ricing rovided that all nonshoers have the same search cost. Weitzman [] formulates the otimal search strategies given firms (asymmetric) rice distributions in a general setting. Arbatskaya [] is among the few who study ordered search and rice cometition. By considering cost distributions atomless at the zero oint, that article focuses on the ure-strategy equilibrium. There are other studies that analyze mixedstrategy equilibrium in duoolistic cometition in different contexts. For examle, Cambell et al. [4] consider the effect of shoers in a symmetric duoolistic setting, and Narasimhan [4], Weber and Zheng [], and Xu et al. [] consider asymmetric duoolistic mixed strategies. In contrast to these works, we consider the existence of shoers in the ordered search market, and exlicitly derive asymmetric mixed-strategy equilibrium in oligoolistic rice cometition with both exogenous and endogenous consumer search. The rest of the aer is organized as follows. Next, we start with a model in which consumers search behavior is exogenously given. The third section details the analysis and shows the main results. In the fourth section, we endogenize the consumer search such that consumers rationally decide what search order to follow and when to sto. We consider both cases of osition-invariant and osition-deendent insection costs. We show that the local-cometition attern may arise in the fully rational framework when consumers have higher insection costs for inferior ositions. The aer concludes with a discussion on managerial imlications. Model Th e r e a r e n ( ) f i r m s s e l l i n g h o m o g e n e o u s ro d u c t s and cometing for consumers in a roduct market. These firms have the same marginal roduction cost, which is normalized to zero without loss of generality. A continuum of consumers with unit mass exists in the market. Each consumer has a unit demand of the roduct and realizes a unit utility from consuming the roduct. Therefore, consumers will buy the roduct only if its rice does not exceed. Firms are identical excet for their rank

5 Oligoolistic ricing with online search 5 in the search ordering, and consumers are identical excet for their search behavior. By eliminating differentiation in all other dimensions, we are able to show that the distinctive features of consumers online search behavior alone could drive an interesting rice disersion attern. Consumers obtain roduct information through an information ortal with a list of hyerlinks directed to firms Web sites, where urchases can be made directly. Firms are laced in different ositions in the list, which can be viewed as the outcome of a regame cometition, such as a bidding cometition. Because all firms are identical ex ante, the location cometition outcome is irrelevant for analyzing the rice cometition. Any assignment of ositions becomes identical after relabeling firms by their osition rank. Therefore, we do not include the location cometition in the model but start from after firms get laced at different ositions. Different ositions have different rominence levels, which can be strictly ordered. Without loss of generality, we call the most rominent osition the first osition, the second most rominent osition the second osition, and so on. For convenience, we call the firm at the ith osition firm i (i =,..., n). We start with the case in which consumers search strategies are exogenously given, in a way reflecting the two unique features of online search atterns: first, there exists a common search ordering so that all consumers start searching from the first osition and may continue to the second, then the third, and so forth. Second, consumers search costs are highly diversified so that they may sto searching at different stages. In articular, there exists a certain ortion of shoers who have a nonositive search cost and samle all ositions before making the urchase decision. Secifically, we assume that after samling the ith osition, a ortion of α i (0 < α i < ) sto searching, while the other α i continue to samle the next osition. Therefore, the ortion i of consumers who visit the ith (i ) osition is P j= ( a j ). To simlify notation, i we denote b i = P j= ( a j ), and let β = and α 0 = 0. To rule out violent fluctuation in the attention-declining rates α i s, we assume that a i a i+ ( a i ) ( i < n). This condition requires that the rate of decline in attention (i.e., α i s) does not increase dramatically from one osition to the next, which can be easily satisfied (e.g., the same rate of decline across ositions (α i = α i+ ) satisfies this condition). The timing of the game is as follows. Firms at different ositions rice their roducts simultaneously. Consumers samle the osition(s), learn the rice(s), and make the urchase decision. For those who samle at least two ositions, they urchase from the firm with the lowest rice. When there is a tie in the lowest rice, they randomly ick one of the firms, with equal robability. A brief summary of notation is resented in Aendix A. Equilibrium with Exogenous Search We first d e r i v e f i r m s equilibrium r i c i n g s t r at e g i e s and then analyze the attern of equilibrium rice disersion. In deriving the equilibrium ricing, we notice that any static ricing is unstable due to the existence of shoers.

