DOI 10.1111/poms.13583
© 2021 Production and Operations Management Society
Optimal Pricing and Overbooking of Reservations
Jaelynn Oh*
David Eccles School of Business, The University of Utah, Salt Lake City, Utah 84112, USA, jaelynn.oh@eccles.utah.edu
Xuanming Su
The Wharton School, University of Pennsylvania, Philadelphia, Pennsylvania 19104, USA, xuanming@wharton.upenn.edu
e study the optimal design of reservations for a firm with limited capacity. The firm faces a random number of customers, each of whom has a random valuation for service. The reservation policy has two components: pricing and
overbooking. For the former, the firm charges a reservation fee (at the time of reservation) and a service price (at the time
of service). For the latter, the firm imposes a booking limit that caps the number of reservations it sells. Given the firm’s
reservation policy, customers make reservations in advance and later decide whether to show up. Denying service to
reservation holders is costly. We obtain the following equilibrium results. First, when demand is small relative to capacity,
the firm’s pricing structure relies on reservation fees prepaid in advance, but when demand is large relative to capacity, it
relies on payment received upon service. Second, when demand is low and/or predictable, the firm accepts all reservation
requests, but when demand is high and/or variable, the firm uses a booking limit.
W
Key words: pricing; overbooking; reservations; revenue management; advance selling; capacity
History: Received: May 2020; Accepted: August 2021 by Dan Zhang, after 2 revisions.
*Corresponding author.
deposit, and some hotels require a full and nonrefundable payment upon reservation. Second, on overbooking, firms have to decide how many excess
reservations to accept beyond the number capacity
can accommodate. Denying service to reservation
holders can be costly; for example, the Department of
Transportation mandates that up to 400% of the oneway fare must be compensated to each involuntarily
bumped passenger (DOT 2020). Yet, bumping rates
differ across US airlines, ranging from 0.02 to 6.28 per
100,000 passengers in 2018 (McCarthy 2019).
By studying the pricing and the overbooking decisions jointly, we hope to answer the following
research questions. First, how does the market condition affect the optimal overbooking policy? Second,
how can the firm choose the optimal pricing policy
among the wide range of reservation pricing policies
observed in practice dependent on the market condition? From the answers of the above research questions, we hope to provide recommendations on the
optimal overbooking and pricing policies to firms that
take reservations while facing the uncertainties on the
market size and customer valuations.
We analyze the following stylized model. First, the
firm determines its reservation policy, which comprises of two prices (a reservation fee and a service
price) and a booking limit on the maximum number
of reservations to take. Then, the market size is realized; customers decide whether to make reservations
1. Introduction
Reservations are ubiquitous for firms such as car rentals, airlines, and hotels. These firms have limited
capacity and may take reservations in advance in
order to subsequently guarantee service. Because of
the time gap between the point reservation is made
and the point service is rendered, both the firm and
the customers face uncertainty. Firms give out reservations without knowing the precise number of
patrons, and customers make reservations before
learning their precise need for service. This study
intends to study how firms can set the optimal overbooking and pricing policies to deal with the uncertainties the firm and the customers face when
reservations are made. As we study the optimal reservation policies, we provide explanations on how certain market conditions, characterized as the average
market size and the market size variability, lead to
certain equilibrium outcome.
In this study, we focus on two specific aspects of
reservation policies: pricing and overbooking. First,
on pricing, firms generally have to make two decisions: how much to charge for making the reservation
and how much to charge for exercising it; these are
analogous to the purchase price and strike price of
financial options. There is a wide variation observed
in practices. For example, some hotels give out reservations for free, others charge a nonrefundable
928
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Vol. 31, No. 3, March 2022, pp. 928–940
ISSN 1059-1478|EISSN 1937-5956|22|3103|0928
Production and Operations Management 31(3), pp. 928–940, © 2021 Production and Operations Management Society
(paying the reservation fee) and if so, the firm accepts
reservations up to its booking limit. Finally, individual customer valuations are realized; customers
decide whether to show up (paying the service price)
and the firm serves customers up to its capacity limit.
This model admits a wide range of reservation policies that are chosen by firms. We summarize our
results below.
First, the optimal overbooking policy has a simple
structure and takes one of two forms. In some cases, it
may be optimal to accept all reservation requests and
impose no booking limit whatsoever. In other cases, a
booking limit is needed: this caps the number of reservations given out so that the number of customers
who are expected to show up is aligned with the units
of capacity available. Note that the booking limit may
exceed capacity, so some overbooking may be practiced. In summary, the optimal overbooking policy
may be an “accept all policy” or a “booking limit
policy.”
Second, the optimal pricing policy often involves a
combination of reservation fees and service prices.
The former dominates in small markets while the latter dominates in large markets. In extreme cases (i.e.,
when average market demand is extremely small or
extremely large, relative to capacity), it may even be
optimal to charge only the reservation fee (i.e., fully
prepaid reservations) or only the service price (i.e.,
free reservations). As the average market size
increases, the firm relies less on advance pre-payment
at the time of reservation but more on spot payment
at the time of consumption.
Third, by running numerical simulations, we map
the optimal reservation policies with market conditions characterized as a combination of the mean and
the standard deviation of the market size. We find
that accepting all reservations is an optimal overbooking policy for a market with small mean and/or small
variance. On the other hand, setting a booking limit is
optimal when the average market size is large and/or
the standard deviation of the market size is large.
From the numerical results we can also see that the
optimal prices alone can effectively regulate the customer traffic when the market size is less variable, but
booking limits become essential as the market size
variability increases to avoid customer bumping.
