Vol. 29, No. 4, July–August 2010, pp. 650–670issn 0732-2399 �eissn 1526-548X �10 �2904 �0650

informs ®

doi 10.1287/mksc.1090.0546©2010 INFORMS

Limited Memory, Categorization, and Competition

Yuxin ChenKellogg School of Management, Northwestern University, Evanston, Illinois 60208,

yuxin-chen@kellogg.northwestern.edu

Ganesh IyerHaas School of Business, University of California at Berkeley, Berkeley, California 94720,

giyer@haas.berkeley.edu

Amit PazgalJesse H. Jones Graduate School of Business, Rice University, Houston, Texas 77005, pazgal@rice.edu

This paper investigates the effects of a limited consumer memory on the price competition between firms. Itstudies a specific aspect of memory—namely, the categorization of available price information that the con-

sumers may need to recall for decision making. This paper analyzes competition between firms in a market withuninformed consumers who do not compare prices, informed consumers who compare prices but with limitedmemory, and informed consumers who have perfect memory. Consumers, aware of their memory limitations,choose how to encode the prices into categories, whereas firms take the limitations of consumers into accountin choosing their pricing strategies. Two distinct types of categorization processes are investigated: (1) a sym-metric one in which consumers compare only the labels of price categories from the competing firms and (2) anasymmetric one in which consumers compare the recalled price of one firm with the actual price of the other.We find that the equilibrium partition for the consumers calls for finer categorization toward the bottom of theprice distribution. Thus consumers have a motivation to invest in greater memory resources in encoding lowerprices to induce firms to charge more favorable prices. The interaction between the categorization strategies ofthe consumers and the price competition between the firms is such that small initial improvements in recallmove the market outcomes quickly toward the case of perfect recall. Even with few memory categories, theexpected price consumers pay and their surplus is close to the case of perfect recall. There is thus a suggestionin this model that market competition adjusts to the memory limitations of consumers.

Key words : bounded rationality; memory; behavioral economics; categorization; competitive strategy; pricingstrategy

History : Received: November 19, 2008; accepted: September 30, 2009; processed by Miklos Sarvary. Publishedonline in Articles in Advance December 30, 2009.

1. IntroductionA common and implicit assumption in the litera-ture on imperfect price competition is that consumerswho compare prices across different firms are able toperfectly recall all the prices they encountered anduse them in their decision making. However, thereis a substantial body of psychological research thatexamines the effect of memory limitations on con-sumer choice among available alternatives.1 Limita-tions on short-term memory mean that consumerswould not be able to perfectly recall relevant priceinformation, and consumers are more likely to facegreater short-term memory constraints in environ-ments with higher levels of information. Althoughimperfect short-term memory on prices is well doc-umented for consumers buying routinely purchased

1 See Alba et al. (1991) for a survey of the information processingliterature that examines the effect of memory on consumer choice.

products or products with low involvement (Dicksonand Sawyer 1990, Monroe and Lee 1999), such limi-tations would also be significant in complex productmarkets where consumer comparisons across firmsare based on the recall of not just the posted numer-ical prices but on the full price that includes otherassociated monetary aspects such as payment terms,financing, delivery, warranty offers, or other optionalfeatures. Faced with memory constraints, consumersmake decisions using heuristics that help them toform suitable price impressions.An important heuristic to deal with the abun-

dance of information is the grouping of events, objects,or numbers into categories on the basis of theirperceived similarities (see Rosch and Mervis 1975).Accordingly, memory limitations in our model per-tain to the categorization scheme that consumers useto encode and recall prices. The purpose of this paperis then to highlight the effect of a specific aspect oflimited consumer memory on the pricing strategies

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of competing firms—namely, the fact that consumersare unable to recall the exact prices encountered andinstead recall only the categories to which the pricesbelong. Specifically, we develop a model of marketcompetition between firms that face consumers whocompare prices but have the ability to recall previ-ously encountered market prices only as categories.This paper highlights the following aspects of mem-

ory limitations: First, we characterize memory limi-tations as the ability to recall only the category ofa previously observed price rather than the actualprice realization. Second, we take the analysis a stepfurther by embedding consumers with limited mem-ory in a market setting involving price competitionbetween firms. In equilibrium, the pricing strategiesof the firms take into account the memory limitationsof consumers. Finally, following Dow (1991), we alsoconsider the problem of consumers who have limitedmemory but who are aware of this limitation and takeit into account to make the best possible decision. Inother words, we explicitly consider the “encoding”decision of consumers given the information environ-ment.2 In doing so, we provide a framework to inves-tigate how memory limitations are affected by and, inturn, influence the market equilibrium.Consumers in our model can only use some limited

number of categories to construct their optimal mem-ory structure of prices charged by the firms. Considera consumer visiting a store and classifying a prod-uct’s price as “expensive” or “inexpensive.” When thesame consumer compares the price to that at a dif-ferent store, she can only recall that the price in thefirst store was “high” or “low” but not its exact value.Knowing that she will only remember the category,the consumer will optimally choose the threshold thatpartitions the prices at the first store into the highor low categories. Firms that compete in the marketknow that consumers can only remember if a productwas expensive or not and take that into account whenmaking their pricing decisions.The model consists of a duopoly market in which

firms compete for consumers who have limited mem-ory. A limited-memory consumer compares the pricesof the firms before purchase. The consumer does notrecall the actual prices at the firm but instead dividesthe price distribution into categories and recalls thecategory that contains the actual price. We considerthe effects of two distinct types of categorization pro-cesses to represent memory limitations, which areparamorphic representations of the price recall prob-lems that consumers face in actual markets. First, we

2 Thus the model is in the spirit of bounded rationality as definedin Simon (1987), where the decision maker makes a rational choicethat takes into account the cognitive limitation of the decisionmaker.

consider a symmetric categorization process in whichthe limited-memory consumers compare the categorylabels of the prices of the firms. This represents a sit-uation in which consumers can only compare imper-fect impressions of the total prices offered by bothfirms. Next, we consider an asymmetric categoriza-tion process in which consumers compare the actualprice realization at one firm to the recalled category ofthe price at the other firm. This process is akin to Dow(1991) and highlights the value of remembering priceinformation. A consumer with limited memory con-tacts both firms in sequence to compare their prices.The consumer observes the actual price charged atthe final firm and compares it to the expected priceof the recalled category of the first firm.3 We con-sider heterogeneity by allowing for several segmentsof consumers that differ in their memory recall abili-ties. Besides the limited-memory consumers, we con-sider a group of uninformed or loyal consumers forwhom memory limitations do not matter because theyconsider shopping only at their favorite firm (as longas the offered prices are below their common reserva-tion price), as well as a group of informed consumerswith perfect memory for prices. Both the uninformedand the informed consumers with perfect memory areas in Varian (1980) or Narasimhan (1988).We find the Nash equilibrium of firms’ pricing

strategies and the surplus maximizing categoriza-tion strategy for limited-memory consumers. Severalkey results about the effect of limited memory areremarkably consistent across the categorization pro-cesses and market conditions. For both the symmet-ric and the asymmetric categorization processes, wefind that the equilibrium categorization structure forthe consumers calls for finer categorization towardthe bottom of the price distribution. This implies thatconsumers should devote more memory resources toencoding lower prices to induce firms to charge morefavorable prices. The interaction between the cate-gorization strategies of the consumers and the pricecompetition between the firms is such that small ini-tial improvements in recall move the market out-comes quickly toward the case of perfect recall. Thus,even with few memory categories, the expected priceconsumers pay and their surplus are close to the caseof consumers having a perfect recall. There is thusa suggestion in this model that market competitioncompensates for the imperfect recall of consumers.We show that the presence of limited-memory con-

sumers along with the perfect-memory consumers can

3 As in Dow (1991), this process is not to be interpreted as one inwhich consumers search across the firms. However, in this paper,we also consider the case in which the decision process of thelimited-memory consumers involves the initial decision of whetherto compare prices at all, after observing the price at the first firm.

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soften price competition between the firms. In theasymmetric categorization case, equilibrium profitsare higher in a market with both limited-memory andperfect-memory consumers (for a given number ofuninformed consumers) than in a market in which allthe informed consumers either have perfect memoryor have limited memory. In the symmetric categoriza-tion case, for a given level of uninformed consumers,the increased presence of limited-memory consumersamong the informed consumers leads to greater equi-librium profits. Finally, across the different catego-rization processes, the expected profits of the firmsalways increase when there are a greater number ofuninformed consumers.Comparison across the two types of categorization

processes reveals some interesting results. Becausein the asymmetric categorization process consumerscompare a price conditional on a recalled cate-gory to an actual price, consumers in this casehave better price information than in the symmet-ric categorization case. Consequently, the asymmetriccategorization process may create a more competitiveand undifferentiated environment. Consistent withthis intuition, we find that the asymmetric catego-rization process, in general, leads to lower equilib-rium profits. We also investigate the robustness ofour results to different representations of the categorywithin the asymmetric categorization case. The keyresults of this paper are robust to different endoge-nous representations of the category such as themedian (rather than the mean), as well as exoge-nous representations such as the top or the bottomof the category. Indeed, we establish that the modelwhere consumers remember the top of the category inthe asymmetric categorization model is equivalent tothe symmetric categorization model where consumerscompare the price labels from both firms. Interest-ingly, when consumers use the mean of the category,which is endogenous to the equilibrium firm actions,their equilibrium surplus is actually higher than whenconsumers use exogenous rules such as the bottom orthe top of the category. Thus, if consumers in a marketwere to learn (for example, through experience overtime) to do the best for themselves and were moti-vated to make the best possible purchase decisions,it would be optimal for them to recall the categorymean price rather than to use any exogenous rule.We also extend the asymmetric categorization modelto the case in which consumers choose whether toobtain prices from both firms. We show that all thekey equilibrium results pertaining to firm pricing andthe categorization by consumers are also robust to thisextension.

1.1. Related ResearchA useful way of modeling bounded rationality inthe literature has been to enforce limitations on the

information processing of the decision maker. Thedecision maker cannot perfectly convert the inputsshe receives to the optimal outputs she needs tochoose. Our paper can be seen as related to a classof such models that involve the specific modelingof partitions or categories. The consumer receives asan input a signal (e.g., prices) that she cannot per-fectly recall. The consumer partitions the entire set ofpotential signals and classifies the one received into apartition. The action taken by the consumer is iden-tical for all signals that fall within a category. In theliterature, exogenously given partitions are the mostcommon. Along these lines there is research in gametheory that models the use of finite automata mech-anisms in repeated games (see Kalai 1991). In con-trast to the literature on exogenous partitions, Dow(1991) investigates the question of the optimal choiceof the partitions by consumers. However, in Dow,consumers face exogenously fixed price distributions.In our paper, the price distributions are a result ofthe competitive market equilibrium. In addition, toexamine the case that the partitions are exogenouslygiven, we are also interested in the optimal choice ofthe partition structure by consumers who face pricedistributions offered by firms in a market equilib-rium. Rubinstein (1993) analyzes a model in whicha firm price discriminates between consumers whohave different memory capacities by choosing a ran-dom lottery of prices, which consumers categorize. InRubinstein’s paper, the prices are a result of the choiceof a monopoly firm rather than the result of competi-tion between firms.Our paper is also related to the recent literature

on the effect of bounded rationality on market inter-action. For example, Basu (2006) characterizes theequilibria of pricing competition between firms thatface consumers that do not try to remember the fullprices they observe but rather round them to thenearest dollar. He shows that in equilibria (some)firms will always use prices ending in “9,” whichmay be seen as equivalent to pricing at the top ofthe category in our model. Spiegler (2006) developsa model in which firms are fully rational as in thispaper but where consumers are limited in their abilityto process information about firm characteristics andform boundedly rational expectations about the firm.4

Competitive firms in such markets choose actions thatobfuscate consumers leading to a loss of consumerwelfare. Camerer et al. (2004) present an approach to

4 Another recent paper by Iyer and Kuksov (2010) examines therole of consumer feelings in quality evaluations and the supply ofquality by firms. Consumers are not able to separate the effect ofenvironmental variables from the true quality offered by firms, butthey rationally try to infer the true equilibrium quality from theirquality perception.

