Endogenous Differentiation of Information Goods under Uncertainty

Transcription

1 Endogenous Differentiation of Information Goods under Unertainty Robert S. Gazzale Department of Eonomis Jeffrey MaKie-Mason Shool of Information and Department of Eonomis University of Mihigan Ann Arbor, MI ABSTRACT Information goods an be reonfigured at low ost. Therefore, firms an hoose how to differentiate their produts at a frequeny omparable to prie hanges. However, doing so effetively is ompliated by unertainty about ustomer preferenes, ompounded by the fat that the searh for a good produt nihe is arried out in ompetition with other searhing firms. We study two firms that differentiate their information goods. The firms simultaneously ompete in produt onfiguration and prie. We assume a non-uniform distribution of onsumers:the largest number prefer a produt loated at a sweet spot, but the rate at whih the ustomer density falls off away from this produt onfiguration is unknown. Our haraterization reflets the standard tradeoff between exploitation (urrent profit) and exploration (learning to enhane future profit). In our model firms balane urrent profits from ompeting for a mass and a nihe market, while learning about the profitability of these alternative strategies. We show that the amount of learning that firms will undertake depends on the onvexity or onavity of the profit funtion in the rate of demand fall-off. In our model firms have an inentive to learn, and an use both prie and produt onfiguration in order to explore. We show that the ability to explore in produt harateristi spae leads to a previously unidentified onsequene of learning:attenuation of ompetition. The inentive to learn indues firms to differentiate their produts more than they would if the value of learning were ignored. This leads to dereased diret om- MaKie-Mason gratefully aknowledges support from an IBM Partner Fellowship, and from NSF grant CSS The authors appreiate helpful omments from audienes at Johns Hopkins University, the University of Maryland Robert H. Smith Shool of Business, and Ford Motor Researh. Permission to make digital or hard opies of all or part of this work for personal or lassroom use is granted without fee provided that opies are not made or distributed for pro t or ommerial advantage and that opies bear this notie and the full itation on the rst page. To opy otherwise, to republish, to post on servers or to redistribute to lists, requires prior spei permission and/or a fee. EC 01 Otober 14-17, 2001, Tampa, Florida, USA Copyright 2001 ACM /01/ $5.00. petition with rivals, and thus higher pries and profits than if the firms were ating myopially. Thus, we might expet that when firms are not well informed about onsumer preferenes for information goods as might be espeially true in new markets for innovative produts produt diversity will be higher and diret ompetition will be smaller than might otherwise be expeted. 1. INTRODUCTION Information goods an be reonfigured at low ost. For example, information aggregators (newspapers, databases) an unbundle and re-bundle information objets in a variety of ways. In the print-on-paper world, low-prie bundles (like daily newspapers) generally are offered in one standard edition (with perhaps a small number of minor variants). Extensive ustomization is provided by information servies at a high ost. With eletroni publiation, the ost of ustomizing a standard edition an approah zero. There has been little researh on how firms hoose to differentiate their information goods. This problem is espeially hallenging beause firms rarely have omplete information about the preferenes of potential ustomers over produt harateristis. Thus, over time they make their prie and produt onfiguration deisions based not only on expeted urrent profits but also based on the value of the learning they expet from eah period s offering. To further ompliate things, this searh for a good produt nihe is arried out in ompetition with other searhing firms. We onsider two firms ompeting in two dimensions:produt onfiguration and prie. We model produt onfiguration as a one-dimensional spae:a line on whih firms hoose a loation. In ertain markets it is learly tehnologially feasible, and perhaps optimal, for a provider of information goods to ustomize its offerings so that it in effet oupies multiple loations in produt spae. 1 We limit, however, the firms in our model to hoosing one loation in any period for a few reasons. First, we do so in order to fous on the ability of firms to ontrol the degree of produt differentiation in an environment where firms need to learn about the attrativeness of differentiation. Seond, even if firms ould ompletely ustomize their offerings based on ertain ustomer harateristis, it remains quite likely that firms will attempt to differentiate their offerings from those of its ompetitors in other ways. Thus, our model might be 1 See, for example, Farag and Van Alstyne [7].