6 6 Xu, Chen, and Whinston Lemma (Lack of Pure-Strategy Equilibrium): There is no ure-strategy equilibrium in the rice cometition. Proof: We rove the above result by contradiction. Suose there is a ure-strategy ricing equilibrium. If there is no tie in the lowest rice, then it is rofitable for the firm with the lowest rice to deviate by increasing its rice closer to but still lower than the second-lowest rice. If there is a tie in the lowest rice that is strictly ositive, then any of the firms with the lowest rice has rofitable deviation by slightly cutting the rice. If there is a tie in the lowest rice that is zero, then among all these firms with zero rice, the one with the highest rominence level can achieve ositive rofit by slightly increasing its rice. Therefore, urestrategy ricing cannot be an equilibrium. Q.E.D. Because there exists a certain ortion of consumers who samle all ositions to look for the lowest rice, a slight cut in rice to offer the lowest can lead to a significant increase in market share by caturing this ortion of consumers. As a result, cometing firms tend to lower their rices relative to the rivals. However, once the rice is ushed to a certain low level, the firm at a better osition in terms of rominence can be better off by charging a higher rice and exloiting those consumers who sto searching right there. Therefore, any ricing strategy in which firms statically charge one rice cannot be stable. Clearly, the driving force here is the resence of shoers and the locational asymmetry created by the search ordering. We next examine the mixed-strategy ricing equilibrium. We use F i (), i =,..., n, to describe firm i s mixed strategy of ricing. Like regular cumulative distribution functions, F i () measures the robability that firm i charges a rice less than or equal to. Proosition (Equilibrium Pricing and Local-Cometition Pattern): The equilibrium mixed strategy of ricing from osition i is as follows: i+ Fi = αi ( i ) αi( αi ) αn ( n ) Fn = α n i+, i) i n i, i n, n, () where \ i s are recursively defined as = = 0 i = ki i i =,..., n, and the coefficient k i s are recursively defined as k k n n i = α αi = α + α α k i i+ i i+ ( i = n,..., ). () ()

7 Oligoolistic ricing with online search 7 Proof: All roofs are resented in Aendix B, unless indicated otherwise. We next use an examle to illustrate the attern of the equilibrium mixed-strategy ricing. Examle : Consider a case of four ositions with the same declining rate. Secifically, n = 4 and α = α = α = /. According to the recursive definition in Equation (), k = α = /, and we can then derive that k = 4/5 and, further, that k = 5/7. Thus, according to Equation (), \ =, \ = k \ = 5/7, \ = k \ = 4/7, and \ 4 = k \ = /7. Notice that by definition, the sequence of rice suort bounds {\i }n is monotonically decreasing, so that \ > \ > \ > \. i= 4 The ricing strategies of the four firms are as follows: 5 5 F = 7 7, = F, 7 7 = ( ) 5, 7 (4) 4 7, 7 7 F = ( ) , =,. 7 F 7 7 Figure illustrates the suorts and distributions for the ricing strategies of the firms in these four ositions. Figure deicts the simulated results of the equilibrium in Examle with each oint reresenting an indeendent draw from the rice distributions in Equation (4). Notice that the dotted square areas that the diagonal asses through in (a), (b), and (d) indicate the direct rice cometition between two adjacent firms. In equilibrium, each firm achieves a constant exected ayoff by charging any rice within its rice suort. On the one hand, the constant ayoff makes each firm willing to randomize rice over its entire suort. On the other hand, each firm randomizes its rice in a way that gives its cometitors a constant ayoff within their rice suorts. Secifically, firm i s exected rofit can be written as πi = αiβi + αi+ βi+ Fi+ i+, i) πi = αiβi Fi i, i. When firm i ( i n ) charges a rice within the lower half of its rice suort [\ i+, \ i ), the firm catures all consumers who samle its site and sto searching there (i.e., a ortion of α i β i ). This is because its rice is lower than firm (i ) s rice for (5)

