MANAGEMENT SCIENCE
Vol. 56, No. 3, March 2010, pp. 430–448
issn 0025-1909 eissn 1526-5501 10 5603 0430
inf
orms
®
doi 10.1287/mnsc.1090.1125
© 2010 INFORMS
Revenue Management with Strategic Customers:
Last-Minute Selling and Opaque Selling
Kinshuk Jerath
Tepper School of Business, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, [email protected]
Serguei Netessine, Senthil K. Veeraraghavan
The Wharton School, University of Pennsylvania, Philadelphia, Pennsylvania 19104
C
ompanies in a variety of industries (e.g., airlines, hotels, theaters) often use last-minute sales to dispose
of unsold capacity. Although this may generate incremental revenues in the short term, the long-term
consequences of such a strategy are not immediately obvious: More discounted last-minute tickets may lead
to more consumers anticipating the discount and delaying the purchase rather than buying at the regular
(higher) prices, hence potentially reducing revenues for the company. To mitigate such behavior, many service
providers have turned to opaque intermediaries, such as Hotwire.com, that hide many descriptive attributes of
the service (e.g., departure times for airline tickets) so that the buyer cannot fully predict the ultimate service
provider. Using a stylized economic model, this paper attempts to explain and compare the benefits of last-
minute sales directly to consumers versus through an opaque intermediary. We utilize the notion of rational
expectations to model consumer purchasing decisions: Consumers make early purchase decisions based on
expectations regarding future availability, and these expectations are correct in equilibrium. We show that direct
last-minute sales are preferred over selling through an opaque intermediary when consumer valuations for
travel are high or there is little service differentiation between competing service providers, or both; otherwise,
opaque selling dominates. Moreover, contrary to the usual belief that such sales are purely mechanisms for
disposal of unused capacity, we show that opaque selling becomes more preferred over direct last-minute selling
as the probability of having high demand increases. When firms randomize between opaque selling and last-
minute selling strategies, they are increasingly likely to choose the opaque selling strategy as the probability of
high demand increases. When firms with unequal capacities use the opaque selling strategy, consumers know
more clearly where the opaque ticket is from and the efficacy of opaque selling decreases.
Key words: distribution channels; competition; revenue management; strategic consumer behavior;
rational expectations
History: Received March 18, 2008; accepted November 6, 2009, by Ananth Iyer, operations and supply chain
management. Published online in Articles in Advance February 12, 2010.
1. Introduction
Firms in the travel industry (e.g., airlines, hotels, and
car rentals) face the problem of uncertain demand for
their services. Although these firms typically begin
by selling regularly priced products through direct
channels as well as through a variety of interme-
diaries, later in the selling cycle for the product,
they often utilize last-minute sales discounts to sell
off their leftover capacity (in the airline industry,
often termed “distressed inventory”). Because service
capacity in these industries is hard to adjust in the
short term and the marginal cost of providing ser-
vice is negligible, such dynamic discounts are perva-
sive. One common practice that firms adopt is to sell
last-minute tickets at low prices through their own
websites. For instance, in the case of US Airways,
consumers can visit http://www.usairways.com/
awa/faresale/eSaver.aspx and find the current week’s
discounted fares. Unfortunately, last-minute sales
directly to consumers are dangerous in that they con-
dition potential consumers to expect that there might
be a deal available at the last moment at a much
lower price. As a result, some consumers may pre-
fer to wait for last-minute sales rather than purchase
earlier at a higher price. Many industry analysts,
observers, and executives have questioned the last-
minute sales approach altogether because it “starts
a cycle of price degradation that will eventually lead
to destroying the airlines” (Sviokla 2004).
Another mechanism for selling distressed inventory
appeared more recently under the name of opaque
selling. Before purchasing the opaque product from
an intermediary, a consumer does not know which
company will ultimately provide the service, when
exactly will the service commence, and how long will
it take (for air tickets), etc. In the travel industry, firms
430
Jerath, Netessine, and Veeraraghavan: Revenue Management with Strategic Customers
Management Science 56(3), pp. 430–448, © 2010 INFORMS 431
like Hotwire.com, CheapTickets.com, Priceline.com,
1
OneTravel.com, and many others have popularized
this selling mechanism in which the intermediaries
hide some attributes of the product from the con-
sumers and reveal them only after the purchase has
been made.
This is the opposite of transparent sales, discussed
above, whereby all attributes are observable up front.
One frequently cited reason for the existence of
the opaque channel is that it “enables airlines to
generate incremental revenue by selling distressed
inventory cheaply without disrupting existing distri-
bution channels or retail pricing structures” (Smith
et al. 2007, p. 75).
Using opaque sales can have two opposing effects:
By hiding key attributes of the product, the firms may
persuade some consumers to buy regularly priced ser-
vice (and to some extent mitigate the “cycle of price
degradation”), but there is also a chance that con-
sumers indifferent among multiple service providers
may be diverted to the opaque channel if “the price
is right.” Given that 60% of consumers shopping
online buy the lowest fare available (PhoCusWright
2004) and that opaque selling of products has been
widely prevalent in recent years among travel com-
panies (Lambert 2006, Harrison 2006), it is important
to understand how channel choice affects profitability,
prices, and consumer segmentation.
The goal of this paper is to understand the dynam-
ics of last-minute discounts and to shed some light
on the relative merits of last-minute sales directly
by the firm versus through an opaque intermediary.
To analyze these issues, we propose a stylized eco-
nomic model in which two firms sell horizontally
differentiated products to consumers on a Hotelling
line in two time periods. The firms have fixed capac-
ities but can adjust prices from period to period.
The firms use transparent sales in the first period
and may use either transparent last-minute sales or
opaque last-minute sales in the second period, if there
is leftover inventory. Our model yields the following
major findings:
1. When firms sell through opaque channels, con-
sumers (who prefer a ticket from one firm over
the other) form expectations about the availability
of the tickets from either firm and factor this into
their purchase decisions. When firms have symmet-
ric capacities, we demonstrate that in equilibrium
the consumers expect that the opaque product comes
from either of the two firms with equal probability
1
In addition to opacity, “Name-Your-Own-Price
®
” is an interesting
parallel concept that Priceline.com uses (i.e., consumers can haggle
for ticket prices). In this paper, we focus on studying opacity and
therefore abstract away from the bargaining/haggling process; see
Terwiesch et al. (2005) for further discussion of online haggling.
and the two firms supply equal quantities of their
products to the opaque channel; i.e., the firms achieve
“perfect masking” of the product identity.
2. When there is no uncertainty in demand, sell-
ing through the opaque channel weakly increases
the firms’ profits compared to selling only through
the direct channel. This is because using the opaque
channel as a “clean-up” mechanism by charging a
low opaque price but masking the product attributes
leads to ex ante expected surplus of zero for all
consumers—who, therefore, purchase the product.
This does not distort the regular pricing structure (i.e.,
in the absence of the opaque channel), whereas con-
sumers who would not have purchased otherwise, if
any, now purchase the opaque product.
3. When demand is uncertain, both opaque last-
minute sales and direct last-minute sales can lead
to higher profits under different conditions. Under
uncertain demand, consumers trade off the possibil-
ity of buying later at a lower price with the risk
of not buying at all (if demand turns out to be
high and inventory runs out). In opaque selling, the
identity of the product is masked, which makes the
ex ante expected surplus from an opaque product
zero. Hence, consumers do not benefit from waiting,
and the firms use this to charge high prices in the first
period, even when consumer valuations for tickets
are low. In direct last-minute sales, the firms price in
the first period to extract the surplus from consumers
who have a high preference for purchasing, whereas
low-preference consumers choose to wait and buy at
low prices through the last-minute sales if any prod-
ucts are left over. In this strategy, the firms make
the bulk of their profits through first-period prices.
However, if consumer valuations are low, these first-
period prices are low; as valuations increase, these
prices (and, correspondingly, firm profits) increase.
Hence, opaque sales are preferred over transparent
last-minute sales when consumer valuations are low,
and vice versa.
4. We demonstrate that, as the probability of
having high-demand realization increases, opaque
selling becomes more and more preferred over
direct last-minute selling. This finding is contrary
to the traditional understanding of opaque selling
as a mechanism to clear leftover inventories when
demand is low. The intuition behind this result is
again that masking the product identity leads to
ex ante expected surplus of zero from an opaque
product, so that consumers do not have a benefit
from waiting and they prefer to purchase in the first
period. As the probability of high demand (and there-
fore the possibility of not getting a product) increases,
the competition (or “clamor”) among consumers for
products in the first period increases, which enables
the firms to charge higher first-period prices and
increase profits.
Jerath, Netessine, and Veeraraghavan: Revenue Management with Strategic Customers
432 Management Science 56(3), pp. 430–448, © 2010 INFORMS
5. We analyze a situation in which firms random-
ize between selling through opaque or through their
own channels in the second period. We find that
as probability of high demand increases, firms are
more likely to adopt opaque selling strategies (i.e.,
they sell through opaque intermediaries with higher
probability).
6. When firms are asymmetric, with one firm hav-
ing larger capacity than the other, they will not be
able to achieve perfect masking of the opaque prod-
uct because consumers will rationally expect that
the probability that it is from the larger firm is
higher. Hence, because of the imperfect masking in
the opaque channel, the pricing power of the opaque
intermediary and the two firms will diminish. We find
that the firm with a larger capacity is at a greater dis-
advantage. In spite of this finding, using opaque sales
still helps asymmetric firms to increase their profits.
To summarize, our study is the first to shed light
on the comparative advantages of direct selling ver-
sus opaque selling as last-minute sales strategies.