2 interpreted as one in whih firms hoose a brand identity. Generalizing the model to firms that offer multiple produt onfigurations is a worthwhile task for future researh. 2 The largest number of ustomers most prefer a produt loated at a sweet spot, with the density of ustomers preferring other produts falling off with distane from the sweet spot. The firm s optimal produt onfiguration needs to balane the rewards from selling to the many ustomers near the sweet spot against the dual osts of losing ustomers in the less densely populated tails and of lower pries due to fierer ompetition near the sweet spot. This is intended to suggest the hoie between ompeting for a mass market and a nihe market. 3 To introdue unertainty about onsumer preferenes we assume that the firms know the loation of the sweet spot, but not the rate at whih demand falls off with distane from the sweet spot. We use a two-period model to allow the firms an opportunity to learn about preferenes from their experiene. Now we have a problem of exploitation versus exploration:the loations and pries firms hoose eah period will determine urrent profits, but (in the first period) will also reveal information that might inrease their ability to extrat profits in future periods. The most informative loation/prie ombination will not generally yield the highest expeted urrent profits. Therefore, the optimal produt onfiguration and priing deision generally balanes the value of learning against the ost of foregone urrent profits. Grossman et al. [9] are among the first to study have identified the exploration versus exploitation tradeoff in an eonomi problem. 4 As an example, they onsider an individual s onsumption of an item whose value is unknown. Eah time the onsumer tries the item, the value she reeives is equal to the underlying value plus a stohasti shok. Thus the more she experiments with an item, the better she knows its true value. Under the onditions outlined, the non-myopi onsumer makes larger purhases of this item in order to learn its value and make better deisions in future periods. Subsequent authors, suh as MLennan [17] and Aghion et al. [1] study experimentation by a monopolist unertain about the demand for its produt, and derive onditions under whih there will be adequate learning. In a related paper, Harrington [10] onsiders duopolists ompeting in prie in a differentiated produts market with firms unertain about the degree of substitutability among produts. However, in ontrast to our model with endoge- 2 Some authors studied firms in Hotelling models that an sell more than one produt, eah with a different loation or onfiguration. These authors make the extremely limiting assumption that prie is fixed exogenously, so that ompetition is only in loation, as well as the other restritive assumptions of the Hotelling models identified above. Even in these highly stylized models results are hard to obtain and are inonsistent. For example, Gabszewiz and Thisse[8] find that two firms spread their produts aross the spae but loate eah of their varieties right next to the ompeting firm s most similar variety. But Martinez-Giralt and Neven [16], with only one minor hange in assumptions, finds that firms loate all of their produts in a luster, yet loate those lusters as far from the ompetitor s luster as possible. 3 MaKie-Mason et al. [15] analyze the effet that Internet servie arhiteture an have on the hoie between mass market and nihe produt onfiguration. 4 [11] presents an early disussion of exploration versus exploitation in his formalization of the adaptive learning problem. nous produt differentiation, Harrington s firm loations are fixed. He shows that under ertain demand onditions firms wish to learn in the first period, while under other onditions they do not wish to learn. With prie the only strategi variable in his model, greater learning follows from a greater prie differene between the two firms. In our model, with firms hoosing both prie and produt onfiguration, learning an be inreased by lowering prie (thereby attrating more nihe ustomers far from the sweet spot) or by differentiating produts. Our model also differs beause Harrington s firms are unertain about the degree of differentiation between their produts, whereas ours are unertain about the distribution of onsumer preferenes. An impliation of this differene is that, for a given prie derease (holding everything else onstant) a firm in Harrington s model knows the number of new ustomers who enter the market, but not how many ustomers the firm takes from its rival. In our model, neither the number of new ustomers in the market nor the number of ustomers taken from its rival is known with ertainty. Our model is also related to the Hotelling literature on endogenous produt differentiation. 5 The standard model in that large literature has two firms loating on a line, but onsumer preferenes are distributed uniformly on a segment rather than more densely around a sweet spot. We work with a riher model of onsumer preferenes beause the uniform distribution has only one parameter (the width of the line segment), and to model unertainty would have required that we suppose firms did not know how far uniform onsumer preferenes over produt onfigurations extended, whih does not readily map to familiar information goods markets. Our approah allows for unertainty in a natural way:firms are not sure how rapidly onsumer demand falls off away from the sweet spot. The standard Hotelling model also fixes prie and lets firms ompete only in loation (in ontrast to Harrington, who reverses the state and ontrol variables). We endogenize both prie and produt onfiguration. Finally, although some work on the Hotelling problem inorporates firm unertainty, to our knowledge we are the first (other than Harrington) to study learning in a loation model of endogenous produt differentiation. Our work is also related to the growing literature, using both empirial methods and simulations, that studies the produt positioning of information goods. Clay et al. [6] find that as new firms entered online book selling, pries remained flat or rose. They doument a wide degree of heterogeneity produt and priing strategies. They onlude that the real puzzle is the stores with wide seletion and average pries, but in a new market with substantial learning, our model suggests that experimenting with this and various other onfigurations may not be so puzzling after all. Segev and Beam [18] report on some of the praties of eletroni brokerages, who provide pries for other goods or servies, and potential mathes to trading partners. They find tremendous unertainty about profit maximizing strategies, and that in response experimentation with pries and produt onfigurations is greater than might be expeted. Through a simulation they find that in this environment brokers will do best to differentiate widely, for example by 5 See Anderson et al. [2] for a thorough survey.