8 8 Xu, Chen, and Whinston Figure. Price Suorts and Cumulative Distributions for Different Positions sure (recall that firm (i ) s rice suort is [\i, \ ]), which accounts for the first term i of the right-hand side of the first equation in Equation (5). Firm i can cature those who continue to samle the (i + )th osition and sto searching there (i.e., a ortion of α i+ β i+ ) only when its rice is lower than firm (i + ) s rice, which accounts for the second term. Naturally, firm i forgoes all those consumers who continue to samle the (i + )th osition, since firm (i + ) s rice suort is [\ i+, \ i+ ] and its rice is lower for sure. The second equation in Equation (5) is the exected rofit when firm i rices within the uer half of its rice suort, which can be interreted in a similar way. Substituting in the firms equilibrium ricing strategies and by simle algebra, we can show that by charging any rice within its suort, firm i achieves a constant exected rofit a i b i \ i. The most interesting feature of the equilibrium ricing is the local-cometition attern; that is, firms comete directly with their neighbors only. There is no overla between the rice suorts of any two firms more than one osition aart and thus no direct rice cometition between firms distant from each other. The driving forces of such a attern are the two features of consumers search behavior. Because of the existence of shoers, firms do not statically charge one single rice but have to mix their rices to comete for consumers. Nevertheless, such cometition is localized because of the decrease in visits along the common search order. The firm at a lower osition cannot be better off by entering a higher rice range, because it would then lose its already quite limited customer base. The firm at a higher osition will not undercut its rice, because in doing so it would only entangle itself in a fiercer rice cometition against the lower-ranked firms, which would result in little gain in extra

9 Oligoolistic ricing with online search 9 (a) Simulated Prices at Positions and (b) Simulated Prices at Positions and (c) Simulated Prices at Positions,, and (d) Simulated Prices at Positions and 4 Figure. Simulation of the Equilibrium in Examle (0,000 Simulated Points) demand but significant loss in the rofit from the catured demand. As a result, to comete locally is the equilibrium outcome. Another interesting asect of the local-cometition attern is that each firm s equilibrium ricing strategy only involves local information. According to Equation (), firm i s rice distribution F i only contains consumer search arameters α i, α i, and α i+. Also, firm i s rice suort is determined by (the lower bound of) firm (i ) s suort and (the uer bound of) firm (i + ) s suort; within its suort, firm i s rofit is determined only by firm (i + ) s and firm (i ) s ricing strategies, according to Equation (5). In this sense, although the formal equilibrium analysis needs to be based on the whole icture of the game, to formulate otimal ricing strategy in ractice, decision makers can simly focus on the traffic information at the adjacent

10 0 Xu, Chen, and Whinston ositions and the ricing strategies of the neighboring firms (i.e., firms that are adjacent and one osition aart). It is worth noting that the local-cometition attern is the result of global consideration. Although no direct rice cometition is exlicitly observed in equilibrium, even firms distantly above or below have an imact on a articular firm s ricing strategies. In fact, although firm i + k (i k) does not comete directly with firm i, it affects firm i s ricing through a chain effect, from firm i + k (i k + ) through firm i + (i ). To see this, reconsider Examle. When the fourth firm is eliminated, for examle, although the first two firms still have the same neighbors as before, the rice suort of the first firm shifts toward the right from [5/7, ] to [4/5, ], and the second firm s suort shifts to the left from [4/7, ] to [/5, ]. In fact, when the cometitor from below disaears, the third firm tends to increase its rice (and shifts its rice suort from [/7, 5/7] to [/5, 4/5]). In resonse, the second firm lowers its rice suort to cature more demand. Meanwhile, the more intense cometition between the second and the third firms drives away the first firm s interest, which leads to the increase of the lower bound of its rice suort. As we can see, behind the local-cometition henomenon actually lies the global mutual deendence among all firms. Several other features of equilibrium ricing are also worth noting. First, excet for the first one, all firms equilibrium ricing strategies are atomless within their entire suorts, including the uer and lower bounds. This is because a mass oint in one firm s rice distribution would result in a downward jum of another firm s exected demand at that oint and, consequently, lower rofit levels in a contiguous region right to that oint. For this reason, the only ossible lace where a mass oint may occur is the common uer bound of the first two firms rice suorts : although the mass \ oint in F ( ) causes a downward jum in firm s exected rofit at = \, firm s actual exected rofit is not affected because F ( ) laces a nonositive robability measure on that articular oint. This feature under our oligoolistic model is in line with the results derived from duoolistic cometition in other settings []. The kinks in firms ricing distributions can be exlained by the localized cometition. The ricing distributions in the first and second arts of the suort for firm i ( i n ) are determined by the cometition against its direct neighbors, firms i and i +, resectively. As the cometition against the firm above and the firm below are generally different, naturally a kink arises in firm i s ricing distribution at. The shae of ricing distributions for the first and last firms is distinctive \i from the rest because those two firms have one direct neighbor only. The next corollary reveals the monotonic decrease of the exected rofit of firms at different ositions, which exlains why the to osition of a sonsored list in online search advertising is usually the most oular and engenders fierce bidding cometition. Corollary (Decrease of Exected Profits): The firms equilibrium exected rofits decrease monotonically from the first toward the last; that is, π i > π i+, i =,..., n. Proof: Note that i = a i b i \ i. Because \ i > \ i+ and a i a i+ ( a i ), i > i+. Q.E.D.