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modeling bounded rationality in the form of cogni-tive limitations of players that leads to lack of ratio-nal expectations of the beliefs that agents have aboutother agents.In addition, this paper is related to the recent

research direction that examines how common biasesin consumer decision making affect strategic interac-tion. For example, Amaldoss et al. (2008) show thatthe psychological bias called the asymmetric dom-inance effect may facilitate coordination in games.Another recent example of psychological biases instrategic decision making is Lim and Ho (2007), whostudy the effect of counterfactual thinking-relatedpayoffs when retailers in a channel are faced withmultiblock tariffs.A different approach to modeling imperfect recall

involves modeling the impact of past decisions takenby a consumer on current decision faced by the sameindividual when the memory recall of the past is lim-ited and consumers cannot recall the exact decisiontaken (Hirshleifer and Welch 2002, Ofek et al. 2007).Along similar lines, Mullainathan (2002) models somespecific psychological aspects of human memory foreconomic behavior, namely, rehearsal and associative-ness, and in doing so provides a structure to under-stand when individuals will under- or overreact tonews.5 The aim of these papers then is to use memoryloss as a basis to explain psychological biases suchas inertia in decision making. In contrast, our paperfocuses on a specific aspect of memory limitations—namely, the ability of consumers to recall informationas categories and investigates its impact on the com-petitive market equilibrium.The rest of this paper is organized as follows. Sec-

tion 2 presents and analyzes the model with symmet-ric categorization process, and §3 analyzes the modelwith asymmetric categorization process. In §4, we dis-cuss the robustness of our results across different cate-gorization schemes and consumer decision processes,as well as the external validity of our results. Finally,§5 concludes and provides a summary.

2. The ModelWe start by analyzing the simplest possible model oflimited memory and categorization to highlight theirjoint effects on firm competition. Consider two sym-metric competing firms indexed by j (where j = 1�2)

5 The literature on dynamic models of learning also captures therecall aspects through modeling the forgetting of information (seeCamerer and Ho 1999). On the empirical side, Mehta et al. (2004)develop a structural econometric model to analyze the role ofimperfect recall and the forgetting of consumers for choice deci-sions of frequently purchased products. Their aim is to characterizethe role of the forgetting over time on the brand choice decisionsof consumers.

selling a homogenous product. Let the marginal costof production of each firm be constant and assumeit is equal to zero. The market is comprised of aunit mass of consumers with each consumer requir-ing at most one unit of the product. Consumers havea common reservation price, which is normalized toone without loss of generality. The consumers wouldlike to purchase the lowest-priced good and thuscompare the prices offered by both firms prior tomaking any purchase decision. However, consumers’recall of market prices are imperfect in that they can-not remember the exact prices they encountered butonly the category to which they belong. Specifically,assume that the consumers have limited memory inthe sense that they are endowed with n+ 1 memorycategories, which they use to encode the price infor-mation. Consumers encode all prices as long as theyare at or below their reservation price and reject anyprice that is above and opt out of the market. Theentire range of price information is thus divided inton+1 mutually exclusive and exhaustive categories orpartitions such that any observed market price is clas-sified into one (and only one) category. When makingtheir purchasing decisions, consumers cannot recallthe exact prices offered by the firms, but only the cat-egory to which the prices belong. Consumers havememory limitations but are aware of that fact and actoptimally given their memory constraints.Explicitly, let the set of memory categories be

�Ci�n+1i=1 and assume without loss of generality that

they are indexed in increasing order of prices (j > kimplies that p > q for every p ∈Cj� q ∈Ck). Denote theset of cutoff prices as �ki�

ni=1, where ki separates cate-

gory i from category i+1. From our previous assump-tion on the order of the categories, we have that k1 <k2 < · · · < kn Because the categories are exhaustiveand mutually exclusive, ki belongs to one and onlyone of them; we assume that each category is a setthat is open to the left, which implies ki ∈ Ci for i =1� �n Therefore ki−1 < p ≤ ki ⇒ p ∈ Ci Finally, forthe sake of completeness, we define kn+1 and k0 to bethe highest and lowest possible prices charged by afirm, respectively (clearly, kn+1 = 1). Therefore the cat-egory Ci is defined as the set of prices �ki−1� ki� fori= 1�2� �n+ 1 Suppose n= 1; then one might think of consumers

partitioning the observed prices into an “inexpensive”or an “expensive” category, with k1 being the priceabove which a product is considered to be expen-sive. Bounded rationality on the part of the con-sumers implies that the division into inexpensive andexpensive categories is chosen to help the consumersdeal with their limitations in recalling the exact priceinformation and aid in making the optimal purchasedecision.

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Consumers form the categorization scheme bychoosing the set of cutoff prices between adjacentcategories in a way that maximizes their expectedsurplus. Because consumers are aware of their limi-tations regarding the recall of prices, they will striveto devise the optimal system (categorizations) to dealwith it. Thus the model represents agents whose cog-nitive capacities are bounded but who nevertheless arerational in recognizing this limitation and accountingfor it in making the optimal decision. This is exactlyin the spirit of bounded rationality as defined bySimon (1987), where the consumer makes a rationalchoice that takes into account her cognitive limitation.Our model of consumer categorization also reflects animportant aspect of human cognition: although con-sumers are limited in short-term capacity for remem-bering market information, they can be sophisticatedin identifying patterns and forming optimal decisionrules.6

The categorization process can also be interpretedas follows: when consumers deal with firms in a mar-ket, they would have encountered many prices (andthe total price faced by consumers may have sev-eral aspects). This makes price comparisons imperfect,and the categorization heuristic captures this imper-fection. Given this, we can appeal to the finding incognitive psychology that shows that despite the factthat individuals have limited short-term memory, themind is sophisticated in constructing heuristics tooptimally react to the limitations in information pro-cessing. Indeed, in the specific context of the cate-gorization heuristic, Rosch (1978) provides the bestsupport when she argues that individuals aim for”cognitive efficiency” by minimizing the variationwith each category. Furthermore, the findings of themore recent experimental literature on the automatic-ity of categorical thinking is also consistent with ourmodel in that subjects may develop and use complexcategorization rules without even being consciouslyaware of computing it (see Bargh 1994, 1997), whileat the same time, their ability to recall informationfrom short-term memory might be limited. This char-acterization is consistent with our model and pricecategorization is akin to a long-term heuristic thatmay be automatically formed by the mind of the con-sumer because of experience over time with the mar-ket prices, which then helps the consumer to makeoptimal decisions during a specific purchase occasion.Finally, in the context of behavioral pricing, Monroeand Lee (1999) argue that consumers may not be ableto perfectly recall the price of a product, but at thesame time, they are likely to tell if the product is “too

6 For example, an analyst is unlikely to be able to remember num-bers in a data set but can be good at detecting patterns in the dataand at forming rules or heuristics and in using those rules.

expensive,” “a bargain,” or “priced reasonably.” Thisalso provides support to our model of price catego-rization for consumers with limited memory.

2.1. Comparing Category Labels: SymmetricCategorization Process

We first consider the case in which the limited-memory consumers encode the prices from both firmsin categories and compare only the labels of the cat-egories that the prices from both the firms fall into.In other words, the categorization process is symmet-ric across the firms. This represents the situations inwhich consumers have an imperfect impression aboutthe prices from both firms. This case arises in environ-ments that require consumers to process and comparenot only the posted price offered by the competingfirms but also numerous other informational detailsthat are relevant for the full price faced by consumers.Thus the decision process should not be interpreted asone of search but rather one that highlights price com-parisons when there are imperfections in the recall ofconsumers.The specific process of the limited-memory con-

sumers is as follows. Consumers encode the pricesposted by the firms into categories. To make a pur-chase decision, they recall and compare the categoriesassociated with the prices of both the firms. Con-sumers buy from the firm whose recalled price wasin a lower category. In the case of a tie, when therecalled prices of the two firms are in the same cate-gory, consumers purchase randomly from either firmwith equal probability. The parameter n is exoge-nously given and captures the precision of consumers’price recall or the degree of memory.7 If n = 0� thenconsumers have no memory and all prices fall in asingle category. If n= 1� consumers divide the firms’prices into two categories: high prices or low prices.As n increases, consumers categorize the firms’ pricesinto finer and finer partitions. Consequently, con-sumer recall improves, and the recalled price willmore closely reflect the actual price charged by thefirms.We analyze the price equilibria for a given cate-

gory structure and then examine the endogenous casewhere the consumers optimally choose the categoriza-tion to maximize their surplus. The following are thesequence of decisions: given the number of categoriesn + 1� consumers with limited memory optimallychoose the categorization cutoffs �ki�

ni=1 in an opti-

mal way. In the next stage, the two firms decide on

7 We can endogenize n if we assume a cost function that is increas-ing for n. We choose not to do so because adding an extra stageto the model in which n is chosen by consumers does not affectthe results. However, at the end of §3.3, we discuss how n, if it isendogenously chosen, can be affected by the competitiveness of themarket environment.

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Chen, Iyer, and Pazgal: Limited Memory, Categorization, and CompetitionMarketing Science 29(4), pp. 650–670, © 2010 INFORMS 655

their pricing strategies given the consumers’ cutoffs.Note that we can also consider the timing in whichconsumers and firms move simultaneously to, respec-tively, choose categories and prices without chang-ing the results of the paper.8 In the last stage, allconsumers make their purchase decisions based onprice realizations and the decision process describedearlier. We solve for the symmetric subgame-perfectNash equilibrium of the game, which consists of theset of optimal cutoffs chosen by the consumers andthe pricing strategies chosen by both firms. The opti-mal cutoffs chosen by the limited-memory consumers(�k∗

i �ni=1) satisfy the requirement that the equilibrium

surplus of the limited-memory consumers is maxi-mized. In cases where there are multiple equilibria forfirms’ pricing strategies given a set of cutoffs �ki�

ni=1�

we use the selection criterion that firms will play theequilibrium strategies with Pareto-dominant payoffs.We identify the symmetric pure-strategy equilibriumwhen it exists.

2.2. Analysis and ResultsIn this section, we present the equilibrium solution ofthis simple model of symmetric categorization whenconsumers compare the labels of the categories. Notethat consumers in choosing to buy from a firm recallthe labels of the categories in which two firms’ actualprices occurred. Because consumers can only recallthe category labels, the firms will have the incentiveto price at the top of each category.It is useful to begin by considering the simplest pos-

sible case of n = 1, where consumers can only recallhigh versus low prices. If the cutoff between the cate-gories is at k, each firm will optimally use at most twoprices k or 1. It is easy to see that both firms pricing atk always constitutes a pure-strategy equilibrium. Bothfirms pricing at 1 constitutes an equilibrium as longas 1

2 ≥ k and this equilibrium is the Pareto-dominantone for firms. This implies that when consumers canchoose k to maximize their surplus, they will, in equi-librium, choose it to be slightly above 1

2 to strate-gically induce the firm to charge their equilibriumprices in the low category.In general, given the cutoff points k1 ≤ · · · ≤ kn ≤

kn+1 = 1 and the corresponding n+ 1 categories, anyequilibrium pricing strategy for the firms can assigna positive probability only to prices at the top of eachcategory. The following proposition shows that thereis a unique symmetric equilibrium in pure strategies.The proofs for all the propositions and lemmas can befound in the appendix.