3 either fousing on serving buyers (harging high fees to sellers and low fees to buyers), or fousing on serving sellers. In setion 2 we present our model, with details on the information goods market, firm behavior and onsumer behavior. We then solve for the subgame perfet equilibrium of the two-stage game in setion 3. We disuss the results and possible generalizations in setion 4. Our primary result is that firms will use first-period prie and produt onfiguration in order to inrease learning. However, in ontrast to standard models of firm learning, this is not at the expense of first-period profits. Firms are able to inrease learning by inreasing the level of differentiation between their produts. This redution of ompetition enables firms to inrease pries and thus inrease short-term as well as long-term profits. 2. THE MODEL 2.1 The Market We onsider a market for an information good that an be differentiated in one dimension. An example would be Web sites that provide news ontent, differentiated by the ratio of national to international news. A more general model would permit differentiation in multiple dimensions. We represent this dimension as a line on the real numbers. The produt offered by eah firm is haraterized as a loation on this line. + l # of onsumers l 1 0 l 2 - l 2 Produt onfiguration Figure 1: Distribution of onsumers most preferred produt onfigurations over range of produt possibilities We haraterize a onsumer by the produt onfiguration (loation) she most prefers. We then map the distribution of onsumers over the produt spae line, with the vertial height above the line representing the number of onsumers for whom that loation represents their most preferred produt onfiguration (see figure 1). We assume that there is a single produt onfiguration that is most preferred by the largest number of onsumers. We all the loation of these α onsumers the sweet spot, normalized to be the zero on the horizontal axis. Along the produt spae axis, the distane from the most popular loation is represented by l, whih an be either positive or negative. The number of onsumers dereases as one moves away from the sweet spot at a rate of γ. Thus the number of onsumers loated at l is α γ l. 2.2 Consumer Behavior We assume onsumers purhase at most one of the two ompeting goods in eah period. There is no ost to evaluating the options and hoosing a provider. A onsumer reeives a utility of r if she onsumes her most preferred good, and an amount that dereases at rate the further the onsumed good is from her most preferred onfiguration. Letting (l, p) represent a produt s onfiguration and prie, a onsumer of type t reeives r p t l. Consumers selet the good that provides the greater utility, or neither if utility would be negative. That some onsumers may hoose to purhase nothing is another fator that distinguishes our model from the standard Hotelling approah, whih assumes that all onsumers purhase. In our model there is both an intensive and extensive margin:a firm an lose (nihe) ustomers to the outside option or (mass market) ustomers to head-to-head ompetition with the other firm. We assume that the distane ost is linear for analyti onveniene. The onstant ost ould be interpreted as the loss in utility per artile as a bundled information good offers fewer artiles of the type the onsumer wishes to read (e.g., less national news). In a more general representation of preferenes the distane ost might be nonlinear. The density of onsumers who purhase a given firm s good aording to the behavioral rule above onstitute that firm s demand. For two reasons we add a stohasti omponent to eah firm s demand. First, it is unreasonable to model a world with firm unertainty but to then assume that every onsumer makes exatly the right deision every period. The seond reason we provide below, after we explain the information available to the firm and its behavior. To implement stohasti demand we assume that eah firm i s demand is subjet to an additive random variable, ɛ i, whose df G i() has a mean of zero and variane of σ ɛ. 2.3 Firm Behavior Two firms ompete in this market for two periods. The firms are ex ante idential. In eah period, at zero ost, eah firm an differentiate its produt by hoosing a loation on the line, at the same time announing a prie. The firm s objetive is to maximize the sum of disounted profits, whih are equal to revenues beause we assume that loation and prodution osts are zero to apture the easy reonfigurability and reprodution of information goods. We assume the values α,, r, and the distributions of the ɛ i and of γ are known to both firms. The need for learning arises beause they do not know the value of γ. However, the firms have the same distribution of prior beliefs over γ, denoted by the CDF F (γ), and thus the same expeted valuation (ˆµ 0). After the first period of trade, the pries, loations and number of onsumers served by eah firm is ommon knowledge. Conditional on this knowledge and the prior belief ˆµ 0, firms update their beliefs about the value of γ. Our primary goal is to investigate how the opportunity to learn about the value of γ affets the ondut of the firm in the first period. Above we gave one reason why we assume eah firm s demand has an additive stohasti omponent, ɛ i. The seond reason is that given the ommon knowledge assumed above, almost any loation and prie in the first period would re-

4 veal the true value of γ to eah firm. The intuition is that the density funtion is pieewise linear with eah slope having the same absolute magnitude, and firms already know one point on the funtion (α). If they ould observe demand from this density perfetly then they ould solve for a seond point on the density funtion and ould perfetly alulate the slopes. In no realisti problem an firms perfetly infer all relevant onsumer preferene information from a single experiment, so we add a noise term to ensure inomplete inferene. t l = l 1 + p1 r, t r = l 2 + r p2. The onsumer type whih is indifferent between the offerings of the two firms, t m, will be See figure 2. t m = l1 + l2 2 + p2 p1. 2 # of onsumers 3. SUBGAME PERFECT EQUILIBRIUM In this setion we solve the model for a subgame perfet equilibrium. Our two-period subgame perfet framework does enable us to draw valuable inferenes about the more realisti ase where the number of periods is larger. First, we use seond period behavior as a no learning or myopi benhmark against whih to ompare the ations of firms who take into aount the onsequenes of urrent period ations on subsequent period profits. Seond, adding additional periods does little alter the inentives of the game. We an thus view the first period as representing periods under whih the firms at under unertainty and the final period as the limiting ase as the value to learning goes to zero. We believe there is substantial value to the study of the equilibria of a tratable but reasonably realisti model of the dynamis of learning and produt onfiguration. First, by knowing the equilibria of the game, we will know something about the behavioral inentives faing firms that find themselves out of equilibrium in a more realisti setting. Seond, we are able to obtain expliit analyti results, whih enables us to establish general preditions about the omparison firms that strategially learn and those that do not. In future researh these preditions an then be tested against empirial data. Further, the preditions of the game-theoreti equilibria an be used as a guide to the design of intelligent heuristis; we disuss this possibility in relation to our own researh on software agent heuristis in setion 4. Sine the game is finite, we use bakward reursion:we first solve for optimal play by the two firms in the seond period, onditional on their updated expetation, ˆµ 1,from the first period. In the subgame we look for Nash equilibria, in whih if eah firm makes the best play onditional on the hoies of the other firm, the hoies will be mutually onsistent. Then, given the solutions for pries and loations in the seond period as a funtion of ˆµ 1, we solve for the optimal prie and loation hoies by the firms in the first period. Sine their objetive is to maximize the sum of disounted profits over the two periods, their first period hoies will take into aount not only profits in the first period, but also the effet of these first period ations on expeted seond period profits due to their learning about the slope of the ustomer preferene density. We denote the leftmost firm as firm 1, and the rightmost as firm 2, and their loations as l 1 and l 2 respetively. Given the onsumer hoie rule, for any l 1 and l 2 we an identify the leftmost and rightmost onsumer types who purhase one of the goods as follows: Figure 2: firms t l Market for firm 1 Market for firm 2 l 1 0 l 2 t m t r Produt onfiguration Illustration of market division between Proposition 1. In a pure strategy equilibrium, all ustomers loated between the two firms are served. Thus there exists a unique t m. (Proofs for the results are given in an appendix.) Without loss of generality, assume that t m 0. Then demand for eah firm is 0 t m D 1 (p 1,p 2,l 1,l 2)= (α + γl) dl + (α γl) dl + ε 1, (1) t l 0 D 2 (p 1,p 2,l 1,l 2)= Profits for eah firm are: t r t m (α γl) dl + ε 2. (2) π 1 (p 1,p 2,l 1,l 2) = p 1D (p 1,p 2,l 1,l 2) π 2 (p 1,p 2,l 1,l 2) = p 2D (p 1,p 2,l 1,l 2). 3.1 Seond Period Equilibrium Given ˆµ 1, their expetation of γ after period 1, firms maximize total expeted profit. 6 Taking the other firm s prie andloationasgiven,eahfirmalulatesthefirstorder onditions for its profit funtion subjet to two onstraints. 6 The profit funtions are linear in γ, so we an replae γ by its expeted value ˆµ when alulating expeted profits.