11 Oligoolistic ricing with online search Corollary shows that location advantage is rewarding in the sense that the firm in the advantageous location earns a higher rofit. It is worth ointing out that the rofit difference between a higher-ranked osition and a lower-ranked one should dissiate in the regame location cometition if all firms are ex ante identical. That is, a firm has to ay a higher rice for a suerior osition, which counterbalances its rofit advantage. There is a rich literature on the cometition for better locations or exosure, from the classical advertising literature [6] to the recent work on online advertising and osition auctions []. The next corollary indicates that the equilibrium ricing under the ordered search with shoers exhibits two levels of disersion: not only are the realized rices at different ositions different, but the exected rices are also different across ositions. Corollary (Decrease of Exected Prices): The exected rice decreases monotonically from the first osition toward the last one; that is, E( i ) > E( i+ ), i =,..., n. Notice that firms adjacent to each other adot similar ricing strategies over the overlaed interval of their suorts. In fact, according to Equation (4), the conditional robability density functions of their ricing strategies over the overlaed interval are the same, which imlies the same conditional exectations; that is, E( i [\i+, \ i ]) = E( i+ [\ i+, \ i ]). Because firm i rices within the uer half of its rice suort [\ i, \ i ] with ositive robability and firm i + rices within the lower half of the suort [\ i+, \ i+ ] with ositive robability, the unconditional exectation of firm i s rice is strictly higher than that of firm (i + ). Corollary shows that the equilibrium rice exectation decreases monotonically along the direction of consumers search ordering. As a result, search is rewarding in the sense that those who kee searching are more likely to find a lower rice. Equilibrium with Endogenous Search In t h e r e v i o u s a na ly s i s, w e ta k e c o n s u m e r s s e a r c h b e h av i o r, including the search order and stoing rules, as exogenously given. In this section, we extend the analysis to endogenize consumers search strategies. The focus is to exlore whether and under what conditions a similar equilibrium ricing attern continues to arise in the fully rational framework. As we show, the inherent difference among ositions is necessary for the local-cometition attern to arise in equilibrium ricing. When consumers are free to samle any osition with no articular ordering constraint and insecting different ositions incurs the same cost, such a attern disaears. The attern arises only if the more rominent osition incurs a lower samling cost than the less rominent osition, which reflects the underlying distinction between ordered search and random search. In this case, we further show that under certain arametric conditions, equilibrium ricing with the same local-cometition attern can be derived in the fully rational framework. We now consider consumers as active layers in the game. We assume that all consumers are fully rational and decide their search order and stoing rules strategically. We consider the rational-exectations equilibrium (REE), in which consumers search