2. Literature Review
This study is closely related to the advance selling literature. The key feature in common is that customers
make purchase decisions while facing uncertainty
over their consumption valuations. Despite this
uncertainty, DeGraba (1995) finds that limiting supply induces customers to buy in advance, and the
threat of unavailability even allows prices to be set
929
above market-clearing levels. Gale and Holmes (1993)
show that offering a discount can also induce customers to purchase in advance; this helps shift
demand from peak to off-peak periods. Dana (1998)
finds that such advance purchase discounts can generate competitive advantages. Xie and Shugan (2001)
develop a framework for comparing advance selling
to spot selling (i.e., selling to customers after their valuation uncertainty is resolved) and also study
whether to advance sell at a discount or at a premium;
Shugan and Xie (2005) generalize the earlier framework to competitive settings. Prasad et al. (2011)
embed a classic newsvendor model into the advance
selling framework to study the firm’s decision of how
many units to advance sell; they also compare
advance selling and spot selling. Nasiry and Popescu
(2012) study how anticipated regret can affect customer decisions and firm profits in an advance selling
context. Yu et al. (2015) study the pricing and capacity
decision of an advance selling firm when customer
valuations are correlated. In this stream of work, the
focus is on pure advance selling and pure spot selling,
that is, customers make purchase decisions either
before or after resolving uncertainty over their valuations. In contrast, we offer a general framework that
integrates these two modes of operation: customers
pay a reservation fee in advance and/or pay another
spot price when they show up for their reservations.
Furthermore, we incorporate overbooking into
advance selling by allowing the firm to sell more
reservations than its capacity can serve.
There is a series of related papers that focus on
reservations. An early study by Png (1989) finds that
optimal reservation policies should provide compensation to reservation holders who show up but are
denied service. Bertsimas and Shioda (2003) develop
a decision tool for managing restaurant reservations,
taking into account the possibility of no-shows. Chen
et al. (2017) studies long-term revenue-maximizing
admission policy when different types of customers
arrive to make reservation requests with services that
have heterogeneous duration and start time and the
sellers decide which requests to accept or deny.
Alexandrov and Lariviere (2012) point out that reservations help attract demand when customers incur a
travel cost to request service. Cil and Lariviere (2013)
study how to allocate capacity between reservation
holders and walk-in demand. Elmaghraby et al.
(2009) and Osadchiy and Vulcano (2010) study binding reservations in retail environments: customers
may reserve a unit for purchase but are obligated to
buy if there is availability at the end of the selling season. In the papers above, the firm sells reservations at
a single price. However, we consider a more general
mechanism that involves two prices: a reservation fee
and a service price.
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Oh and Su: Optimal Pricing and Overbooking of Reservations
Oh and Su: Optimal Pricing and Overbooking of Reservations
Production and Operations Management 31(3), pp. 928–940, © 2021 Production and Operations Management Society
Our two-price modeling setup has alternative interpretations that have been used in some earlier papers.
For example, charging a reservation fee in advance
and a service price later is equivalent to collecting a
partially refundable payment in advance. In the economics literature, Courty and Li (2000) show that a
menu of partial refund contracts can be used to screen
multiple customer segments: customers who face
greater uncertainty are willing to pay more in
advance for more generous refunds. Similarly, Akan
et al. (2015) apply partial refund contracts to develop
a continuous time sequential screening mechanism
when selling to customers who realize their valuations at different times. The above papers focus on
price discrimination and do not consider the role of
capacity constraints, which play a critical role in our
study. In this sense, our paper is more closely related
to the following revenue management models. Gallego and Sahin (2010) study capacity options: customers pay an option price for the right to purchase a
unit of capacity and may later pay the strike price to
exercise that right; the option and strike prices here
are analogous to our reservation and service fees. The
authors focus on profit and welfare implications of
capacity options, but we are more interested in understanding the structure of optimal reservations policies
(e.g., whether reservations should be fully or partially
prepaid). More recently, Georgiadis and Tang (2014)
study reservation policies consisting of a nonrefundable deposit and a retail price. They consider four customer segments with low/high service valuations and
low/high no-show probabilities, and study the optimal subset of customers to sell reservations to. In contrast, we consider customers who endogenously
choose whether to show up for service based on their
realized valuations, which follow general distributions. This key difference allows pricing to play a
stronger role in regulating customer arrivals in our
model. With our model, we distinguish between the
following regimes: no reservations (i.e., spot selling),
fully refundable reservations, partially refundable
reservations, and nonrefundable reservations (i.e.,
advance selling). While considering advance deposits
and spot prices for reservation customers, Oh and Su
(2018) study capacity allocation between reservations
and walk-ins. This study, on the other hand, studies
how to use booking limits to discipline overbooking
practices.
Finally, our study is related to the literature on
appointment scheduling and overbooking. Cayirli
and Veral (2003) and Gupta and Denton (2008) provide a comprehensive review of the literature, most of
which is in the health care context. Kim and Giachetti
(2006) develop an overbooking model that considers
both no-shows and walk-ins to obtain the optimal
number of appointments. LaGanga and Lawrence
(2007), Zeng et al. (2010), and LaGanga and Lawrence
(2012) study overbooking models that balance the
benefit of expected revenue increase and the cost due
to increased customer waits and provider overwork.
Robinson and Chen (2010) and Liu et al. (2010) compare appointment overbooking to an open access policy where patients can come right away on the day
they want to be seen. Zacharias and Pinedo (2014)
find an optimal overbooking policy that minimizes a
weighted sum of customer wait time, provider idle
time, and provider overtime. Liu and Ziya (2014) look
for the optimal panel size (total number of patients
the provider commits to provide service for) and
study overbooking decisions. The study mentioned
above focus on appointment systems with fixed
prices. In contrast, we study joint overbooking and
pricing decisions.
3. Model
The Firm: There is a monopolist firm with a fixed
capacity. This capacity may be cars in a car rental
company or rooms in a hotel. When capacity is full,
no more customers may be served. Let μ denote the
firm’s capacity; in other words, the firm can serve no
more than μ customers. The firm sells its capacity
through reservations. The firm’s reservation policy
specifies two prices: a reservation fee ϕ and a service
price p. The reservation fee is collected when a reservation is made, and the service price is charged when
a reservation holder shows up and receives the service. In addition, the firm sets a booking limit B,
which is the maximum number of reservations the
firm takes. Note that if the firm takes too many reservations and too many of them show up, the firm will
be unable to honor all these reservations. For each
reservation holder the firm turns away, the firm
arranges a substitute service while incurring a bumping cost C > 0. The firm’s bumping cost C may
include a goodwill cost, any cost related to arrangements made to a bumped customer, and a compensation the firm may have to give to a denied customer.