5 The first is that all onsumers who purhase reeive nonnegative utility The seond is that l 2 l 1. We assume for the moment that neither onstraint binds. This yields four best response funtions in four unknowns: l 1(p 1; p 2,l 2) = 0 p 1(l 1; p 2,l 2) = 0 l 2(p 2; p 1,l 1) = 0 p 2(l 2; p 1,l 1) = 0. whih are then solved to find the Nash equilibrium. 7 Only one of the sixteen solutions to this system satisfies the seondorder onditions, so in the unique equilibrium firms set prie and loation as follows: p 1 = 3α 8ˆµ (3) p 2 = 3α 8ˆµ (4) l1 = r 7α 8ˆµ (5) l2 = r + 7α 8ˆµ, (6) whih will yield the following expeted profit E [π i ˆµ] = p i l 2 + r p 2 0 = 9α3 64ˆµ 2. (α ˆµl) dl We an provide some eonomi interpretation to the best response funtions and the resulting equilibrium. If we look solely at the loation deision of firm 1, setting marginal benefit equal to marginal ost implies that for an inremental move loser to its opponent, the number of ustomers that firm 1 gains from its rival equals the number lost on the outside margin. By differentiating the bounds of integration of the demand funtion with respet to loation, we see that the former is equal to 1 the height at tm and the latter is 2 equal to the height at t r or t l. The best response in terms of prie alone is more ompliated, but still involves balaning the internal and external margins. For our symmetri equilibrium where the middle indifferent onsumer is at the sweet spot, there are a ontinuum of prie and loation pairs that satisfy the ondition that the external and internal margins must be equal. Whether the firms loate near the middle at a relatively low prie or loser to the midpoint of 0 and t r (or t l ) at a relatively high pries depends on how muh firms desire to fight for the mass market. The desirability of the mass market in turn depends on the expeted slope of onsumer density, ˆµ:the lower is ˆµ, the more valuable are the nihe markets relative to the (more ompetitive) mass market. This an be seen from the effet of ˆµ on equilibrium pries and loations in equations (3)-(6). We next examine at how expeted profit depends on ˆµ: 7 The best response funtions are extremely long so we do not reprodue them here. They are available from the authors upon request. E [π i ˆµ] / ˆµ = 9α3 < 0 ˆµ >0 (7) 32ˆµ 3 2 E [π i ˆµ] / ˆµ 2 = 27α3 > 0 ˆµ >0. (8) 32ˆµ 4 Expeted profits are dereasing in ˆµ. This is due to the fat that, when we inrease γ, demand falls off more sharply as we move away from the sweet spot. Equation (8) implies that expeted profits are onvex in ˆµ. That expeted profits are onvex in ˆµ implies the firms behave as if risk-loving, i.e., they prefer more variability in their posterior mean on γ. That is, they prefer to learn more information about theatualvalueofγ, as this permits them to do a better job optimizing in period 2. We disuss this point in further detail in Setion 4. At this point, we must return to the negleted onstraints, whih fore us to put bounds on the range of beliefs for whih the above equilibrium holds. As shown in Proposition 1, in any pure-strategy equilibrium there are no unserved onsumers between the two firms. This will only be true in a symmetri equilibrium if a onsumer loated at the sweet spot has non-negative utility. From the onsumer utility funtion and the equilibrium pries in (3)-(4) this will be true in our equilibrium only if ˆµ 5α. Likewise, to ensure 8r that l 2 l 1, it is neessary that ˆµ 7α 8r.8 The above symmetri equilibrium thus holds for ˆµ [ 5α, ] 7α 8r 8r. 9 We an show that for values of ˆµ outside of this range, the unique equilibrium will be asymmetri. The reasonable width of this range for ˆµ is an empirial question, sine there are no onstraints other than nonnegativity on the free parameters. The important question is whether this region aptures an eonomially interesting set of problems. We believe that it does. This region is important beause it is preisely the region in whih firms desire to ompete both for the mass market and remain attrative to many in the nihe market. This is an aurate desription for many markets of interest. Outside of this range for ˆµ the model orresponds to different types of markets. For example, onsider the effet of dereasing ˆµ inside the symmetri range:the nihe markets beome more attrative. As one would expet, our equilibrium onditions show that the firms are less inlined to ompete for the mass market:the firms inrease produt differentiation and pries inrease as firms at more like loal monopolists in the nihes. As we ontinue to derease ˆµ past the symmetri range, an interesting thing happens. We an show that if one firm is serving the sweet spot, the best response of the other firm is to not ompete at all for onsumers served by the former. In short, the nihe beomes so attrative that it is preferable to be a loal monopolist for this nihe. In this situation there is no head-to-head ompetition for the mass market, and thus the firms do not balane their appeal to mass market and nihe ustomers. 8 That l 2 l 1 is arbitrary, but we used it to speify the demand funtions faing eah firm in our solution for the equilibrium. The same parameter restrition would hold if we reversed the firms and imposed l 1 l 2. More generally, what is required is that firms ompute their expeted demand onsistently with their equilibrium loation. 9 In order to ensure that ˆµ is in this range in the seond period, it suffies that the firms plae zero probability on any γ outside of this range in the first period.