8 The equilibria of sequential move game are also equilibria in thesimultaneous move game as well. This is because the subgame-perfect Nash equilibrium in the sequential game is also a Nashequilibrium of the simultaneous game with the selection criterion,involving equilibrium strategies with Pareto-dominant payoffs forconsumers.

Proposition 1. When consumers optimally choose thecutoffs, there is a unique pure-strategy equilibrium.9 Theoptimal cutoffs are k∗

i = � 12 �n+1−i+� for every i= 1� �n,

where � � �1/2�n ∀n and � → 0 Both firms chargep∗

j = k∗1 = � 12 �

n + �. Each firm makes positive equilibriumprofits �∗

j = � 12 �n+1+ �/2, which are decreasing in n.

Competition in a market with consumers who com-pare the price category labels results in a symmetricpure-strategy equilibrium in which firms compete bycharging prices, which are at the top of lowest cat-egory. When consumers choose the categories thatmaximize their surplus, the categorization schemeconsists of a unique set of cutoffs. The first point isthat the equilibrium prices are strictly greater thanmarginal costs and the firms earn positive profitsdespite the fact that they are undifferentiated andface a single homogenous segment of consumers.Bounded rationality of consumers who can recallprices only as the category labels moves firms awayfrom the Bertrand competition outcome. Because con-sumers do not distinguish between all prices within acategory, firms have the incentive to charge only thehighest price within a category. However, the compe-tition between the firms induces them to price only inthe lowest category.10

Ever since Edgeworth (1925), there has been a lit-erature on possible resolutions to the Bertrand para-dox, the idea that undifferentiated firms facing ahomogenous consumer market might still be able toprice above marginal costs and earn positive equilib-rium profits. These resolutions have typically focusedon supply-side factors such as capacity constraints(Levitan and Shubik 1972, Kreps and Scheinkman1983, Iyer and Pazgal 2008) or the nature of cost func-tions (Baye and Morgan 2002). Proposition 1 adds tothis literature by showing the role of bounded ratio-nality on the consumer side as a means to resolve theBertrand paradox.An interesting corollary of Proposition 1 is that for

i = 1� �n + 1, the differences between consecutivecutoffs is given by

k∗i − k∗

i−1 =(12

)n+2−i

The difference decreases geometrically as i decreases.This implies that the categorization becomes finertoward the lower end of price range. In fact, whenmoving from low to high prices, each successive

9 Optimal choice by the consumers involves each individual con-sumer setting cutoffs that maximizes her own surplus.10 Note that in this pure-strategy equilibrium, while the firms’ strat-egy is to price in the lowest category, it is the best response to theconsumer’s categorization strategy, and, in turn, consumers chooseall the n partitions optimally given the firms’ pricing strategies.

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Chen, Iyer, and Pazgal: Limited Memory, Categorization, and Competition656 Marketing Science 29(4), pp. 650–670, © 2010 INFORMS

category is exactly double the size of its lowerneighbor (�Ci� = 2�Ci−1�). This is intuitively appeal-ing because it suggests that consumers pay moreattention to and invest more memory resources inencoding lower prices rather than higher prices. Thereason behind this is the consumers’ strategic goalto induce the firms to lower prices as much as pos-sible. The consumer’s equilibrium cutoffs strategyk∗

i = � 12 �n+1−i + � for every i = 1� �n� is designed

such that at each of the higher price levels, the firmsare induced to undercut each other. When both firmscharge high prices close to the reservation price, adeviation by a firm to a price in a lower categorycan be profitable even if the undercutting amountrequired for winning consumers away from the rivalfirm is of a relatively large amount. However, whenboth firms charge lower prices, the discount needed toundercut and switch consumers away from the com-petitor has to be smaller to make such a move prof-itable. Thus, it is optimal for consumer categorizationto have wider partitions at the upper end of the pricerange. Moreover, as the prices decrease, the partition-ing becomes successively finer such that in equilib-rium firms are induced to price only in the lowestcategory with the lowest possible k∗

1 Furthermore, the equilibrium prices and profits

of the firms decrease with the number of cate-gories. Thus as consumers’ recall of the market pricesimproves, the equilibrium prices charged by the firmsmove toward marginal cost. Indeed, as the degrees ofmemory increase beyond bound (n →�), the modelrepresents the standard model of perfect recall. Inthis case, the equilibrium prices converge to marginalcost, and thus we are able to recover the standardBertrand competition outcome as the limiting case ofthis model with infinite degrees of memory.Proposition 1 also reveals an important conver-

gence property of the market outcome as the degreesof consumer memory increase toward perfect mem-ory. As n increases, the equilibrium prices and profitsconverge to the Bertrand outcome of marginal costpricing at a decreasing rate. Thus, additional cate-gories have a smaller effect in changing the equilib-rium price as compared to the initial few categories.Overall, this simple categorization model of limitedrecall suggests that given strategic market interac-tions, small amounts of initial improvements in recallcan lead to equilibrium choices that are close to theperfect recall outcome.We can also examine the consumer surplus and the

loss of consumer surplus because of limited recall.In the perfect-recall Bertrand outcome, the consumersurplus will be at its maximum of one, and from theProposition 1, the equilibrium surplus of the limited-recall consumers is S�n� = 1− p∗

j = 1− � 12 �n Thus the

loss in consumer welfare relative to the case of perfect

recall is decreasing in the degrees of memory and themarginal effect is also decreasing. This again impliesthat the small amounts of initial improvements inmemory will lead to substantial gains in consumerwelfare.

2.3. Introducing Consumer Heterogeneity:Adding Uninformed Consumers

We now extend the model to investigate the effect ofconsumer heterogeneity in firm preference as well asin memory capacity. We first extend the basic modelto include a group of uninformed or loyal consumers.Let there be a group of uninformed consumers ofsize 2� who randomly purchase from either firm withequal probability as long as the price is below theirreservation price.11 Note that these consumers do notcompare prices across the firms, and therefore thereis no role for memory in facilitating price compar-isons for these consumers. The remaining group of�1 − 2�� consumers are limited-memory consumerswith n degrees of memory as in §2.2. The followingproposition establishes the equilibrium:

Proposition 2. When consumers optimally choose thecutoffs and �1/�2�1− ����n > 2�, there is a unique sym-metric pure-strategy equilibrium. The optimal cutoffs arek∗

i = �1/�2�1− ����n+1−i + � for every i = 1� �n, where� � �1/�2�1 − ����n ∀n and � → 0 Both firms chargep∗

j = k∗1 = �1/�2�1 − ����n + �. Each firm makes positive

equilibrium profits �∗j = 1

2 �1/�2�1−����n+�/2, which aredecreasing in n.

Even in a heterogenous market with uninformed orloyal consumers and consumers with limited memory,there exists a unique symmetric pure-strategy equi-librium in which firms price at the top of the lowestpartition as long as � is not too large. It can be notedthat the presence of uninformed consumers in themarket who consider buying from only one firm cre-ates market differentiation between the firms. Thus,the higher level of � represents a more differentiatedmarket with less intense price competition. Therefore,as expected, the equilibrium price and profit of thefirms increase with � It is also interesting to notethat in this equilibrium, the firms’ profits are �∗ =12 �1/�2�1 − ����n, which are, in fact, greater than ��the maximum profit that can be attained in a stan-dard model of competition in which consumers areable to compare the actual prices (Varian 1980). All theother key results of §2.2 are preserved with the addi-tion of uniformed consumers. For example, as in thebasic model, the categorization becomes finer towardthe lower end of price range. Furthermore, in a more

11 Alternatively, � of these consumers can also be assumed to con-sider purchasing only from one of the two firms, whereas theremaining � consider the other firm.

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competitive market with a smaller �, the partitions atthe lower end of the price range become even finer.In the appendix, we characterize a symmetric

mixed-strategy equilibrium in which the categoriza-tion by consumers are finer toward the lower end ofthe price range when the condition �1/�2�1− ����n >2� does not hold. Interestingly, in this mixed-strategyequilibrium, firms charge a price in every categorywith positive probability. Furthermore, as the num-ber of categories increases beyond bound, this equi-librium also converges to one in which the informedconsumers have perfect memory as in the standardmodel of Varian (1980).

2.4. Adding Heterogeneity in Memory CapacityConsider now a market that also consists of a group ofsize 2� fully informed consumers with perfect mem-ory for market prices who can therefore compare theactual prices offered by both firms. As before, thereare 2� uninformed or loyal consumers and, conse-quently, a group of 2�= 1− 2� − 2� consumers withlimited memory. We can now investigate the effect ofconsumer heterogeneity in memory capacity as dis-tinct from heterogeneity in consumer firm loyalty.

Proposition 3. When consumers optimally choose thecategories and ��� + ��/�1 − ���n > �/�� + ��, thereis a unique symmetric mixed-strategy equilibrium. Theoptimal cutoffs are k∗

i = ��� + ��/�1 − ���n+1−i + � forevery i = 1� �n, where � � ��� + ��/�1 − ���n ∀nand �→ 0 Both firms price at the top of the lowest cate-gory according to the cumulative distribution functionFj �p� = ��� + ��/�2����k∗

1 − p�/p, where p ∈ ���� + ��/�1− ��+����k∗

1� k∗1�.

With a segment of consumers with perfect mem-ory, there is no longer a pure-strategy equilibrium.There is, however, a unique symmetric mixed-strategyequilibrium in which firms charge prices only in aninterval extending from the top of the lowest parti-tion. Once again, the key results of §§2.2 and 2.3 arepreserved as the categorization becomes finer towardthe lower end of price range in this case. Firms’profits are �∗

j = �� + ����� + ��/�1− ���n, which aregreater than � This result is also similar to that ofProposition 2.We can now compare the results of Proposition 3

with those of Propositions 1 and 2 to better under-stand the effect consumer heterogeneity in this model.First, notice that when � → 0, the price support ofthe equilibrium price distribution shrinks to a singlepoint, k∗

1 = �1/�2�1− ����n + �, and so we recover theequilibrium price in Proposition 2. On the other hand,if � → 0, then the market consists of only limited-memory and perfect-memory consumers, and we geta market with consumer heterogeneity in memorycapacity. In this case, the equilibrium will be one in

mixed pricing strategies in the lowest category and inthe interval p ∈ ���/�1− ���k∗

1� k∗1� Firms make posi-

tive profits �k∗1 For a fixed proportion of uninformed

or loyal consumers, the increased presence of perfect-memory consumers leads to lower firm profits com-pared to the one in Proposition 1. Conversely, for agiven �� a greater proportion of limited-memory con-sumers increases equilibrium firm profits.

3. Asymmetric Categorization ProcessIn §2, the categorization scheme of the limited-memory consumers was symmetric across the firms.Consumers compared only the labels of the cate-gories that represented the prices charged by boththe firms. In this section, we analyze an alterna-tive categorization process along the lines of Dow(1991) and Rubinstein (1993), where consumers com-pare the actual price at a firm they are at with therecalled category for the price that they encounteredat the other firm. As in these earlier papers, thisasymmetric categorization process is intended as aframework designed to highlight the value of recalledinformation.Specifically, consider a three-stage decision process

that is similar to Dow (1991) and in which the limited-memory consumers contact both firms in the follow-ing manner:12 In the first stage, consumers observe theprice at a firm and encode this price (half of the con-sumers observe Firm 1 while the other half observeFirm 2). In the second stage, the consumers observethe exact price at the other firm, compare it with theencoded price recalled from their memory, and decidewhether to buy the product at the current firm. Ifthe actual price observed is lower than or equal tothe price recalled from memory, the consumer willpurchase one unit of the good at the second firm (pro-vided that the price is not higher than the reservationprice).13 Finally, in the third stage, if the consumersdid not purchase at the second stage from the secondfirm, they purchase from the original firm providedthat the price there is below the reservation price.14

12 We can extend our model to allow consumers to decide whetherto compare prices at all after observing the price at the first firm.The results of this paper are unaffected in this extension. Detailsare provided in §4.2 and the appendix.13 Note that we assume that if the observed price at the second firmis exactly identical to the price that is recalled from memory, theconsumer will buy at the second firm.14 This decision process can also have other interpretations pertain-ing to the broader class of problems of communication constraintsbetween agents in organizational settings. For example, it can beseen as reflecting communication constraints when the decision-making team consists of two agents. Agent 1 observes the pricein the first firm and sends a message to agent 2, who then hasthe discretion to decide after observing the price at firm 2. Here,the constraint can be interpreted as a limit on the set of words ormessages that can be sent.