We explicitly model the effect of consumers’ strate-
gic behavior regarding product availability on firms’
sales strategies to dispose of inventories, and we
characterize conditions under which firms prefer one
strategy over the other. The rest of this paper is orga-
nized as follows. In §2, we review related literature
from economics and operations and revenue man-
agement. In §3, we describe our model; in §4, we
analyze the case of deterministic demand; and in §5,
we analyze the case of stochastic demand. In §6, we
extend our basic model to the case of asymmetric
firms; in §7, we consider the consequences of relaxing
several other assumptions. We conclude in §8 with a
discussion.
2. Related Literature
The work on intertemporal sales started with the sem-
inal Coase conjecture (Coase 1972), which postulates
that given an infinite number of selling opportuni-
ties over time, a monopolist will eventually decrease a
product’s price to its marginal cost, and all consumers
will anticipate this decrease and delay their pur-
chases. Numerous subsequent papers modeled sce-
narios in which the Coase conjecture may not hold
(Stokey 1979, Besanko and Winston 1990, DeGraba
1995). Specifically, DeGraba (1995) suggested that
under uncertain demand, if product availability is
limited, consumers might not have an incentive to
wait for a lower price because of the threat of unavail-
ability. We build on this trade-off between price
and availability in the context of opaque sales when
consumers form rational expectations about future
availability.
Our paper contributes to the small but growing lit-
erature in operations management that models strate-
gic consumer behavior. Recent papers in this stream
that explicitly incorporate product availability con-
siderations and demand uncertainty into last-minute
sales models include Cachon and Swinney (2009), Lai
et al. (2010), Su (2007), Su and Zhang (2008), and
Yin et al. (2009). For instance, Cachon and Swinney
(2009) demonstrate that the value of quick response
strategies is higher in the presence of consumers
who strategically wait for sales. However, none of
the aforementioned papers considers competition and
opacity in product attributes. For a rich compilation
of recent work in operations management that incor-
porates strategic consumer behavior, the reader is
referred to Netessine and Tang (2009).
There is a rich literature studying revenue man-
agement practices in the travel industry (Talluri and
van Ryzin 2004), which is the primary, but not the
only, adopter of opaque selling. This literature usu-
ally assumes that availability of products or compe-
tition does not affect consumer demand (see Liu and
van Ryzin 2008 and Netessine and Shumsky 2005
for exceptions). Koenigsberg et al. (2008) consider the
impact of airline capacity and the number of customer
segments (differing in price sensitivity) on the pat-
tern of sales by airlines. However, strategic waiting
by consumers for low prices, an extremely important
issue that airlines face every day, has not been con-
sidered in the revenue management literature. This
paper is an attempt to unravel the impact of strategic
consumers on airline firms’ selling strategies.
2
To our knowledge, only a few very recent papers
explicitly consider opaque selling mechanisms. Jiang
(2007) considers opaque selling for two flights owned
by the same monopolist firm but scheduled at dif-
ferent times throughout the day. Fay and Xie (2008)
study “probabilistic selling,” under which a monop-
olist creates a probabilistic good by clubbing sev-
eral distinct goods together. Fay (2008) considers a
model of competition with deterministic demand and
shows that the opaque channel increases the degree
of price rivalry and reduces industry profits unless
firms have very loyal customers. Shapiro and Shi
(2008) model competing firms selling opaque prod-
ucts through a passive intermediary that posts prices
2
Xie and Shugan (2001) consider the strategy of advance selling of
tickets to strategic consumers. Here consumers’ valuations for the
product are uncertain in advance and are realized after the pur-
chase decision has been made. In our model, both consumers and
firms are certain about consumers’ valuations for the product; the
uncertainty is on the realization of demand and correspondingly
on future product availability. Su (2009) considers a two-period
model where firms face inertial and rational customers, assuming
that inertial customers have a tendency to refrain from purchasing.
Levin et al. (2009) consider dynamic pricing of limited capacity sold
over multiple periods to strategic customers.
Jerath, Netessine, and Veeraraghavan: Revenue Management with Strategic Customers
Management Science 56(3), pp. 430–448, © 2010 INFORMS 433
dictated by firms while hiding the identity of the
products. There are two segments of consumers with
different strengths of brand preference. They find that
opaque sales intensifies competition for customers
with low brand preferences and enables firms to com-
mit to high prices for customers with high brand pref-
erence so that the total profit can increase.
All of these papers utilize single-period models,
in which opaque sales occur simultaneously with
transparent sales. Our model is very different in
many ways. First, we explicitly recognize that the
opaque sales strategy is a last-minute sales mecha-
nism and we study it using the resulting dynamic
model. The dynamic aspect is an important prac-
tical consideration for opaque selling (Elkind 1999,
Harrison 2006): in the travel industry, opaque prod-
ucts are sold only a few days before service deliv-
ery,
3
whereas transparent products are sold at regular
prices up to a year in advance. Second, we also allow
for demand uncertainty, which is only resolved very
late in the selling horizon. By virtue of the above two
aspects, consumers face a trade-off between buying
at a higher price early versus at a lower price (under
the threat of stockout) later, which is the key trade-off
of our model. Third, we compare opaque selling with
another commonly observed last-minute sales mech-
anism, direct last-minute sales, and characterize the
conditions under which one is better than the other
for firms involved.
Versioning (Varian 2000) and “damaged goods”
(Deneckere and McAfee 1996) are related price dis-
crimination strategies in which a high-quality prod-
uct is sold in its original form and also in an inferior
version with some of its features disabled. Opaque
selling is related to these strategies because (i) the
same product is sold both through a transparent
channel and through an opaque channel with some
attributes hidden and (ii) airline reservations made
through opaque channels cannot typically be modi-
fied or exchanged without an extra fee. However, the
key difference for opaque selling is because it requires
the availability of an alternative competing product.
Furthermore, all consumers who purchase the inferior
product in the form of versioning or damaged goods
receive lower utility from consuming it (relative to
consuming the original product), whereas a fraction
of consumers that purchases the opaque product end
up receiving the same utility (and higher net surplus)
3
To be precise, opaque products are sometimes available in
advance, but they are priced at the same level as transparent prod-
ucts, which makes them unattractive to any consumer. This can
be observed through a simple experiment on Hotwire.com, which
starts selling deeply discounted opaque tickets only a few days
before the service date.
relative to what they would have received by pur-
chasing a transparent product. Finally, rational expec-
tations regarding product availability (because of lim-
ited capacity and demand uncertainty) never come
into consideration in either the versioning or the dam-
aged goods literature.
3. The Model
Two competing firms, A and B, each hold a quan-
tity K/2 of inventory (later, we consider firms with
asymmetric capacities). The inventories can be service
capacities, e.g., seats on flights operated by the firms
on a particular date or rooms for a particular day in
similar hotels run by the firms. The products are per-
ishable in the sense that they have to be sold before a
certain time and have no value if they remain unsold.
For travel services, this is a reasonable assumption
(e.g., products have no value after the flights take off,
and hotel rooms have no value if they remain unfilled
by the day under consideration).
Consumers have heterogeneous preferences be-
tween firms. The reason might be loyalty to the firm,
preference for a brand, or simply an established rela-
tionship with the company (e.g., through rewards
programs). We capture this consideration by invoking
a horizontal differentiation model similar to that of
Fay (2008) and other papers on opaque selling.
4
We
assume that the two competing firms A and B are
located at each end of a Hotelling line of length 1, and
a continuum of consumers is spread on the horizontal
line over the interval 0 1 with uniform density. A
population of J consumers is spread uniformly over
the entire line. We consider cases of deterministic low
demand (J<K), deterministic high demand (J>K),
and random demand (J = H>K with probability ,
and J = L<K with probability 1 − ).
Each consumer has a valuation V for the product
and purchases at most one unit. The brand preference
of every consumer is completely characterized by his
location x ∈ 0 1 on the line, which influences the
utility a consumer derives when he purchases a prod-
uct from a firm. The parameter t denotes the strength
of brand preference in the market. A consumer at x
incurs a disutility tx when buying a product from
firm A and a disutility t1 − x when buying a prod-
uct from firm B. Thus, the customers have varying
preferences toward the competing firms. If firms A
and B charge prices p
A
and p
B
, respectively, then a
consumer located at x receives net utility V − tx − p
A
when purchasing a product from firm A and receives
net utility V −t1− x− p
B
when purchasing a product
from firm B. We assume that V/t≥ 1/2 so that every
4
We assume that there is no vertical differentiation between prod-
ucts of the two firms; i.e., one product is not inherently superior to
the other for all customers.
Jerath, Netessine, and Veeraraghavan: Revenue Management with Strategic Customers
434 Management Science 56(3), pp. 430–448, © 2010 INFORMS
consumer would receive nonnegative utility from the
product from at least one of the firms if it were offered
for free by both firms.
We will encounter the ratio V/t frequently in the
analysis to follow. This ratio can be interpreted as a
“brand preference-adjusted valuation” for a product
and it reflects the degree of competition between the
firms. If V is large, the valuation for a product in the
market is high and the market will be competitive,
and vice versa. Further, if t is small, the consumers
do not care about the firm from which they buy and
competition will be high, and vice versa. Overall, as
V/t increases, the market becomes more competitive.
We divide the selling horizon into two periods
so that each firm has two pricing opportunities. We
assume no discounting between the two periods. The
firms choose one of the following strategies for selling
products:
1. The firms can sell the products through their
own channels in both periods and offer different
prices in each period of sale.
