probabilistic goods: a creative way of selling products and services

Vol. 27, No. 4, July–August 2008, pp. 674–690issn 0732-2399 �eissn 1526-548X �08 �2704 �0674

informs ®

doi 10.1287/mksc.1070.0318©2008 INFORMS

Probabilistic Goods: A Creative Way ofSelling Products and Services

Scott Fay, Jinhong XieDepartment of Marketing, University of Florida, Gainesville, Florida 32611

{[email protected], [email protected]}

This paper defines a unique type of product or service offering, termed probabilistic goods, and analyzes a novelselling strategy, termed probabilistic selling (PS). A probabilistic good is not a concrete product or service but

an offer involving a probability of getting any one of a set of multiple distinct items. Under the probabilisticselling strategy, a multi-item seller creates probabilistic goods using the existing distinct products or services andoffers such probabilistic goods as additional purchase choices. The probabilistic selling strategy allows sellersto benefit from introducing a new type of buyer uncertainty, i.e., uncertainty in product assignments. First,introducing such uncertainty enables sellers to create a “virtual” product or service (i.e., probabilistic good),which opens up a creative way to segment a market. We find that the probabilistic selling strategy is a generalmarketing tool that has the potential to benefit sellers in many different industries. Second, this paper showsthat creating buyer uncertainty in product assignments is a new way for sellers to deal with their own marketuncertainty. We illustrate two such benefits: (a) offering probabilistic goods can reduce the seller’s informationdisadvantage and lessen the negative effect of demand uncertainty on profit, and (b) offering probabilistic goodscan solve the mismatch between capacity and demand and enhance efficiency. Emerging technology is creatingexciting (previously unfeasible) opportunities to implement PS and to obtain these many advantages.

Key words : probabilistic selling; probabilistic goods; opaque goods; pricing; product differentiation;e-commerce; yield management; inventory management; product line; price discrimination

History : This paper was received August 31, 2006, and was with the authors 3 months for 2 revisions;processed by Greg Shaffer. Published online in Articles in Advance March 31, 2008.

1. IntroductionIn this paper, we define a unique type of prod-uct or service offering, termed probabilistic goods, andanalyze a novel selling strategy, termed probabilis-tic selling. To understand these concepts, considerthe recently observed “opaque sales” offered by twoonline travel intermediaries, Priceline and Hotwire.For example, Priceline offers “opaque” hotel roomsin which a buyer specifies dates, city, and approx-imate quality (e.g., star rating), but the particularhotel property is not revealed until after payment hasbeen made. Priceline requires buyers to bid for price(known as “name your own price” or NYOP). If thebid is accepted, the buyer’s credit card is charged.A no-refund policy is strictly enforced.Priceline’s business model has attracted increasing

attention from consumers, the media, and academia.Consumers share their experiences in specializedweb discussion forums (e.g., BetterBidding.com).Many articles and books offer tips to help travelerstake advantage of Priceline’s pricing strategy (e.g.,“Priceline.com for Dummies” (Segan 2005)). Profes-sional reviewers gather and disperse informationabout such “blind” sites. Marketing scholars havealso shown interest in such new business models,

and several recent studies have examined the NYOPmechanism from the perspectives of service providers,intermediaries, and consumers (Chernev 2003; Hannand Terwiesch 2003; Fay 2004, 2008; Spann et al.2004; Ding et al. 2005; Spann and Tellis 2006; Wanget al. 2006). This research stream has provided insightson various important theoretical and empirical issuessuch as online bidding, channel coordination, andthe impact of an opaque intermediary on traditionalchannels.While the current academic attention has mainly

focused on the various aspects of the business modelof Priceline, our interest is to explore the fundamen-tal product/market conditions required for the bene-fit of introducing uncertainty in product assignmentsby offering “probabilistic goods,” which we define asa gamble involving a probability of getting any oneof a set of multiple distinct items. We use probabilis-tic selling to denote the selling strategy under whichthe seller creates probabilistic goods using the seller’sexisting distinct products or services (referred to here-after as component goods) and offers such probabilis-tic goods to potential buyers as additional purchasechoices. For example, a retailer selling two differ-ent colors of sweaters, red and green, may offer

674

Fay and Xie: Probabilistic Goods: A Creative Way of Selling Products and ServicesMarketing Science 27(4), pp. 674–690, © 2008 INFORMS 675

an additional “probabilistic sweater,” which can beeither the red or green sweater. A theatre that offerstwo different shows on a given weekend can sell anadditional probabilistic ticket, “Saturday or Sundayperformance.” We use the term traditional selling todenote the conventional selling strategy under whichthe seller only offers the component goods for sale.It is important to note that Priceline operates in a

complicated market environment with many uniqueindustry characteristics: It is an online intermediarythat depends on multiple suppliers (e.g., different air-lines, hotel chains, and car rental firms) and alsocompetes with these suppliers’ direct channels; it issubject to special characteristics of the travel indus-try (e.g., a nonstorable, perishable good, capacity con-straints, and demand uncertainty); and it sets pricesvia a complicated buyer bidding system (i.e., NYOP).It is unclear what are the key factors that motivatethe seller to introduce uncertainty in product assign-ments and if such uncertainty can benefit firms thatare not subject to these specific industry characteris-tics. Thus, our primary objective is to uncover the fun-damental factors required for probabilistic selling tobe advantageous to a seller. Specifically, we examinewhy, when, and how a seller can benefit from intro-ducing a probabilistic good. We provide answers toseveral important questions: What are the conditionsunder which offering a probabilistic good improvesprofit? How does the degree of horizontal productdifferentiation between the component goods impactthe profit advantage of probabilistic selling? Is prob-abilistic selling more or less profitable for sellers fac-ing demand uncertainty? How do capacity constraintsaffect the profit advantage of probabilistic selling? Wedevelop a formal model to address these issues.Our results reveal that probabilistic selling can

improve profit without requiring specific industrycharacteristics such as multiple suppliers, an interme-diary market structure, selling perishable goods, anda buyer bidding system. First, we find that proba-bilistic goods have a fundamental advantage of allow-ing the seller to benefit from a special type of buyerheterogeneity—differentiation in the strength of buy-ers’ preferences (e.g., some vacationers may have astrong preference but others may have a weak pref-erence about the two different bus tours offered ata national park). A discounted probabilistic good,because it attracts consumers with weaker preferenceswhile allowing the seller to obtain high margins onsales to those consumers with strong preferences, canenhance price discrimination and expand the totalmarket. We illustrate that offering a probabilistic goodcan improve profit even if the seller achieves full mar-ket coverage under the traditional selling strategy andeven if all consumers prefer the same product. Given

that in almost all markets with multiple product offer-ings, consumers differ in the strength of these prefer-ences, probabilistic selling can be a general marketingtool that potentially benefits many sellers.Second, we find that when there is an advantage

from introducing a probabilistic good, it is gener-ally optimal to assign an equal probability to eachcomponent product as the probabilistic good. Sucha symmetric assignment maximizes the profit underprobabilistic selling even if the seller faces asymmet-ric demand (i.e., consumers on average prefer oneproduct over the other). While this result may notbe intuitive, as we will explain later, deviating fromsuch an equal probability will diminish the two pos-itive effects of probabilistic selling: price discrimina-tion and market expansion.Third, our analyses reveal an intriguing benefit

of offering a probabilistic good: It can provide abuffer against a seller’s own demand uncertainty.A seller offering multiple products often has uncer-tainty about which product will turn out to be thehigh-demand (or low-demand) product. For example,a toy manufacturer may not be sure which of the twonew toys introduced for the holiday season will be“hot” at the time of product launching. Under tradi-tional selling, such uncertainty reduces profit becausethe firm is unable to tailor prices to demand con-ditions (i.e., charging a higher price for the “hot”toy). However, we find that introducing a probabilis-tic good can reduce, and sometimes even eliminate,the need to make prices depend on the demand foreach individual product, which reduces the negativeeffect of demand uncertainty on profit. As a result,demand uncertainty increases the profit advantage ofprobabilistic selling.Fourth, our model reveals that probabilistic selling

may be particularly beneficial in industries or mar-kets subject to both demand uncertainty and capacityconstraints, because probabilistic selling can increasecapacity utilization and enhance efficiency via reduc-ing the mismatch between capacity and demand. Formany industries, mismatch between demand andcapacity can occur because one product may turnout to be more popular than the other products (e.g.,one of the two shows offered in the same week mayhave an ex post higher demand than the other one).Under the traditional selling strategy, the seller fac-ing both demand uncertainty and capacity constraintscan not use price to shift demand to match capacitybecause the seller is not certain which product willhave binding capacity and which will have excesscapacity.1 However, we show that under probabilistic