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Chen, Iyer, and Pazgal: Limited Memory, Categorization, and Competition658 Marketing Science 29(4), pp. 650–670, © 2010 INFORMS

Unlike the case where the consumers compare theprice labels of both firms (symmetric categorizationcase), this decision process allows us to representprice comparisons between the two firms when a con-sumer has better information about one of the twofirms. In actual markets, this would represent sit-uations where the consumer has better informationabout the price offer at the current firm than theprice that was previously observed (and encoded) atthe competing firm. We have two specific objectivesin considering this alternative categorization process.First, we aim to investigate the robustness of the equi-librium consumer and firm strategies to the alter-native forms of categorization. Furthermore, we canexamine whether the categorization process, consid-ered in this section leads to a relatively more com-petitive market. Second, as opposed to the symmetriccategorization process, in the asymmetric categoriza-tion process, the consumer has to compare an actualprice to a recalled category. Consequently, the con-sumer has to assign some number in the category torepresent its prices. This then allows us to comparethe effects of exogenous representations of the cate-gory (such as the top of the category) versus endoge-nous statistics (such as the mean or the median ofprices charged in the category).As in §2, consumers with n degrees of memory are

endowed with n+1 categories �Ci�n+1i=1 that are indexed

in increasing order of prices. They may have to opti-mally select n cutoff prices �ki�

ni=1 that separate the

categories. Consumers remember the expected pricecharged in a category as the representative price inthat category.15 Let �mi� j �

n+1i=1 be the set of the mean of

the prices that each firm j charges in each category;then mi� j = E�pj � pj ∈ Ci� Clearly, kn+1 ≥ mn+1� j > kn ≥mn�j > · · ·> k1 ≥m1� j > k0 for i= 1� �n and j = 1�2 Figure 1 shows the categorization scheme with thecategories, cutoffs, and mean prices.Except for the categorization process of the limited-

memory consumers, we maintain exactly the samefeatures of the market as in §2.4. In other words,the segment of uninformed and perfect-memory con-sumers are exactly as previously described.

3.1. The General CaseWe start by presenting the most general case of n+ 1categories in a market with all three segments of con-sumers. It is immediate that the symmetric equilib-rium of this model involves mixed strategies. Prior tothe explicit derivation of the equilibrium price sup-port, note that the potential price range of each firmis �b�1�, where b = �/�1− ��. A firm will never set aprice pj below b because the maximum profit it can

15 In §4.1, we discuss the effects of other representations of the cat-egory such as the median or the top of the category.

obtain is pj�� + 2�+ 2��, which is lower than �� theguaranteed profit it can obtain by setting pj = 1 andselling only to its uninformed consumer group. Hencethe n + 1 categories that consumers use to classifyprices are all within �b�1� Given our notation, wehave k0 = b = �/�1−�� The following lemma identifies the equilibrium

price support.

Lemma 1. The support of the price distribution usedby the firms in a symmetric mixed-strategy equilibriumcontains no atoms and is comprised of a union of intervals:⋃n+1

i=1 ��bi�mi� ∪ �vi� ki��, where ki > vi > mi > bi > ki−1(i.e., there are two “holes” in the support of the pricingdistribution in each category).

Let Wj�p� be the probability that firm j pricesabove p, and denote wi = W�vi� and si = W�bi� fori = 1� �n+ 1 (and by definition sn+2 = 0). Note thatsi is just the probability of pricing in any of the cate-gories between i� �n+ 1 Note that the equilibriumfirm strategies will make the rival indifferent betweenany of its strategies. For the extreme points of the dis-tribution, we get the following profit expressions fori= 1� �n+ 1:

pj = ki! "= �� +wi�+ si+1�+ 2si+1��ki�

pj = vi! "= �� +wi�+ si+1�+ 2wi��vi�

pj =mi! "= �� +wi�+ si�+ 2wi��mi�

pj = bi! "= �� +wi�+ si�+ 2si��bi

(1)

When pricing at ki, a firm will get four groups of con-sumers: (1) all of its uninformed consumers; (2) theinformed consumers with perfect memory who find ahigher price at the other firm; (3) the limited-memoryconsumers who started with it, recall mi (rather thanthe actual price), and encounter a higher price thanmi at the other firm; and (4) finally, the limited-memory consumers who begin with the other firm,observe a price above bi+1 and remember a price mi+1.When charging vi, a firm will get, in addition to theabove consumers, all the informed perfect-memoryconsumers who find a higher price at the other firm.A price of mi will get a firm the obvious uninformedand informed perfect-memory consumers, as well asall the limited-memory ones that started with theother firm and saw a price higher then bi (as theyremember mi but will not purchase from the first firmeven in a case of a tie) and the limited-memory con-sumers that started with it and compare to a priceabove mi in the other firm. Finally, by pricing at thelower end of the support, a firm will get additionalinformed consumers with perfect memory as well. Tosolve for the equilibrium, recall that mi is the mean ofthe price distribution within category i and given bymi =

∫ ki

bi�p�d/dp��1−W�p��� dp

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Figure 1 The Categorization Scheme

kn + 1 = 1kn – 1 knk0 mn mn + 1m1

Cn Cn + 1C1

k1

…..

A firm pricing at kn+1 = 1 guarantees itself a prof-its of " = � +wn+1�, which is the profit in the sym-metric mixed-strategy equilibrium. Therefore, firmsmake profits higher than �, which is the profit in theextreme case where all the informed consumers haveperfect recall (� → 0 and � > 0). This result is simi-lar to that of Proposition 3 for the case in which thelimited-memory consumers recall and use the cate-gories from both firms.

3.2. The Equilibrium as �→ 0The effect of limited memory on competition can beclearly seen from the analysis of the limit market inwhich all the informed consumers have limited mem-ory of degree n. This also recovers the case that isanalogous to that in Proposition 2 of a market withuninformed or loyal and limited-memory consumers.If we take the limit of � approaching zero, then weget vi = ki, mi = bi� and wi = si+1� i= 1� �n (as wellas wn+1 = 0� In other words, as � approaches zero, thetwo price support intervals in each category i shrinkto two points, p= ki and p=mi, where the probabilityof charging prices at the kis approach zero. Conse-quently, the set of equations in (1) reduces to

pj =mi! � = ��+ si+1�+ si��mi ∀ i= 1� �n+1 (2)A few comments about the nature of the equilib-

rium price support are in order. In the models ofcompetitive price promotions in which the informedconsumers have perfect recall, and therefore com-pare the actual prices of both the firms, the equilib-rium price distribution is continuous (for example, seeNarasimhan 1988, Raju et al. 1990, or Lal and Villas-Boas 1998). However, limited rationality in the formof the asymmetric categorization process in which thelimited-memory consumers compare an actual priceto a recalled category leads to firms choosing fromonly a finite set of possible prices. In empirical stud-ies, it has been observed that the distributions ofprices are typically such that most of the probabilitymass is concentrated around a small number of pricepoints (see, for example, Villas-Boas 1995, Rao et al.1995). Furthermore, the number of prices chargedgoes up with the improvement in the degrees of mem-ory. Behavioral studies have pointed out that highinvolvement environments lead to greater attentional

capacity being devoted to encode a relevant piece ofinformation in memory (e.g., Celsi and Olson 1988).Thus one can expect to observe more prices beingcharged in high-involvement product markets withgreater degrees of consumer memory. Proposition 4states the equilibrium of this limit market case withuninformed and limited-memory consumers.

Proposition 4. In the limit market (as � → 0) withuninformed consumers and limited-memory consumerswho have n degrees of memory, a symmetric mixed-strategy equilibrium will involve each firm charging pricesat pi =mi = 2/�1/ki−1+ 1/ki� with probabilities Pr�mi�=��/2���1/ki−1−1/ki� for i= 1� �n+1, where the �ki�

ni=1

is the set of cutoffs for the limited-memory consumers suchthat 1= kn+1 > kn > · · ·> k1 > k0 = �/�1−��

Proposition 4 shows that the prices charged in eachcategory are the harmonic means of the category cut-offs. Note that the profits for the case where all theinformed consumers had perfect memory was � Asseen in the proof of Proposition 4 in the appendix, theequilibrium profits in a market where all the informedconsumers compare prices with limited memory isalso � Furthermore, in this market with only limited-memory consumers, the number of prices charged byfirms is equal to the number of categories. Therefore,in the limit market equilibrium, the available memorycapacity is aligned with the price information that isrequired to be recalled, and it is as if a market withperfect recall is mimicked. This results in the firmscompeting away all but the guaranteed profits thatcan be made from their uninformed consumers.Proposition 4 identifies the equilibrium firm strate-

gies as function of the cutoffs of the consumers.We now characterize the equilibrium of this modelif the cutoffs satisfy the requirement that the equi-librium surplus of the limited-memory consumersis maximized. This requirement results in a uniquePareto-optimal equilibrium. Note that each firm’sequilibrium profits are always � Therefore therequirement that at the optimal cutoffs the equi-librium surplus is maximized results in a uniquePareto-optimal equilibrium that is best for the limited-memory consumers. This is also consistent with theidea of boundedly rational consumers doing the bestfor themselves given the constraints that they face.Proposition 5 identifies these optimal cutoffs k∗

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Chen, Iyer, and Pazgal: Limited Memory, Categorization, and Competition660 Marketing Science 29(4), pp. 650–670, © 2010 INFORMS

Proposition 5. As � → 0, the optimal cutoff pricesthat maximize the equilibrium surplus of the limited-memory consumers are k∗

i = ��/�1 − ����n+1−i�/�n+1�. Forthese cutoffs, each firm’s equilibrium prices are m∗

i =2/��1 + ��1 − ��/��1/�n+1����/�1 − �����n+1−i�/�n+1� withprobability Pr�m∗

i �= ��/�1−2������1−��/��1/�n+1�−1�×��1−��/���n+1−i�/�n+1� for i= 1� �n+ 1 We can now summarize the main findings from the

analysis of this limit market consisting of only unin-formed or loyal and limited-memory consumers andcompare these results with those of Proposition 2,where consumers categorize the prices from bothfirms before comparing them. In Proposition 5, eventhough consumers have limited memory, the pricingstrategies of the firms adjust so that the number ofprices charged is aligned with the degrees of con-sumer memory, and so it is as if consumers can per-fectly recall the actual prices that are charged. Thusthe market equilibrium adjusts to the memory capac-ity of consumers, and each firm charges only a singleprice with positive probability in each category.Next, we can see from Proposition 5 that for

i = 1� �n + 1, the differences between consecutivecutoffs are

k∗i −k∗

i−1 =(

�

1−�

)�n+1−i�/�n+1�−(

�

1−�

)�n+2−i�/�n+1�

=(

�

1−�

)�n+1−i�/�n+1�(1−

(�

1−�

)1/�n+1�) (3)