2. The firms can sell in the “transparent” channel
in the first period and sell opaque products through
an intermediary (such as Hotwire.com) in the sec-
ond period. The intermediary, denoted by I, makes its
own pricing decision p
I
. It keeps a fraction 1 − of the
revenue it makes. The remaining fraction of the rev-
enue from each product goes to the airline whose
product the intermediary sells (see Elkind 1999, Price-
line.com 2006, and Phillips 2005 for a description of
such arrangements in the travel industry).
Because the firms sell products over two periods,
possibly at different prices, the consumers strate-
gically time their purchases based on their valua-
tions, expectations of future inventory availability,
and the firms’ pricing strategies. In other words, a
consumer might decide to purchase products in the
second period rather than in the first period. A con-
sumer purchasing from the opaque intermediary does
not know from which firm the product is coming,
but develops expectations about the service provider
based on expectation of inventory supplied to the
intermediary by the two competitors. All consumers
have the same beliefs. To differentiate from the actual
equilibrium outcomes, we represent beliefs (or expec-
tations) by using the superscript e.
We assume consumers are forward looking and
have rational expectations; i.e., the availability of tick-
ets from each firm in every period matches con-
sumers’ expectations on the availability.
5
Amaldoss
5
The rational expectations concept assumes strong rationality on
the part of the agent (Muth 1961); i.e., an agent can correctly expect
the future equilibrium path in a multiperiod game and acts on
these expectations, so that this equilibrium indeed arises. There is
a debate in the literature around this strong rationality assumption.
and Jain (2008) have a similar model with two time
periods, where consumers in the first time period
form expectations about the future popularity of a
product and make their purchase decisions accord-
ingly. Diamond and Fudenberg (1989), Stokey (1981),
and several others solve similar multiperiod games
using the rational expectations concept.
We first consider the case of deterministic demand,
with both low demand (demand is lower than
capacity) and high demand (demand is higher than
capacity), and then consider the case of uncertain
demand. In all cases, the firms and consumers know
the values of all parameters. The above is a general
description of the model. We provide specific details
for each case as appropriate.
4. Deterministic Demand
In this section, we explore the two strategies of the
firms when demand is deterministic: (i) the firms
can sell through their own channels and have the
option of offering different prices in the two periods
of sale and (ii) the firms can sell opaque products
in the second period after sales in the first period
have concluded. We consider two possible scenarios
for each strategy: (i) low-demand scenario (J<K)
and (ii) high-demand scenario (J>K). The determin-
istic demand model helps us gain insights into the
players’ decisions when demand is lower or higher
than capacity and it serves as a logical building block
for the more complex model with demand uncer-
tainty (§5).
4.1. Selling Through Firms’ Direct Channels
When firms sell only through their direct channels
and demand is deterministic, we find that each firm
charges the same price in the two periods.
6
Intuitively,
if the firms were to try to charge a higher price in the
first period and a lower price in the second period,
the consumers, being strategic and having full infor-
mation about demand, would wait to buy products
until the prices were lowered. (In case of uncertain
demand, we will see that this result changes.) We pro-
vide the detailed analysis in §A1.1 in the technical
Experimentally, Amaldoss and Jain (2005), Sunder (1995), and oth-
ers show that markets can converge to predictions of rational expec-
tations equilibria, even though Garner (1982), Smith et al. (1988),
and others show that expectations at the individual level may not
always correspond to theoretically predicted individual-level ratio-
nal expectations. Liu and van Ryzin (2007) show that adaptive
learning mechanisms can also lead to an equilibrium identical to
the rational expectations equilibrium.
6
Prices would not be identical across periods if consumers dis-
counted their second-period utility. However, the discount-adjusted
prices would be identical across periods. Introducing discounting
makes the analysis more tedious, although all our insights continue
to hold.
Jerath, Netessine, and Veeraraghavan: Revenue Management with Strategic Customers
Management Science 56(3), pp. 430–448, © 2010 INFORMS 435
appendix (provided in the e-companion)
7
and provide
only the main insights here.
The prices charged and market covered depend on
whether demand is low or high and on the value of
the quantity V/t. In the case of low demand (J<K),
when V/t is small (1/2 ≤ V/t < 1), the firms act as
local monopolies, and each firm prices at V/2 to cover
the length V/2t on its side of the Hotelling line,
which is less than half. As V/t increases, the market
covered increases, until the firms start to compete—
specifically, when 1 ≤ V/t <3/2, each firm prices at
V − t/2, covers half the market and sells J/2 <K/2
tickets; and when V/t ≥ 3/2, each firm prices at t,
covers half the market, and sells J/2 <K/2 tickets.
In the case of high demand (J>K), the same
pattern of coverage that increases with V/t holds.
However, in this case, the firms never get into com-
petition and always act as local monopolies because
each firm can sell a maximum of K/2 tickets. Hence,
if V/t is large enough (specifically, V/t ≥ K/J), the
firms charge a price of V − K/2J t and cover exactly
K/2J<1/2 on their side of the market.
4.2. Opaque Selling
As we described in the introduction, firms often sell
products and services through opaque intermediaries
very close to the terminal time, i.e., after consumers
have bought in the transparent channel, but the firms
still have some inventory of products left over. In this
case, we assume that the firms declare that they might
sell through an opaque channel late in the selling
horizon (e.g., airlines selling opaque tickets through
Hotwire declare this by having their names listed
on the website Hotwire.com). Firms will engage in
opaque selling only if there are products that are left
unsold through their own direct channels. More for-
mally, the game proceeds in the following manner:
1. Every consumer is endowed with expectations,
given by
e
A
and
e
B
, about the probabilities that the
product he will obtain from the opaque seller will be
from firm A or firm B, respectively. In the first period,
firms A and B set prices p
A
and p
B
in the direct-to-
consumers channel (transparent channel).
2. Given prices p
A
and p
B
and his expectations
about availability in the opaque channel from both
firms, every consumer makes a purchase decision in
the transparent channel.
3. After the transparent channel sales are over, the
leftover products are made available to the opaque
intermediary I by both firms. The opaque intermedi-
ary sets a price p
I
for the opaque product. Consumers
who did not purchase in the transparent channel now
make their purchase decisions in the opaque channel.
7
An electronic companion to this paper is available as part of the on-
line version that can be found at http://mansci.journal.informs.org/.
A consumer may not obtain an opaque product if the
number of leftover products is less than the number of
consumers who are willing to purchase at price p
I
.We
denote the probability that the consumer can obtain
an opaque product by so that each consumer desir-
ing a product is equally likely to obtain it.
4. The opaque intermediary keeps a fraction 1 −
of the revenues from the opaque channel. The remain-
ing fraction is distributed between firms A and B in
proportion to the products sold for each firm.
8
Consequently, for the consumer at x
A
who is indif-
ferent between buying from firm A and buying in
the opaque channel, the following condition holds in
equilibrium:
V − p
A
− tx
A
= V − p
I
−
e
A
tx
A
−
e
B
t1 − x
A
We now characterize the equilibria under low- and
high-demand cases. Without loss of generality, we
focus on = 1; any ∈ 0 1 yields the same insights.
We provide the detailed analysis in §A1.2 in the tech-
nical appendix and provide the main insights here.
First, consider the case of low demand (J<K). To
solve for the equilibrium, we use backward induction.
We start with the opaque intermediary’s problem in
the second period—the intermediary has access to
consumers on the line segment x
A
x
B
(x
A
and x
B
are
arbitrary, and the intermediary can choose to cover
this market fully or partially). This determines the
actual number of leftover tickets from each firm and
therefore the realized probability of obtaining tickets
from firm A in the opaque channel through the expec-
tation function as
A
= K/2 − x
A
J /K/2−x
A
J +K/2−
1 − x
B
J . We solve for the equilibrium through the
following procedure.
The opaque intermediary sets the price p
I
it will
charge to sell the leftover tickets from the two firms.
Because this is the case of low demand, there is no
shortage of tickets, so that = 1. Based on the above,
the intermediary sets the price p
I
as function of loca-
tion of the indifferent customers x
A
, x
B
, and first-
period prices p
1
A
and p
1
B
. In the first period, both firms
set their prices, based on the prices set by the opaque
intermediary under all second-period possibilities.
As we demonstrate in §A1 in the technical
appendix, in equilibrium, the rational expectations of
the fraction of opaque products from each firm are
e
A
=
e
B
= 1/2. This implies that if the firms have equal
capacities, then it is rational for consumers to expect
that in the opaque channel half of the products come
from one firm and the other half come from the other.
8
This revenue sharing contract with opaque intermediaries is con-
sistent with observations and industry practice (see Elkind 1999,
Priceline.com 2006, Phillips 2005).
Jerath, Netessine, and Veeraraghavan: Revenue Management with Strategic Customers
436 Management Science 56(3), pp. 430–448, © 2010 INFORMS
This result further implies that, in equilibrium, at
price p
I
, the ex ante utility of each consumer from pur-
chasing in the opaque channel is V − p
I
− t/2, so the
intermediary charges a price of p
I
= V − t/2 to make
every consumer’s ex ante surplus equal to zero. The
benefit from the opaque channel lies in the fact that by
masking the identity of the product, the ex ante sur-
plus from purchasing an opaque ticket is zero for all
consumers in x
A
x
B
, and hence they purchase tick-
ets. Recall that in the transparent channel, the surplus
from purchasing a ticket is zero only for the customers
at x
A
and x
B
who are indifferent between purchasing
from firms A and B, respectively. For every other cus-
tomer in x
A
x
B
, the surplus is negative and they do
not purchase tickets in the transparent channel.