6 Whether we are in this ase, the ase where ˆµ is so steep as to make the nihes unattrative, or the intermediate ase in whih firms balane between selling to both mass and nihe markets of ourse depends on the market in question. In this paper, we limit our attention to the ase where ˆµ is inside the speified range, and leave an analysis of the dynamis of the other two ases to future work. 3.2 First Period Equilibrium Having solved for the equilibrium in the last period, we now haraterize the Nash solution for the first period, taking into aount the effet of first period hoies on expeted seond period pries, loations and profits. The link between periods is through learning:realized first period demand provides information about the value of γ, so that generially ˆµ 1 ˆµ 0, and seond period pries and loations are funtions of ˆµ 1 (see equations (3)-(6)). Firms update their beliefs on γ based on the impliation of their loations and the total realized demand for the preferene density slope, taking into aount that demand has a mean zero stohasti omponent. Making use of the symmetry of the demand density and that firms loate equidistant from the sweet spot, we derive the following total demand equation for D(p 1,p 2,l 1,l 2) D 1(p 1,p 2,l 1,l 2)+ D 2(p 1,p 2,l 1,l 2): D(p 1,p 2,l 1,l 2) = 2 t r 0 (α γl) dl + ε (9) = 2αt r γt 2 r + ε ( = 2α l2 + r p2 Rearranging equation (9) gives: 2α l 2 + r p 2 D ( (p1,p2,l1,l2) ) l 2 + r p 2 2 = γ ) ( γ l 2 + r p2 ε ( l2 + r p 2 ) 2 + ε. ) 2. (10) Proposition 2. The left-hand side of (10) is an unbiased σε estimator for γ with variane equal to 2 ( l 2 + r p 2 After observing first period total demand, firms apply Bayes rule to ombine their prior beliefs with this new unbiased estimate to obtain updated beliefs on γ. We an view the hoie of pries and loations followed by a demand observation as an experiment. An experiment is more informative if it redues the variane of the estimator. As the denominator of the variane is equal to t 4 r = t 4 l, reduing the variane is aomplished by inreasing the number of nihe onsumers served. The intuition is that as the firms inrease their reah (i.e., by moving t l and t r further away from the sweet spot), demand is more affeted by γ, andthe relative effet of ε diminishes. Firms inrease their reah by either lowering their pries or moving away from the sweet spot That the distribution of ε is independent of total demand is an analyti onveniene. Inreasing reah will assuredly provide a more informative experiment if the ratio of variane of ε to total demand is non-inreasing as t r = t l inreases. ) 4. We define the posterior df of γ as F ( D(p 1,p 2,l 1,l 2)). 11 We apply Bayes Rule to get: ( ) F (γ D(p 1,p 2,l 1,l 2)) = G D 2αt r + γt 2 r F (γ), H (D) where H (D) = G ( D 2αt r + γt 2 r) F (γ)dγ. Beause both firms have aess to the same information about the outome of the first-period experiment, they arrive at the same updated distribution of beliefs about γ and thus the same and expeted value, ˆµ 1.Wehaveshownpreviously that with the same bounded beliefs about γ, the unique seond-period equilibrium is symmetri. Therefore, for any first-period equilibrium the two firms have the same expeted seond-period profit equal to [ ] W (p 1,p 2,l 1,l 2)= E π i F (γ D(p 1,p 2,l 1,l 2) dγ H (D) dd. Both firms maximize umulative profits disounted at rate δ, so the first-period value funtion for firm 2 is t r (α γl) dl t m +δw (p V 2 (p 1,p 2,l 1,l 2) if t m 0 1,p 2,l 1,l 2)= 0 t r (α + γl) dl + (a γl) dl t m 0 +δw (p 1,p 2,l 1,l 2) if t m 0. A firm s best reply funtion, φ i, is a pair of prie-loation values defined by φ i (p j,l j) arg max V i (p i,l i; p j,l j). p i,l i Proposition 3. φ i exists. A symmetri subgame perfet equilibrium exists iff there exists ˆl 1 = ˆl 2 and ˆp 1 =ˆp 2 suh that {ˆp 1, ˆl 1} = φ (ˆp 2, ˆl ) 2 {ˆp 2, ˆl 2} = φ (ˆp 1, ˆl ) 1 Proposition 4. A symmetri subgame perfet equilibrium exists. We now establish our main result. We wish to establish the effet that the opportunity to learn has on first-period prie and produt differentiation deisions. To do this, we ompare the equilibrium first-period pries and loations to those that would be an equilibrium if both firms ignored the value of learning. Sine both pries and loations an be hanged ostlessly between periods, without learning there is no link between the periods, and optimal first-period behavior is purely exploitative:that is, the maximizing the sum of disounted two-period profits degenerates into separately maximizing 11 To redue lutter we do not label variables with a period index. In this setion pries and loations refer to first period ativity.