There are different practices regarding the compensation in different industries. The hotel industry
requires accommodations to be provided at a neighboring hotel at an equal or complementary rate, and
they maintain partnerships with neighboring hotels
to utilize as alternative arrangements for overbooked
customers (Wikipedia 2021). Airlines, however, are
mandated to pay an involuntarily bumped customer
an amount equal to 200% of the one-way fare upon
arranging substitute transportation that is scheduled
to arrive within 2 hours after the original arrival time
(DOT 2020). The firm does not divulge overbooking
practices and the booking limit B is not observable.
However, the prices p and ϕ are observable.
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930
Production and Operations Management 31(3), pp. 928–940, © 2021 Production and Operations Management Society
The Customers: The total number of customers
(i.e., market demand) is Λ ≥ 0. Here, Λ is a random
variable with density g and distribution G; let
¼ 1 G. Each customer is a strategic utility maxiG
mizer and makes two decisions sequentially: first,
whether to make a reservation (paying the reservation fee), and second, whether to exercise the reservation (paying the service price) if one has been
made. When making the former decision, customers
are uncertain about how much value they will derive
out of the firm’s service. This valuation u is independent and identically distributed across customers; it
follows distribution F and density f, and let
¼ 1 F. Based on the distribution of valuations,
F
customers choose whether to make a reservation. If a
customer chooses not to make a reservation, she
leaves the market and receives zero utility. If a customer chooses to make a reservation, she faces the
second decision, that is, whether to show up for
the reservation and pay the price for service, after
the service valuation u is realized. However, due to
overbooking, it is possible that a reservation holder
shows up for service while the firm does not have
available capacity to serve the customer Figure 1. In
this case, the customer incurs a denial cost c. We
assume that the inconvenience of being bumped
exceeds any potential compensation paid by the
firm, so our analysis excludes customers who prefer
to be bumped in hopes of earning the compensation,
that is, c ≥ 0.
Sequence of Events: The game chronology can be
summarized as follows. First, the firm sets the
reservation fee ϕ and the service price p, which are
both publicly observable. In addition, the firm sets
its booking limit B. Next, the market size Λ is realized; the firm observes the market size, but customers only know its distribution. Then, customers
decide whether to make reservations and pay the
reservation fee. The firm then accepts reservation
requests up to the booking limit B. Finally, customers’ service valuations u are realized, and reservation holders decide whether to show up and pay
the price of service. The firm serves customers up to
Figure 1 Customers’ Decisions and Payoffs [Color figure can be
viewed at wileyonlinelibrary.com]
931
its capacity limit μ. Figure 2 summarizes the
sequence of events described above. The figure also
labels the continuation equilibrium, which is the set
of outcomes that follow a particular set of prices ϕ
and p, and the full equilibrium, which consists of
equilibrium prices and their corresponding continuation equilibrium.
Utility Function: We now study customers’ utility
function. Using backward induction, we first examine a reservation holder’s decision of whether to
exercise the reservation (i.e., the second to last box in
Figure 2). The decision depends on two factors: first,
the customer’s valuation u, which has already been
realized, and second, the customer’s probability q of
receiving the service if she shows up. (We will derive
q later in this section.) The customer’s expected utility is
ðu pÞq cð1 qÞ ϕ,
if show up,
ϕ,
if no show:
Note that the customer pays the price of service p if
service is rendered, but she incurs the denial cost c
if capacity is unavailable despite securing the reservation; the former occurs with probability q. A reservation holder will thus show up when doing so
yields a higher expected utility than not showing up
and losing the reservation fee ϕ. In other words, to
maximize expected utility, customers with realized
1q
valuations u above the cutoff v ¼ p þ c q will
show up, while other customers will not. Therefore,
the expected utility from making a reservation is
νðqÞ ¼ ϕ þ Eu maxfðu pÞq cð1 qÞ, 0g:
When this expected utility is positive, customers
will choose to make reservations.
Profit Function: Next, we turn to the firm’s
profit function in the continuation equilibrium.
Having already chosen the prices ϕ and p, the
firm is now choosing the booking limit B. We
assume that customers make reservations; if they
do not, the firm earns nothing and the booking
limit plays no role. The firm’s expected profit as a
function of the booking limit and customers’ cutoff
valuation v is
πðB; vÞ ¼ EΛ p min RðBÞ FðvÞ,
μ
þ ϕ RðBÞ C max RðBÞ FðvÞ
μ, 0 ,
where R(B) ≡ min(B, Λ) is the number of reservations given out. There are three terms in the expectation. The first term is total revenue from service
provision since a fraction FðvÞ
of the R(B) reservation holders show up for service, and the firm may
serve up to μ of them. The second term is the total
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Oh and Su: Optimal Pricing and Overbooking of Reservations
Production and Operations Management 31(3), pp. 928–940, © 2021 Production and Operations Management Society
Figure 2 Sequence of Events [Color figure can be viewed at wileyonlinelibrary.com]
revenue from reservation fees. The third term is the
total bumping cost from turning away reservation
holders. Based on this function, the firm chooses the
optimal booking limit that maximizes the expected
profit.
Continuation Equilibrium: We are now ready to
analyze the continuation equilibrium. Here, the
prices ϕ and p have already been set. Based only
on these prices, customers choose whether to make
reservations. If they do not, the game ends and
everyone earns nothing. In the more interesting
case, customers make reservations, and the continuation equilibrium outcomes are described as what
follows.
DEFINITION 1. Given (ϕ, p), a continuation equilibrium
ðBðϕ;pÞ , vðϕ;pÞ , qðϕ;pÞ Þ, in which customers make reservations, must satisfy the following conditions:
(i) Bðϕ;pÞ ∈ arg maxB πðB; vðϕ;pÞ Þ,
(ii) vðϕ;pÞ ¼ p þ c
1qðϕ;pÞ
qðϕ;pÞ
(iii) qðϕ;pÞ ¼ EΛ min
,
RðBðϕ;pÞ
μ
ÞFðv
ðϕ;pÞ
,
1
:
Þ
Condition (i) states that the booking limit Bðϕ;pÞ
maximizes the firm’s expected profit, given the
customers’ cutoff strategy vðϕ;pÞ . Condition (ii) states
that the cutoff vðϕ;pÞ maximizes customers’ expected
utility given the fill rate qðϕ;pÞ . Condition (iii) calculates
the fill rate qðϕ;pÞ based on the firm’s and customers’
decisions Bðϕ;pÞ and vðϕ;pÞ . Together, these conditions
characterize continuation equilibrium outcomes
following prices ϕ and p, when customers make
reservations.