12 Xu, Chen, and Whinston strategies are rational given firms ricing strategies and firms have no rofitable deviation in ricing given consumers search strategies. For shoers with zero search cost, it is always otimal for them to samle all ositions before making a urchase decision. For nonshoers, we consider a sequential search rocess: the consumer insects one osition and learns the rice, and then he or she decides whether to continue searching or to sto and, if to continue, which osition to insect next. In other words, an individual consumer s search strategy consists of a sequence of decisions; each decision d(z, C) can be to sto, to insect a osition, or to randomly insect several ositions with a robability distribution; d(z, C) deends on the lowest rice z from all the insected ositions, and the choice set C, which contains all the uninsected ositions. To determine d(z, C), the consumer needs to calculate the net exected gain from all ossibilities for the next ste. The net exected gain from insecting the ith osition, given the lowest samled rice z and the choice set C, EG(i; z, C), equals the exected decrease in urchase rice lus the net exected gain from another rational search afterward, minus the insection cost. Notice that when no osition has been insected yet, we let z =, and we thus only need to consider z throughout the rest of the aer. Formally, for an individual consumer with ositive insection costs k i, given firms ricing strategies F i, similar to the formation in Weitzman [], the net exected gain can be formulated recursively as = z + { } {} EG izc ;, z dfi ki EG min z,, C \ i dfi, (6) where = { } EG z, C max EG j; zc,, 0. j C To determine the rational search decision is to comare the net exected gain of all the otions in the choice set. If further search yields no ositive net exected gain, stoing is the rational decision and EG = 0. Otherwise, the rational search decision is to continue to insect the osition that generates the highest net exected gain. In the case of a tie, randomly insecting any of them is rational. We next use an examle to illustrate consumers rational search strategy. Examle : Consider two firms. Firm sets its rice equal to or 0 with equal robability. Firm rices uniformly over [0, /]. Consumers insection costs are the same for both firms, and let k = k = /8. The rational search strategy in this case is to insect the first firm at first: if the quoted rice is zero, sto searching; otherwise, continue to insect the second firm. In this examle, the exected rice of the first firm (/) is higher than the exected rice of the second firm (/4). This examle shows that it may not be rational to start searching from the osition with a lower exected rice even if the insection costs are the same. In fact, consumers rational search strategies generally deend on the full distributions of equilibrium rices, which in turn are determined by consumers rational search strategies. Such interdeendence makes the analysis comlex, esecially when we consider asymmetric oligoolistic cometition and heterogeneous consumer

13 Oligoolistic ricing with online search search costs. For this reason, equilibrium analysis of oligoolistic ricing with rational search generally results in no closed-form solution. In this section, we seek to derive exlicit equilibrium under certain conditions. We consider the simlest oligoolistic case of three firms. The three firms are located in three different ositions. Again, we refer to the firm located in the ith osition as the ith firm or firm i, i {,, }. Consumers are different in their search costs. To be consistent with the revious settings, we assume that among all consumers with total mass, α of them have the highest search costs and are referred to as tye consumers; α ( α ) of them have lower search costs and are referred to as tye consumers; the rest ( α )( α ) are shoers. For the sake of simlicity, we let α = α = α (0 < α < ). Assume tye consumers incur a cost k i to insect the ith osition, while tye consumers incur a cost k i, i {,, }, 0 k i k i. All other settings follow the revious model setu. We first study the case in which consumers insection costs are osition-invariant; that is, k = k = k and k = k = k. In this case, there is essentially no difference among ositions, and firms thus are symmetric. We derive symmetric equilibrium under certain conditions and uncover an interesting equilibrium ricing attern that involves segmentation of rice suorts. We also show that the local-cometition attern from the revious analysis does not hold in the case of osition-invariant insection costs. We then allow insection costs to vary across different ositions and give a necessary condition and a sufficient condition for the local-cometition attern to arise in equilibrium. Position-Invariant Costs In this subsection, we consider the case that the insection costs are the same for all ositions. We let k = k = k k and k = k = k k, and we assume 0 < k < k <. Because we now do not imose any ordering constraint, when there is no difference in the insection cost for different ositions, all ositions, and thus all firms, are essentially the same. Naturally, this case reduces to the symmetric random search setting. When nonshoers have the same search cost, the symmetric equilibrium rice distribution can be exlicitly derived and analyzed [8]. When nonshoers have heterogeneous search costs, like in our setting, the equilibrium analysis is more comlicated because nonshoers may adot different search strategies, which in turn comlicates firms ricing decisions. Stahl [9] shows that with continuous cost distribution, a symmetric mixed-strategy equilibrium always exists, although the actual distribution atterns deend on the cost distribution and generally have no closed-form solutions. To better contrast with the local-cometition attern from the revious analysis, we next exlicitly derive the symmetric equilibrium under certain conditions. In the symmetric case, because firms adot the same ricing strategies in a symmetric equilibrium, the search order becomes trivial: in each ste, the rational strategy is either to sto or to randomly ick one of the uninsected firms. In other words, the search strategies simly reduce to the stoing rules. In addition, to determine the stoing rules, the exected gain in Equation (6) is easier to calculate. As we can show, when