7 profits in eah period based on the prior expetation ˆµ 0 on the unknown slope γ. Consequently, the best response funtions in the first period are the same as in the seond period. Denoting first-period equilibrium prie and loation values for a firm that ignores learning by { p i, l i}, thesevalues are the same as the seond-period values: { p 1, l 1, p 2, l 2} = {p 1(ˆµ 0),l 1(ˆµ 0),p 2(ˆµ 0),l 2(ˆµ 0)}. Therefore, from the results of setion 3.1 we know that a unique and symmetri firstperiod equilibrium exists for these non-learning firms. Proposition 5. A firm that takes the value of learning into aount in the first period will hoose a loation further from the sweet spot, and a prie higher than would a firm that ignores the value of learning. That is, for i =1, 2, ˆp i > p i ˆl i > l i. The onsequene is that onsumers will fae more produt diversity, but higher pries, in an information goods market desribed by our assumptions. 3.3 Consumer Welfare We analyze the effet of the learning proess on onsumers for two reasons. First, while the proess results in a short run inrease in market power for the firm, it also results in an inrease in produt diversity and in the number of onsumers served, so that aggregate onsumer welfare may atually inrease. Seond, understanding the effet on onsumer welfare sheds light on the manner in whih firms ondut their learning. We find that even within the range of beliefs about γ where the symmetri equilibrium holds, their beliefs about the attrativeness of the nihe will determine how muh ompetition is relaxed for a given experiment. The effet of the learning proess on onsumer welfare is a ompliated affair. Looking solely at the first period effet, how an individual onsumer fares will depend on her type. Those loated near the sweet spot will fae higher pries and produts less tailored to their tastes when firms loate further apart but raise their pries. Consumers loated to the outside of the firm loations will reeive a more desirable produt, albeit at an inreased prie. Finally, as the number of onsumers served inreases in a learning environment, these new onsumers learly benefit. In this setion, we look to resolve some of this ambiguity. Using notation developed in Setion 3.2, we define the no-learning equilibrium pries and loations as p i and l i. Making use of the symmetry of equilibrium demands, we look solely at the expeted surplus of onsumers to the right of the sweet spot. Their expeted surplus in the no-learning ase, CS is: CS = t r 0 (α ˆµl)(r l 2 l p 2)dl. (11) We now look at how onsumer surplus hanges as firms inrease their reah. As we show in setion 5 at equations (17)-(19), for any given reah, t = t r = t l, we an write the profit maximizing pries and loations as follows: p 2 = t t 2 ˆµ (12) 2α l 2 = r +2 t t 2 ˆµ 2α. (13) Armed with a haraterization of the equilibrium pries and loations as firms inrease their reah, we an now gauge their effets on onsumers. In equation (11), we hange the outer bound of integration to t, and substitute for pries and loations as detailed in equations (12) and (13). Differentiating with respet to t gives us: CS/ t = ( 16rα + 64r2 ˆµ 47α2 ˆµ ). (14) There is thus a region where onsumer surplus is inreasing in expetation, and one in whih it is dereasing. We an solve equation (14) to find the threshhold, whih we shall all µ, µ = α(4 3 1), (15) 8r where onsumer surplus is inreasing in expetation for ˆµ > [ µ. Our threshhold is approximately at the midpoint of 5α, ] 7α 8r 8r. We an gain some insight towards interpreting this ondition by looking at how a firm hanges prie as it hanges its reah. In Setion 5, we show that p 2/ t = ( 1 tˆµ/α ). Thus the greater ˆµ, the smaller any prie inrease for a given unit of learning (i.e. hange in reah). Likewise, the greater ˆµ, the less firms move their loations towards thetails. Weanthusseehowthemannerinwhihfirms experiment is affeted by their beliefs. Even though firms desire to explore the nihes, if their beliefs about the attrativeness of the tail are pessimisti enough, the mass market is more worth fighting over, and this moderates their move towards the nihe for the sake of learning. Thus, for low ˆµ (more valuable nihes) firm learning redues onsumer welfare, while for higher ˆµ (more ompetition for the mass market) learning inreases onsumer welfare. 4. DISCUSSION Rather than harge pries or differentiate goods to maximize urrent expeted profits, firms may hoose different pries or produt onfigurations in order to reate better experiments to improve their estimates of onsumer preferenes. Experimentation is usually thought to be undertaken at the expense of short-run profits. We have shown that this need not be the ase. In a model of ompetition under unertainty, in whih firms have the ability to derease diret ompetition, firms desire to resolve unertainty an lead to to short-run profits higher than would be the ase if firms did not are about subsequent periods. In our model of endogenous produt differentiation with unertainty about onsumer preferenes, firms are trying to learn the rate at whih onsumer preferenes fall off away from the sweet spot, in order to hoose the right balane between ompeting with low pries for the mass market of onsumer and ompeting with higher pries for a nihe market. What firms learn in the first period about the distribution of onsumer preferenes hanges their seond-period prie and produt onfiguration hoies. Thus, first-period prie and onfiguration hoies affet expeted seond-period profits. The amount of learning that a firm desires to undertake (at the ost of foregone urrent profits) depends on the onvexity or onavity of the profit funtion in the firm s belief about the unknown parameter, ˆµ. Due to Jensen s In-