1 Using price to shift demand may also not be practical for otherreasons, e.g., menu costs.

Fay and Xie: Probabilistic Goods: A Creative Way of Selling Products and Services676 Marketing Science 27(4), pp. 674–690, © 2008 INFORMS

selling, the seller can smooth demand across the prod-ucts without knowing the direction of the mismatchbetween demand and capacity for each product. As aresult, introducing a probabilistic good helps improvecapacity usage rates and also ensures that capacitywill be available to serve those consumers with strongpreferences for the high demand product.This paper contributes to marketing theory by

proposing and illustrating the benefit of introducinga new type of buyer uncertainty, i.e., uncertainty inproduct assignments. We show that introducing suchuncertainty enables sellers to create a “virtual” prod-uct or service (i.e., probabilistic good), which opensup a new dimension to segment a market. Sellers cur-rently carrying any number of component products orservices may benefit from adding probabilistic goodsto their product lines. Such additions would be par-ticularly valuable when adding additional new con-crete products or services is too costly or logisticallyimpossible. For example, a theatre already open everynight of the week may not be able to introduce anadditional performance, but it could offer a numberof different probabilistic tickets using its current per-formances as the component goods (e.g., “Tuesday orWednesday” and “Wednesday or Thursday”).These contributions complement extant research on

buyer uncertainty. Recent papers (Shugan and Xie2000, Xie and Shugan 2001) suggest that firms canbenefit from creating buyer uncertainty by selling inadvance, i.e., completing transactions with consumersbefore they learn their valuations. Our paper sug-gests that firms can profit from creating a differenttype of buyer uncertainty, i.e., offering a probabilis-tic good that can turn out to be any one out of aset of multiple items. The two strategies not only dif-fer in the nature of buyer uncertainty created (i.e.,uncertainty about one’s own consumption state ver-sus uncertainty about the specific product one willreceive) but also differ in the fundamental sources oftheir profit advantages. In contrast to advance sell-ing where the seller benefits by reducing consumerheterogeneity, i.e., by selling to consumers at a timebefore idiosyncratic differences are known to con-sumers, probabilistic selling, as shown in our analysis,instead capitalizes on idiosyncratic differences by sell-ing the probabilistic product to consumers with weakpreferences and selling the specified products to con-sumers with strong preferences.2

2 Several other papers also provide important insights on the linksbetween buyer uncertainty and marketing strategies. For example,Venkatesh and Mahajan (1993) show that bundling services mayallow a seller to profit from buyer uncertainty about the avail-ability of their future spare time. Xie and Gerstner (2007) illus-trate that when buyers are uncertain if they will find alternativeoffers, the sellers can benefit by offering partial refunds for service

This work also augments the recent literature onstrategies that can mitigate the negative effect of selleruncertainty on profit in industries with capacity con-straints (e.g., travel related industries).3 For example,Biyalogorsky and Gerstner (2004) propose that sellersfacing demand uncertainty and capacity constraintscan benefit from a contingent pricing strategy, i.e.,canceling the sales to low-paying advance buyers ifhigh valuation customers show up later. Biyalogorskyet al. (2005) illustrate that providers with multiclassservices can increase capacity utilization by introduc-ing “upgradeable tickets,” which upgrade the ticketholders to a higher class service at the time of ser-vice delivery only if the reserved higher class capacityremain unsold. Recent research on revenue manage-ment (Gallego and Phillips 2004) suggests that sell-ers can overcome the difficulty of mismatch betweendemand and capacity by selling a flexible product, i.e.,assigning one out of a set of alternative products tothe prepaid buyer only after the seller has learnedwhich product has excess capacity. Each of theseinnovative strategies addresses potential mismatchesbetween capacity and demand by reserving sellerflexibility. Our paper offers a different approach. Inour model, the allocated product is confirmed imme-diately after the completion of each transaction, i.e.,before the seller has acquired any additional infor-mation about demand. We show how, under certainconditions, probabilistic selling enables the seller tospread demand more evenly across the products andthus improves capacity utilization without delayingthe confirmation of product assignments.This research is particularly vital and urgent to

practitioners because advances in new technology aremaking implementation of a probabilistic selling strat-egy much more efficient and practical. In the past,displaying and selling probabilistic goods could bevery costly and inefficient. For example, a consumermay balk at having to carry two blouses up to thecash register in order for the random draw, say, bya coin flip to take place. However, the Internet iscreating a more efficient shopping environment forselling probabilistic goods. In an online setting, theseller could easily create a split-screen, illustratingthe two possible component goods from which theprobabilistic good is drawn, using existing productdescriptors. The random draw could be implemented

cancellations. Guo (2006) shows that buyer uncertainty about theirproduct valuation motivates them to reserve “consumption flexi-bility” by purchasing multiple items, which can create a “flexibilitytrap” and reduce profit.3 There is an extensive literature that considers some general strate-gies that can reduce a seller’s uncertainty about demand uncer-tainty without capacity constraints, such as acquiring the ability totarget individual customers (Chen et al. 2001) and sharing informa-tion with a retailer (Kulp et al. 2004).


via integrated software and data communication net-works. New retailing technologies, such as “AnyplaceKiosk” (recently developed by IBM) and radio fre-quency identification (RFID) technology, are also eas-ing the implementation of probabilistic selling even inan offline setting. As the costs of these technologiesfall and consumers become more familiar with andaccepting of them, the probabilistic selling strategywill become increasingly feasible for a wider array ofindustries.The remainder of this paper is organized as follows:

In the next section, we present our basic model andoffer several generalizations of this basic model. Sec-tion 3 focuses on demand uncertainty and capacityconstraints. Finally, §4 discusses managerial implica-tions and future research. Sketches of proofs of allpropositions are in the appendix. The full proofs arecontained in the Technical Appendix online at http://mktsci.journal.informs.org.

2. The ModelIn this section, we start with a standard Hotellingmodel (referred to hereafter as the “basic model”) toexamine conditions required for the profit advantageof probabilistic selling. Our objective is to first use asimple model to illustrate the basic economic forcebehind the probabilistic selling strategy and offer keyinsights without introducing unnecessary mathemati-cal complexity. We then examine the robustness of ourfindings and offer additional insights by relaxing sev-eral of the assumptions of the basic model and adoptsome alternative demand structures (see §2.3).

2.1. AssumptionsSeller behavior. Consider a seller offering two com-

ponent products, j = 1�2, which have symmetric pro-duction costs: c1 = c2 = c. To ensure the seller can havea positive demand at a price above marginal cost, weassume 0≤ c < 1. In the basic model, we assume thatthe seller is aware of the demand and able to sat-isfy all demand (if it so desires). We extend the anal-ysis in §3 by considering the case where the sellerfaces capacity constraints and demand uncertainty.The seller considers two strategies: traditional selling(TS) and probabilistic selling (PS). Under the traditionalselling strategy, the seller sells each component prod-uct j at a price, PTSj . Under the probabilistic sellingstrategy, the seller sells each component product j ata price, P PSj , and also a probabilistic good, which hasa probability to be either component product, at aprice of P PSo . Let � be the probability that product 1 isallocated as the probabilistic product, which is deter-mined by the seller and is naturally restricted to theinterval [0�1].Buyer behavior. Let vji be the value of product j to

consumer i. The basic model assumes that valuations

for the two component products follow a Hotellingmodel in which the value for one’s ideal product isnormalized to one, the fit-cost-loss coefficient equals twhere 0 < t ≤ 1, and the consumer’s location on theHotelling line is xi. The valuations are given in Equa-tion (1):

v1i = 1− txi

v2i = 1− t�1− xi where xi ∼U�0�1�� (1)

Each consumer buys at most one unit of one good,i.e., there is no value from consuming a second prod-uct. Each consumer chooses the product offering thatmaximizes her expected surplus. For example, whenthe seller adopts the probabilistic selling strategy, con-sumer i has four choices: (a) buy product 1, (b) buyproduct 2, (c) buy the probabilistic good, and (d) buynothing. She chooses the option that leads to the high-est expected surplus.The consumer’s surplus from buying the proba-

bilistic good is the difference between her expectedvaluation and the price of the probabilistic good. Itis important to notice that a consumer’s valuation forthe probabilistic good depends on her valuations forproducts 1 and 2 and her expectation about the prob-ability that the probabilistic good will turn out to beproduct 1. We assume that consumers are rational andforward-looking, i.e., their expectations are confirmedin equilibrium. Thus, a consumer’s valuation for theprobabilistic good is �v1i + �1−� v2i.

2.2. Optimal StrategyLet DTS

j �PTS1 �PTS2 represent the demand for the com-ponent product j under the traditional selling strat-egy, and let DPS

j �P PS1 �P PS2 �P PSo �� and DPSo �P PS1 �P PS2 �

P PSo �� represent the demand for the componentproducts and the probabilistic good, respectively,under the probabilistic selling strategy. The profitfunctions under the two strategies are given in Equa-tion (2):

Traditional selling strategy:

�TS =2∑

j=1�PTSj − c DTS

j

Probabilistic selling strategy:

�PS =2∑

j=1�P PSj − c DPS

j + �P PSo − c DPSo

� (2)

The seller maximizes profit by choosing prices (PTSj )under the traditional selling strategy and both prices(P PSj � P PSO ) and probability (�) under the probabilisticselling strategy.The detailed solutions to the seller’s profit max-

imization problem are given in the appendix (seeTable A.1). Lemma 1 reports the important relation-ships we derive from these solutions.


Lemma 1 (Advantage of Probabilistic Selling).The following relationships hold for the two sellingstrategies:

Comparison Interpretation

(a) ��=�PS−�TS > 0 Conditions required forif 0< � < 1� c < c the advantage of PS

(b)

�pj > 0� �D = 0if c < ¯c

�pj ≥ 0� �D > 0if ¯c < c < c

Sources of profitadvantage of PS, where

�pj = P PSj − PTSj and

�D = �DPS1 +DPS

2 +DPS0

− �DTS1 +DTS

2

(c)��PS

��

≥ 0 if �≤ 1/2≤ 0 if �≥ 1/2

The impact of probabilitythat assigns a componentproduct as theprobabilistic good

(d)��pj

�t> 0�

��PSo

�t< 0,

��

�t

≥ 0 if t ≤ t

< 0 otherwise�

The impact of degree ofhorizontal productdifferentiation, where

�PSo = �P PSo − c DPS

o

where c = 1− t/2, ¯c = 1− t, and t is definedin the appendix.