Full equilibrium: We are now ready to describe the
full equilibrium of the game. Let π ðϕ, pÞ and
ν ðϕ, pÞ, respectively, denote the firm’s expected
profit and customers’ expected utility corresponding
to the outcomes ðBðϕ;pÞ , vðϕ;pÞ , qðϕ;pÞ Þ described above.
We have
π ðϕ, pÞ ¼ πðBðϕ;pÞ ; vðϕ;pÞ Þ,
ν ðϕ, pÞ ¼ νðqðϕ;pÞ Þ:
A customer will make a reservation if the expected
utility ν ðϕ, pÞ is greater than the outside option of
zero. Therefore, the firm solves the following profit
maximization problem:
max
ϕ, p
π ðϕ, pÞ
s:t: ν ðϕ, pÞ ≥ 0:
The full equilibrium of the game is thus defined as
follows.
DEFINITION 2. The full equilibrium ðϕ , p , B , v , q Þ
satisfies the following conditions.
(i)
(ii)
(iii)
(iv)
ðϕ , p Þ ∈ arg maxðϕ;pÞ: ν ðϕ;pÞ≥0 π ðϕ, pÞ,
B ¼ Bðϕ ; p Þ ,
v ¼ vðϕ ; p Þ ,
q ¼ qðϕ ; p Þ .
Condition (i) requires that the firm’s pricing policy
ðϕ , p Þ maximizes its expected profit in the
continuation equilibrium while ensuring that customers are willing to make reservations. Conditions (ii),
(iii), and (iv) ensure that ðB , v , q Þ is a continuation
equilibrium given the prices ðϕ , p Þ. All our results
below are based on analyses of the full equilibrium.
4. Analysis
4.1. The Overbooking Policy
We begin by showing that there are two possible
types of equilibria: the accept-all equilibrium and the
booking-limit equilibrium.
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Production and Operations Management 31(3), pp. 928–940, © 2021 Production and Operations Management Society
PROPOSITION 1. In the full equilibrium, ðB , v , q Þ
must take one of the following two forms.
(i) (Accept-all equilibrium) B ¼ ∞, v ¼ p þ c
n
o
μ
q ¼ EΛ min ΛFðv
,
1
.
Þ
(ii) (Booking-limit
q ¼ 1.
equilibrium)
μ
B ¼ Fðp
Þ,
1 q
q ,
v ¼ p ,
In the first type of equilibrium shown in Proposition 1(i), the firm effectively accepts all reservation
requests because the booking limit is B ¼ ∞. However, when market demand is too high, the firm does
not have enough capacity to honor all the reservations
it gives out. In other words, the probability that a
reservation holder will be served is only q < 1. Customers anticipate this, so their cutoff valuation,
1 q
v ¼ p þ c q , is strictly greater than the service
price p to account for the risk of being bumped. We
call this the accept-all equilibrium.
In the second type of equilibrium shown in Propoμ
sition 1(ii), the firm uses a booking limit B ¼ Fðp
Þ.
Specifically, the firm accepts up to B reservations, at
which point it stops accepting any more that might be
requested. The booking limit is chosen so that when it
is reached, the expected number of reservation holders who show up for service is aligned with the units
of capacity available. (Customers who realize that
their valuation is below the price will not find it
worthwhile to show up for service, so only a fraction
Þ of reservation holders will show up.) Using the
Fðp
booking limit B , the firm ensures that all reservation
holders who show up will be served in expectation
(i.e., q ¼ 1). Consequently, the corresponding cutoff
valuation v equals the service price p . We call this
the booking-limit equilibrium.
The two candidate equilibria reflect different priorities in reservation management. To explain this point,
we present the following proposition.
PROPOSITION 2. Consider some fixed (ϕ, p) with two
continuation equilibria, one satisfying the conditions of
the accept-all equilibrium and the other satisfying the
conditions of the booking-limit equilibrium. Then, we
have the following.
(i) The accept-all equilibrium is more profitable if p is
sufficiently small, but the booking-limit equilibrium
is more profitable if p is sufficiently large.
(ii) The accept-all equilibrium is more profitable if ϕ is
sufficiently large, but the booking-limit equilibrium
is more profitable if ϕ is sufficiently small.
With two continuation equilibria following the
same set of prices, a meaningful comparison between
933
the accept-all and the booking-limit reservation formats is possible. Note that the full equilibrium is
uniquely determined since the firm chooses the format of continuation equilibrium that brings a higher
expected profit. Even though these continuation equilibria may not occur in the full equilibrium, comparing them is instructive nonetheless. According to
Proposition 2 above, if the firm relies heavily on reservation fees (i.e., high ϕ but low p), then the accept-all
equilibrium is more profitable. This is reasonable
since the accept-all equilibrium gives out more reservations and thus collects more reservation fees. On
the other hand, if the firm relies more on the revenue
collected from service, then the booking-limit equilibrium is more profitable. In this case, the booking
limit helps to maintain high service levels, which supports higher service prices that contribute to higher
revenue. Therefore, the two candidate equilibria
reflect different priorities in reservation management:
the accept-all equilibrium focuses on volume, while
the booking-limit equilibrium focuses on service.
We now study the market condition under which
one equilibrium overbooking format outperforms the
other. We focus on how market variability influences
the optimal overbooking policy. In the following proposition, we assume that the market size Λ follows a
normal distribution and study the effect of the standard deviation parameter, holding the mean fixed.
PROPOSITION 3. Let market demand Λ follow a normal
distribution Nðλ, σ 2 Þ truncated at zero.
(i) If σ is sufficiently small, the equilibrium must be an
accept-all equilibrium.
(ii) If σ is sufficiently large, the equilibrium must be a
booking-limit equilibrium.
The result above shows that when the market variability is low, the firm accepts all reservation requests.