The difference decreases exponentially as idecreases. This implies that the categorizationbecomes finer toward the lower end of price range.Thus, this result is robust across different types ofcategorization processes. Furthermore, the probabil-ity of charging a particular price is proportional to��1 − ��/���n+1−i�/�n+1� and thus is also exponentiallydecreasing in i As in §2, these results are intuitivelyappealing, suggesting that the consumers pay moreattention in encoding lower prices than higher prices,and this induces firms to respond by charging lowerprices with higher probabilities. Also, we get that$k∗

i /$� > 0 The values of the cutoffs increase inless competitive markets with more uninformedconsumers, and this result again is robust acrossthe different types of categorization processes. Asexpected, an increase in the size of the uninformedgroup of consumers increases each price that thefirms will charge and also shrinks the price range1 − b From Proposition 2, we can see that firms’profits in the symmetric categorization model whereconsumers compare the price category labels fromboth firms are higher than that in the asymmetriccategorization model of this section. This is quiteintuitive because consumers in the asymmetric cate-gorization model have better price information than

in the symmetric categorization model as the actualprice from one firm is used in making purchasedecisions when categorization is asymmetric. Thisinduces more intense competition between firms andthus reduces their profits.Finally, another interesting result of this limit mar-

ket case is the behavior of the expected prices chargedby the firms.Result 1. The expected equilibrium price charged

by the firms, as well as the price variance, increaseswith the degree of consumer memory.With greater n� consumers with limited memory

become more sensitive to price differences betweenfirms. This increased price sensitivity reduces firms’expected profits from the limited-memory consumersbecause with greater n, those consumers are less likelyto make mistakes and more likely to end up buy-ing from the firm that (actually) has the lower price.The strategic responses of the firms are therefore toincrease the prices charged, on average, to extractgreater surplus from their uninformed consumers.Thus, in the equilibrium, the average market pricecharged by the firms goes up with improvementsin memory and is the highest when the informedconsumers have perfect recall. This result may beseen as interesting in that the average market priceincreases even as the consumer cognition for pricecomparisons improves.16 It is important to note thatthis actually implies that the expected price paid bylimited-memory consumers decreases with n becausefirms profits are invariant with regard to n, and theloyal consumers do pay higher prices, on average.This is because the limited-memory consumers paythe average of the minimum price charged by thefirms. In contrast, the uninformed or loyal consumerspay the expected prices charged by the firms, whichincreases with the degrees of memory.The above results on equilibrium prices is dif-

ferent from the pure-strategy equilibrium result inProposition 2, where both uninformed and informedconsumers pay the same price and that pricedecreases with n. Therefore, although in the symmet-ric categorization case the improvement of memoryfor the informed consumers provides a positive exter-nality to the uninformed consumers, it leads to a neg-ative externality to the uninformed consumers in theasymmetric categorization case.

3.3. Comparing Limited Memory toPerfect Memory

We now turn to the comparison of this model to thestandard model where all the informed consumers

16 This result is similar in flavor to those previously presented inthe literature. For example, in Rosenthal (1980), the average marketprice increases even as the number of sellers increases; in Iyer andPazgal (2003), the expected market prices increase with the numberof retailers who choose to join a comparison shopping agent.

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who compare prices are assumed to have perfect recall.In the standardmodel (Varian 1980, Narasimhan 1988),W�p� = Pr�pj ≥ p� = �/��1− 2���× �1/p − 1�, b ≤ p ≤ 1and E�p�= ��/�1− 2���× ln��1−��/��. As the degreeof memory increases, the price distribution resemblesthe standard model, and in the limit as the degree ofmemory increases beyond bound, it is identical to thedistribution in standard model. Formally,

Proposition 6. Given an arbitrarily small % > 0� ∃ Nsuch that for every degree of memory n > N and everyprice p ∈ ��/�1 − ���1�, we have (i) ∃ mj�n� such that�p−mj�n��< % and (ii) �Wn�p�−W�p��< %, where Wn�p�is the equilibrium probability of each firm pricing above pand mj�n� is a price charged with positive probability whenthe limited-memory consumers use n cutoff prices.

Thus we recover the standard model of perfectrecall competition as the limiting case of our modelwith infinite degrees of memory. Next, consider a con-vergence measure that is based on the expected priceconsumers pay. Define the degree of convergence by'En�p�= �E�p�−En�p��/�E�p��, where E�p� is the aver-age equilibrium price in the case of perfect memory:

'En�p�= 1−2�n+ 1�

ln��1−��/��

���1−��/��1/�n+1� − 1����1−��/��1/�n+1� + 1� (4)

Smaller values of this measure imply greater con-vergence of the expected price to the case of per-fect memory. It is straightforward to verify that$'En�p�/�$n� < 0 and $2'En�p�/�$n

2� > 0. Thereforethe convergence is increasing in n, but the marginalgain in convergence is decreasing in n Thus addi-tional categories have a lower effect in changingthe expected price compared with the first fewcategories. Thus once again, this result is similarto the convergence result that we established withthe symmetric categorization process. Also, we havethat $'En�p�/�$�� < 0 and $2'En�p�/�$�

2� > 0� whichimplies that the convergence is faster (with themarginal gain in convergence decreasing) in a lesscompetitive market that has a greater proportion ofuninformed consumers. An increase in the proportionof the uninformed consumers means fewer compari-son shoppers who have limited memory. This impliesthat the expected prices in the case of limited-memoryapproach that for the case of perfect memory. In theextreme case, at 2� = 1, the difference disappears. Itis easy to also verify that limn→��'En�p��= 0. Overall,both the symmetric and asymmetric categorizationmodels suggest that given strategic market interac-tions, small amounts of initial improvements in recallcan lead to equilibrium choices that are close to theperfect recall outcome.We can also examine the expected consumer wel-

fare loss because of limited memory. Since S��n� =S − S� − 2" = 1 − 4� + 2�En�p�� we can define the

relative loss in total surplus that all the comparisonshopping consumers incur using n + 1 categoriza-tions versus perfect memory as (n = �S� − S��n��/S� =2��E�p�−En�p��/�1−4�+2�E�p��. Again, it is straight-forward to show that $(n/�$n� < 0� $2(n/�$n

2� > 0�$(n/�$�� > 0� and $2(n/�$�

2� > 0 Therefore the rel-ative surplus loss for all the limited-memory con-sumers is decreasing in n� and the rate at which thesurplus decreases is decreasing in n Finally, the rel-ative surplus loss for all the limited-memory con-sumers increases with the proportion of uninformedconsumers in the market. Because $2(n/�$n

2� > 0,the relative marginal gain in surplus of the limited-memory consumers from an increase in memory isdecreasing with n In this model, even with just onecutoff (n= 1), we get (1 < 7% for every feasible valueof � ∈ �0�0 5�. Overall, this suggests that the expectedwelfare loss to consumers from limited memory ofprice recall is attenuated when we account for marketcompetition between firms for these consumers, andthis result again is similar to what was obtained inthe case of the symmetric categorization.In addition, note that $S��n�/�$�� < 0 in both

the symmetric and asymmetric categorization cases.Thus, for any given degree of memory, the totalsurplus from all the limited-memory consumersincreases as the market becomes more competitive.Hence, when the market is more competitive, con-sumers have greater value in allocating more memoryresources for encoding and storing price information.Interestingly, this is also precisely the situation forwhich firms will be using more prices in the equi-librium support. This implies that in more compet-itive product markets, consumers would use moredegrees of memory, which should then result in moreobserved prices being used by firms.

4. Robustness of Findings4.1. Comparisons of the Categorization SchemesIn this section, we compare and contrast the resultsacross the symmetric and asymmetric categorizationprocesses to highlight the similarities and explainthe differences. Although the actual pricing strate-gies used by firms can obviously be driven by thespecific features of the categorization process or bythe nature of consumer heterogeneity, we uncover aremarkable degree of robustness in the general eco-nomic effects of categorization in a market settings.The first robust effect is about the manner in whichconsumers categorize price. Across the different cat-egorization processes, the different consumer hetero-geneity conditions, and irrespective of whether theequilibrium is one in pure or mixed strategies, we findthat the categorization is finer toward the lower end

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of the price range. Intuitively, this suggests the gen-eral point that consumers who are bounded in theirrecall have a strategic motivation to invest memoryresources in encoding lower prices to induce the firmsto charge favorable prices with higher probability.Next, we recover a convergence property of the

market across the different categorization processes.We find that the interaction of categorization and mar-ket competition is such that small initial improve-ments in recall move the market outcomes quicklytoward the perfect recall outcome. Thus the initialfew increases in categories lead to equilibrium pricingchoices and consumer surplus, which are close to thecase where the number of categories are infinite. Asexpected, we also find that consumer heterogeneityand the presence of uninformed or loyal consumersin the market increase the equilibrium profits of thefirms. Finally, we also find that the effect of increas-ing n is, in general, to weakly reduce the equilibriumprofits of the firms.There are also interesting differences between the

effects of the two types of categorization processes,which are intuitively appealing. The asymmetric cat-egorization process in which consumers compare acategory to actual prices can be seen as creating amore competitive and undifferentiated environmentfor firms than the symmetric categorization process.Consistent with this intuition, our analysis shows thatthe asymmetric categorization process can lead tomore competitive markets and lower equilibrium firmprofits. This is evident from the comparison of thebasic model of a market with only limited-memoryconsumers (or the model with both limited-memoryand uninformed or loyal consumers) across the twocases. For example, in the symmetric categorizationcase when all consumers compare labels of categories,as can be seen from Proposition 1, the firms make pos-itive equilibrium profits. However, in the analogouscase, where consumers compare a recalled category toactual prices, we get the Bertrand outcome, and firms’equilibrium profits are zero. As we mentioned earlier,the intuition behind this result is that consumers inthe asymmetric categorization case have better priceinformation than in the symmetric categorization casebecause the actual price from one firm is used in pur-chase decisions in the former case.We also investigated whether the findings of our

model with asymmetric categorization are robust todifferent representations of the category. We haveassumed that consumers represent the recalled cate-gory by the mean of the category and compare themean to the actual price at the current firm. However,we have also analyzed the model in which consumersrepresent the category by the median price in the cat-egory. We find that all the qualitative results of §3

are preserved even if consumers use the median. Sim-ilarly, the main results of §3 continue to hold evenif consumers use exogenous markers as representa-tion of the category such as the top, middle, or bot-tom of the category. Indeed, we find that the modelwhere consumers remember the top of the categoryin the asymmetric categorization model is mathe-matically equivalent to the symmetric categorizationmodel of §2. Finally, it is interesting to note that whenconsumers use the mean of the category, their equilib-rium surplus is actually higher than when consumersuse any exogenous rule to represent the category. Thisis because the mean of the category is endogenous tothe equilibrium actions of the firm. Thus, if consumersin a market were to learn through experience overtime to do the best for themselves and were motivatedto make the optimal long-term purchase decisions, itis likely that they would learn to recall the categorymean price rather than using an exogenous rule.