Proposition 1 summarizes the key results inLemma 1.

Proposition 1 (Advantage of Probabilistic Sell-ing).(a) Adding a probabilistic good to the seller’s product

line strictly improves profit if marginal costs are suffi-ciently low (i.e., c < c).(b) Such profit improvement comes from enhanced price

discrimination alone if the cost is sufficiently low, but fromboth price discrimination and market expansion if cost isin a midrange.(c) The profit advantage of probabilistic selling does not

require assigning an equal probability to each componentproduct as the probabilistic good, but reaches the maximumunder such an equal probability (i.e., �∗ = 1/2).(d) The profit advantage of probabilistic selling is high-

est when the horizontal differentiation of the componentproducts is at an intermediate level.

First, Proposition 1 reveals that the profit advan-tage of probabilistic selling does not require specificcharacteristics of the travel industry such as capac-ity constraints and demand uncertainty, an interme-diary channel structure, or particular features of someonline travel agencies (e.g., Priceline.com) such as aNYOP pricing structure. As shown in Proposition 1,when buyers differ in the strength of their product

preferences, offering a probabilistic good can improveprofit as long as the marginal cost is sufficiently small.This result is important because it suggests that prob-abilistic selling may be a more general marketing toolto improve seller profit than currently thought.Second, Proposition 1 shows that offering a prob-

abilistic good can improve profit even if the marketis already fully covered under the traditional sell-ing strategy (i.e., �� > 0, �D = 0 when c < ¯c). Inthis case, probabilistic selling improves profit simplybecause it allows the seller to raise the prices of tra-ditional goods (i.e., �pj > 0). Introducing the proba-bilistic good enables the seller to separate consumerswith strong preferences (who buy the componentgoods) from consumers with weak preferences (whobuy the probabilistic good), thus enhancing price dis-crimination. For sellers facing unfilled demand undertraditional selling (i.e., ¯c < c < c), in addition to thediscrimination effect, probabilistic selling also leads tomarket expansion (�D > 0), i.e., allows the seller tosell the probabilistic good to low valuation consumerswho would not have bought under the traditionalselling strategy.Third, it is interesting to learn that probabilistic

selling can be profitable for ANY arrangement inwhich each component good has a positive prob-ability of being assigned as the probabilistic good(i.e., 0< � < 1). However, although an equal probabil-ity is not required for the profit advantage of prob-abilistic selling, it is in the seller’s best interest toassign such an equal probability, i.e., �∗ = 1/2. This isbecause moving � away from 1/2 negatively impactsprofit in two ways. Consider an increase in � above1/2. Increasing � makes the probabilistic good moreattractive to consumers who prefer product 1 butless attractive to consumers who prefer product 2.The former implies a cannibalization effect: Some con-sumers who would have bought product 1 at a higherprice under � = 1/2 now buy the discounted prob-abilistic product under � > 1/2. The latter implies ademand reduction effect: Some consumers who wouldhave bought the probabilistic product under � = 1/2now do not buy anything under � > 1/2. Can theseller reduce the two negative effects by adjustingthe price of the probabilistic good? The answer is nobecause increasing the price of the probabilistic goodwould reduce the attractiveness of the probabilisticgood to all consumers, which weakens the cannibaliza-tion effect but intensifies the demand reduction effect. Asa result, it is optimal for the firm to assign an equalprobability to each product as the probabilistic good.When � = 1/2, consumers have the same expected

value for the probabilistic good. However, when � =1/2, consumers vary in the expected surplus that theyreceive from purchasing the probabilistic good. Thus,while identical valuations for the probabilistic good


are not a necessary condition for probabilistic sellingstrategy to be advantageous, the seller will maximizeprofits by choosing � such that consumers will haveidentical valuations for the probabilistic good. Inter-estingly, as we will show later, such symmetric assign-ments (i.e., equal probability) are optimal even if theproducts have asymmetric demand (see §2.3.1).Finally, Proposition 1 reveals that the profit advan-

tage of probabilistic selling is maximized at an inter-mediate value of t, which represents the degree ofhorizontal differentiation of the component products.Notice that the price discrimination effect is increas-ing in t (i.e., ��pj/�t > 0), but that the profit obtainedfrom the probabilistic good is decreasing in t (i.e.,��PS

o /�t < 0) due to increasing expected fit-cost lossesfrom consuming the probabilistic good. If product 1and product 2 are near-perfect substitutes (t is closeto zero), then the advantage of probabilistic selling issmall because the price discrimination effect is small(�pj → 0 as t → 0). The seller is unable to extract a siz-able premium for the component products by intro-ducing a probabilistic good because no consumerviews the probabilistic good as being very inferiorto their most preferred product. On the other hand,if products 1 and 2 are extremely differentiated (t islarge), then the expected value of the probabilisticgood is very small, which implies that the price ofthe probabilistic good has to be very low to inducesales. Therefore, probabilistic selling is most advan-tageous when the products that comprise the proba-bilistic good have moderate differences—enough thatsome consumers will still be willing to pay a signifi-cant premium for their preferred product, but not toolarge that all consumers view the probabilistic goodas being of very low value.

2.3. Model Robustness and Additional InsightsThe basic model discussed above helps to under-stand the fundamental reasons why probabilistic sell-ing can improve profit. In the rest of this section,we demonstrate that these benefits apply to moregeneral settings and offer additional insights aboutthe probabilistic selling strategy. In particular, each ofthe following two subsections considers one specificextension of the basic model. Subsection 2.3.1 relaxesthe assumption of uniform distribution by assum-ing a general distribution of consumers and §2.3.2increases the number of component products from 2to N . Other assumptions remain the same as in thebasic model.

2.3.1. A General Distribution of Consumers. Thebasic model assumes that consumers are uniformly-distributed along the Hotelling line. In this subsec-tion, we relax the uniform-distribution assumption byallowing a general distribution f �x , 0≤ x ≤ 1. Propo-sition 2 summarizes our results in this more generalsetting.

Proposition 2 (Probabilistic Selling with aGeneral Distribution). Consider a general distributionof consumers f �x � 0≤ x ≤ 1�(a) Adding a probabilistic good to the seller’s product

line strictly improves profit in the following situations:1. The market is uncovered under the traditional sell-

ing strategy and costs are sufficiently low (i.e., c < c, wherec is given in Lemma 1).

2. The market is covered under the traditional sellingstrategy and some customers are indifferent between thetwo component products (i.e., 0< prob�xi = 1/2 < 1).(b) When probabilistic selling improves profit, the profit

advantage of probabilistic selling reaches its maximum ifthe seller assigns an equal probability to each componentproduct as the probabilistic good (i.e., �∗ = 1/2).Corollary 1 (When All Consumers Prefer the

Same Product). Even if all consumers prefer the sameproduct (e.g., f �x = 0, ∀x ≥ 0�5),(a) The seller can benefit from probabilistic selling.(b) It is optimal to assign an equal probability (i.e.,

�∗ = 1/2).Proposition 2 shows that the primary advantages

of probabilistic selling do not require several featuresof the standard Hotelling model. In particular, theredoes not need to be a constant density of consumersalong the Hotelling line nor is a symmetric distri-bution of customers required. In fact, from Proposi-tion 2(a) we see that there does not need to be anyconsumer located exactly at the midpoint of the line.Any consumer that is not served under TS has weakerpreferences between the two component goods thana consumer who purchases under TS. Thus, any het-erogeneity in the strength of preferences that is strongenough to induce the seller to exclude some con-sumers under TS (i.e., uncovered market under TS) isalso enough heterogeneity to allow PS to be advan-tageous as long as such market expansion can beobtained at a price that exceeds the marginal costof production (c < c). Furthermore, Proposition 2(a)verifies that the price discrimination effect will makePS advantageous if, under the TS strategy, the sellerwould choose to sell to weak-preferenced consumers(i.e., full market coverage under TS).Equally important, Proposition 2(b) reveals that for

any general distribution such that probabilistic sell-ing is advantageous, it is still optimal for the sellerto assign an equal probability to each componentproduct as the probabilistic good. Corollary 1 furtheremphasizes that an equal probability is optimal evenif all consumers prefer the same product. This resultmay not be intuitive because one would wonder ifall consumers prefer product 1, would it not be moreprofitable if the seller announces and assigns a higherprobability to product 1 as the probabilistic good? To


understand why �∗ = 1/2 regardless of the distribu-tion of consumer valuations, we need to notice thatthe probabilistic good targets those consumers withweak tastes, while the component products are tar-geted to those with stronger preferences. Deviatingfrom �= 1/2 reduces the seller’s efficiency of achiev-ing this segmentation. For example, suppose the firmskews assignments such that product 1 is more likelyto be assigned as the probabilistic good. This devia-tion will increase the expected value of the probabilis-tic good for all consumers. However, the consumersthat experience the greatest increase in expected valueare those who have the strongest preferences for prod-uct 1. Thus, skewed assignments make it more diffi-cult to attract a premium price for product 1.The following numerical example helps to under-