The price charged by the opaque intermediary, p
I
=
V − t/2, is always lower than the price charged in the
transparent channel in the first period, which is equal
to p
A
= p
B
= V/2 for 1/2 ≤ V/t≤ 1. Hence, the opaque
channel serves as a clean-up mechanism to increase
profits by selling leftover tickets at a low price to con-
sumers who would otherwise not have purchased at
all, and masking the identity of the tickets enables
this. When V/t≥ 1, there are no opaque sales because
each firm covers half the market directly with the
transparent sales. Hence, firms use opaque selling for
the range (1/2 ≤ V/t<1).
In the case of high demand (J>K), the
same insights hold in equilibrium,
A
=
B
= 1/2,
p
I
= V − t/2, and the ex ante expected surplus of con-
sumers in the opaque channel is zero because of
masking the product identity. However, in this case,
one difference is that as V/t increases and firms
increase their market coverage, they stock out when
they cover a length of K/2J, which occurs for V/t≥
K/J. Hence, firms use opaque selling for a smaller
range (1/2 ≤ V/t<K/J). The second difference is that
when firms do use opaque sales, all the consumers
in x
A
x
B
are willing to purchase opaque tickets, but
only a fraction = K − V /tJ /J − V /tJ < 1 actu-
ally obtain opaque tickets.
4.3. Comparison of Strategies Under
Deterministic Demand
In both high- and low-demand scenarios, the opaque
channel acts as a clean-up mechanism to dispose of
unsold products by selling them to consumers who
would otherwise not have purchased at all. Hence, if
opaque selling is used (when the market is not fully
covered by the transparent channels), it will strictly
improve firm profits.
Figure 1 depicts the optimal strategies for the
firms given different values of consumer valuations
(the ratio V/t) and inventory availability relative
to demand (the ratio K/J). Under both high and
low demand, firms sell products through the opaque
Figure 1 Equilibrium Outcomes Under Deterministic Demand for
Different Valuations and Capacity and Demand Scenarios
V/t
K /J
0.5 1.0 1.5
1
0
Opaque Competitive
equilibrium
Noncompetitive
equilibrium
Low demandHigh demand
channel only if V/t is small enough because in this
case the firms do not cover the full market in the
transparent channels and they use opacity as a mech-
anism to dispose of unsold products. As the ratio
V/t increases above a threshold, the firms have the
option of using an opaque channel, but they price in
the transparent channels to cover the market anyway
and do not need to resort to selling cheaper opaque
products. Figure 1 also shows that if demand is high,
opaque sales will be seen less frequently (for a smaller
range of V/t) than if demand is low. This is consis-
tent with the notion that the opaque channel is used
to dispose of distressed inventory (Harrison 2006).
5. Uncertain Demand
Uncertainty in demand volume is a pervasive fea-
ture in many industries. In the travel industry, for
example, firms usually can estimate the demand dis-
tribution for a given airline route or hotel using his-
torical records, but the precision of such estimates
is quite limited (see Talluri and van Ryzin 2004).
As the departure date approaches, the firms can
improve the forecast and therefore project with a
higher degree of confidence whether the demand for
the route is higher or lower than the available capac-
ity. Building on the analysis in previous sections, this
section extends our model to incorporate demand
uncertainty.
Due to the presence of demand uncertainty, con-
sumers cannot always credibly adopt a strategy of
waiting in the early stages of the game because mar-
ket demand could be high and products could be
unavailable later. However, a consumer can form
expectations about future availability and buy early if
the expected utility from doing so is higher than the
expected utility from waiting. These dynamics cap-
ture the practical consideration that not all consumers
wait for last-minute discounts, and they allow us to
Jerath, Netessine, and Veeraraghavan: Revenue Management with Strategic Customers
Management Science 56(3), pp. 430–448, © 2010 INFORMS 437
derive several insights beyond the model with deter-
ministic demand.
The specifications of the model remain the same,
except that the level of demand is now variable. We
assume that with probability , the total number of
consumers in the market is H (>K), and with proba-
bility 1− , the total number of consumers in the mar-
ket is L (<K). As before, each firm has capacity K/2.
The parameters , L, H, and K are common knowl-
edge. The selling horizon is divided into two periods.
In the first period, the firms and the consumers know
the distribution of demand but do not know its real-
ization (whether demand is H or L). At the end of the
first period, but before the second period begins, the
realization of demand is observed by the firms and
the consumers.
9
We assume that in any selling period, if the num-
ber of consumers who are willing to buy a product is
higher than the capacity available, products are allo-
cated randomly to the consumers. In other words, if a
certain number of consumers desires products at the
announced price, but the number of products avail-
able is lower than the number of products demanded
(which can be the case if demand is high), it is
possible that consumers with a lower expected (but
positive) surplus obtain products at the expense of
consumers with a higher expected surplus. In the fol-
lowing sections, we analyze the two strategies of sell-
ing through the firms’ direct channels (“last-minute
sales strategy,” or LMSS) and opaque selling (“opaque
sales strategy,” or OpSS).
5.1. Selling Through Firms’ Direct Channels
The following is the order of events in the game when
firms adopt an LMSS.
1. In the first period, firm A prices its products at
p
1
A
, firm B prices its products at p
1
B
, and both firms
declare that there might be last-minute sales.
2. All consumers form expectations about the num-
ber of consumers purchasing in each period (and
therefore the corresponding future availability) and
strategically make or postpone their purchase.
3. At the end of period 1 and before period 2
begins, demand uncertainty is fully resolved. The
level of demand is determined as H or L and is
observed by both the firms and the consumers. The
firms then set their prices (firm A sets price p
2L
A
if
demand is low and p
2H
A
if demand is high, and simi-
larly for firm B).
4. The consumers who postponed their purchase in
the first period decide to purchase or not in the sec-
ond period at the announced prices.
The equilibrium solution LMSS is provided in the
following proposition.
9
In practice, some residual uncertainty in demand would remain.
Proposition 5.1. When the firms sell products
through their own channels, the following equilibrium
always exists: In the first period both firms set prices to
cover x
A
= 1 − x
B
= K/2H of the market. If demand is
high, no products are sold in the second period because
the firms stock out in the first period. If demand is
low, consumers located between x
A
= K/2H and
x
B
= 1 − K/2H buy in the second period. The first-
and second-period prices are as follows:
Second-period
First-period prices prices when demand
p
1
A
= p
1
B
is low p
2L
A
= p
2L
B
V
t
1
2
≤
V
t
< 1 −
K
2H
1 +
2
V −
K
2H
t
1
2
V −
K
2H
t
1 −
K
2H
≤
V
t
V −
K
2H
t
V −
t
2
<
3
2
−
K
H
+ 1 −
V −
t
2
V
t
≥
3
2
−
K
H
V −
K
2H
t
t
1 −
K
H
+ 1 − t
1 −
K
H
In the equilibrium, all consumers who attempt to
buy a product in the first period obtain a product
but pay the high price p
1
A
or p
1
B
as in the proposition
above. Note that the first-period prices are such that
the customer who is indifferent between purchasing
in the first period from firm A and waiting for the
second period to purchase from firm A has a posi-
tive (and of course equal) surplus from purchasing the
ticket in either period. If demand is high, firm A sells
to HK/2H= K/2 consumers in the first period and
thus exhausts its capacity, so there are no products
sold in the second period through last-minute sales. If
demand is low, firm A sells to K/2H· L< K/2 con-
sumers in the first period and will have some prod-
ucts left over to sell in the second period. Moreover,
there are more of these leftover products than the
number of unserved consumers in the market in the
second period. Therefore, the consumers who waited
for the “last-minute” products obtain them at lower
prices only if demand is lower than capacity. The sit-
uation is symmetric for firm B.
To summarize, in the first period all consumers
with “high brand preference” (located in the interval
0K/2H) purchase at a high price from firm A.
If there are any leftover products, the consumers
with “low brand preference” (located in the interval
K/2H1/2) purchase from firm A during the last-
minute sales at lower prices. If there are no leftover
products, there are no sales in the second period. In
effect, the firms are separating out consumers who are
ready to pay a higher price under the threat of stock-
out and making most of their profits from the higher
Jerath, Netessine, and Veeraraghavan: Revenue Management with Strategic Customers
438 Management Science 56(3), pp. 430–448, © 2010 INFORMS
prices charged to the high-preference consumers in
the first period.
5.2. Opaque Selling
The following is the order of events in the game when
the firms adopt an opaque sales strategy.
1. Every consumer is endowed with expectations
about the probabilities that the product he will obtain
from the opaque seller in the second period will be
from firm A or firm B for both high- and low-demand
realizations. In the first period, firm A prices its prod-
ucts at p
1
A
, firm B prices its products at p
1
B
, and both
firms declare intention of sales through an opaque
channel.
2. Consumers strategically purchase or postpone
purchasing based on expectations about availability in
the second period () and about the probability with
which they will obtain the opaque product from each
firm (
He
A
He
B
L e
A
L e
B
).
3. At the end of period 1 and before period 2
begins, demand uncertainty is resolved; the level of
demand is determined as H or L and is observed by
the firm and the consumers. The leftover products, if
any, are made available to the opaque intermediary I,
who then sets a price p
H
I
if the demand realization is
H or a price p
L
I
if the demand realization is L.
4. Consumers who have not purchased in the trans-
parent channel now make their purchasing decision
in the opaque channel. The intermediary commits to a
credible opaque strategy and sells products from both
firms at price p
I
. For every product sold, it keeps a
fraction 1− of the revenue accrued from the opaque
channel and distributes the remaining fraction to
firm A or B whose product it sold.