14 4 Xu, Chen, and Whinston firms adot the same ricing strategy, the otimal stoing decision based on the exected gain from all the future searches is exactly the same as the decision based simly on the exected gain from the next one search. Therefore, in the symmetric equilibrium where each firm rices according to F( ), consumers search strategies (with search cost k) can be characterized by an otimal stoing rice r, which is determined by r ( r ) df k = 0. (7) When the current lowest rice exceeds r, it is worthwhile to conduct an additional search; otherwise, it is otimal to sto. Thus, in the following symmetric equilibria, we consider consumers rational search strategies as follows: shoers always insect all ositions before making a urchase, and nonshoers randomly insect one firm with equal robability. If the rice does not exceed their otimal stoing rice (denote as r for tye consumers and r for tye consumers), they sto searching and urchase the roduct. Otherwise, they continue to randomly insect one of the firms left with equal robability, until the rice is no greater than their otimal stoing rice or there is no firm left to insect. They then buy from the one with the lowest rice. Therefore, a comlete descrition of the symmetric equilibrium only needs to secify firms common ricing strategy (reresented by the cumulative distribution function F ) and nonshoers otimal stoing rices (r and r ). Proosition (Symmetric Pricing with Continuous Suort): There exists a mixedstrategy equilibrium in which all firms rice according to the distribution = F ( α) ( ) ( α) [ ],, (8) where \ = r and α = + α r, and the otimal stoing rices r = r + k k and r are determined by = F d k, if the insection costs k and k are not too different such that 0 <r <r < and r > ( / ( a))r. The next examle illustrates the equilibrium described in Proosition. Examle : When α = 0., k = 0., and k = 0., we can calculate nonshoers otimal stoing rices r = 0.6 and r = 0.5 (r > ( / ( a))r ) and firms

15 Oligoolistic ricing with online search 5 (a) F() in Examle (b) F() in Examle 4 Figure. Examles of Price Distributions in the Symmetric Equilibrium rice suort bounds \ = 0.5 and \ = 0.. The rice distribution is deicted in Figure a. When the search cost of the tye consumers is not too high relative to that of the tye consumers, firms set the uer bound of the rice suort equal to the otimal stoing rice of the tye consumer so that all nonshoers sto searching after samling once. In this case, the firms forgo the otion of charging a higher rice to take advantage of the high-cost tye consumers because the benefit from exloiting the high-cost consumers cannot counterbalance the loss of business from the low-cost consumers. Proosition (Symmetric Pricing with Segmented Suort): There exists a mixedstrategy equilibrium in which all firms rice according to the distribution = F α α α α φ φ α( ) α + + ( α), 4 [ ],, (9) where

16 6 Xu, Chen, and Whinston = r α = r α+ α φ = r α = α+ α α α φ φ + ( ) r and α φ = α α( 9α 9α 4)+ 4α α r + + α α / r, and the otimal stoing rices r and r are determined by 4 F d = k + ( )( ) = F d φ k k, (0) if the insection costs k and k are different enough such that 0 < r < r < and α r < r < r α α. The next examle illustrates the equilibrium described in Proosition. Examle 4: When α = 0., k = 0.5, and k = 0., we can calculate nonshoers otimal stoing rices r = 0.88 and r = 0.7 ([a/( a)]r < r < [ / ( a)]r ) and firms rice suort \ = 0.88, \ = 0.56, \ = 0.7, and \ 4 = 0.. Also, f = 0.9. The rice distribution is deicted in Figure b. Intuitively, when the search cost of the tye consumers is significantly higher than that of tye consumers, the otimal stoing rice of the tye consumers is thus considerably higher than that of the tye consumers. As a result, charging a higher rice to exloit the high-cost consumers could be as rofitable as charging a lower rice to comete for more market share, which exlains the rightward exansion of the rice suort comared to the revious result. Particularly, it is worth noting that segmentation of rice suorts arises in equilibrium in this case. The ga between \ and \ results from the dro of exected demand for rices right above \. Because \ equals the tye consumers otimal stoing rice r, when ricing below \, a firm can sto all tye consumers who insect its osition from further searching. However, once its rice exceeds \, the tye consumers will continue to insect other ositions and are very likely to urchase from elsewhere (unless all the other ositions charge even higher rices). Therefore, the exected demand dros substantially at \. As a