stand Corollary 1. Suppose a seller with two compo-nent products (j = 1�2) has two potential customers(A and B) and has no marginal cost of production. Theconsumer valuations for products 1 and 2 are A 10�0!,B 6�4!. Clearly, in this case, all consumers preferproduct 1. We first show �PS > �TS when � = 1/2.Under TS, the seller prices product 1 at $10 and prod-uct 2 at $4, making a profit of $14 (by selling prod-uct 1 to A and product 2 to B). Under PS, the sellerprices product 1 at $10 and sells a probabilistic good(which is equally likely to be product 1 or product 2,i.e., � = 1/2) for $5. Customer A purchases product 1and B purchases the probabilistic good, thus generat-ing a profit of $15, i.e., 7.1% higher than under TS.Next, let us see why �∗ = 1/2. Asymmetric assign-ments diminish the profit under PS. For example, if� = 0�4, then B is only willing to pay $4.80 for theprobabilistic good and, thus, profit under PS wouldonly be $14.80. If �= 0�6, then B’s expected value forthe probabilistic good is $5.20 but A’s expected valuefor the probabilistic good is even higher ($6). So, forP PSO = $5�20, P PS1 must be no larger than $9.20 in orderto induce A to purchase product 1. Thus, profit underPS with � = 0�6 is only $14.40, which is lower thanthat with � = 0�5. One might wonder if �∗ = 1/2 stillholds if there are more type B consumers (becausethe seller gains from type B but loses from type Awhen increasing �). It is easy to verify that PS is stilladvantageous and �∗ = 1/2 continues to hold for thecase with two or three type B consumers. When thenumber of type B consumers is higher than three, PScan no longer outperform TS. For instance, considerthe case with four type B consumers. Under TS, theseller would price product 1 at $6 and make a profitof $30. PS (for any level of �) can not improve uponthis. Hence, as stated in Proposition 2(b), when prob-abilistic selling improves profit, it is optimal to assignan equal probability �∗ = 1/2.Notice from these examples that PS can be advan-

tageous even though the market is covered under TS

and no customer is completely indifferent betweenthe two component products. Thus, the conditions inProposition 2(a) are sufficient but not necessary toobtain an advantage under PS.

2.3.2. N Component Goods (The Circle Model).Thus far, we have assumed that the seller’s initialproduct line consisted of two goods prior to the intro-duction of the probabilistic good. In practice, mostsellers have more extensive product lines. In this sub-section, we extend the basic model to allow for N > 2component goods.We assume a standard circular spatial market, a

model which was first introduced by Salop (1979)and has been used and revised to fit many otherresearch settings (e.g., Balasubramanian 1998, Dewanet al. 2003). Each consumer’s ideal product, repre-sented by a location xi, is valued at one and theseideal points are distributed uniformly across a circleof unit circumference. Each consumer incurs a linearcost t per unit distance for consuming a product thatdiffers from her ideal. The seller offers N componentgoods which are equally spaced along the circumfer-ence of this circle. In particular, product j (j = 1 to N )is located at j/N .Under the traditional selling strategy, the seller

chooses the component good prices PTSj in orderto maximize the sum of profits across the N goods.Under the probabilistic selling strategy, the seller setscomponent good prices P PSj and also determineswhich probabilistic goods to offer (if any) and at whatprices. Many different probabilistic goods are feasibleby using two or more of the component goods andsetting the probability that each will be assigned asthe probabilistic good. In particular, the probabilityassigning each component good as the probabilisticgood is defined as a vector �k

1��k2� � � � ��

kN !, where∑N

j=1 �Kj = 1, �k

j ≥ 0, �kj < 1, and �k

j > 0 for at least twoof the j’s. Proposition 3 summarizes our results forthis model.

Proposition 3 (Probabilistic Selling for Sellerswith N Component Goods).(a) Adding probabilistic goods to the seller’s product line

strictly improves profit if costs are sufficiently low (i.e.,��=�PS−�TS > 0 if c < c, where c = 1− t/�2N ).(b) The required cost condition for probabilistic selling

to be advantageous is weaker as the number of componentgoods increases (i.e., �c/�N = t/�2N 2 > 0).(c) The optimal probabilistic selling strategy is to offer

N probabilistic goods, each one consisting of two adja-cent component goods, with both of these component goodsbeing equally likely to be assigned as the probabilistic good.Formally, �PS is maximized by introducing N probabilisticgoods (k= 1 to N ), where{

�kk = �k

k+1 = 1/2� �kj = 0 ∀ j k� k+ 1! if k < N

�N1 = �N

N = 1/2� �kj = 0 ∀ j 1�N! if k=N�


Proposition 3 shows that the benefits of probabilis-tic selling extend to a seller with numerous compo-nent products. As in the two-good Hotelling model,some consumers strongly prefer one of these goodswhile others have weaker preferences. By introduc-ing a probabilistic good that randomly selects fromthe two component goods that are most near to aweak-preferenced consumer’s ideal point, the sellercan price discriminate on the basis of this differingstrength of preferences. Furthermore, it is importantto note that probabilistic selling is advantageous overa wider range of costs when there are more than twogoods. As the number of component products grows,the maximum distance from one’s ideal product tothe two most suitable component products falls. Thus,larger N allows the seller to set higher prices for theprobabilistic good, thereby achieving positive profitmargins over a wider range of costs. Finally, it isclear that the benefits of probabilistic selling could beextended to many other spatial models, e.g., if con-sumer ideal points are uniformly distributed within acube. In such a three-dimensional space, the numberof probabilistic goods would grow exponentially, i.e.,a probabilistic good could profitably be introducedfor any two adjacent products (as long as costs aresufficiently low).The results of these two model extensions show

the robustness of the findings derived from the basicmodel, suggesting that these findings can hold in var-ious more general settings. In the remainder of thispaper, we examine several important market/productfactors that may affect the profit advantage of proba-bilistic selling. In the process, we discover additionaladvantages of PS.

3. Impact of Demand Uncertainty andCapacity Constraints

In this section, we examine the impact of two impor-tant factors, (1) demand uncertainty and (2) capacityconstraints, on the profit advantage of probabilisticselling. Such an examination is important becausein many markets, e.g., travel services, entertainmenttickets, educational classes, and holiday gift buying,demand uncertainty and capacity constraints playpivotal roles in a seller’s pricing decision. Problemsassociated with excess demand or idle capacity can beparticularly acute in service industries. For instance,consider a professional career development centerthat offers a particular computer training class at bothof its locations within a city. Prior to the start of thesession, the center may be uncertain how many peo-ple will show up to register at each location. Thus,there is a possibility that one session will be morepopular than the other. Yet, not knowing which classwill be more popular, it is difficult for the center to

set prices appropriately and to ensure efficient use ofits facilities.In this section, we ask whether probabilistic sell-

ing exacerbates or lessens the problems that may becaused by demand uncertainty and/or limited capac-ity. To foreshadow our results, the answers to thesequestions may be somewhat surprising. Notice that inthe previous section, we showed that market expan-sion is one potential benefit of the probabilistic sellingstrategy. At first glance, one would question whethera seller with limited capacity would ever benefit fromintroducing a probabilistic good since increasing thetotal quantity demanded does not seem to be partic-ularly helpful. However, we will soon show that inthe presence of demand uncertainty, the advantage ofPS can be even larger when capacity is limited thanwhen it is unlimited.

3.1. Model of Asymmetric Product DemandIn order to address the issues related to demanduncertainty and capacity constraints, we now focuson a model of asymmetry in product popularity. Weadopt the same assumptions as in the basic model butallow asymmetric distribution of demand. In addi-tion, to sharpen the focus and simplify the presenta-tion, we assume the fit-cost-loss coefficient (t) is one,i.e., t = 1, and a zero marginal cost, c = 0.4The demand given in Expression (1) can be rewrit-

ten as Equation (3):1/2 of customers prefer product 1:

v1i = 1− xi v2i = xi xi ∼U�0�1/2�

1/2 of customers prefer product 2:v1i = xi v2i = 1− xi xi ∼U�0�1/2�

� (3)

To systematically incorporate asymmetric demandfor the component products, we assume that $ of cus-tomers (1/2 ≤ $ ≤ 1) prefer product 1 and 1 − $ ofcustomers prefer product 2.5 This implies the follow-ing distribution of consumer valuations:

$ of customers prefer product 1:

v1i = 1− xi v2i = xi xi ∼U�0�1/2�

1−$ of customers prefer product 2:v1i = xi v2i = 1− xi xi ∼U�0�1/2�

� (4)

Note that our basic model is a special case ofEquation (4) because distribution (4) reduces to Equa-tion (3) when $ = 1− $ = 1/2. The boundary case of

4 Allowing a positive cost does not qualitatively affect our resultsbut does significantly complicate the analysis.5 Our analyses focus on asymmetry in product popularity. We haveproved that the same results hold qualitatively in a model wherethe products differ in quality. Detailed analysis of this model canbe obtained by contacting the authors.


$ = 1 is a standard vertical differentiation model inwhich all consumers prefer product 1 to product 2although they differ in the magnitude in which theyprefer product 1 over product 2. Proposition 4 sum-marizes the impact of asymmetry in demand on theprofit advantage of probabilistic selling.