4.2. The Decision Whether to Compare PricesUnder Asymmetric Categorization

The asymmetric decision process of the limited-memory consumers, which we have considered in §3,implies as in Dow (1991) that the consumers neces-sarily have contact with both firms but that the priceinformation from one of the firms is imperfect. Here,we provide an extension that allows consumers todecide whether to compare prices at all after observ-ing the price at the first firm. This extension can beviewed as being consistent with the interpretation ofthe decision process as search with optimal stopping.After observing the price at the first firm, consumersdecide whether to search or to stop (with zero incre-mental search cost) and obtain the price at the secondfirm. We show that all the results of §3 are robust tothis extension.If the decision process involves the choice of

whether to go to the second firm, then the consumerupon observing the price at the first firm will haveto decide whether to stop and buy at that firm orto compare prices by obtaining the price at the sec-ond firm. The consumer might optimally decide notto compare prices if the benefit of obtaining the sec-ond price is sufficiently small. This can occur in thismodel if the limited-memory consumers encounter asufficiently low price at the first firm so that (evenwith zero search costs) they would be worse off goingto the second firm because they might not recall thelow price at the first firm precisely.In the limit market case of �→ 0, it follows imme-

diately that the consumer is never worse off by decid-ing to obtain the price from the second firm afterhaving observed a price at the first firm. Note fromProposition 4 that firms charge only the prices mi

with positive probability in each category. Thus, if the

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limited-memory consumer encounters a price in anycategory i > 1, she will be strictly better off going tothe second firm to obtain its price given zero incre-mental cost of search. In addition, for the lowermostcategory i = 1, the consumer will be no worse off.Therefore the consumer will always have the incen-tive to search and obtain the price at the second firmin the decision process after having observed the priceat the first firm, and consequently, all the results dis-cussed above for the limit market case are not affectedeven if the consumer explicitly chooses whether tocompare prices.Consider now the market with all three consumer

segments. For the case of no categories �n= 0�, whichis analyzed in the appendix, there is a thresholdprice u below which the limited-memory consumerswill not compare prices with the second firm afterobserving the price at the first firm. If they indeeddo decide to obtain the price at the second firm, theywill encode the first price as a higher price �m, whichis the mean price conditional on p > u If the con-sumer is at the second firm, then she will recall thefirst firm’s price as the conditional mean �m The equi-librium support will be �b�u�∪ �d� �m�∪ �v�1� (whereb < u < d < �m < v < 1). From the profit expressions atthe extreme points of the distribution and from thedefinition of the conditional mean, we can derive theequilibrium of this model. Finally, for the general caseof n categories and � > 0, in the lowermost categoryi = 1, the equilibrium price support is similar to thatdescribed above with a threshold price u above whichthe consumer will obtain the price at the second firm.Then, as in the case of n= 0, the consumer will com-pare the price at the second firm with �m the recalledprice at the first firm. Interestingly, for this generalcase, we can show that equilibrium consumers willuse the threshold price u for only the lowest categoryi = 1, and therefore the main results of §§3.1 and 3.2will hold. For example, as before the equilibrium prof-its with consumer heterogeneity in memory capac-ity are higher than when all the informed consumersare homogenous in their memory capacity (i.e., onlyperfect-memory or only limited-memory consumers).

4.3. External Validity and Implicationsof the Results

Our analysis suggests that firms would use a finitenumber of prices in markets where consumers havelimited memory and that consumers should devotegreater memory resources to encoding lower pricesresulting in finer categorization toward the bottomof the equilibrium price distribution. Accordingly, theinterval between two adjacent prices charged by afirm with positive probability in equilibrium shrinkstoward the lower end of the price distribution.There are empirical findings that are consistent with

the above implications of our model. For example,

Villas-Boas (1995) and Rao et al. (1995) found that thedistributions of prices are such that most of the prob-ability mass is concentrated around a small numberof price points, and a bimodal distribution is typical.Their findings are also consistent with the implicationof our results, which suggest that using a very smallnumber of categories (e.g., n= 1) would often be suf-ficient for consumers as this leads to small surplusloss in a competitive marketplace. As a reaction to asmall n� firms would also only adopt a small numberof price points.Furthermore, Krishna and Johar (1996) show

through a series of experiments that lower-priceddeals are more easily recalled by consumers thandeals involving higher prices and that firms haveincentive to offer deals that have smaller price dif-ferences. In their study of consumer perceptions ofpromotional activity, Krishna et al. (1991, p. 8) col-lected price information in a New York supermarketfor a period of 12 weeks on nine brand-size combina-tions with “considerable variance in terms of productclass purchase frequency, market share, and frequencyof promotion combinations.” Among the nine brand-size, seven used only two price points and the remain-ing two used only three price points. Interestingly,for both the two brand size combinations that usedthree price points, the gap between the highest priceand middle price was larger than the gap betweenthe middle price and the lowest price (Krishna et al.1991, Table 3). These findings offer further evidencethat supports the results of our model.The implications of our model are applicable across

different product classes. On the one hand, for fre-quently purchased products and products purchasedwith low involvement and time pressure, consumersare likely to rely on nonconscious, automatic process-ing in making purchase decisions and thus allocatelimited-memory resources in recalling prices (Monroeand Lee 1999). Not surprisingly, researchers havefound exact price recall to be low for small-ticketsupermarket items (Dickson and Sawyer 1990). Onthe other hand, in complex purchasing situations (e.g.,for automobiles or appliances) the full price facing theconsumer consists of not only the quoted posted pricebut also various types of discounts, trade-in payment,financing offers, warranty, delivery schedule, serviceand shipping fees, assembly charges, etc. When con-sumers compare across firms, it is the impression of thecomplex price comprised of several facets in additionto the posted product price that is relevant, makingprice comparisons imperfect. In these situations, thesimple strategy of noting down the prices on a pieceof paper rather than (imperfectly) recalling them will

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not resolve the consumers’ problem because what isrelevant is not only the numerical posted price butthe full price impression, which includes all the otherinformational details and specifications that are perti-nent for comparison across the firms.17

5. Summary and DiscussionIn this paper, we take an initial step toward under-standing the effects of limited memory and catego-rization by consumers on price competition betweenfirms. We focus on a specific aspect of memorylimitations—namely, the inability of consumers torecall exact price information. Limited-memory con-sumers can only recall the category to which a par-ticular price belongs to. This paper investigates pricecomparisons with limited memory in a competitivemarket and analyzes the interaction between con-sumer price categorization and the equilibrium pric-ing strategies used by firms.The analysis establishes several effects of limited

consumer recall that are remarkably consistent acrossthe different categorization processes and market con-ditions. When consumers compare either categorylabels (symmetric categorization) or a label to anobserved price (asymmetric categorization), we findthat the optimal strategy for the consumers calls forfiner categorization toward the bottom of the equi-librium price distribution. This implies that in equi-librium, consumers should devote greater memoryresources to encoding lower prices to induce firmsto put more emphasis and charge more favorableprices. We establish a robust convergence result thatemerges from the interaction between the catego-rization strategies of the consumers and the pricecompetition between the firms: i.e., for both catego-rization methods, small initial improvements in mem-ory capacity shift the equilibrium market outcomesquickly toward the perfect recall outcome. So evenwith a few memory categories, the expected price con-sumers pay and their surplus are quite close to caseof perfect recall. There is thus a suggestion in ourmodel that the existence of market competition mit-igates the negative consequences of imperfect recallfor consumers.There are several interesting questions that are

related to our investigation of limited memory. Theproblem of allocation of limited-memory resourcesto different tasks, such as recalling several product

17 Indeed, the possibility of consumers being able to costlessly notedown the price provides a partial justification for the considerationof the perfect-memory consumer segment. The perfect-memoryconsumers can be seen as those consumers who can costlessly andperfectly note down and codify all the relevant full price informa-tion that is necessary for across firm comparisons. On the otherhand, the limited-memory consumers find it costly (or are unable)to note down all the relevant information and therefore rely ontheir limited recall of the information.

attributes or the prices of different products that theconsumer buys, seems to be an interesting one to pur-sue. In this paper, we model memory limitations asthe inability of consumers to recall exact price infor-mation; instead, they only recall the category to whichthe price belongs. Alternatively, imperfect memoryrecall can be modeled as consumers recalling the pricewith an added random noise, consumers recallingonly the nearest round amount (as in Basu 2006), orconsumers recalling a price distribution instead of theexact realization. The last approach will be analogousto that used in the search and consideration set for-mation literature (Mehta et al. 2003). It might alsobe useful to explore other memory mechanisms andtheir effects on a firm’s decision making. Memory canalso be thought of as a device to carry informationover time. The information-theoretic characterizationof memory is especially relevant in markets for fre-quently purchased goods across different shoppingoccasions. It would also be interesting to consider theanalysis of our paper as it would unfold in a multi-period setup in which firms repeatedly set prices andconsumer categorization strategies evolve as a result.Finally, on the experimental side, it would be inter-esting to understand how the distributional charac-teristics of market variables such as price or productquality affect their encoding into the consumer mem-ory. Overall, the analysis of limited recall in marketsettings can be a fruitful area for future investigation.

AcknowledgmentsThe authors thank the editor, the area editor, two anony-mous reviewers, and Dmitri Kuksov, Glenn MacDonald,Jesse Shapiro, Jose Silva, Marko Tervio, and J. MiguelVillas-Boas for helpful comments. The authors also thankseminar participants at the Berkeley Micro Theory Semi-nar, Carnegie Mellon University, Harvard Business School,University of Michigan, University of Minnesota, NewYork University, University of Southern California, Whar-ton School, and Yale University.

AppendixProof of Proposition 1. Each firm can potentially use

exactly n+ 1 prices conditional on the choice of the parti-tions by the consumers. Firm 1 will have demand only ifits price is the same or lower than firm 2’s price. Clearly,both firms charging at the top of the lowest partition(p1 = p2 = k1) is an equilibrium. A firm that raises prices willlose all consumers, whereas lowering the price will onlybring lower revenues from half of the market. The payoffmatrix for firm 1 is given by (the payoff for firm 2 can bespecified in an analogous way)

"1�p1 = kr� p2 = kt�=

0 if r > t�

kr

2if r = t�

kr if r < t�

for r� t = 1� �n+ 1 (5)

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Consumers will optimally choose their cutoffs so that theunique equilibrium will be at lowest partition and that thelowest cutoff will be at the lowest possible value. We startby considering the highest possible prices. For p1 = kn+1 = 1,p2 = kn+1 = 1 not to be an equilibrium, the cutoffs mustsatisfy kn+1/2 < kn; similarly, for p1 = kn� p2 = kn not to bean equilibrium, we need kn/2 < kn−1 Similarly, we needkr+1/2 < kr for r = 1� �n Iterated substitution leads tothe following condition on kr (r = 1� �n):

kr >( 12

)n+1−r

Clearly, the best choices for consumers are k∗r = � 12 �

n+1−r +� for any infinitesimal � > 0. Both firms price at the top ofthe lowest partition and split the market generating profitsof �∗

j = 12 �12 �

n + �/2. Furthermore, given the choice of allconsumers, no single consumer can benefit from unilaterallychanging her cutoff points. Q.E.D.