Let the locations of consumers indifferent between
buying in the first period and buying opaque tick-
ets in the second period be x
A
and x
B
. Then we have
the following two cases based on the realization of
demand.
1. If the level of demand is low, then leftover
products for firm A must be l
L
A
= maxK/2 − x
A
L 0
and leftover products for firm B must be l
L
B
=
maxK/2 − 1 − x
B
L 0. In equilibrium, the customers
will obtain a product from firm A with probability
L
A
= l
L
A
/l
L
A
+l
L
B
and from firm B with probability
L
B
=
l
L
B
/l
L
A
+l
L
B
. (If there are no leftover products, the prob-
abilities are set to zero.)
2. If the level of demand is high, then the leftover
for firm A is l
H
A
= maxK/2 − x
A
H0 and the leftover
for firm B is l
H
B
= maxK/2 − 1 − x
B
H 0. In equi-
librium, if the opaque channel exists, the customers
will obtain a product from firm A with probability
H
A
= l
H
A
/l
H
A
+ l
H
B
and from firm B with probability
H
B
= l
H
B
/l
H
B
+ l
H
B
, if defined.
Based on the demand realization and availability,
the opaque intermediary fixes opaque ticket prices.
Based on the second-period decision of the opaque
intermediary, the firms optimally fix the first-period
prices. The equilibrium of the above game is char-
acterized in Proposition 5.2. (To keep results simple,
we present the case with = 1. The analysis for any
∈ 0 1 yields the same insights.)
Proposition 5.2. When the firms sell products
through the opaque intermediary, the following rational
expectations equilibrium exists:
First-period prices Opaque prices
p
A
= p
B
(p
L
I
p
H
I
)
V
t
1
2
≤
V
t
<
K
H
V
2
V −
t
2
V −
t
2
K
H
≤
V
t
<
K
H
+
1 −
K
2L
V −
K
2H
tV−
t
2
,–
K
H
+
1 −
K
2L
V
2
+
1 −
K
4L
tV−
t
2
,–
≤
V
t
< 1 +
1 −
K
2L
1 +
1 −
K
2L
V −
t
2
–, –
≤
V
t
<
3
2
+
1 −
K
L
V
t
≥
3
2
+
1 −
K
L
1 +
1 −
K
L
t –, –
Note that in equilibrium, consumers have the same
probability of obtaining a ticket from either firm in the
opaque market (if it exists), irrespective of whether
the demand realization is high or low. This also
implies that the consumer who is indifferent between
purchasing a transparent ticket in the first period and
an opaque ticket in the second period has the ex ante
expected surplus of zero.
There is, however, a critical difference from
the deterministic demand case. Under deterministic
demand consumers know the state of demand, and
the opaque channel is primarily a clearance mech-
anism used when the entire market could not be
covered by the firms through transparent prices. In
contrast, when demand is uncertain, there is an addi-
tional factor—because the consumers do not know the
state of demand in the first period, they face the pos-
sibility of the firms stocking out if demand is high.
In other words, if a consumer has nonnegative util-
ity in the first period at the price offered by a firm,
then he will purchase the product, inferring that he
might not obtain it at all later if the demand turns
out to be high. This consideration allows the firms to
charge higher prices in the first period as compared
to the prices in the deterministic case. Consequently,
if demand is low, only a few products will be sold
in the first period. However, in this eventuality, the
Jerath, Netessine, and Veeraraghavan: Revenue Management with Strategic Customers
Management Science 56(3), pp. 430–448, © 2010 INFORMS 439
firms can use the opaque channel to “clean up” the
leftover products if any. Selling to a smaller popula-
tion at higher prices in the first period helps the firms
to increase the expected profit across two periods.
The above argument naturally leads to the inter-
esting insight that as the probability of high demand
increases, the firm will rely more and more on the
opaque channel (the formal proof is straightforward
and is therefore omitted): If there is a greater chance
that demand is high, the “competition for products”
among consumers in the first period will be higher,
which means that the firms will be able to raise the
first-period prices. If demand turns out to be high, the
firms will exhaust their capacities. On the other hand,
even if demand turns out to be low, there will still
be some consumers left in the market because of high
first-period prices. Consequently, there will be some
leftover products, and firms will sell them through
the opaque channel.
5.3. Opaque Sales vs. Last-Minute Sales
We saw in the previous two sections that both LMSS
and OpSS can increase firms’ profits. In this sec-
tion, we can use our analytical results to answer
the following question: When should firms employ
LMSS versus OpSS? To illustrate our insights better,
we compare the profits of the firms for these two
strategies for a representative set of parameter values
( = 1/2, K = 1, L = 1/2, H = 3/2, t = 1) in Figure 2.
We use numerical and graphical illustrations for expo-
sitional simplicity. Qualitatively, the results do not
change for other values of the parameters. From Fig-
ure 2, we obtain the insight that if V is low, the
profits from OpSS are higher than the profits from
LMSS which is consistent with the intuition stated
above. As V increases, the profits from OpSS flatten
out, whereas the profits from LMSS keep increasing.
Figure 2 Profits Accrued by a Firm (
A
B
) Under Uncertain Demand
When the Firm Employs the Last-Minute Sales Strategy vs.
the Opaque Sales Strategy
0.5 4 6
Profits
V
0.5
1.0
0
2
Opaque
sales
Last-minute
sales
Note. For the figure, we use = 1/2, K = 1, L = 1/2, H = 3/2, t = 1, and
= 1.
Above a certain threshold for V , LMSS profits become
higher than OpSS profits.
To see why the above result holds, note that under
LMSS the bulk of a firm’s profits comes from products
sold in the first period to the consumers that are closer
to the firm on the Hotelling line. If the valuation for
products in the market is high (i.e., V is high), the
first-period prices are high. However, if the valuation
for products is low, the first-period prices are low, the
second-period prices are even lower, and hence prof-
its from LMSS are low. In OpSS, on the other hand,
the first-period prices are higher than in LMSS for
low V because each firm is choosing to cover only a
small portion of the market in the transparent chan-
nel by charging a price that makes the surplus of the
marginal consumer equal to zero and covers the rest
using the opaque channel. Moreover, note that the
second-period prices in the opaque channel (if opaque
sales are present) are equal to or higher than the
second-period prices for LMSS (except when V/t <
1 − K/2H). As in the deterministic demand case, by
masking the identity of the product, the opaque inter-
mediary can sustain relatively higher second-period
prices. Hence, for low V/t, OpSS yields higher profits.
As V increases, the revenue from LMSS increases
faster, because the firms are able to separate out the
consumers with a high preference for a particular
firm and charge these consumers higher prices even if
demand is low. In OpSS, on the other hand, prices are
such that the firms cover a large portion of the market
at lower prices if demand is low. In fact, if V is high
enough, the firms are in a competitive equilibrium in
the first period itself under OpSS when demand is
low so that prices are very low. (In Figure 2, this is the
region where the OpSS profits level off.) Hence, when
V is high, LMSS yields higher profits because it allows
the firms to “milk” the high-preference consumers in
the first period, even if it has to charge lower prices
in the second period when demand turns out to be
low.
What happens as the probability of high-demand
realization increases? As we discussed earlier for
OpSS, as the probability of high-demand realization
increases, consumers are under a higher threat of
stockout in the first period. Thus, many more con-
sumers prefer to buy in the first period, and therefore
the firms increase prices. In other words, not only is
there a higher chance that demand is high, the prices
are also high. If demand turns out to be low, the first-
period sales suffer, but the leftover capacity is cleared
through the opaque channel. Over the two periods,
expected profits increase.
In LMSS, however, the firms charge a first-period
price that increases at a slower rate with an increase
in the probability of high demand. Further, consumers
with low firm preferences buy only if demand is low,
Jerath, Netessine, and Veeraraghavan: Revenue Management with Strategic Customers
440 Management Science 56(3), pp. 430–448, © 2010 INFORMS
Figure 3 Strategies the Firms Should Adopt for Different Consumer
Valuations (V/t) and Different Probabilities of High
Demand ()
Consumer valuations (V/t )
Probability of high demand ()
0.5 108642
1.0
0.8
0.6
0.4
0.2
0
Last-minute
selling optimal
Opaque selling
optimal
Notes. The shaded area denotes the region where the opaque selling market
exists for deterministic low demand (i.e., when V/t ≤ 1). For the figure, we
use K = 1, L = 1/2, H = 3/2, t = 1, and = 1. Qualitatively, the results do
not change for other values of the parameters.
which now happens with lower probability. Hence,
even though expected profits increase (because there
is a higher chance of high demand), the increase is
slower than in OpSS. Figure 3 summarizes the com-
parison between the opaque strategy and the last-
minute direct sales strategy for various probabilities
of high demand ( ∈ 0 1 on the y-axis) and con-
sumer valuations (V on the x-axis).
5.3.1. Results for Subgame Perfect Nash Equi-
librium.
10
In the rational expectations equilibrium
that we have derived above, consumers are assumed
to rationally predict the full equilibrium path of
the dynamic game; i.e., in the beginning of the
first period, they develop correct point expectations
regarding ticket availability (in equilibrium) from
each firm in the second period. In contrast, in a sub-
game perfect Nash equilibrium, expectations on ticket
availability from each firm are developed at the end
of the first period as a function of the prices charged
by the firms in the first period. Both rational expecta-
tions equilibrium and subgame perfect Nash equilib-
rium have been used as solution concepts for dynamic
games in the extant literature. The formulation using
the subgame perfect Nash equilibrium concept is
10
We thank an anonymous reviewer and the associate editor for
suggesting this line of analysis.
algebraically intractable for our model, so we con-
duct a numerical study to analyze this equilibrium.