The intuition is as follows. When market demand is
relatively predictable, customers are in a good position to regulate their own arrival patterns and ensure
that just the right amount of demand shows up for
service. In the extreme case where demand is deterministic, customers can even ensure 100% service
level by independently choosing whether to show up
for service (cf. Lariviere and Van Mieghem 2004). For
example, if everyone knows that there are 500 customers in the market and the firm has ample capacity
to serve 400, an equilibrium will emerge whereby the
400 customers with the highest valuations will show
up for service and all of them will be served. This outcome is possible because customers respond strategically to prices, that is, only customers with realized
valuations above the service price will show up. Ultimately, it is the firm’s responsibility to set the right
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Oh and Su: Optimal Pricing and Overbooking of Reservations
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service price so that just the right amount of demand
shows up to fully utilize capacity. If the price is too
high, too few customers show up and capacity is
wasted; if the price is too low, too many show up and
some of them are “bumped.” With the right prices,
the accept-all equilibrium can indeed be optimal in
predictable markets.
On the other hand, when market variability is high, a
booking limit must be used in equilibrium. This is
because customers are no longer sufficiently wellinformed to regulate their own arrivals. Since customers are not privy to demand realizations, they
decide whether to show up based only on their valuations. In equilibrium, the same proportion of reservation holders will show up for service, regardless of
whether demand is high or low. To avoid having too
many customers show up for service, a booking limit is
needed. When demand is low, reservations can be
given out without any restriction because the firm has
ample capacity to honor them. However, when
demand is high, the firm must limit the number of
reservations in order to maintain reasonably high service levels. This role is served by the booking limit,
which is necessary when the market variability is high.
4.2. The Pricing Policy
In the previous section, we studied how booking limits can help firms hedge against the risk of bumping
customers when the market size is uncertain. We now
study how the firm can deal with customer valuation
uncertainties using the reservation pricing policy. We
begin by stating the following proposition, which
shows the equilibrium pricing policies and their
dependence on the average market size. To solicit the
effect of the average market demand while keeping
the analysis simple, we first study the case where the
market size Λ is deterministic.
PROPOSITION 4. When the market size Λ is deterministic,
the equilibrium price ðϕ , p Þ satisfies the following.
R∞
(i) If Λ ≤ μ: p ¼ 0 and ϕ ¼ 0 ufðuÞdu.
R
1 ð μ Þ and ϕ ¼ ∞ ðu p ÞfðuÞdu.
(ii) If Λ > μ: p ¼ F
Λ
p
Therefore, the service price p increases as the market size
Λ increases, and the reservation fee ϕ decreases and converges to zero as the market size Λ increases.
Proposition 4(i) shows that when the market size is
smaller than the capacity, the service price p becomes
zero, so the firm’s revenue comes primarily from
reservation fees. Since the bulk of the payment is
received upfront, this scheme protects the firm
against no-shows. In addition, this scheme leads to
efficient consumption: since the firm essentially does
not charge for service, all reservation holders with
positive valuations will receive the service. In contrast, if the firm charges p > 0 for service, some customers with positive valuations below p will not
show up, leading to lost surplus and hence lost profits
(since the firm extracts full surplus). Therefore, the
firm adopts a full prepayment scheme when average
demand is sufficiently small.
When the market size is larger than the capacity, on
the other hand, Proposition 4(ii) shows that the firm
charges service price so that customers can selfregulate to avoid being bumped. The fraction of customers having realized valuation greater than the
price of service is set equal to the fraction of capacity
over the market size. Hence, as the market size
increases, the firm charges a higher service price p,
but needs to correspondingly decrease the reservation
fee ϕ to make reservations attractive ex ante. When
the market size becomes significantly large, reservation fee ϕ becomes negligibly small, and reservations
are offered for free.
The results of Proposition 4 follow from the fact
that the reservation price is paid before customers
know their valuations, whereas the service price is
paid after customers’ valuations are realized. When
the firm is charging a price to customers who are unaware of their valuations, the firm can only set the
price that is attractive to the customers in expectation.
Therefore, the reservation deposit is set based on the
customers’ expected valuation. In contrast, the service
price that is charged after customers learn their valuations can actively depend on the market size. To illustrate, consider the case where a firm has 20 units of
capacity to sell to the customers, and customers’ service valuations follow Uniform distribution with the
support of [$0, $100]. If the firm sells the capacities to
customers who have not yet seen their realized valuations, the only price the customers are willing to pay
is less than or equal to their expected valuation of $50.
However, if the firm sells the capacities to customers
who already know their valuations, it can sell them to
20 customers who have the highest realized valuations. For example, if there are 100 customers in the
potential market, the firm can sell its capacities to the
top 20% of the population by setting the service price
equal to $80. If there are 1000 customers in the potential market, the firm can now sell its capacities to top
2% of the population with the price of $98. As it can
be seen in the above exercise, the market size does not
have a direct effect on the prepaid reservation
deposit, whereas the service price increases as the
market size increases. When the firm charges both the
reservation deposit and the service price as in our
model, the firm increases the service price as the
potential market size increases, and the prepayment
needs to be lowered so that customers are willing to
pay the deposit based on their expected utility.
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Oh and Su: Optimal Pricing and Overbooking of Reservations
934
Production and Operations Management 31(3), pp. 928–940, © 2021 Production and Operations Management Society
Proposition 4 also implies that the firm’s optimal
pricing strategy for reservations diverges from the
common belief that reservations are more useful
when capacity is scarce. When the market size is
large, this is when customers would value reservations more. However, our results show that the firm
should give out reservations for free when the market
size becomes significantly large. This is because customers view reservations as a means to secure capacity. However, to the firm reservations can be a tool to
change the point of payment to maximize revenue.
When the market size is small, the firm is better off
charging the reservation deposit in advance while
customers are still uncertain about their valuations.
On the other hand, when there are more potential customers, the firm can sell its capacities at a higher price
when the price is charged after customers find out
their realized valuations.
A key question in the advance selling literature is
when one pricing scheme dominates the other
between advance sales and spot sales. It is generally
recognized that as the market size increases, the optimal policy switches from advance to spot sales (Xie
and Shugan 2001). In comparison, with a generalization of advance selling in the form of reservations, we
find that pure advance selling and pure spot selling
are optimal only in extreme cases. In all other cases, it
is important to consider a “hybrid” reservation system with two prices, ϕ and p. With two prices
charged, we see that the market size has implications
on pricing.