Proposition 4 (Impact of Demand Asymmetry).(a) Differentiation in product popularity weakly

reduces the profit advantage of probabilistic selling, i.e.,��/�$≤ 0.(b) Introducing a probabilistic good reduces price dif-

ferentiation between component goods, i.e., pTS1 > pTS2 �pPS1 = pPS2 .

Proposition 4(a) suggests that the seller gains lessfrom introducing a probabilistic good when the com-ponent products differ in their popularity than in theabsence of such asymmetry. This is because demandasymmetry allows the seller to price discriminateeven under traditional selling (i.e., pTS1 > pTS2 ) andhence reduces the value of introducing a probabilis-tic good. However, it is important to notice that, asexemplified by the example in §2.3.1, introducing aprobabilistic good can improve profit even if all con-sumers prefer the same product (i.e., $= 1).Proposition 4(b) reveals an interesting result: The

prices of the component products are less differenti-ated under PS than TS (i.e., pTS1 > pTS2 � pPS1 = pPS2 ).

6 Tounderstand this, please notice that under TS, the onlyway for the seller to segment the market is to adoptdifferentiated prices. However, under PS, the sellercan segment the market by selling a discounted prob-abilistic good, i.e., sell the probabilistic good ratherthan the less popular component product (at a dis-counted price) to those consumers with weak pref-erences for the more popular item. This allows theseller to charge relatively high prices for both com-ponent products and, thus, maintain high margins onsales to consumers that strongly prefer the less pop-ular product. This finding that the prices of the com-ponent products are less differentiated under PS thanTS implies that knowing demand for individual prod-ucts can be more critical under TS than under PS.The importance of this finding will soon become moreapparent when we discuss the impact of demanduncertainty below.

3.2. Demand Uncertainty (DU)In the preceding section, we assumed that product 1is more popular than product 2 (i.e., $ > 1/2). In thissection, we incorporate seller uncertainty by assum-ing that product 1 and product 2 are equally likely

6 In the alternative model of demand based on asymmetric quality,the component good prices under PS are not identical. However,we still have the result that the prices of the component productsare less differentiated under PS than TS (i.e., pTS1 − pTS2 > pPS1 − pPS2 ).

to be the more popular good and that the seller doesnot know which will be the more popular productwhen making its pricing decision. Specifically, theseller does not know if $ > 1/2 will apply to product 1or product 2. Other assumptions remain the same asin §3.1.Let L (where L=DU�NDU) denote the type of the

market, i.e., with and without demand uncertainty.Let �s

L denote the profit under strategy s (s = TS�PS)in market L. Note that the difference �s

NDU − �sDU

measures the profit loss caused by uncertainty understrategy s (s = TS�PS), and the difference �PS

L − �TSL

measures the profit advantage of probabilistic sellingin market L, respectively. Proposition 5 summarizesour key results.

Proposition 5 (Impact of Demand Uncertainty).(a) Sellers facing demand uncertainty benefit more from

probabilistic selling than sellers without such uncertainty(i.e., �PS

DU−�TSDU ≥�PS

NDU−�TSNDU).

(b) Demand uncertainty increases the profit advantageof probabilistic selling because introducing a probabilisticgood weakens or eliminates the dependence of pricing deci-sions on the identity of the more popular product, whichreduces the profit loss caused by demand uncertainty (i.e.,�TSNDU−�TS

DU > �PSNDU−�PS

DU).

The finding that demand uncertainty increases theprofit advantage of probabilistic selling is intriguing.Such a positive effect of demand uncertainty on theprofit advantage of probabilistic selling occurs be-cause probabilistic selling can buffer against demanduncertainty, thus offering an additional reason forintroducing a probabilistic good. In the absence ofdemand uncertainty (see §3.1), under the traditionalselling strategy, it is optimal to charge a price pre-mium for the more popular product (i.e., PTS2 < PTS1 ).However, in the presence of demand uncertainty,without knowing which good will be more popular,the seller is unable to set prices correctly based on the(unknown) demand for each product. This informa-tion disadvantage leads to a lower profit compared tothe case without demand uncertainty. However, thisinformation disadvantage can be removed by intro-ducing a probabilistic good. First, the price of theprobabilistic good is not affected by the identity ofthe “hot” product because the value of a probabilis-tic good is determined by the expected value of itscomponent products (e.g., the price of a “probabilis-tic toy” is the same regardless if toy A or toy B willturn out to be the hot toy). Second, introducing aprobabilistic good also makes the prices of the com-ponent goods less sensitive to the demand of individ-ual products. As we pointed out in the discussion ofProposition 4(b), by introducing a probabilistic good,the seller has less need to use differentiated prices


because she can segment the market using the prob-abilistic good. Furthermore, a probabilistic good cancushion the damage caused by suboptimal prices. Forinstance, under traditional selling, if a price is set toohigh due to the seller’s failure to perfectly forecast thedemand for its product, the firm loses an entire salewhereas, under probabilistic selling, a missed con-sumer will not be lost altogether but will instead pur-chase the probabilistic good (albeit at a lower marginthan the traditional product).

3.3. Capacity Constraints and DemandUncertainty (CC and DU)

We now consider a seller with both limited capacityand demand uncertainty. Let K denote the total num-ber of units of each product that the seller has thecapacity to produce. As in §3.2, the seller is uncertainabout which product will be the more popular prod-uct. As the degree of the product asymmetry is higher(i.e., $ is larger), the seller faces greater uncertainty.Let �s

DU&CC denote profit of strategy s (s = TS�PS)with DU and CC. The seller faces the following max-imization problem:

Traditional selling

�TSDU&CC = max

PTS1 �PTS2

PTS1 DTS1 + PTS2 DTS

2

s.t. DTS1 ≤K� DTS

2 ≤K� PTS1 = PTS2

(5)

Probabilistic selling

�PSDU&CC = max

PPS1 �PPS2 �PPSO ��Xo

P PS1 DPS1 + P PS2 DPS

2 + P PSO XO

s.t. DPS1 +�XO ≤K� DPS

2 + �1−� XO ≤K�

XO ≤DPSO � P PS1 = P PS2 �

(6)

where XO in Equation (6) is the number of probabilis-tic good sales. The first two constraints in Equation (6)ensure that total sales of each product (including eachproduct’s share of the probabilistic good sales) donot exceed capacity constraints. The third constraintensures that the number of probabilistic good saleschosen by the seller does not exceed the demand forthe probabilistic good. The last constraint restricts theseller from charging a price premium for the morepopular product because it does not know whichproduct will be more popular.Recall that �s

DU denotes the profit under strategy sin market with DU but without CC. It is obvious thatimposing an additional constraint on the seller cannot improve profit under either strategy, i.e., �s

DU&CC−�sDU ≤ 0, s = TS�PS. Notice that (�PS

DU&CC − �TSDU&CC)

and (�PSDU − �TS

DU) measure the profit advantage ofprobabilistic selling in markets with and withoutcapacity constraints, respectively, when the markethas demand uncertainty.The impact of capacity constraints in the presence

of DU is summarized in Proposition 6.

Proposition 6 (Effect of Capacity Constraints,with DU). In the presence of demand uncertainty, capac-ity constraints(a) Increase the profit advantage of probabilistic selling

when capacity is in a midrange and demand uncertainty issufficiently large.(b) Decrease the profit advantage of probabilistic selling

when capacity is very small.(c) Have no effect on the profit advantage of probabilistic

selling when capacity is sufficiently large.Formally,

��PSDU&CC−�TS

DU&CC − ��PSDU−�TS

DU

·

> 0 if K < K < �K and $ > $

= 0 if K ≥ �K< 0 if K ≤K�

Proposition 6 shows an important result: Capac-ity constraints can be a favorable factor for the profitadvantage of probabilistic selling when a seller facesdemand uncertainty. Demand uncertainty and capac-ity constraints combine to create two problems for theseller: (1) demand-capacity mismatch, and (2) ineffi-ciency. Introducing a probabilistic good can help theseller to overcome both.First, the coexistence of demand uncertainty and

capacity constraints can create mismatch betweendemand and capacity (i.e., one product has unfulfilleddemand but the other has excess capacity). Such amismatch problem is hard to resolve under the tra-ditional selling strategy because, without knowingwhich product will be perceived more favorably byconsumers, the seller is unable to use prices to opti-mally shift demand towards the product with under-utilized capacity. However, when the seller offers aprobabilistic good, all consumers with weak pref-erences (although a majority of which may preferthe higher demand product) will buy the probabilis-tic good, but only some of these consumers will beassigned to the ex post high demand product. Asa result, the seller, without knowing which productwill have excess capacity, is able to spread aggregatedemand across the two goods more evenly, whichincreases capacity utilization.Second, the co-existence of demand uncertainty and

capacity constraints can create inefficiency (i.e., con-sumers with weak preferences may take the capac-ity of the preferred product away from consumerswith strong preferences). Such inefficiency is hard toresolve under the traditional selling strategy becausethe seller is unable to use prices to ration consumers.However, under probabilistic selling, the seller canuse the probabilistic good to effectively move someconsumers with weak preferences to consume the


“inferior” product, which saves capacity of the “supe-rior” product for consumers with strong preferences.As shown in Proposition 6, the positive effect of

capacity constraints requires a midrange capacity anda sufficiently high level of demand uncertainty. Whendemand uncertainty is very low, neither mismatchnor inefficiency is a major problem for the seller.When capacity is sufficiently large (K ≥ �K), capacityconstraints do not affect the profit advantage of PSbecause, in this case, the seller has sufficient capac-ity to implement the unconstrained solution for eitherPS or TS. When capacity is sufficiently small (K ≤K),capacity constraints reduce the profit advantage ofintroducing a probabilistic good because the sellerwould allocate few (if any) units of its scarce capacityto the lower-margin probabilistic good, which weak-ens the benefit of PS.