Proof of Proposition 2. If both firms price at the lowestcutoff p1 = p2 = k1, each makes a profit of k1/2. For anypair of prices, the payoff matrix for firm 1 is given by (andanalogously for Firm 2)

"1�p1=kr� p2=kt�=

�kr if r >t�

kr

2if r= t�

�1−��kr if r <t�

for r�t=1� �n+1 (6)

The firm with the lower price gets all the consumersexcept those loyal to its rival. When both firms have equalprices, they split the market. For r = 1� �n, the purestrategies �p1 = kr� p2 = kr � constitute a strict equilibrium ifno firm wants to deviate. The most profitable deviation for,say, firm 1 is to charge the highest possible price of p1 = 1,making a profit of �1 = � (and the same holds for firm 2).Therefore, no firm will have the incentive to deviate andcharge the reservation price if

"j�p1 = kr� p2 = kr �=kr

2> � (7)

Moreover, no firm has an incentive to lower the price to thecategory immediately below if

"j�p1 = kr� p2 = kr �=kr

2> �1−��kr−1 (8)

For the topmost category, the equilibrium condition isonly

"j�p1 = kn+1� p2 = kn+1�= 12 > �1−��kn (9)

For the bottom category, it is

"j�p1 = k1� p2 = k1�=k12

> �

If k1/2 ≥ � and for every r = 2� �n conditions (8) arenot satisfied and condition (9) is not satisfied as well, we getthat the unique symmetric equilibrium in pure strategies isfor both firms to price at k1 If conditions (8) and (9) are notsatisfied, we have for r = 2� �n+ 1,

kr

2≤ �1−��kr−1

This condition implies 12 = kn+1/2 ≤ �1 − ��kn or kn ≥

1/�2�1 − ��� Applying the above condition iteratively, weget kr ≥ �1/�2�1− ����n+r−1, and finally k1 ≥ �1/�2�1− ����n Adding the condition k1/2≥ � guarantees that pricing at k1is an equilibrium for both firms. Thus, we get the condition

k1 ≥(

12�1−��

)n

≥ 2�

Because � < 12 , we have 2� ≥ �/�1− ��; so we are guaran-

teed that k1 > k0 = �/�1−�� We still need to check that no pure-strategy nonsymmet-

ric equilibrium exists. Consider the payoff matrix (6) if r >t + 1; then the payoff to firm 2 is �1− ��kt , but a deviationto pricing at kt+1 will yield a profit of �1− ��kt+1, which ishigher, so in any potential pure-strategy equilibrium, firmsmust price at most one category apart. Now, assume thatfirm 1 prices at kt and firm 2 prices at kt+1 then firm 2 makesa profit of �kt+1 If it lowers its price to kt , it would makekt/2 If kt/2 > �kt+1 then pricing at kt and kt+1 cannot bean equilibrium. However, we know even more; we knowkt/2> � (kt ≥ k1 > 2�), and thus we are done.Consumers will try to choose their strategies to induce as

low a k1 as possible, so we get k∗r = �1/�2�1−����n+r−1+� for

a very small � > 0. Firms price at p∗1 = p∗

2 = �1/�2�1−����n+�and make a profit of 12 �1/�2�1−����n + �/2 Q.E.D.

Proof of Proposition 3. Suppose that firms price onlyin the lowest partition; namely, in the interval �l1� k1� Theequilibrium condition requires firms to get the same profitfrom each potential price (note that the � consumers see nodifference between prices in the same partition):

l1�2�+�+��="�

k1��+��="

A firm that decides to price in a higher partition willsee its demand shrink to �; hence the best potential devi-ation is to price at kn+1 = 1 To guarantee no deviation,then we must have that " ≥ �� which is equivalent tok1�� + �� ≥ � or, as presented in the proposition, k1 ≥�/��+�� For completeness, we can solve for the entire equilib-

rium and get l1�2�+ �+ �� = k1��+ �� or l1 = k1���+ ��/�1−�− ���� and for every p ∈ �l1� k1�, the probability W�p�of each firm charging below p is given by

p�2�W�p�+�+��= k1��+���

W�p�= ��+��

2�

(k1p

− 1)

(10)

When consumers choose their partitions optimally, theyaim to minimize k1 while making sure that pricing in anyother partition is not an equilibrium. Assume that bothfirms pricing in the same partition, say, �li� ki�, constitutesan equilibrium. Then we must have i= 2� �n+ 1:

li�2�+�+��= ki��+��="i

Firms will not increase their price if " ≥ �, but thisis true since k1 ≥ �/�� + �� implies ki ≥ �/�� + �� or"i = ki��+��≥ �. Hence the only way to prevent both firms

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from pricing in the same partition is to ensure that firmswill have the incentive to deviate to a lower partition

"i < ki−1�1−���

ki��+�� < ki−1�1−��

Thus the conditions ki��� + ��/�1 − ��� < ki−1, i =2� �n+ 1, guarantee that both firms pricing in the samepartition (other than the lowest one) will not constitute anequilibrium. In other words, kn > �� + ��/�1 − ��� kn−1 >��� + ��/�1− ���2 and generally ki > ��� + ��/�1− ���n+1−i

guarantee that pricing only at the lowest partition constitutethe unique symmetric equilibrium. Because the consumershave the incentive to induce a k1 as low as possible, theywould choose k∗

i = ���+ ��/�1− ���n+1−i + �, and the firmswould price according to W�p�= ���+ ��/2���k∗

1/p− 1� forp ∈ ����+ ��/�1− �− ���k∗

1� k∗1� To complete the proof, we

need to show that the firms will not want to price in two dif-ferent partitions. Following the logic of the proof of Propo-sition 2, if firm 1 prices in the interval �lr � kr � while firm 2prices in �lt� kt� such that r > t+1. Then the payoff to firm 2is at most �1− ��kt , but a deviation to pricing at kt+1 willyield a profit of �1− ��kt+1, which is higher so firms mustprice at most one category apart. Now, assume that firm 1prices at �lt+1� kt+1� and firm 2 prices at �lt� kt�. Then firm 1makes a profit of at most �kt+1 if it lowers its price to kt ,it would make at least kt�� +�� Because we assumed k1 ≥�/��+��, we have kt > k1 ≥ �/��+�� > ��/��+���kt+1, andthus kt�� + �� > �kt+1 and the deviation by firm 2 is prof-itable. Hence, in equilibrium, both firms will charge pricesin the same category. Q.E.D.

Proof of Lemma 1. Let Wj�p� = Pr�pj ≥ p� be the prob-ability that firm j = 1�2 charges a price above p In asymmetric equilibrium, Wj�p� = W�p� As in Lemma 1,the equilibrium price support will have no mass points.The demand for a firm whose price approaches mi frombelow in a symmetric equilibrium will be � + �W�ki−1� +�2�+ ��W�mi� Next, for the price mi + /� (for / → 0), thefirm’s demand changes discontinuously to � + �W�ki� +�2� + ��W�mi� Therefore, any such price will be domi-nated by mi� implying that the equilibrium distribution willhave a hole from mi up to some vi > mi Define the mini-mum price charged in the category i to be bi Therefore, bythe definition of the mixed-strategy equilibrium, we shouldhave

��bi� = bi�� +�W�vi�+�W�bi�+ 2�W�bi��

= ��ki−1�= ki−1�� +�W�vi−i�+�W�bi�+ 2�W�bi��

Now, because W�vi−i� > W�vi�� it follows that bi > ki−1 Therefore, there is a hole in the distribution between bi

and ki−1 The remaining prices in �bi�mi� and �vi� ki� are partof the equilibrium price support because of standard argu-ments as in Varian (1980) and Narasimhan (1988). Q.E.D.

Proof of Proposition 4. Note that for every positive �,the support of the equilibrium price distribution is com-prised of two intervals in each category and each price inthe support leads to the same profit. Thus, at the limit as� → 0, the profit from charging mi must equal the profitfrom charging ki ∀ i = 1� �n + 1 Charging a price of

p = kn+1 = 1 gives a profit of �, which is the equilibriumprofit. When charging p= ki, the expected profit for a firm is

�i = �� + 2si+1��ki = � i= 1�2� �n (11)

From this, we can show that

si =��1− ki−1�2�ki−1

i= 2�3� �n+ 1� (12)

and s1 = ��1− k0�/�2�k0� = 1 by definition. By noting thatPr�mi�= si − si+1, we get

Pr�mi�=��1− ki−1�2�ki−1

− ��1− ki�

2�ki

= �

2�

(1

ki−1− 1

ki

) (13)

To calculate the values of the prices charged in each cat-egory, we use (2):

mi =�

�+si�+si+1�= �

�+��1−ki−1�/�2ki−1�+��1−ki�/2ki

= 21/ki−1+1/ki

(14)

Q.E.D.Proof of Proposition 5. We select the optimal cutoffs

k∗i that satisfy the following conditions: (i) the equilibriumconsumers’ surplus are at the maximum, and (ii) no indi-vidual consumer will have the incentive to deviate from k∗

i

given that the other consumers are also using these cut-offs. The first condition implies identifying the cutoffs thatmaximize the surplus of the limited-memory consumersS�

18 Clearly, the total surplus is S = 1 (a unit mass of con-sumers with a common reservation price of 1) and the totalproducer surplus of the two firms is 2� = 2� The con-sumer surplus of the uninformed consumers is given byS� = 2��1−∑n+1

i=1 Pr�mi�mi�. Thus the limited-memory con-sumers’ surplus will be S� = S − S� − 2� = 1 − 4� +2�

∑n+1i=1 Pr�mi�mi. From (13) and (14), we can see that the

expected price paid by the uninformed consumers is justEn�p� =

∑n+1i=1 Pr�mi�mi = ��/��

∑n+1i=1 ��ki − ki−1�/�ki + ki−1��

Hence the necessary condition for maximizing S� is that fori= 1� �n+ 1, we have

$S�

$ki

= 2��

�ki−1− ki+1��−ki−1ki+1+ k2i �

�ki + ki−1�2�ki+1+ ki�2

= 0 19

This implies that the condition for a maximum is −k∗i−1k

∗i+1+

k∗2i = 0 Thus the cutoff prices �k∗

i �ni=1 form a geometric

sequence with the boundary conditions k∗n+1 = 1, k∗

0 = �/�1−�� Hence, for i= 1� �n, we have the optimal cutoffsto be

k∗i =

(�

1−�

)�n+1−i�/�n+1� (15)

Substituting Equation (15) into (14) and (13) yields thedesired expression in the proposition for m∗

i and Pr�m∗i �

Next, we have to show the second condition that no con-sumer has the incentive to deviate from the optimal �k∗

i �ni=1

18 Note that this is equivalent to the maximization of each limited-memory consumer’s surplus.19 It is tedious but straightforward to calculate the Hessian,$S2�/�$kj$ki�� and show that the necessary conditions are indeedsufficient.

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Chen, Iyer, and Pazgal: Limited Memory, Categorization, and CompetitionMarketing Science 29(4), pp. 650–670, © 2010 INFORMS 667

given that other consumers choosing �k∗i �

ni=1 as given above.

The reasons are as follows: if a consumer deviates from the�k∗

i �ni=1 given above, firms’ pricing strategy will not change

because of the assumption that there is a large number ofconsumers in the market. Then, if such a deviation resultsin Ci containing either one or two prices from �m∗

i �n+1i=1 being

charged at equilibrium, the consumer will have the samesurplus. However, if such deviation leads Ci to contain threeprices from �m∗

i �n+1i=1 (denoting them as mh > mm > ml), the

consumer can never be better off. Denote the new meanprice of Ci to be �m. If �m=mm, the consumer is indifferent.If �m > mm, the consumer who observes ml at the first firmwill recall it as �m and will buy from the second firm if theprice there is mm. If �m < mm, the consumer who observesmh at the first firm will recall it as �m and will not buy fromthe second firm if the price there is mm. In both cases, theconsumer is worse off. Similarly, the consumer can neverbe better off if a deviation leads to Ci to contain more thanthree prices from �m∗

i �n+1i=1 . Hence, an individual consumer

has no incentive to deviate given other consumers’ strategy.Therefore the �k∗

i �ni=1 given above are indeed the optimal

cutoff prices in the symmetric equilibrium. Q.E.D.Proof of Result. We can explicitly calculate the

expected equilibrium price of each firm, which is

En�p�=n+1∑i=1Pr�m∗

i �m∗i =

2�n+1���1−2��

���1−��/��1/�n+1�−1����1−��/��1/�n+1�+1� (16)

Direct calculation shows that $En�p�/�$n� > 0 A similar cal-culation shows that $12

n�p�/�$n� > 0, where 12n�p� is the vari-

ance of the equilibrium price distribution. Q.E.D.Proof of Proposition 6. When consumers’ degree

of memory is n� the maximum size of a category is�1− kn�n�� = 1− ��/�1− ���1/�n+1� (recall that the categoriesare finer toward the lower end of the price range). Thus,for n > N1 = ln��/�1− ���/ ln�1− %� − 1, we have that thesize of the largest category is smaller than % Let p ∈ Cj�n�be the mean of the prices in this partition; mj�n� ∈ Cj�n� ischarged with positive probability and is % close to p Recallfrom Equation (12) that the probability of a firm pricingin category Cj�n� or higher is just sj �n� = ��/�1 − 2��� ×�1/�kj−1�n��− 1�, where kj−1�n� is the lower cutoff bound ofCj�n�. Hence, if p < mj�n�, we have Wn�p�= sj �n�; otherwise,Wn�p�= sj+1�n� Recall that for the standard model of Varian(1980) or Narasimhan (1988), which is equivalent to the caseof perfect recall, the probability of charging a price above pis W�p�= ��/�1−2����1/p−1�; thus we have (for p < mj�n�)

�Wn�p�−W�p�� = �sj �n�−W�p�� = �

1− 2�∣∣∣∣ 1kj−1�n�

− 1p

∣∣∣∣= �

1− 2��p− kj−1�n��

kj−1�n�p<

�

1− 2�%

p�p− %�

<�

1− 2�%

��/�1−�����/�1−��− %�

If p > mj�n�, then

�Wn�p�−W�p�� = �sj+1�n�−W�p�� = �

1− 2�∣∣∣∣ 1�kj �n�

− 1p

∣∣∣∣<

�

1− 2�%

p× p<

�

1− 2�%

p�p− %�

Define 2 such that�

1− 2�2

�1−�

� �1−�

−2�< %�

so for any n > N2 = ln��/�1−���/�ln�1−2��−1, we get that�Wn�p�−W�p��< % Finally, N =max�N1�N2� Q.E.D.