Proposition 5 extends the results of Proposition 4 to
the case where Λ is uncertain.
PROPOSITION 5.
Let λ = EΛ.
(i) Suppose λ is sufficiently small such that the market
demand Λ is smaller than μ with probability 1 − ɛ,
for some small ɛ > 0. Then, p < ɛ.
(ii) Suppose λ is sufficiently large such that the market
demand Λ is greater than 1ɛ with probability 1 − ɛ,
for some small ɛ > 0. Then, ϕ < ɛ.
(iii) There is an increasing function p(λ) and a decreasing function ϕ(λ) such that for any λ, p ≥ pðλÞ and
ϕ ≤ ϕðλÞ.
Proposition 5 shows that (i) the service price p
becomes negligible when the average market size is
small, (ii) the reservation fee ϕ becomes negligibly
small with a large enough market size, and (iii) the
firm depends more heavily on the price of service and
less so on the price of reservation as the market size
increases.
Our results are opposite from the findings of Georgiadis and Tang (2014), who find that prepayment
plays a more important role in large rather than small
935
markets. The reason is as follows. In their model, the
no-show probability is exogenous, so the number of
reservation holders willing to show up cannot be
manipulated using the service price. As market size
increases, the opportunity cost of wasted capacity
increases. Hence, it becomes more important to use
prepayment to combat no-shows. In contrast, our
model endogenizes customers’ decisions to show up
for service. As market size increases, a greater number of reservation holders are willing to show up at
each given price, so we can increase service price
while fully utilizing capacity; the reservation fee correspondingly decreases. Consequently, there is less
reliance on prepayment in larger markets.
4.3. The Overbooking and the Pricing Policies
So far, we have seen how booking limits can be useful
to deal with the market size uncertainty, while the
pricing policy uses customer valuation uncertainty to
maximize firm profit. As we studied the overbooking
policy and the pricing policy separately, we showed
how the format of the optimal overbooking policy
depends on the standard deviation of the market size
and how the pricing policy changes with the average
market size. Now, to study how the mean and the
standard deviation of the market size conjointly determine the optimal overbooking and the pricing policies, we run a numerical study using simulation
method. While keeping the distribution of customer
valuation as normal distribution with mean 200 and
standard deviation 10, we vary the customers’ denial
cost c and the firm’s bumping cost C where c =
[50, 75, 100, . . ., 300] and C = [50, 75, 100, . . ., 300].
For each pair of c and C values, we vary the average
market size/capacity ratio from 0.8 to 2 and the standard deviation of the market size from 0 to 100. We
then compute the optimal pricing and the overbooking policy for each combination of mean market size/
capacity ratio and the standard deviation of market
size.
We first discuss the simulation results that show
the market conditions that result in the accept-all
equilibrium over the booking-limit equilibrium as
shown in Figure 3. In Figure 3, the x-axis represents
the values of average demand over capacity ranging
from 0.8 to 2, whereas the y-axis shows the standard
deviation of the market size ranging from 0 to 100.
Figure 3 shows the results for the case where both the
customers’ denial cost c and the firm’s bumping cost
C are 100. The yellow squares represent the (average
demand/capacity, standard deviation of market size)
pairs that result in the accept-all equilibrium format,
whereas the blue/green squares represent those pairs
that result in the booking-limit equilibrium as the
optimal overbooking policy. Among the squares that
are of the booking-limit equilibrium, darker blue
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Oh and Su: Optimal Pricing and Overbooking of Reservations
Oh and Su: Optimal Pricing and Overbooking of Reservations
Production and Operations Management 31(3), pp. 928–940, © 2021 Production and Operations Management Society
Figure 3 Accept-all Equilibrium vs. Booking-limit Equilibrium. The Yellow Squares Represent the (average demand/capacity, standard deviation of market size) Pairs that Result in the
Accept-all Equilibrium, Whereas the blue/green Squares Represent the Pairs that Result in the Booking-limit Equilibrium.
Darker Blue Squares Represent Less Capacities being Overbooked; Squares that are Closer to Yellow in Color Represent Market Conditions Resulting in Larger Capacities being
Overbooked
[Color
figure
can
be
viewed
at
wileyonlinelibrary.com]
squares represent the cases with less fraction of capacity being overbooked, whereas the squares that are
closer to yellow in color represent booking limits that
allow more capacity to be overbooked.
From Figure 3 we confirm that the theoretical
results of Proposition 3 hold: for a fixed value of average market size, the accept-all equilibrium is optimal
when the standard deviation of the market size is
small and the booking-limit equilibrium is optimal
when the standard deviation of the market size is
large. Figure 3 also shows that for a fixed value of
market size standard deviation, the accept-all equilibrium is more profitable for smaller average market
size values and the booking-limit equilibrium is more
profitable for larger average market size values. From
the above two facts, one can see that the threshold
standard deviation of the market size below which
the accept-all equilibrium is optimal decreases as the
average market size increases. In other words, when
the demand is low and/or predictable, the firm
accepts all reservation requests, but when the demand
is high and/or variable, the firm uses a booking limit.
By the colors of Figure 3, one can also see that the fraction of capacity that is overbooked increases with
average demand but decreases with the standard
deviation of demand.
From the simulation results, we can also see how
the (average market size/capacity, standard deviation of the market size) pair affects the optimal
reservation pricing policy as shown in Figure 4.