4. Discussion and Conclusions4.1. Managerial ImplicationsThis paper proposes and analyzes a creative wayof selling products and services: probabilistic selling.Our analyses illustrate that this is a general mar-keting tool that can benefit many sellers in broadindustries because its advantage is fundamentallydriven by buyer heterogeneity in the strength oftheir preferences, which occurs in almost all markets.The generality of probabilistic selling’s advantage hasimportant managerial implications, especially becauseprobabilistic goods are not concrete but “virtual”goods. Offering probabilistic goods can be an innova-tive way to extend one’s product line without incur-ring the cost of developing new products or services,which can be a significant benefit in situations wherenew product development or provision is difficultor impractical due to resource constraints. The “vir-tual” nature also allows flexibility in adjusting thelength of product line (e.g., a cruise line can flexi-bly offer a four-day “Eastern or Western Bahamas”cruise only in the peak season but not in the off-peak season). Some examples of probabilistic sellingalready exist in practice. For example, Circus Circusin Las Vegas sells “run-of-the-house” rooms at a dis-counted rate through travel intermediaries (such asvacationstogo.com) but charges a premium to guar-antee a room in a certain tower; many restaurantsoffer a house wine (from an undisclosed winery) ata discounted price but customers can pay a higherprice to select a certain label. However, the numberof additional potential applications for manufacturers,service providers, and retailers is almost limitless. Forinstance, a hotelier could charge a premium to guar-antee a room on a specific floor; a university couldset higher tuition for students who want to assurethemselves a particular slate of courses rather than be

subject to uncertainty about which sections or coursesmay be open this semester; or a dentist could chargean additional fee if the patient wants to receive ser-vice from a particular dental hygienist.7

The results in this paper also provide insight intowhich markets are likely to see the greatest bene-fit from introducing probabilistic goods. In particu-lar, our findings suggest that factors which enhancethe profitability of the probabilistic selling strat-egy include low marginal costs, buyer heterogeneityin their tastes, moderate horizontal differentiationbetween the component goods, similar aggregatedemand across component products, and demanduncertainty (especially when combined with capac-ity constraints). A multiproduct seller should seek toimplement the probabilistic selling strategy for prod-ucts or services with the above characteristics. Con-sider a stage theatre that offers the same show overthe course of a week. The probabilistic selling strat-egy could be applied by raising the current ticketprices and then offering a discounted probabilisticgood(s), being careful to avoid grouping nights thatare too asymmetric in popularity. For example, thetheatre could offer a “weekend ticket” (i.e., the con-sumer is assigned to either a Friday or Saturday nightperformance) or a “midweek ticket” (i.e., the con-sumer is assigned to either a Tuesday or Wednesdaynight performance). The probabilistic selling strategycould be particularly appealing for service providerswith high variability in capacity usage (e.g., an autorepair shop or an ice skating rink for which its con-sumers may face long waits on some days or timesbut capacity and the staff are underutilized on someother days or times), especially when these patternsare unpredictable or the service provider faces otherbarriers to adjusting prices in response to variations indemand.It is important to point out several issues related

to the implementation of probabilistic selling. Likeseveral other marketing strategies (e.g., advance sell-ing and quantity discounts), probabilistic sellingcan suffer from arbitrage. Sales of the probabilisticgood need to be nonrefundable, nontransferable, andnonexchangeable so that a purchaser of the proba-bilistic good faces some risk, else the probabilisticgood would cannibalize all component product sales.Recent developments in technology are increasing theseller’s capability to meet these required conditionsfor implementing probabilistic selling. For example,new technologies such as smart cards, RFID chips,biometric palm readers, and electronic tickets enablethe seller to successfully attach a particular prod-uct or service to a particular consumer, which drasti-cally increases the seller’s capability to limit arbitrage

7 We thank Steven Shugan for suggesting the house wine and uni-versity examples.


(Xie and Shugan 2001, Shugan and Xie 2005) and toenforce the special policies required for the sales ofprobabilistic goods (e.g., nonrefundable). Even with-out advanced technologies, arbitrage is easily avoidedfor service industries in which services can be deliv-ered at the time of purchase. Furthermore, advancedtechnologies enable sellers to implement more com-plex pricing schemes (Rust and Chung 2006), such asthose required under probabilistic selling.

4.2. Limitations and Future ResearchIn this paper, we assumed symmetric costs. Withasymmetric costs, consumers may expect that once apayment is made, the seller may have an incentiveto avoid assigning the high-cost good as the proba-bilistic good (or the good that is in short supply ifthe firm faces asymmetric capacity constraints). Thiscan undermine the profitability of probabilistic sell-ing since it is essential that consumers are uncertainabout the assignments of the probabilistic good atthe time of purchase and believe that it could turnout to be either product. Although this will not be aproblem for sellers with endogenous credibility (i.e.,due to a long-term reputation effect), it would beuseful to develop possible mechanisms to endoge-nously establish a seller’s “uncertainty credibility.”Recent increased popularity of various independentproduct information created by professional productreviewers and consumers due to advances in tech-nology (Chen and Xie 2005, 2008) may provide waysfor sellers to obtain such “uncertainty credibility.” Forexample, Consumer Reports WebWatch spent approx-imately $38,000 booking airline seats, hotel rooms,and rental cars in order to report differences betweenopaque travel web sites with nonopaque sites (McGee2003). The existence of such independent informationhelps to establish sellers’ credibility. Recent devel-opments of various new forms of consumer socialinteractions also create increasing new opportunitiesfor the seller to overcome the commitment problems.For example, biddingfortravel.com, which has expe-rienced over 100 million visits, has a section devotedexclusively to posting “winning bids” in which Price-line users report what bids were accepted and whatflight itinerary, hotel property, or car rental com-pany was received. When random past assignmentsare reported in such an online forum, consumersmay believe that future assignments are also random.Finally, an intermediary such as online travel agentmight also serve as a commitment mechanism.Another important dimension would be to incorpo-

rate risk aversion in the analysis. Attitudes toward theprobabilistic good depend not only on the strengthof one’s preferences (as accounted for in this cur-rent paper), but also on one’s disposition toward risk.Probabilistic selling may enable the seller to discrim-inate according to variation in risk aversion. A third

important dimension is more elaborate market set-tings such as those with competition or an interme-diate channel structure. As Liu and Zhang (2006) doin the context of personalized pricing, it would beinteresting to consider how the probabilistic sellingstrategy would alter the market interactions betweenmultiple sellers or channel members. A fourth poten-tial extension would be to study settings where thereare more complex interactions in the demand forthe component products (e.g., variations in the valueof each consumer’s ideal product and/or nonlineartransportation costs).8

Finally, future research could generalize the prob-abilistic selling strategy as a hybrid with bundling.Venkatesh and Kamakura (2003) demonstrate thatmixed bundling (i.e., selling a bundle in addition tothe component products) is not strictly more prof-itable than pure component selling (i.e., selling onlycomponent products) if the component products aretoo close of substitutes. In particular, when consumersare interested in consuming only a single good (e.g.,only want to see a single movie tonight—“HannibalRising” or “The Hitcher”), as is true for the Hotellingand Salop circle model, there is no advantage frommixed bundling. However, we demonstrate that theprobabilistic selling strategy can be strictly advanta-geous to pure component selling in this setting. Thissuggests that it may be desirable to generalize thedefinition of a bundle. Traditionally, in the bundlingliterature, it is assumed that each component of thebundle is included with probability one. In contrast,a probabilistic good, as defined in the current paper,could be thought of as a bundle where the proba-bilities of getting each particular component sum toone. More generally, it might be possible to offer a“probabilistic bundle” where each component goodhas a probability less than one of being allocated tothe buyer but the sum of the probabilities across allthe component products is not necessarily equal toone, e.g., a book club that sends each out of 10 bookswith probability of 1/2 so that the member receiveson average 5 books per month. It would be interestingto explore when such a probabilistic bundle might beoptimal.9

8 It is unclear whether it would still be optimal for the seller tomake symmetric assignments of the probabilistic good in such set-tings. For example, with extreme risk aversion and/or nonlineartransportation costs, the probabilistic good can be a poor alterna-tive even for consumers with only moderately strong preferences.Thus, the cannibalization effect from asymmetric assignments maynot be as severe. We thank an anonymous reviewer for this insight.9 We thank the area editor for pointing out this interesting potentialextension.