Mixed-Strategy Equilibrium forSymmetric CategorizationConsider the case of a market with limited-memory con-sumers and uninformed consumers and investigate the casewhere �1/�2�1− ����n < 2� For this case, we characterize amixed-strategy equilibrium, which is as follows.Given �/�1− �� ≤ k1 ≤ · · · ≤ kn ≤ kn+1 = 1� we can char-

acterize the completely mixed symmetric equilibrium to begiven by the unique solution to the set of equations thatwill be derived below. Assume that each firm prices at kj

with probability qj j = 1� �n+ 1 Then, the profit for firmj when pricing at kj is given by j = 1� �n:

"i�kj �= kj

(� +�qj + 2�

n+1∑m=j+1

qm

)

Pricing kn+1 = 1 yields"i�kn+1�= �� +�qn+1�

Pricing at the k1 yields

"i�k1�= k1�� +��2− q1��

For a totally mixed-strategy equilibrium, we need "i�kj � =" = constant for j = 1� �n + 1 as well as ∑n+1

m=1 qm = 1 Thus we have n + 2 equations with n + 2 unknowns thatpossess a unique solution. Subtracting the equation for kj+1from the one for kj yields the following set of equations forj = 1� �n:

qj + qj+1 ="

�

(1kj

− 1kj+1

)> 0

The specific solution to this set of equations depends onthe parity of n Thus suppose, for instance, that n is an oddnumber, then a possible solution can consist of consumerschoosing qj+1 = 0, j = n�n − 2�n − 4� The proof is asfollows: if n is odd and qj+1 = 0, j = n�n−2�n−4� � thenwe have "= � +�qn+1 = �, and

qn+1 = 0�

qn ="

�

(1kn

− 1kn+1

)= �

�

(1kn

− 1)�

qn−1 = 0�

qn−2 =�

�

(1

kn−2− 1

kn−1

)�

Then

qn−1+ qn =�

�

(1

kn−1− 1

kn

)= �

�

(1kn

− 1)

→ 12kn

=(1+ 1

kn−1

)

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Chen, Iyer, and Pazgal: Limited Memory, Categorization, and Competition668 Marketing Science 29(4), pp. 650–670, © 2010 INFORMS

Similarly, we have

qn−3+ qn−2 =�

�

(1

kn−3− 1

kn−2

)= �

�

(1

kn−2− 1

kn−1

)

→ 12kn−2

=(1

kn−1+ 1

kn−3

)�

�

where we can define k0 = �/�1−�� (the lower bound).If we solve the whole system of equations with con-

sumers’ surplus maximization, we will get

k∗i =

(�

1−�

)�n′+1−i′�/�n′+1�for i= n+ 1�n− 1�n− 3� �

12k∗

i−2=(1

k∗i−1

+ 1k∗

i−3

)for i= n+ 2�n�n− 2� �

where n′ = �n+ 1�/2 and i′ = �i + 1�/2. We can then noticethat k∗

i �i = n + 1� n − 1� n − 3� � are like the k∗i in the

asymmetric categorization model with limited memory andinformed consumers and k∗

i �i = n� n − 2� n − 4� � arelike the m∗

i (means) in the asymmetric categorization model.It can also be easily noted that this solution gives con-sumers greater surplus than the pure-strategy equilibrium.As we have shown in the asymmetric case, the above equi-librium converges to the perfect recall solution as in Varian(1980) and Narasimhan (1988) as the number of categoriesincreases beyond bound.

The Decision to Compare Prices UnderAsymmetric CategorizationThe asymmetric categorization model in this paper assumesthat the limited-memory consumers have contact with boththe firms by assumption. We now present the analysisof the case in which the decision process of the limited-memory consumers allows them to decide whether to com-pare prices at all after the price at the first firm is observed.Given the price that a consumer encounters at the first firm,the consumer has to decide whether to continue and obtainthe price from the second firm. If the consumer encountersa low enough price at the first firm, then she might decidenot to compare prices at the second firm.

The n= 0 Case. We start with the case of n= 0 Denoteby u the threshold price below in which the limited-memoryconsumers will not obtain the price at the other firm. Define�m to be the mean price conditional on p > u Then the equi-librium support will be �b�u� ∪ �d� �m� ∪ �v�1�, where b <u < d < �m < v < 1. Define W�p� as Pr�p ≥ p� and f �p� as theprobability density function of price. We have that

� = � +�W� �m� �p= 1��� = �� +�W�v�+�W� �m��v �p= v��

� = �� + ��+��W� �m�+�W�u�� �m �p= �m��

� = �� +�W�d�+�W� �m�+�W�u��w �p= d��

� = �� +�W�u�+�+�W�u��u �p= u��

� = �� +�+�+�W�u��b �p= b��

� = �� +�W�p�+�W� �m��p �v < p < 1��

� = �� +�W�p�+�W� �m�+�W�u��p �d < p < �m��

� = �� +�W�p�+�+�W�u��p �b < p < u�

Using the above expressions and the fact that W� �m� =W�v�, W�d� = W�u�, W�1� = 0, and W�b� = 1, and definingW� �m�= h and W�u�= g, we can compute that in the mixed-strategy equilibrium the following hold:

W�p�= � +�h

ap− � +�h

a�v≤ p≤ 1��

W�p�= � +�h

ap− � +�h+�g

a�d ≤ p≤ �m��

W�p�= � +�h

ap− � +�+�g

a�b ≤ p≤ u�

Therefore, from the profit expressions at the extremepoints in the distribution, we have that v = �� + �h�/�� + �h + �h�� �m = �� + �h�/�� + �h + �h + �g��d = ��+�h�/��+�g+�h+�g�� u= ��+�h�/��+�g+�+�g�, and b = �� +�h�/�� +�+�+�g� From the definitionof �m and the expressions for W�·� derived above, we have

�m = � +�h

a

[∫ 1

v

1p

dp+∫ �m

d

1p

dp

]

= � +�h

aln(1v

�md

)

Now, from the definition of u and the expressions forW�·� derived, we can derive the optimal stopping rule thatdetermines the threshold price below which the consumerwill not compare prices:

0 =∫ �m

d�u− p�f �p�dp+

∫ u

b�u− p�f �p�dp

⇒ u�1−h�= � +�h

aln

�md

u

b

Then, using the expressions of v, �m, d, u, and b derivedearlier, we have

� +�h

� +�h+�h+�g

= � +�h

aln(

� +�h+�h

� +�h

� +�g+�h+�g

� +�h+�h+�g

)�

� +�h

� +�g+�+�g�1−h�

= � +�h

aln(

� +�g+�h+�g

� +�h+�h+�g

� +�+�+�g

� +�g+�+�g

)

The equilibrium �h�g� can be solved from the aboveequations given that by definition g ≥ h As �→ 0, we have

� = � +�h� � = �� +�h�v�

� = �� +�h+�g�m� � = �� +�h+�g�d�

� = �� +�+�g�b� � = �� +�+�g�u

Hence b = u� �m = d� v = 1 Also, as � → 0, W�p� =�� + �h�/ap − �� + �h�/a �v ≤ p ≤ 1� leads to h = W�v� =�� +�h�/ap− �� +�h�/a= 0. Therefore, based on

� +�h

� +�g+�+�g�1−h�

= � +�h

aln(

� +�g+�h+�g

� +�h+�h+�g

� +�+�+�g

� +�g+�+�g

)�

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Chen, Iyer, and Pazgal: Limited Memory, Categorization, and CompetitionMarketing Science 29(4), pp. 650–670, © 2010 INFORMS 669

we can show that g = 0 and that this would result in thesame solution for the case of the n = 0 case in the modelof §3. Thus, �m = d = v = 1 and all the probability mass ison u Hence, when � → 0� the profit equation at u is thesame as the profit function at m in this paper. Consequently,all the results of this paper at �→ 0 will be preserved.

The General Case of n Categories. In the generaln-category case, for any � > 0, we can prove that the stop-ping rule only applies to the lowest category in the equi-librium; i.e., i = 1. The proof is as follows. Suppose thereexists some category i > 1 in which there is a thresholdprice ui below which the limited-memory consumers do notcompare prices with the other firm. Because i > 1 if thelimited-memory consumers do not search upon encounter-ing a price below ui� then the firms have the incentive toundercut each other for only the perfect-memory consumersin that category. Therefore, �vi−1�ui� will be on the equilib-rium support. This implies that ki−1 ∈ �vi−1�ui� should be onthe equilibrium support. However, consumers have incen-tive to search and compare prices at ki−1 but not at ki−1+� Thus, there must be a mass point at ki−1 to make the profitat ki−1 equal to the profit at ki−1 + � as � → 0 However, amass point at ki−1 cannot be part of an equilibrium becausethe other firm will have incentive to undercut it with a masspoint at ki−1−�. Hence there is a contradiction, and so thereis not an equilibrium in which the limited-memory con-sumers do not compare prices for i > 1 Therefore, in equi-librium, the threshold only applies to category 1 regardlessof the size of � For category i = 1, the equilibrium price support is

�b1�u�∪ �d� �m�∪ �v1� k1�, where b1 < u < d < �m < v1 < k1. �m isthe mean price and u is the threshold price below whichthe limited-memory consumers will not obtain a price at thesecond firm. Define g = W�d� = W�u�; the profit equationsfor category 1 corresponding to (1) in this paper become

pj = k1! � = �� +w1�+ s2�+ 2s2��k1�

pj = v1! � = �� +w1�+ s2�+ 2w1��v1�

pj = �m! � = �� +w1�+ g�+ 2w1�� �m�

pj = d! � = �� +w1�+ g�+ 2g��d�

pj = u! � = �� + g�+�+ 2g��u�

pj = b1! � = �� + g�+�+ 2��b1

(17)

As in the n = 0 case, when � → 0, we have v1 → k1,�m → v1� d → �m, b1 → u� and only �b1�u� is charged withpositive probability. Therefore we can see that u is also theunconditional mean of category 1; i.e., u=m1. Thus, when�→ 0, Equations (17) here become

pj = k1! � = �� + 2s2��k1�

pj =m1! � = �� + s2�+��m1

The above equations are the same as those for pj = k1 andpj = m1 at � → 0 given in this paper (without the optimalstopping rule). Hence, all results in this paper are preserved.

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