17 Oligoolistic ricing with online search 7 result, the exected rofit jums downward at \ and does not rise back until \. For this reason, charging any rice between \ and \ is subotimal. One question of our articular interest is: could the local-cometition rice attern arise in equilibrium in the case of osition-invariant costs? The answer is negative, as we show in Proosition 4. In fact, we can conclude that the tyical symmetric random search market cannot induce the local-cometition equilibrium ricing attern. Position-Deendent Costs We now allow consumers insection costs to be different for different ositions. Position-deendent insection costs reflect the inherent difference across different ositions, which lies in the difference in terms of visibility and accessibility. Just like bending over to check the bottom-level shelf sace in the suermarket could be costly for seniors, scrolling down and looking for an unhighlighted link on a Web age or switching multile Web ages could be troublesome for non-tech-savvy users or in the case of unsatisfactory network connection. We first give a necessary condition for the local-cometition attern to arise in equilibrium: Proosition 4 (Local-Cometition Pattern Under Endogenous Search: A Necessary Condition): A similar ricing attern as in Equation () may arise in a rationalexectations equilibrium only if at least one grou of consumers have a higher insection cost for osition than for osition (i.e., k > k or k > k ). Proosition 4 shows that for the attern to aear in REE, at least some consumers insection cost for the third osition should be strictly higher than that for the first osition. Otherwise, because the rice at the first osition is always higher than that at the third osition, as in Equation (), insecting the first osition would be dominated by insecting the third one for all consumers. In this case, the rational search decision would be to insect the second and the third ositions only. If this is the case, then both the first and the second firms will deviate from the resumed ricing strategies. As a result, the attern in Equation () cannot hold as an equilibrium. Proosition 4 thus excludes the ossibility that the local-cometition attern is an equilibrium when there is no inherent difference among different ositions. It reveals the fact that the secial attern of equilibrium rice disersion is an outcome of osition-deendent insection costs, which induce certain search ordering of consumers as rational equilibrium behaviors. We next exlicitly derive an equilibrium with the same ricing attern as Equation () under certain conditions: Proosition 5 (Local-Cometition Pattern Under Endogenous Search: A Sufficient Condition): When nonshoers insection costs for different ositions increase with osition ranks and the differences are large enough, recisely when k ( ) α( α) ln + α α < + α α

18 8 Xu, Chen, and Whinston with k k > α ln + + α α ln α α α k = 0 k < ( γ ) γ ln ( ) α α α α α α γ = ( α) + α( α) and k > + αln α, ( α) + α( α) there exists a rational-exectations equilibrium in which firms rice according to F = F = ( ) ( α) α( ) F = α [ ), =, ) [, ],, () where αln α ln + α α = min k + + ( ) α α α ( ), =, + α α and \ = a \ ; consumers adot the search strategies secified in Table.