Appendix

Summary of Notations

s Type of strategy, s = TS (Traditional selling),PS (Probabilistic selling)

J Specified component product, j = 1�2xi A taste parameter that is i.i.d.

across consumers

T A fit-cost-loss coefficient that reflects theloss of utility from not consumingone’s ideal product

vji Value of product j to consumer i

cj Marginal cost of product j

P sj Price of the specified product j under

strategy s; j = 1�2; s = TS�PSPPSo Price of the probabilistic (“opaque”) good

� The probability that product 1 is allocatedas the probabilistic good

x2 The maximum value of xi for whicha consumer will purchase product 2

x1 The minimum value of xi for whicha consumer will purchase product 1

DTSj Demand for the specified product j under TS

DPSj Demand for the specified product j under PS

DPSo Demand for the probabilistic good under PS

F �x Cumulative distribution function inthe general model

N Number of component goods in

the circle model

�s Profit of the seller under strategy s; s = TS�PS$ When there is differentiation in popularity,

the fraction of consumers that prefer thehigh-demand product over thelow-demand product

K The total number of units of each productthat the seller has the capacity to produce

XO The number of units of the probabilisticgood sold

�sDU Profit of strategy s (s = TS�PS) when there is

demand uncertainty and there are notcapacity constraints

�sNDU Profit of strategy s (s = TS�PS) when there is

not demand uncertainty and there are notcapacity constraints

�sDU&CC Profit of strategy s (s = TS�PS) when there is

both demand uncertainty and constrainedcapacity

In what follows, we sketch the proofs for all of the proposi-tions in the paper. Full details are available in the TechnicalAppendix online at http://mktsci.journal.informs.org.

Sketch of Proof of Lemma 1 and Proposition 1

Traditional Selling StrategyFor the value distribution in Equation (1), demand for

each product is given by:

DTSj =

0 if PTSj ≥ 11− PTSj

tif 1> PTSj ≥ 2− t− PTS−j

t− PTSj + PTS−j

2tif PTSj < 2− t− PTS−j �

(7)

Maximizing �TS = ∑2j=1 �P

TSj − c DTS

j yields the profit andprices listed in Table A.1.

Probabilistic Selling StrategyWithout loss of generality, we restrict attention to � ≥

1/2. We can divide the Hotelling line into three segments:consumers with xi ≤ x1 purchase product 1, consumers withxi ≥ x2 purchase product 2, and consumers with x1 < xi < x0purchase the probabilistic good. For the probabilistic goodto have positive sales, then x0 must be strictly larger than x1.There are two possible outcomes: full coverage (i.e., x0 = x2)and incomplete coverage (i.e., x0 < x2).

Full coverage (FC): The seller’s profit is given by

�PSFC = �PPS1 −c �x1 +�PPS2 −c �1− x2 +�PPSO −c �x2− x1 � (8)

For a given x1 and x2, the prices that extract the maximumpossible consumer surplus are

PPSO = 1−�tx2− �1−� t�1− x2 (9)

PPS2 = 1− t�1− x2 (10)

PPS1 = �1−� t�1− 2x1 + PPSo � (11)

Maximizing Equation (8) with respect to x1 and x2 yieldsx1 = 1/4 and x2 = �1+� /�4� , and a resulting profit of:

�PSFC = 1− c− t�5�− 1

8�� (12)

Notice that ��PSFC/�� =−t/�8�2 < 0. Comparing �PS

FC to theprofits under TS:

�PSFC−�TS =

t�1−�

8�> 0 if c ≤ 1− t

1− c− �1− c 2

2t− t�5�− 1

8�

> 0 if c < c

< 0 if c > c

otherwise

�

where c = 1− t+ t√

��1−�

2��

�c

��< 0� (13)

Incomplete coverage (IC): The seller’s profit is given by

�PSIC = �PPS1 −c �x1 +�PPS2 −c �1− x2 +�PPSO −c �x0− x1 � (14)

For a given x1, x2, and x0, the prices that extract the maxi-mum possible consumer surplus are

PPS2 = 1− t�1− x2 (15)

PPSO = 1−�txo − �1−� t�1− xo (16)

PPS1 = �1−� t�1− 2x1 + PPSo � (17)


Table A.1 The Optimal Prices and Profits in the Basic Model

Traditional selling Probabilistic selling Comparison

Specific goods

Price P TSj =

2− t

2if c ≤ 1− t

1+ c

2if c > 1− t

P TSj =

1− t

4if c < c

1+ c

2if c ≥ c

�pj =

t

4> 0 if c ≤ 1− t

21− c− t

4> 0 if 1− t < c < c

0 if c ≥ c

Sales DTSj =

12

if c ≤ 1− t

1− c

2tif c > 1− t

DPSj =

14

if c < c

1− c

2tif c ≥ c

�Dj =

−14< 0 if c ≤ 1− t

−21− c− t

4t< 0 if 1− t < c < c

0 if c ≥ c

Probabilistic good

Price P PSo =

1− t

2if c < c

N/A if c ≥ c

Sales DPSo =

12

if c < c

0 if c ≥ c

∗ =

12

if c < c

N/A if c ≥ c

Total demand DTS =

1 if c ≤ 1− t

1− c

tif c > 1− t

DPS =

1 if c < c

1− c

tif c ≥ c

�D =

0 if c ≤ 1− t

t − 1− c

t> 0 if 1− t < c < c

0 if c ≥ c

Total profit �TS =

1− t

2− c if c ≤ 1− t

1− c2

2tif c > 1− t

�PS =

1− 3t8

− c if c < c

1− c2

2tif c ≥ c

��=

t

8> 0 if c ≤ 1− t

AB

8t> 0 if 1− t < c < c

0 if c ≥ c

Notes. c= 1− t/2, A= 21− c− t , B = 3t − 21− c, �pj = P PSj − P TS

j , �Dj = DPSj −DTS

j , �D = DPS −DTS, ��=�PS −�TS.

Maximizing Equation (14) w.r.t. x1, x2, and xo yieldsx1 = 1/4, x2 = 1 − �1− c /�2t , and xo = �1 − c − t�1 − � /�2t�2�− 1 . This outcome represents positive sales ofthe probabilistic good, i.e., x0 > x1, only if c < c wherec = 1− t/2. Furthermore, the market is incompletely cov-ered only if x0 < x2 or:

c > 1− t�3�− 1 2�

≡ ˆc� (18)

The resulting profit is:10

�PSIC = 4��1− c 2− 4t�1− c �1−� + �1−� t2

8t�2�− 1 (19)

Notice that ��PSFC/�� = −�2�1− c − t 2/�8t�2�− 1 2 < 0.

Assuming c < c and condition (18) is met, i.e., the conditions

10 The incomplete coverage case can only be optimal for a given �if � = 1/2. Thus, we do not need to be concerned that the denomi-nator of Equation (18) is zero when �= 1/2.

under which there is incomplete coverage with positivesales of the probabilistic good, �PS

IC in comparison to �TS is

�PSIC −�TS = �1−� �2�1− c + t 2

8t�2�− 1 > 0� (20)

Under PS, the seller chooses either full coverage or in-complete coverage depending on which outcome yields thehighest profit. Thus, PS is strictly preferred to TS if either ofthese two outcomes yields higher profit. Using the deriva-tions above, this implies

Max��PSFC��

PSIC �>�TS if

(a) c≤1−t,

(b) c>1−t & c<c, or

(c) 1− t

2>c>1−t & c< ˆc�

(21)

Using the definitions in Equations (13) and (18), we haveˆc < c and c < 1 − t/2 ∀� ≥ 1/2, thus proving part (a) ofLemma 1. Part (c) is evident due to demand symmetry andbecause we have shown both ��PS

FC/�� < 0 and ��PSIC/�� < 0


for � > 1/2. Table A.1 records the resulting prices, demand,and profit under PS, taking into account that the seller maywish not to offer the probabilistic good if costs are suffi-ciently large.Part (b) of Lemma 1 involves straightforward compar-

isons of the demand and prices under PS and TS. Part (d) ofLemma 1 records additional comparative statics. For exam-ple, ��/�t is found using �TS from Table A.1 and �PS

FC and�PSIC from Equations (12) and (19), respectively:

��PSFC−�TS

�t

=

1−�

8�> 0 if t ≤ 1− c

18

[4�1− c 2

t2− 1

�− 5

]

> 0 if t < t

> 0 if t > totherwise�

where t = 2�1− c

√�

5�− 1 (22)

��PSIC −�TS

�t=− �1−� �4�1− c 2− t2

8�2�− 1 t2 < 0� (23)

Equation (23) is only valid when PS leads to incompletemarket coverage, i.e., 1−c < t < 2�1− c . Thus, the t referredto in Lemma 1(d) must lie within the interval �1− c� t�.Proposition 1 summarizes the key results.

Sketch of Proof of Proposition 2We start by proving part (b) of Proposition 2, i.e., that sym-metric assignments of the probabilistic good are optimal.Suppose that � ≥ 1/2 so that under PS, the seller parti-tions the market so that any consumer with xi ≤ x1 pur-chases product 1, any consumer with x1 < xi ≤ x0 purchasesthe probabilistic good, and any consumer with xi ≥ x2 pur-chases product 2, where x0 ≤ x2. For any such set (x1� x2�x0),profit under PS is maximized at either �= 1/2 or �= 1.The seller’s profit is given by

�PS�� = �PPS1 − c F �x1 + �PPS2 − c �1− F �x2− -

+ �PPSo − c �F �x0 − F �x1 � (24)

where F �x is the cumulative distribution and - is an arbi-trarily small number.To induce the desired consumer behavior, prices must be

set accordingly:

PPSo = 1−�tx0− �1−� t�1− x0 (25)

PPS1 = �1−� t�1− 2x1 + PPSo (26)

PPS2 = 1− t�1− x2 � (27)

If x0 > 1/2, then �PPSo /�� < 0, �PPS1 /�� < 0, and �PPS2 /�� = 0.Thus, for x0 > 1/2, profit is higher at � = 1/2 than at any� > 1/2.For x0 ≤ 1/2, using the prices given in Equations (25),

(26), and (27), we find

��PS

��= F �xo �1− xo�1+ t − tF �x1 �1− 2x1 ≡.� (28)

. is not a function of �. Thus, if . > 0, profit under PS ismaximized at � = 1, i.e., TS and PS yield equivalent profit.If . < 0, profit under PS is maximized at �= 1/2.Now turn to part (a) of Proposition 2. For part 1, at � =

1/2, by setting a price for the probabilistic good at PPSo = 1−t/2 and keeping the same prices for the component goods asunder TS, the seller will sell the same number of componentgoods sales but will make additional sales to the customerswho were unserved under TS. Such sales enhance profits aslong as c < c.For part 2 of (a) in Proposition 2, the firm could sell the

probabilistic good to any consumer located at xi = 1/2 bysetting a price PPSo = 1− t/2, i.e., a price that generates asleast as high a margin as was obtained under TS. Further-more, the seller could raise prices of at least one componentgood and still make positive sales. Hence, profit under PSis strictly larger than profit under TS.