8 equality, a onave utility funtion indues risk aversion: the deision-maker prefers a given value with ertainty to a gamble with the same expeted payoff. In our model of endogenously-differentiated information goods future profits are onvex in beliefs about γ. Consequently, firms prefer a gamble to that gamble s expeted payoff, and they are willing to alter first period ations to gamble on what they will learn about γ. In order to learn more about γ, firms set first-period pries and produt onfigurations to better explore the tails of onsumer spae than they would if they ignored the opportunity to learn. This risk loving behavior arises even though the firms are by assumption risk neutral. For given pries and loations, profits are linear in γ, and firms thus maximize profits basedontheexpetedvalueofγ. Only when the true state of the world is ˆµ, however, are the firms atually hoosing the optimal ations for that state. The effet of γ on profits is not linear beause if the firms knew that the state of the world was not ˆµ, they would hoose better ations. Firms thus desire better information about what γ atually is, i.e. have a desire to experiment, so that they an better tailor their ations to the atual state of the world. The manner in whih a firm s desire to experiment affets pries and produt onfigurations is relatively straightforward. As a firm s reah inreases its sales are more affeted by the value of γ, and the stohasti omponent of demand beomes relatively less important. Thus, loating further away provides a more informative experiment. While it is true that firms ould derease prie to inrease demand and thus inrease the informativeness of the experiment, we atually see the opposite effet on pries in this model. To understand why, onsider a given experiment, whih is to say an expansion of the outer bounds of onsumers who buy (t l and t r). While a firm ould serve this ustomer base by lowering its prie, it ould also move further away from its diret ompetition, whih allows it to raise prie. Clearly the latter strategy is superior, as it allowsthefirmtoservethesamenumberofustomersata higher prie. Our main result suggests that when there is unertainty about onsumer preferenes for information goods, there will be substantial experimentation in the form of produt diversity. This seems onsistent with asual observation of the past several years of ommere in information and other eletronially transated goods. With many new goods and servies unertainty about preferenes has been high. Correspondingly the rate of introdution of new produts and differentiation amongst them has been quite high. Whether pries have been high or low for new produts is not as obvious. In some markets the desirability of harging higher pries has perhaps been mitigated by other fators, suh as the desire to build a brand reputation or to lok in ustomers. However, evidene from Bailey [3] suggests that as new firms entered in various eletroni ommere markets, pries inreased. Whether our results are robust requires further investigation. For example, we are urrently analyzing the dynamis that arise in the asymmetri equilibria that arise when γ is lower than the range studied in this paper (when nihes are expeted to be either more or less desirable). There are other diretions in whih one ould generalize our model. For example, firms might be heterogeneous in one of several ways: they might start with different beliefs, or they might start at different loations in produt spae and have nonzero osts of re-loation. We also might learn more from a model in whih there are multiple dimensions along whih produts an be differentiated, or in whih there are more than two firms that sell imperfetly substitutable information goods (or in whih eah firm an sell multiple different goods). We also wonder whether the effet of valuable learning opportunities on priing and produt differentiation would be the same if there were more than one unknown parameter of the onsumer preferenes distribution. For example, a firm might not know the slope γ and also might not know the disutility ost onsumers inur as offered produt onfigurations get further away from their most preferred produt. In a series of papers ([12, 4, 14, 13, 5] we and our oauthors have studied the out-of-equilibrium behavior of software agents that searh prie and produt onfiguration spaes under unertainty about onsumer preferenes. In those papers the agents representing firms selling information goods fae environments too omplex to expliitly solve for optimal strategies even in a single firm environment. Instead, they pursue various searh heuristis. We adopted relatively generi (uninformed) searh heuristis due to the relative pauity of prior literature on the theory of optimal produt and prie onfiguration in an information goods environment. The results in the present paper, by haraterizing some of the properties of optimal learning strategies in a partiular setting, provide guidane for the design of informed searh strategies in more omplex (and thus realisti) settings. In separate researh, we are pursuing the impliations of the present paper for omputational analyses of behavior off the equilibrium path. 5. APPENDIX: PROOFS OF PROPOSITIONS Proof. 1 Assume that this is not the ase in the seond period. Then there are unserved onsumers loated between the two firms. Without loss of generality assume that some of these onsumers are of type t>0. Due to the fat that onsumer density dereases as we inrease t, thenumberof onsumers at the right boundary of the leftmost firm (α ˆµ(l 2 r p 2 )) is greater than the number of onsumers at the leftmost boundary of the rightmost firm (α ˆµ(l 2 + r p 2 )) for all γ. Therefore the rightmost firm an profitably deviate by moving its loation to the left, so this annot be a pure strategy equilibrium. The same logi holds if some of these onsumers are of type t<0. The proof is similar for period 1. In addition to the inrease in profits in period 1, the deviation also inreases expeted profits in period 2. To see this, note that from Proposition 2 the deviation dereases that variane of the estimator. As expeted seond period profits are onvex in the expetation of γ, a more informative experiment inreases expeted profits. Proof. 2 The result is transparent:the variane of the estimator is the variane of ε divided by a onstant, whih is σ 2 ε divided by the squared onstant. Proof. 3 From equations (1) and (2) we have that aggregate demand is ontinuous in {p 1,p 2,l 1,l 2}, and thus that the value funtion is ontinuous in the same arguments. Sine p i [0,r]and l i [0, α ], φi exists by the Weierstrass γ Theorem.