19 Oligoolistic ricing with online search 9 Table. Consumers Search Decisions d (z, C) Choice set C Tye consumers Tye consumers {,,} If z =, insect first osition If z =, insect first osition {,} If ( ) z > \, insect second osition; If z > r, insect second osition; If z r, sto If z \, sto {,} If z > r, insect first osition; If z > \, insect first osition If z r, sto {,} If z \, sto If z > \, insect first osition {} If z r, sto If z > \, insect first osition {} If z \, sto If z > r, insect second osition; If z r, sto {} If z \, sto If z \, sto r is defined by r r df k.( Notice that r.) = < < r is defined by r r df k.( Notice that r.) = < < The conditions needed are that the insection costs are higher for the inferior ositions than those of the suerior ones and that they differ in the same order for all nonshoers (i.e., k < k < k and k < k < k ). In addition, the differences between the insection costs for two different ositions are heterogeneous across consumers. In other words, while some consumers (i.e., shoers) are insensitive to location difference, others (i.e., nonshoers) are quite sensitive, and these nonshoers have different levels of tolerance toward locational inconvenience. For examle, some consumers might not mind scrolling down and looking for an unhighlighted link on the same Web age but would not bother to switch to another Web age to continue their search, or it might be fine for some to bend over and check the bottom shelf in the suermarket but it would be too troublesome to get a ladder and check the to shelf. As an equilibrium outcome of such differences, a commonly observed search order arises in equilibrium, and meanwhile different consumers sto at different stages of the search rocess. Anticiating consumers rational search strategies, firms also otimize their ricing strategies by selectively retaining some consumers from further search, cometing for some who samle multile ositions and forgoing the others. As an equilibrium outcome, tye consumers choose to start their search from the first osition because it yields the highest exected gain according to Equation (6). After learning the rice from the first osition, they decide to sto searching because further insection of either the second or the third osition yields a negative exected gain. Similarly, tye consumers sto searching after insecting the first two ositions because of the higher

20 0 Xu, Chen, and Whinston search cost for the third osition. Shoers insect all three ositions before making a urchase, and the order they ursue actually does not matter. Table lists nonshoers search decisions d(z, C) when facing values of {z, C} that are relevant to the equilibrium analysis. From the table, we can identify nonshoers equilibrium search strategies. For instance, the lowest rice is z = before consumers start the first search, and the decision facing consumers is d (,{,, }). For tye consumers, according to Table, d (, {,, }) is to insect the first osition. After that, they learn a rice from the F () so that [\, \ ]. Then the decision d (, {, }) for the tye consumers is to sto, which comletes the tye consumers search strategies in equilibrium. We also secify consumers off-equilibrium strategies. For instance, the tye consumers will not actually face the decision d (z, {, }). Nevertheless, this decision and the associated ayoff should be taken into account when they calculate the exected gain of insecting the second osition in the first lace. Also, the tye consumers face the decision d (z, {, }) after insecting the first osition. Desite the fact that the rice quoted from the first osition always falls in [\, \ ] in equilibrium, we also need to secify the off-equilibrium search decision when z < \ because such decision affects the first firm s rofitability of ossible deviation. As we can verify, all the decisions listed in Table, both in-equilibrium and off-equilibrium, are rational. Conclusion In a d d i t i o n to t h e t h e o r e t i c a l c o n t r i b u t i o n to the literature on search and ricing, this aer has managerial imlications for sellers, consumers, and information systems designers. For sellers seeking the otimal ricing strategy, we emhasize the imortance of recognizing the features of consumers online search behavior and adjusting the rice accordingly. Because the online environment makes it relatively easy to samle around, a cometitive rice could quickly be noticed and result in a surge of sales. Therefore, it would be beneficial to rovide occasional romotions or limited-time deals from time to time, which could not only boost short-term sales but also attract more visits in the long run. Nevertheless, the otimal ricing strategies would avoid being too aggressive or too conservative in rice cometition. The local-cometition attern suggests cometing only against commensurable oonents with similar visibility. In most cases, it would be subotimal to comete against sellers that are much stronger or weaker. Moreover, to form their otimal ricing strategies, sellers only need local information on consumers search behavior at the ositions with similar visibility and on the ricing strategies of the firms at these neighboring ositions. For consumers looking for the best deals, it might sound discouraging that the exact rice at a articular osition is usually difficult to redict. It could be the case that the firm at a rominent osition offers a good deal, or a firm at an inferior osition might not charge as low a rice as exected. Stoing the search rocess early is thus rational for consumers who have high search costs. Nevertheless, since the rice exectation decreases as the location rominence dros, it is generally rewarding to kee searching, esecially for those who are not sensitive to locational inconvenience.