Sketch of Proof of Proposition 3

Traditional Selling StrategyIf each component product is priced at PTSj , profit is

�TS�PTSj

=

PTSj if PTSj ≤ 1− t

2N

2N(1− PTSj

t

)�PTSj − c if 1> PTSj ≥ 1− t

2N

0 if PTSj ≥ 1�

(29)

Maximizing �TS�PTSj w.r.t. PTSj , we find the maximum profitthat is available under TS:

�TS =

1− t

2N− c if t ≤N�1− c

N �1− c 2

2tif t > N�1− c �

(30)

Probabilistic Selling StrategyWe start by assuming the probabilistic goods are created

according to part (c) of Proposition 3 in order to demon-strate parts (a) and (b) of Proposition 3. Then, we show thatthe seller can not improve its profit by creating additionalor alternative probabilistic goods.Since each consumer has an expected value of 1− t/�2N

for the probabilistic good that combines the two componentproducts that lie closest to his or her ideal point, the selleroffers each probabilistic good at the price PPSo = 1− t/�2N .Profit under PS is given by

�PS�PPSj = 2N(1− PPSj

t

)�PPSj − c

+ 2N(12N

− 1− PPSj

t

)(1− t

2N− c

)� (31)

This profit is maximized at PPSj = 1 − t/�4N , yielding aprofit of

�PS = 1− 3t8N

− c� (32)


Comparing the profit under PS and TS:

�PS−�TS =

t

8N> 0 if c ≤ 1− t

N

1− c− 3t8N

− N�1− c 2

2t

> 0 if c < c

≤ 0 if c ≥ c�

if c > 1− t

N

where c = 1− t

2N� (33)

Offering these probabilistic goods strictly increases profit ifc < c, thus proving part (a) of Proposition 3. Furthermore,�c/�N = t/�2N 2 > 0, thus proving part (b) of Proposition 3.Now we verify part (c) of Proposition 3. Under the pro-

posed equilibrium, there is complete market coverage andthe probabilistic goods are priced so that no consumer earnsa strictly positive surplus from purchasing a probabilisticgood. Thus, the only way an alternative probabilistic goodcould increase profit is if it has a higher expected value tosome consumer. Without loss of generality (due to symme-try), consider the segment of consumers located betweentwo component products j and j + 1. The proof consists oftwo parts: First, we show that it is not optimal to sell theseconsumers a probabilistic good that consists of any com-ponent good other than j or j + 1. Second, we show thatsymmetrically assignments of j and j + 1 are optimal. Seethe Technical Appendix for further details online at http://mktsci.journal.informs.org.

Sketch of Proof of Proposition 4The analysis assumes that c = 0 and t = 1. Profit under TSand PS are given in Table A.2. Proposition 4 summarizesthese results.

Traditional Selling StrategyThere are two potential pricing strategies: (a) PTS1 ≥ 1/2,

PTS2 ≥ 1/2 and (b) PTS1 ≥ 1/2, PTS2 < 1/2. In scenario (a), themaximum obtainable profit of �TS = 1/2. In scenario (b)there are “crossover” sales, i.e., some consumers who preferproduct 1 will buy product 2. Profit is maximized at a cor-ner solution in which there is full market coverage. Here,profit is �TS = �1+ 2$ 2/�16$ which is strictly larger than1/2 if $ > 1/2.

Probabilistic Selling StrategyWe derive the equilibrium profit under PS in two parts.

First, we assume �∗ = 1/2. Then, we verify that � = 1/2

Table A.2 Profits with Product Differentiation in Popularity

Strategy Price Profit

Traditional selling (TS) P TS2 < P TS

1 �TS = 1+ 2�2

16�

Probabilistic selling (PS) P PS2 = P PS

1 �PS = 58

∗ = 12

P PSo = 1

2

Comparison ��=�PS −�TS = −4�2 + 6�− 116�

> 0,

��

��= 1− 4�2

16a2≤ 0

maximizes profit. If �= 1/2, then PPSo = 1/2. The profit func-tion is:

�PS = 2$PPS1 �1− PPS1 + 2�1−$ PPS2 �1− PPS2

+ �1− 2$�1− PPS1 − 2�1−$ �1− PPS2 12 � (34)

The maximum obtainable profit is �PS = 5/8. Comparingthis to the profit earned under TS:

��=�PS−�TS = −4$2+ 6$− 116$

> 0� (35)

The comparative statics in Table A.2 immediately follow.To verify that �∗ = 1/2, we show that for � ≥ 1/2, profit

under PS is convex in � and strictly decreasing in � at �=1/2. Thus, profit under PS is maximized either with sym-metric assignments (i.e., �∗ = 1/2) or not selling the proba-bilistic good at all (i.e., �∗ = 1).Sketch of Proof of Proposition 5Table A.3 gives the profit under TS and PS with and with-out demand uncertainty. Proposition 5 summarizes theseresults.

Traditional Selling StrategyWithout demand uncertainty, Table A.2 shows that opti-

mal profit is achieved with prices PTS1 > PTS2 . With demanduncertainty, the seller can not sell the popular good at aprice premium because it does not know whether product1 or product 2 is the popular one. Thus, the seller maxi-mizes its profit such that PTS1 = PTS2 . This yields a profit of�TS = 1/2.Probabilistic Selling StrategyUnder PS, �∗ = 1/2 and PPSO = 1/2. With demand uncer-

tainty, the seller maximizes (34) s.t. PPS1 = PPS2 . This leads tothe same prices and profit (�PS = 5/8) as in the no demanduncertainty case.

Sketch of Proof of Proposition 6Here, the seller faces both capacity constraints and demanduncertainty. Starting with the case where the seller facesonly demand uncertainty (studied above), we add the addi-tional constraints that production of each good can notexceed K. If K is sufficiently large, these constraints are notbinding under either TS or PS. Thus, the profit under eachstrategy is unaffected by capacity constraints, and we have�CC−NCCDU = 0. Furthermore, for moderately large K, capacityconstraints force a deviation from the unconstrained out-come under TS but do not affect PS. Thus, in this range, it

Table A.3 Profit with and Without Demand Uncertainty

Uncertainty about relative product popularity

No uncertainty With uncertainty Impact of uncertaintyStrategy (NDU) (DU) ��t

DU =�tDU −�t

NDU

Traditionalselling (TS) �TS

NDU =1+ 2�2

16��TS

DU =12

��TSDU =− 2�− 12

16�< 0

Probabilisticselling (PS)

�PSNDU =

58

�PSDU =

58

��PSDU = 0

Comparing (a) �PSDU >�TS

DU, (b) �PSDU −�TS

DU > �PSNDU −�TS

NDU,

(c) ��TSDU <��PS

DU


Figure A.1 Impact of Capacity Constraints When There Is DemandUncertainty

CC–NCC∆DU =CC–NCC∆DU > 0

CC–NCC∆DU < 0

0.5

K

1.0

1.0

0

0.5

α

must be the case that �CC−NCCDU > 0. We then identify addi-tional regions for which �CC−NCCDU > 0.

Traditional Selling StrategyThe profit function under TS is given in the proof of

Proposition 4. Facing capacity constraints and demanduncertainty, Equation (36) gives the maximum obtainableprofit:

�TS =

K�2$−K

2$2if Kf ≤K <

2$+ 14

K�2�1−$ −K

1−$if K < Kf

�

where Kf =2$�1−$

1+$� (36)

Probabilistic Selling StrategyWe focus only on identifying potential scenarios for

which �CC−NCCDU > 0. Using the �TS defined in Equa-tion (36), we look at any probabilistic selling strategy thatincreases the magnitude of (�PS−�TS) (versus the value of(�PS

DU−�TSDU) as reported in Table A.3).

If K ≥ 1/2, the optimal strategy is to cover the entiremarket and set prices so that the capacity constraint forthe popular good is just binding. Here, �CC−NCCDU > 0 ifK > $�3− 2$ /2≡K.If K < 1/2, full market coverage is not feasible. The seller

has two feasible strategies in which positive sales of theprobabilistic good would be sold: (a) sell all three productsor (b) only sell the probabilistic product. Under each option,we find that �CC−NCCDU < 0. Proposition 6 summarizes theseresults. Figure A.1 presents a graphical illustration.

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probabilistic goods: a creative way of selling products and services

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