We find that the results are qualitatively the same;
i.e., as the probability of high demand () increases,
opaque selling is preferred for larger values of con-
sumer valuation of the product (V/t). Details on the
numerical analysis are available in §A2.4 in the tech-
nical appendix.
5.4. Mixed-Strategy Equilibria
11
So far, we modeled a problem with two firms that
announce their strategies in the first period (OpSS or
LMSS), implicitly assuming that both firms select the
same strategy. This is a natural assumption because
if one firm does not select the opaque selling strat-
egy, the other firm cannot implement it alone. One
way to circumvent this difficulty and allow firms
to pick different selling strategies is to consider
mixed strategies—both firms randomize between the
strategies under consideration and the relevant strat-
egy is chosen in the second period, with some
probability.
Suppose firm A plays OpSS with probability
q
A
∈ 0 1 and LMSS with probability 1 − q
A
, and
firm B plays OpSS with probability q
B
∈ 0 1 and
LMSS with probability 1 − q
B
. Let
1
A
denote firm
A’s profit in the first period,
O2
A
denote firm A’s
expected profit in the second period when opaque
tickets are sold in the second period (so that total
expected profit is
O
A
=
1
A
+
O2
A
), and
T2
A
denote
firm A’s expected profit in the second period when
transparent tickets are sold in the second period (so
that total expected profit is
T
A
=
1
A
+
T2
A
). If firm A
plays OpSS, with probability q
B
opaque tickets will be
sold in the second period (if firm B also plays OpSS)
and with probability 1 − q
B
transparent tickets will be
sold in the second period (if firm B plays LMSS), firm
A’s expected profit will be equal to q
B
O
A
+ 1 − q
B
T
A
.
If firm A plays LMSS, only transparent tickets can be
sold in the second period, and its expected profit will
be
T
A
. Hence, for firm A, we obtain the condition for
mixing between OpSS and LMSS as
Expected Profit from OpSS
= Expected Profit from LMSS,
q
B
O
A
+ 1 − q
B
T
A
=
T
A
⇒
O
A
=
T
A
⇒
1
A
+
O2
A
=
1
A
+
T2
A
⇒
O2
A
=
T2
A
(1)
Similarly, for firm B, we obtain
O2
B
=
T2
B
. The above
result provides the interesting insight that each firm
will adopt a mixed strategy when the profits from
11
We thank the associate editor for suggesting this line of analysis.
Jerath, Netessine, and Veeraraghavan: Revenue Management with Strategic Customers
Management Science 56(3), pp. 430–448, © 2010 INFORMS 441
opaque sales and transparent sales are equal in the
second period (condition (1)). We obtain this condi-
tion because one firm alone cannot implement the
opaque selling strategy. We provide the rest of the
analysis in §A2.2 in the technical appendix and pro-
vide two salient insights here.
First, we obtain the mixing probabilities for both
firms as q
A
= q
B
=
H − K/1 − H + K. This
expression implies that as the probability of high
demand (given by ) increases, the firms choose
the opaque selling strategy with higher probabil-
ity. This is in line with the insight from Fig-
ure 3, which shows that firms prefer opaque sell-
ing as increases. Furthermore, as capacity becomes
more constrained (H increases or, alternatively, K
decreases), the firms choose the opaque selling strat-
egy with higher probability. Note, however, that the
mixing probabilities do not depend on L because
L<K and the market is always unconstrained in
the low-demand state. Second, when the firms mix
between opaque and transparent sales, the prices
charged for transparent tickets in the second period
are equal under high and low demand (and both
are equal to V − t/2). This is derived indirectly
from condition (1), which states that in both high-
and low-demand cases, the second-period profits
from transparent sales and opaque sales are equal.
Under opaque selling, in turn, prices are equal to
V − t/2 in both the high- and low-demand cases
(because for symmetric firms, tickets come from either
firm with equal probability). This leads to the result
that the prices charged for transparent tickets in the
second period are equal in high- and low-demand
cases.
6. Asymmetric Firms with Unequal
Capacities
12
So far, we assumed the two firms to be symmet-
ric in all respects, which leads to “perfect masking”
in the opaque channel; i.e., all consumers obtain a
ticket from either firm with equal probability. If firms
are asymmetric, with one firm having larger capacity
than the other, then consumers will expect that the
probability that an opaque ticket is from the larger
firm is higher. In this section, we extend the basic
model to investigate the implications of the inabil-
ity to achieve perfect masking in the opaque channel.
Broadly, we find that using opaque sales still helps
firms to increase their profits. However, the pricing
power of the opaque intermediary and of the two
firms indeed diminishes, and the firm with a larger
capacity is at a greater disadvantage.
12
We thank an anonymous reviewer for suggesting this analysis.
6.1. Opaque Selling with Deterministic Demand
Consider first the case of deterministic low demand;
i.e., the total capacity of the firms is more than the
total demand in the market. Here we generalize the
model in §4.2 by assuming that the capacities of
firms A and B are given by K
A
and K
B
, respectively,
and K
B
≥ K
A
. Our solution approach is also the same
as in §4.2. We provide the details in §A3.1 in the tech-
nical appendix and discuss some salient insights here.
Because firm A has smaller capacity, consumers
rationally expect that the probability that an opaque
ticket is from firm A is less than half; i.e.,
e
A
< 1/2.
13
This implies that in the opaque channel, the surplus
for a consumer located at x, given by V − p
I
−
e
A
tx −
1 −
e
A
t1 − x, increases with x. In other words, the
leftmost consumer on the Hotelling line who pur-
chases an opaque ticket will have the lowest surplus,
and all other consumers will have surplus greater
than the surplus of this consumer. The opaque inter-
mediary prices such that the consumer at x
A
, who
is indifferent between purchasing a transparent ticket
from firm A and an opaque ticket, has zero surplus.
(This gives us the equalities V − p
A
− tx
A
= V − p
I
−
e
A
tx
A
− 1 −
e
A
t1 − x
A
= 0, which also implies that
firm A prices its transparent ticket to make the sur-
plus of its marginal consumer equal to zero.)
The above arguments also imply that all other con-
sumers closer to firm B will have positive surplus
in the opaque channel. Now consider the consumer
at x
B
who is indifferent between purchasing a trans-
parent ticket from firm B and an opaque ticket. This
consumer prefers a ticket from firm B, and there is
a higher chance that the opaque ticket is from firm
B. Hence, this consumer has a positive surplus in
the opaque channel; i.e., V − p
I
−
e
A
tx
B
− 1 −
e
A
·
t1 − x
B
>0 because x
B
>x
A
. Hence, firm B will
charge a price p
B
such that V − p
B
− t1 − x
B
= V −
p
I
−
e
A
tx
B
− 1 −
e
A
t1 − x
B
>0. This implies that,
unlike firm A, firm B cannot price in the first period
to extract full surplus from its marginal consumer and
will therefore charge a lower price for its transparent
ticket than firm A.
The above discussion also clarifies why “imperfect
masking” in the asymmetric-capacities case reduces
the profits in opaque selling as compared to the
symmetric-capacities case. Under imperfect masking,
the larger-capacity firm cannot extract full surplus
from its indifferent consumer. In contrast, recall that
with perfect masking, in the symmetric-capacities
case, both firms were able to extract full surplus from
their indifferent consumers. Hence, in opaque sell-
ing with asymmetric capacities, the larger firm has to
13
Note that we also check that the case when the firm with the
larger capacity has fewer tickets in the opaque market (i.e., the
probability of obtaining an opaque ticket from the larger-capacity
firm is < 1/2) is off the equilibrium path.
Jerath, Netessine, and Veeraraghavan: Revenue Management with Strategic Customers
442 Management Science 56(3), pp. 430–448, © 2010 INFORMS
Figure 4 Trends in Equilibrium Expectations of Product Availability
and Prices in the Opaque Channel as the Capacity of
Firm B Increases
K
B
A
0.5
0.6
0.7
0.8
0.9
1
10
100
1,000
10,000
0.0
0.1
0.2
0.3
0.4
0.5
K
B
0.5
0.6
0.7
0.8
0.9
1
10
100
1,000
10,000
0.20
0.25
0.30
0.35
0.40
0.45
0.50
0.55
p
A
p
B
p
I
(a) Plot of
A
e
(=
A
) with K
B
(b) Plots of the prices p
A
, p
B
, and p
I
with K
B
Notes. Note the abrupt change in scale on the x-axis. For the plots above,
the values of the other parameters are chosen as V = 08, t = 1, J = 1,
= 1, and K
A
= 05.
leave some “surplus on the table” for its indifferent
consumer; whereas when the firms have symmetric
capacities, neither firm leaves any surplus on the table
for its indifferent consumer.
Figure 4(a) shows how the probability of obtain-
ing a ticket from firm A in the opaque channel
varies with the capacity of firm B. (Other choices of
the parameters yield qualitatively the same insights.
Notably, if V/t is higher, the prices are higher but fol-
low the same patterns.) When firms have equal capac-
ity, the probability is equal to half, and as firm B’s
capacity increases, the probability goes down. At the
extreme, as firm B’s capacity becomes much larger
than firm A’s capacity, this probability tends to zero.
Figure 4(b) shows the corresponding prices. When
the capacities are equal, firms A and B charge equal
prices. As firm B’s capacity increases and consumers
become more certain about the identity of the prod-
uct in the opaque channel, all prices decline. Firm B’s
price is always lower than that of firm A, and as its
capacity becomes much larger than the capacity of
firm A, its price approaches the opaque channel price.