9 Proof. 4 We first prove that if an equilibrium exists it is symmetri. Note that the expeted seond-period profit funtion δw(p 1,p 2,l 1,l 2) is the same in the value funtion for both firms, and symmetri beause seond-period profits are symmetri. Other than this additive expression, the first-period value funtions are idential to the seond period expeted profit funtions. Thus, a firm s first-order onditions are idential in the two periods exept for the addition of a partial derivative of δw with respet to the hoie variable of interest; that partial derivative will be symmetri for the two firms beause the funtion W is symmetri. Therefore, if a solution to the system of first-order onditions exists, a symmetri solution must exist. We now show the existene of the equilibrium. We first note that firms affet expeted seond period profits, W ( ), solely through their hoie of t r and t l. As any pries and loations yielding the same t l and t r are equally informative, pries and loations will be suh that they maximize urrent profits for a seleted t r and t l. For any t = t r = t l,there exists a unique prie and loation pair ( p i, l 1 = l 2)that maximizes urrent period profits. These values are given by 12 : p 1 = t t 2 ˆµ (16) 2α p 2 = t t 2 ˆµ (17) 2α l 1 = r 2 t + t 2 ˆµ (18) 2α l 2 = r +2 t t 2 ˆµ 2α. (19) We an thus haraterize maximal one-period expeted profits for any t as follows: E[π i t, ˆµ] = t 2 (ˆµ t 2α) 2 4α The firms maximization problem thus redues to finding ˆt to maximize total disounted expeted profits. A firm s first period value funtion is thus: V ( t) = t 2 (ˆµ t 2α) 2 + δw( t) 4α and symmetri equilibrium is ˆt arg max t V ( t). Existene of ˆt follows the same reasoning as presented in preeding proof. Proof. 5 Define t as the reah that maximizes expeted first period profits, i.e. t = l 2 + r p 2,andˆt as t arg max t V ( t). The informativeness of first period pries and loations are inreasing in t. Thus, due to the onvexity of E[π ˆµ] inˆµ, W ( ) is inreasing in t. This ombined with the fat that expeted first-period profits are less than 9α3 (i.e. the best 64ˆµ 2 0 myopi profits) for all t< t implies that ˆt t. To see the diretion in whih pries and loations move as we inrease t, we differentiate equations (17) and (19) with respet to t and get 12 Simple algebrai substitution reveals that the solution to the one-period maximation problem given by equations (3)- (6) is the solution to equations (16)-(19) for t = l 2 + r p 2. ( p 2/ t = 1 tˆµ ) > 0 α t < αˆµ (20) l 2/ t =2 tˆµ α > 0 t <. αˆµ (21) Thus, ˆp i p i and ˆl i l i. 6. REFERENCES [1] Philippe Aghion, Patrik Bolton, Christopher Harris, and Bruno Jullien. Optimal learning by experimentation. Review of Eonomi Studies, 58(4):621 54, July [2] Simon P. Anderson, Andre de Palma, and Jaques-Franois Thisse. Disrete Choie Theory of Produt Differentiation. MIT Press, Cambridge, Massahusetts, [3] Joseph P. Bailey. Eletroni ommere:pries and onsumer issues for three produts:books, ompat diss, and software. Tehnial Report OCDE/GD(98)4, Organisation for Eonomi Co-Operation and Development, Available from [4] Christopher H. Brooks, Edmund Durfee, and Rajarshi Das. Prie wars and nihe disovery in an information eonomy. In EC 00: Proeedings of the Seond ACM Conferene on Eletroni Commere. ACM Press, Otober [5] Christopher H. Brooks, Sott Fay, Rajarshi Das, Jeffrey K. MaKie-Mason, Jeffrey O. Kephart, and Edmund Durfee. Automated strategy searhes in an eletroni goods market:learning and omplex prie shedules. In EC 99: Proeedings of the ACM Conferene on Eletroni Commere. ACM Press, November [6] Karen Clay, Ramayya Krishnan, and Eri Wolff. Priing strategies on the web:evidene from the online book industry. In EC 00: Proeedings of the 2nd ACM Conferene on Eletroni Commere. ACM Press, Otober [7] Neveen Farag and Marshall Van Alstyne. Information tehnology a soure of frition? an analytial model of how firms ombat prie ompetition online. In EC 00: Proeedings of the 2nd ACM Conferene on Eletroni Commere. ACM Press, Otober [8] J. J. Gabszewiz and J.-F. Thisse. Spatial ompetition and the loation of firms. In J. J. Gabszewiz, J.-F. Thisse, M. Fujita, and U. Shweizer, editors, Fundamentals of Pure and Applied Eonomis. Volume 5: Loation Theory. Harwood Aademi Publishers, Chur, Switzerland, [9] Sanford J. Grossman, Rihard E. Kihlstrom, and Leonard J. Mirman. A bayesian approah to the prodution of information and learning by doing. Review of Eonomi Studies, 44(3):533 47, Ot [10] Joseph Harrington. Experimentation and learning in a differentiated-produts duopoly. Journal of Eonomi Theory, 66: , [11] John H. Holland. Adaptation in Natural and Artifiial Systems. University of Mihigan Press, Ann Arbor, MI, first edition, Seond edition (1992), published by MIT Press, Cambridge, MA.

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