Figure 4 shows how the fraction of the total payϕ
ment paid as the reservation deposit, ϕ þ
p, changes
as the market condition changes for the case where
both the customers’ denial cost c and the firm’s
bumping cost C are 100. In Figure 4, the x-axes represent the values of average demand over capacity
ranging from 0.8 to 2, and the y-axes represent the
standard deviation of the market size ranging from
0 to 100. In Figure 4a, the height of each bar repreϕ
sents the value of ϕ þ
p for each pair of (average
market size/capacity, standard deviation of market
size). The yellow bars represent the cases where the
firm only charges the reservation deposit with the
service price of zero; the shorter the bars are, the
smaller the fraction of the reservation deposit is in
the total payment. Figure 4b shows the same information as Figure 4a but color codes the proportion
of ϕ over the total payment on the two-dimensional
market condition space. Each square represents an
(average market size/capacity, standard deviation of
market size) pair. The yellow squares show the
market condition pairs that result in full prepayment as the optimal pricing policy. The darker blue
squares represent the market conditions where the
reservation deposits take smaller proportion of the
total payment; the fraction of the prepayment
becomes larger as the squares become closer to yellow in color. From Figure 4, we can confirm the
results of Propositions 4 and 5. For a fixed value of
market size standard deviation, the service price p
equals zero for small markets so that the payment
is purely made through the reservation deposit. As
the average market size grows, the firm charges
higher service price p, and the corresponding reservation deposit ϕ decreases. When the market size
becomes significantly large, reservation deposit
becomes negligible.
By looking at Figures 3 and 4b together, we can
infer how the overbooking policy and the reservation pricing policy interact as the market conditions
change. As it was discussed following Propositions
3 and 4, when the market size is deterministic, the
prices that maximize the profit can also regulate
customer traffic perfectly and ensure 100% service
level. However, as the market size variability
increases, finding the profit-maximizing prices
become a nontrivial task. In this case, optimal pricing policy alone becomes less competent in regulating customer traffic, and the firm needs a booking
limit to avoid customer bumping. When the market
size is highly variable, the actual market size can
turn out to be quite different from the expected
market size. Therefore, the firm should charge a
non-negligible service price even in the case with
small average market size keeping in mind that the
market realization can be large. Likewise, the firm
also has to charge a fair amount as the prepaid
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936
Production and Operations Management 31(3), pp. 928–940, © 2021 Production and Operations Management Society
937
Figure 4 Fraction of ϕ over ϕ + p [Color figure can be viewed at wileyonlinelibrary.com]
ϕ
Notes: (a) The height of each bar represents the value of ϕ þ
p for each (average demand/capacity, standard deviation of market size) pair. (b) The yellow
squares represent the (mean demand/capacity, standard deviation of market size) pairs that result in fully prepaid reservations; the green/blue squares
represent the market condition pairs that result in partially prepaid reservations. The fraction of the prepayment becomes smaller as the squares become
darker blue in color.
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Oh and Su: Optimal Pricing and Overbooking of Reservations
Oh and Su: Optimal Pricing and Overbooking of Reservations
Production and Operations Management 31(3), pp. 928–940, © 2021 Production and Operations Management Society
deposit even when the average market size is large
knowing that the market can turn out to be smaller
than expected. This can be seen from Figure 4b as
the area of green squares become wider when the
standard deviation of the market size grows. Note
also that the (mean market size/capacity, standard
deviation of market size) pairs that are lighter
green in Figure 4b are in darker blue in Figure 3.
This means that as it becomes a more challenging
task to set the profit-maximizing prices, the firm
tends to set more stringent booking limits to avoid
the risk of bumping the customers.
5. Extension: Free Cancelation
So far, we have studied how firms can set the optimal overbooking and pricing policies to deal with
the uncertainties the firm and the customers face
when reservations are made. We have studied how
a firm can deal with the market size uncertainty by
jointly utilizing the overbooking and the pricing
policies. We have also looked at how the firm can
resort to the pricing policies to manipulate the level
of customer valuation uncertainty at the point payment takes place. In this section, we consider an
extension of the base model where customers’ valuation uncertainties are resolved gradually over time.
Through this extension, we intend to better understand how the firm’s pricing decision and its profitability depend on the point of sales as the
customers’ level of knowledge on valuation changes
over time. This extension also serves as a robustness
check when richer customer choice dynamics are
introduced. In our basic model, once customers
book their reservations, their next decision was at
the time of service when they either show up or
not. However, during this interval, customers may
learn more about their valuations and some may
choose to cancel their reservations. This is a common occurrence, and it is useful to check the basic
results in this scenario.
The model is as what follows. In period 0, reservations are offered for free. In period 1, customers
realize their types i ∈ {h, l}, where h represents the
high type and l represents the low type. We assume
that there are αΛ high type customers in the market, whereas the remaining (1− α)Λ are low type.
Type-i customers have valuation for service ui ,
which is independent and identically distributed
across the type-i customers; the valuation follows
i ¼ 1 Fi . We
distribution Fi and density f i with F
assume that the high type customers’ valuation for
service uh has first-order stochastic dominance over
the low-type customers’ valuation ul so that for any
l ðxÞ. Upon observing their types in
h ðxÞ ≥ F
given x, F
period 1, customers decide whether they should
keep their reservations or not. Customers who cancel their reservations need not pay anything,
whereas those who decide to keep their reservations
pay the price of reservation ϕ as was the case in the
base model. In period 2, customers who have kept
their reservations decide if they show up or not. A
customer who shows up for her reservation and
receives the service pays a price p; a customer who
shows up but is denied a service incurs a cost c of
being bumped.
The firm then has two pricing strategies: (i) set
(ϕ, p) so that both high type and low type customers keep their reservations in period 1,
(ii) set (ϕ, p) so that only high type customers keep
their reservations. Note that a strategy where only
low type customers keeping their reservations is not
feasible. We now state the equilibrium outcome as
the following.
PROPOSITION 6.
In equilibrium, the followings are true.
1. When both types of customers keep reservations in
period 1, ðB , v , q Þ must take one of the following
two forms.
(i) (Accept-all equilibrium) B ¼ ∞, v ¼ p þ
1 q
μ
c q , q ¼ EΛ min
,
1
.
l h αΛF ðv Þþð1αÞΛF ðv Þ
(ii) (Booking-limit
μ
,
h ðp Þ þ ð1 αÞF
l ðp Þ
αF
equilibrium)
v ¼ p , q ¼ 1.
B ¼
2. When only high type customers keep reservations in
Step 1, ðB , v , q Þ must take one of the following
two forms.
(i) (Accept-all equilibrium) B ¼ ∞, v ¼ p þ
μ
c 1 q q , q ¼ EΛ min
,
1
.
h αΛF ðv Þ
(ii) (Booking-limit equilibrium) B ¼
q ¼ 1.