Hence, in equilibrium, the firm with the larger capac-
ity is forced to price lower than the firm with the
lower capacity. Furthermore, even though the larger
firm has a greater total sales volume, its profit is lower
than that of the firm with the lower capacity due to
lower prices.
Other specifications, such as when the total capac-
ity in the market is held fixed but the percentage of
capacity held by one firm is increased, yield similar
insights. The case of asymmetric firms when demand
is deterministic and high (i.e., the total demand in the
market is more than the combined capacity of the two
firms) also yields similar insights. Because of space
considerations, we do not discuss these cases and pro-
ceed directly to the more interesting scenario when
demand is uncertain.
6.2. Opaque Selling with Uncertain Demand
We now generalize the model in §5.2 by assuming
that the capacities of firms A and B are given by K
A
and K
B
, respectively, and K
B
≥ K
A
. The details of the
solution are in §A3.2 in the technical appendix.
As in the deterministic-demand case, the mask-
ing of the opaque ticket’s identity is imperfect in
the uncertain-demand case; i.e., consumers know that
there is a higher probability that the opaque ticket
is from the larger-capacity firm (i.e., firm B), which
hurts the profits in the opaque channel (as compared
to profits with perfect masking in the symmetric-
capacities case). Overall, this imperfect masking leads
to lower prices charged by the intermediary in both
the high- and low-demand opaque channels and, in
turn, for both firms in the first period, with firm B
charging a price lower than firm A’s. Furthermore,
unlike in the symmetric case, consumers can have dif-
ferent expectations of product availability in the high-
and low-demand opaque channels, and this can lead
to different prices of the opaque product in the two
cases.
We now develop some salient insights with the
help of an illustrative example. Using other values for
these parameters yields qualitatively similar insights.
Jerath, Netessine, and Veeraraghavan: Revenue Management with Strategic Customers
Management Science 56(3), pp. 430–448, © 2010 INFORMS 443
Note from Figure 5(a) that as the capacity of firm B
increases, the probability that an opaque ticket is from
firm A decreases for both the high- and low-demand
opaque channels. Moreover, this probability is always
lower in the high-demand opaque channel: When
demand is high both firms sell more tickets (com-
pared to the case when demand is low) in the first
period, and therefore the proportion of tickets in the
opaque channel is larger (compared to the case when
demand is low) for the firm with higher capacity.
In other words, the probability that an opaque ticket
(if available) is from firm B (the larger-capacity firm)
is higher when demand is high than when demand
is low, the masking of the opaque product is less per-
fect, and the consumers know more clearly where the
opaque ticket is from. This implies that, ironically, the
opaque intermediary will charge a lower price for
the opaque product in the high-demand opaque chan-
nel than in the low-demand opaque channel. From
Figure 5(b) we see that this is indeed the case. We
also see that the first-period prices for both firms are
higher than the second-period opaque channel prices,
with firm B charging a lower price than firm A.
As the consumers’ valuation for tickets (V ) in-
creases, keeping other parameters constant, all prices
increase (as shown in Figure 5(d)) and firms A and
B cover a larger part of the Hotelling line through
transparent sales. Hence, both firms sell a larger num-
ber of tickets in the first period, the overflow of tick-
ets into the second period decreases for both firms,
and the proportion of tickets in the opaque channel
increases for the larger firm with increasing V . Hence,
as V increases, the probability that an opaque ticket
is from firm A decreases, and this decrease is sharper
in the case of high demand. This is shown in Fig-
ure 5(c). Finally, as the probability of high demand ()
increases, customers are under a larger threat of stock-
out in the second period. Hence, both firms charge
higher transparent channel prices in the first period
(Figure 5(f)) and cover a smaller market. As firms sell a
smaller number of tickets in the first period, the over-
flow of tickets into the second period increases for
both firms and the proportion of tickets in the opaque
channel decreases for the larger firm with increasing
for both high- and low-demand scenarios. Hence, as
increases, masking of the opaque product improves
(Figure 5(e)), which leads to increasing opaque prices
(Figure 5(f)).
To summarize the results above, opaque selling
always increases the profits of both firms in the
asymmetric-capacities scenario as compared to not
using opaque selling at all. However, imperfect mask-
ing of the opaque product reduces the efficacy of the
opaque channel as compared to perfect masking in
the symmetric-capacities scenario. Furthermore, this
imperfect masking hurts the prices and profit of the
larger firm more than it hurts the prices and profit
of the smaller firm, to the extent that the larger firm
makes overall lower profit across the two periods than
the smaller firm.
6.3. Comparison with LMSS
The above analysis leads to a natural question: Will
asymmetric firms prefer OpSS or LMSS? To answer
this question, we solve the LMSS game with asym-
metric firms and obtain the following insights.
14
The main difference between the asymmetric-
capacities case and the symmetric-capacities case is
that because firm B has larger capacity, it charges
lower prices than firm A and sells more than firm A.
Because of its higher sales, it makes a larger profit
than firm A (though this is lesser than the symmetric-
case profit due to reduced prices). However, even
with asymmetric capacities, the nature of the LMSS
equilibrium remains similar to that in the symmetric
case (described in Proposition 5.1). Specifically, both
firms price in the first period such that all consumers
with “high brand preference” purchase at high prices
from their preferred firms in the first period (because
of the threat of stockout later), and if demand is
high, both firms stock out in the first period itself. If
demand turns out to be low, there is leftover capacity
for both firms and they compete and price low in the
second period. The bulk of the profits for each firm
comes from the first period.
From a numerical comparison with OpSS, we find
that asymmetric firms prefer LMSS over a larger
region of the V /t- space. (Specifically, the equal-
profit contours in the V /t- space in Figure 3
shift more and more outward in the top-left direc-
tion as K
B
/K
A
increases.) The reason is that in LMSS
the nature of the equilibrium remains the same, as
described above. In OpSS, however, asymmetric firms
cannot achieve perfect masking in the opaque chan-
nel in the second period. Because the opaque chan-
nel profits hinge upon how well product identity can
be masked, imperfect masking significantly reduces
the efficacy of the opaque channel. As a consequence,
there is reduced preference for opaque selling with
asymmetric firms.
7. Other Modeling Considerations
Opaque sales and last-minute sales are encountered in
a variety of practical situations, many of which are not
fully reflected in our stylized model. Thus, apart from
analyzing the core model above, we point out some
modeling variations that that might form a good start-
ing point for various future research considerations.
14
We characterize the equilibrium in §A3.3 in the technical
appendix but do not provide the details of the solution due to space
constraints and because the derivation is almost exactly as in §A2.3
in the technical appendix.
Jerath, Netessine, and Veeraraghavan: Revenue Management with Strategic Customers
444 Management Science 56(3), pp. 430–448, © 2010 INFORMS
Figure 5 Trends in Equilibrium Expectations of Product Availability in the Opaque Channel and Prices as the Capacity of Firm B Increases
0.5 0.6 0.7 0.8 0.9 1.0
0.0
0.1
0.2
0.3
0.4
0.5
0.5 0.6 0.7 0.8 0.9 1.0
0.2
0.3
0.4
0.5
K
B
K
B
0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
0.0
0.1
0.2
0.3
0.4
V
0.6 0.7 0.8 0.9 1.0 1.1 1.2 1.3
0.0
0.2
0.4
0.6
0.8
V
0.2 0.4 0.6 0.8
0.35
0.40
0.45
0.50
A
H
0.2 0.4 0.6 0.8
0.2
0.3
0.4
0.5
0.6
p
A
1
p
B
1
p
I
L
p
I
H
(e) Plots of the probabilities
A
H
and
A
L
with ;
K
B
= 0.75, V = 0.8
(c) Plots of the probabilities
A
H
and
A
L
with V;
K
B
= 0.75, = 0.5
(a) Plots of the probabilities
A
H
and
A
L
with K
B
;
V = 0.8, = 0.5
(f) Plots of the prices p
A
1
, p
B
1
, p
I
H
, and p
I
L
with ;
K
B
= 0.75, V = 0.8
(d) Plots of the prices p
A
1
, p
B
1
, p
I
H
, and p
I
L
with V;
K
B
= 0.75, = 0.5
(b) Plots of the prices p
A
1
, p
B
1
, p
I
H
, and p
I
L
with K
B
;
V = 0.8, = 0.5
A
L
A
H
A
L
A
H
A
L
p
A
1
p
B
1
p
I
L
p
I
H
p
A
1
p
B
1
p
I
L
p
I
H
Notes. For the plots above, we use t = 1, L = 075, H = 125, = 1, and K
A
= 05. Values of the other parameters are specified next to the corresponding
plots. Other choices of the parameter values yield qualitatively the same plots.
Jerath, Netessine, and Veeraraghavan: Revenue Management with Strategic Customers
Management Science 56(3), pp. 430–448, © 2010 INFORMS 445
7.1. Damaged Opaque Goods
In our basic model, we assume that consumers derive
the same utility from flying with a firm regardless of
whether they buy in the direct channel or the opaque
channel. However, firms sometimes force opaque sell-
ers to sell “damaged” goods. For instance, firms
might disallow re-booking or charge high cancelation
fees for opaque goods. One way to make this consid-
eration consistent with our model is to impose that
“damaging” the goods affects valuations that con-
sumers obtain on opaque purchases. In our model,
this can be admitted by discounting the valuation
in the opaque channel to
o
V , where
o
∈ 0 1 is a
factor that accounts for the possible disutility from
the reduced flexibility in the opaque channel. As a
consequence of reduced valuation, the intermediary
charges a reduced price p
I
=
o
V − t/2 in equilibrium.
We note that in the symmetric model, this change
affects the prices only on the opaque selling chan-
nel, whereas the LMSS prices remain unaffected. As a
result, the revenues for the intermediary are reduced
in the opaque channel, and more consumers purchase
directly from the firms in the first period. Hence,
firms are more likely to sell in their own channels.
Thus, if opaque goods are damaged, LMSS may be
preferred to OpSS over a larger range of problem
parameters.
7.2. Concentrated vs. Monopolistic Markets
In this paper, we analyzed competing firms sell-
ing through an opaque intermediary. In some situa-
tions, however, the same service providers sell opaque
products as well. For example, Norwegian Cruise
Line offers both specific staterooms on their cruise
ships as well as opaque staterooms that guarantee
certain minimal amenities but not a specific location
on the ship. Similarly, one airline might be selling
opaque tickets with different departure times on the
same route (e.g., morning versus evening flights). We
considered a variation of our model in which both
transparent products are managed by the same firm,
which maximizes its total profit. We find that the
monopoly firm is able to derive higher profit from
LMSS (because second-period prices for transparent
tickets are set in a monopolistic rather than a com-
petitive scenario), whereas our other results remain
qualitatively unchanged. Thus, without competition,
LMSS may be preferred to OpSS over a larger range
of problem parameters.
7.3. Heterogeneous Values for the Core Product
In the basic model, we assumed that consumers are
homogeneous in their preference for the core prod-
uct; i.e., valuation V does not vary by consumer. In
practice, some companies (e.g., airlines) derive sig-
nificant profits by discriminating between “business”
and “leisure” travelers, who typically have drastically
different preferences for travel.
We believe that these considerations will not impact
the main insights from the model because consumers
with high utility for product consumption are likely
to purchase the product at full price and would not
participate in either opaque or last-minute sales chan-
nels. Thus, our model focuses exclusively on price-
conscious consumers with relatively low value for the
product itself. It is, however, possible to incorporate
into our model heterogeneous consumers that dif-
fer in their core value for the product. For instance,
we could introduce a second Hotelling line with a
much larger core value
V representing consumers
with high valuation for the product. (We observe
that this approach introduces more complexity to the
model.) Because these consumers have high willing-
ness to pay, the firms will allocate capacity to sat-
isfy these consumers first and then sell to consumers
with lower V . Future research can carefully explore
how this might affect the trade-off between OpSS and
LMSS.
7.4. Multiple Hidden Product Attributes and
Vertical Product Differentiation
In our model, we assume that products are character-
ized by a single attribute: the company that sells it. In
practice, however, products may differ along multiple
dimensions. Hotel rooms purchased on Hotwire.com
differ in size, location, and amenities. Airline tickets
differ in the number of stops, departure times, and
trip lengths. All these different attributes can be hid-
den from (or revealed to) consumers in the opaque
selling channel. Some opaque intermediaries allow
consumers to select the level of opacity. For exam-
ple, Priceline.com lets its consumers specify whether
a “red eye” flight is acceptable and it also allows
users to set the upper bound on the number of
stops. Selecting the optimal level of opacity and the
right attributes to hide provides potential for future
research but is outside the scope of this study. We
believe adding multiple dimensions of opacity might
shift consumer preferences toward the direct last-
minute sales channel. In our model, consumers are
certain that the firms are selling products of identi-
cal valuations in both the channels. However, if there
is additional uncertainty about exact features of the
product purchased from the opaque channel, then the
consumers will be more likely to purchase directly
from the firms.
7.5. Channel Design and Intermediary Selection
Decisions
Our paper models the decision of firms to choose
between LMSS (selling directly) and OpSS (an opaque
intermediary) to sell limited inventories to strategic
Jerath, Netessine, and Veeraraghavan: Revenue Management with Strategic Customers
446 Management Science 56(3), pp. 430–448, © 2010 INFORMS
consumers. However, channel design and choosing
between several available intermediaries remain as
important decisions for a large number of firms. In
a sequence of papers, Rangan (1986) and Rangan
et al. (1986) discuss a prescriptive model for designing
channels (with wholesalers and retailers) and choos-
ing the right intermediary for the channel designed.
Consumer demand is modeled exogenously in those
papers. How strategic consumer behavior affects
airline channel design (e.g., selling online through
Expedia.com versus selling through travel agents) is
a pertinent question that we leave for future research.
8. Discussion and Conclusions
Because of uncertain demand and (short-term) inflex-
ible capacity, firms in travel industries often end up
with one of the two extremes—a shortfall of capacity
due to high demand or leftover unused (and expen-
sive) capacity due to low demand. To deal with the
mismatch between demand and supply, firms have
implemented a variety of strategies; two of the most
prominent strategies are direct last-minute sales at
reduced prices and sales through an opaque interme-
diary. However, consumers are becoming more and
more strategic—they have learned to anticipate this
last-minute distress selling and might decide to post-
pone their purchase in expectation of future lower
prices. The risk the consumers face while making
this decision is not being able to obtain a product if
demand turns out to be high.
In this paper, we model this strategic interaction
between competing firms and consumers and shed
light on the following question: When should firms
offer last-minute sales directly to consumers versus
through an opaque intermediary? We find that the
answer depends on at least three factors: (1) the val-
uations that consumers have for the service, (2) the
strength of brand preference that consumers have for
competing firms (alternatively, the extent of service
differentiation between competing firms), and (3) the
probability that demand in the market exceeds capac-
ity. If consumer valuation for a product is high or
the strength of brand preference of the consumers in
the market is low, or both, firms prefer direct last-
minute sales over opaque sales. Furthermore, as the
probability of high demand increases, firms start to
prefer opaque sales over direct last-minute sales. At
the extreme, if market demand is deterministic, direct
last-minute sales are never offered, whereas opaque
sales can be offered if consumer valuations for travel
are very low. These findings immediately translate
into empirically testable hypotheses.
We find that the dynamics underlying these two
selling strategies are very different. By using direct
last-minute sales, each firm prices in the first period
so that only consumers with high preference for the
firm buy the product. Thus, each firm derives the
bulk of its profits primarily by charging high prices
to these consumers, whereas second-period prices are
very low (however, these cheap products are avail-
able only if demand turns out to be lower than capac-
ity). Quite differently, in the opaque selling strategy,
if the consumer valuations are very low, the firms set
first-period prices to extract maximum profits from
consumers and then clear any remaining products
through the opaque channel. When valuations are
high, the firms price in the first period to ensure that
the number of consumers that wants to buy prod-
ucts exceeds supply if demand is high (which, at
this point in time, is unknown to everybody), intro-
ducing clamor for the limited number of products
and leveraging the risk of product shortage to charge
higher first-period prices. To summarize, the direct
last-minute sales strategy can be construed as extract-
ing profits from high-preference consumers, whereas
the opaque sales strategy can be thought of as creat-
ing a frenzy for products to raise prices. Clearly, both
strategies in our paper are far from simple “inven-
tory clearance mechanisms”—they are indeed firms’
strategic responses to consumers’ strategic purchasing
decisions.
We omitted several considerations from the model
in order to obtain sharper insights, and these consid-
erations pose several interesting questions for future
research. First and foremost, how opaque should the
opaque product be? Should consumers be able to
specify at least some time intervals for the departure
or not? Addressing this important question is beyond
the scope of the current study. Second, we simplified
the decision for the firms by allowing for two sales
opportunities: one “regular” and one “sales.” In prac-
tice, for example, airlines offer many fares, and prices
tend to increase until the very last moment when last-
minute sales are announced.
In our model, all consumers fully know their brand
preference-adjusted valuations. Those customers with
higher valuations naturally buy earlier. However, cus-
tomers may not always know their valuation for fly-
ing. For instance, some customers, despite having
high valuation for flying for an occasion, may realize
their need late in the selling season. This remains a
challenging problem and is explored in a stream of
research by Akan et al. (2008), Courty and Li (2000),
and others, where factors such as product opacity
and competition are not considered. It is fruitful to
combine the research issues to explore the impact of
learning behavior on opaque selling. Because of the
complexities involved with dynamic learning mod-
els, we leave the solution of this problem for future
research.
Jerath, Netessine, and Veeraraghavan: Revenue Management with Strategic Customers
Management Science 56(3), pp. 430–448, © 2010 INFORMS 447
Incorporating the above considerations into mod-
ern decision support systems remains a challenge.
Finally, although numerous studies have modeled air-
line revenue management decisions, there have been
very few attempts to verify these findings empir-
ically (see Koenigsberg et al. 2008 for an excep-
tion). Empirical studies tend to be limited by data
availability—although airlines periodically share data
with regulatory authorities, these data are not precise
enough to distill specific pricing strategies employed
by an airline. All these directions are promising areas
of future research in years to come.
9. Electronic Companion
An electronic companion to this paper is available as
part of the online version that can be found at http://
mansci.journal.informs.org/.
Acknowledgments
This research was supported by the Mack Center for
Technological Innovation at the Wharton School. The
authors thank the faculty of the OPIM and Marketing
departments at the Wharton School and seminar partici-
pants at the University of British Columbia, New York Uni-
versity, Mini-Conference on Customer-Oriented Operations
Models at Washington University in St. Louis, Marketing
Science Conference 2008 Revenue Management and Pricing
Conference 2008, and INFORMS 2009 for their helpful com-
ments. They also thank Wilfred Amaldoss, Krishnan Anand,
James Dana, Burton Hollifield, two anonymous reviewers,
the associate editor, and the department editor for their
extremely valuable suggestions, which improved this paper
significantly.
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