μ
,
h ðp Þ
αF
v ¼ p ,
From the above proposition, we show that there still
exist two types of equilibria regarding the overbooking policy when customers are granted an option to
cancel their reservations for free in advance.
A natural question that follows then is when it is
more profitable for the firm to set the prices (ϕ, p) so
that low type customers take advantage of free cancelation. To simplify the analyses and focus on interesting findings, we assume that the market size Λ is
deterministic for the rest of this section.
PROPOSITION 7.
such that
~ðΛÞ
For a fixed constant Λ, there exists α
~ðΛÞ, both types of customers keep their
1. when α < α
reservations in period 1, and
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938
Production and Operations Management 31(3), pp. 928–940, © 2021 Production and Operations Management Society
1
R ¼ Λ, p ≥ ðFÞ
v ¼ p , q ¼ 1.
μ
Λ
, ϕ ¼
R∞
p
ðu p Þ f l ðuÞdu,
~ðΛÞ, only high type customers keep
2. when α ≥ α
their reservations in period 1, and
1 h Þ μ ,
If αΛ ≥ μ, then R ¼ αΛ, p ¼ ðF
Λ
R∞
ϕ ¼ p ðu p Þ f h ðuÞdu, v ¼ p , q ¼ 1.
R∞
If αΛ < μ, then R ¼ αΛ, p ¼ 0, ϕ ¼ p u f h ðuÞdu,
v ¼ 0, q ¼ 1.
Proposition 7(1) states that when the population of
high type customers is not large enough, the firm
should set the prices (ϕ, p) so that both types of customers keep their reservations. In this case, the prices
are set to extract the surplus of low type customers
while leaving positive surplus to high type customers.
Proposition 7(2) states that when there are enough
high type customers in the market, the firm should
just have high type customers keep their reservations
and low type customers cancel for free in period 1. By
forgoing the profit from low type customers, the firm
is able to charge prices (ϕ, p) so as to extract surplus
from the high type customers. Note that in both cases
of Proposition 7 we observe the pattern where the
firm depends less on prepayment ϕ and more on the
spot price p as the targeted market size grows.
Another question of interest is if granting customers free cancelation is good for the firm, which is
answered by the following proposition.
PROPOSITION 8. For a fixed market size Λ, the following
statements hold in equilibrium.
~ðΛÞ, offering free cancelation to customers
1. If α < α
decreases the firm’s profit.
~ðΛÞ, offering free cancelation to customers
2. If α ≥ α
increases the firm’s profit.
Proposition 8(1) states that when there are not
enough high type customers in the market and thus
the firm sets the price so that both high type and low
type customers keep their reservations in period 1,
then the firm is better off not giving the option of free
cancelation to the customers. This is because offering
free cancelation allows low type customers to pay the
price of reservation ϕ after they realize their types,
whereas not offering free cancelation forces low type
customers to pay the upfront reservation price when
they are uncertain of their types. When customers are
unsure of their types, they are willing to pay the reservation price based on their expected types; this price
charged based on expectation is higher than the reservation price that can be charged to low type customers
who are aware of their types. This shows that the result
of Shugan and Xie (2005) where selling goods in
939
advance while customers are uncertain of their valuation bring higher profit to the seller extends to our case
of two prices and two-stage realization of valuation.
Proposition 8(2) conversely shows that when there
are enough high type customers in the market and
thus the firm sets the prices so that only high type
customers keep their reservations, then the firm is
better off by offering free cancelation to customers.
This is because the firm now targets to sell reservations only to the high type customers with its
increased price.
Note that in Proposition 8(1) when offering free
cancelation decreases firm’s profit, nobody exercises
the option of free cancelation. On the other hand,
when some customers indeed cancel their reservations
for free, that is when offering free cancelation is more
profitable to the firm as shown in Proposition 8(2).
6. Conclusion
In this study, we study the optimal design of reservations. Our results provide two guiding principles for
optimal reservation policies.
• Principle 1: Reservations require a nonrefundable fee when booked and/or a service charge
when used; the former dominates in small
markets and the latter dominates in large markets.
• Principle 2: All reservation requests can be
accepted when demand is relatively low and/or
predictable, but a booking limit is required to
ensure that all reservations can be honored
when demand is large and/or highly variable.
These two principles characterize the optimal design
of reservation dependent on market condition defined
as a pair of mean and the standard deviation of market size.
The reasoning behind our principles can be outlined as follows. The first principle holds because
reservation holders consider prices when choosing
whether to show up for service. Firms facing low
demand should charge low service prices and high
reservation deposit since they can sell their capacities
at a higher price when customers make the payments
in advance without knowing their valuations for service. On the other hand, firms facing high demand do
not rely on the nonrefundable deposit because they
can charge high service prices: even when the service
price is high, there is sufficient demand to fully utilize
capacity. This logic brings us to the second principle.
When aggregate demand is publicly known, all reservations can be accepted because the fixed market of
customers respond to prices and show up in the
appropriate numbers. However, in unpredictable
markets, customers do not know the realized number
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Oh and Su: Optimal Pricing and Overbooking of Reservations
Oh and Su: Optimal Pricing and Overbooking of Reservations
Production and Operations Management 31(3), pp. 928–940, © 2021 Production and Operations Management Society
of reservation requests and too many may show up;
in such cases, a booking limit helps ensure that not
too many reservations are given out so they can all be
fulfilled.
The current work can be extended in three broad
directions. First, we have not examined the role of
competition. When there are multiple firms, price
competition becomes an issue, and moreover, the possibility that customers may take reservations from
multiple firms complicates overbooking decisions.
Second, there may be information asymmetry in the
quality of firms (e.g., good vs. bad restaurants) and
the structure of reservation policies may be an important signaling device. Some progress in this direction
has been made by Yu et al. (2014) in the advance selling literature. Third, when reservations are transferrable, there is the issue of resale and speculation.
For example, restaurant reservations bought for $50
have been resold for up to $600 (McKinley 2011). It
would be interesting to understand the impact of secondary markets on reservations.
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Supporting Information
Additional supporting information may be found online
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Appendix S1: Appendix.
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