Vol. 27, No. 6, November–December 2008, pp. 1097–1110
issn 0732-2399 eissn 1526-548X 08 2706 1097
inf
orms
®
doi 10.1287/mksc.1080.0370
© 2008 INFORMS
Modeling DVD Preorder and Sales:
An Optimal Stopping Approach
Sam K. Hui, Jehoshua Eliashberg, Edward I. George
The Wharton School of the University of Pennsylvania, Philadelphia, Pennsylvania 19104
W
hen a DVD title is announced prior to actual distribution, consumers can often preorder the title and
receive it as soon as it is released. Alternatively, once a title becomes available (i.e., formally released),
consumers can obtain it upon purchase with minimal delay. We propose an individual-level behavioral model
that captures the aggregate preorder/postrelease sales of motion picture DVDs. Our model is based on an opti-
mal stopping framework. Starting with the utility function of a forward-looking consumer, and allowing for
consumer heterogeneity, we derive the aggregate preorder/postrelease sales distribution. Even under a parsi-
monious specification for the heterogeneity distribution, our model recovers the typically observed temporal
pattern of DVD preorder and sales, a pattern which exhibits an exponentially increasing number of preorder
units before the release, peaks at release, and drops exponentially afterward. Using data provided by a major
Internet DVD retailer, we demonstrate a number of important managerial implications stemming from our
model. We investigate the role of preorder timing through a policy experiment, estimate residual sales, and
forecast post-release sales based only on preorder information. We show that our model has substantially better
predictive validity than benchmark models.
Key words: optimal stopping; timing model; online retailing; motion picture
History: This paper was received March 3, 2007, and was with the authors 3 months for 2 revisions; processed
by Gerard Tellis. Published online in Articles in Advance July 31, 2008.
1. Introduction
The home video market is an important ancillary
market for the motion picture industry (Eliashberg
et al. 2006). In 2005, it generated about $23.8 billion
in total revenues (Standard and Poor’s 2006), whereas
theatrical ticket revenues only accounted for about
$9.0 billion (http://www.mpaa.org). Despite its eco-
nomic importance, the motion picture home video
market has received less attention among market-
ing researchers than the theatrical market, where
researchers have offered rich behavioral explanations
(e.g., Jones and Ritz 1991, Sawhney and Eliashberg
1996, Zufryden 1996) as well as systematic explana-
tions (e.g., Jedidi et al. 1998) for box office revenue
patterns. The potential for modeling home video sales
is suggested by remarkably similar temporal patterns
of DVD sales across different titles. For example, the
weekly preorder/sales pattern (from a major Internet
retailer) for the title 24 Hour Party People shown in
Figure 1 is typical—(preorder) sales increase exponen-
tially before the release week, reach their peak dur-
ing the week of release, and then drop exponentially
afterward.
In previous literature, researchers have used
population-level statistical models to describe the
temporal diffusion patterns of media products such
as in Figure 1, which usually differs from the conven-
tional S-shaped product life cycle (e.g., Hauser et al.
2006). For instance, Lehmann and Weinberg (2000)
modeled the temporal patterns of home video sales
revenues after release using an exponential distribu-
tion and obtained a reasonably adequate fit. In a simi-
lar vein, Moe and Fader (2002) modeled the temporal
pattern of music CD sales using a mixture-Weibull
model. Their model is based on two segments of
consumers: innovators, who may preorder CDs, and
followers, whose actions are influenced by word-of-
mouth information and so only purchase CDs after
their release. The purchase timing decisions for each
segment of consumers are then assumed to follow
two separate Weibull distributions. With this spec-
ification, Moe and Fader (2002) reported that their
model was able to capture many different types of
preorder/sales patterns. They further demonstrated
how retailers may use preorder information to predict
the (post-release) CD sales.
In both of the above studies, the proposed mod-
els are not based on individual-level decision mak-
ing, but instead on a prespecified functional form of
the sales curve (e.g., the exponential distribution in
Lehmann and Weinberg 2000) or on a population-
level behavioral model (Moe and Fader 2002). In this
paper, we take a different approach by developing a
1097
Hui, Eliashberg, and George: Modeling DVD Preorder and Sales: An Optimal Stopping Approach
1098 Marketing Science 27(6), pp. 1097–1110, © 2008 INFORMS
Figure 1 Sales Pattern for the DVD Title 24 Hour Party People
200
150
100
50
0
15 20 30 3525
Notes. The dotted line denotes the release date of the DVD. The y-axis
denotes weekly unit sales, whereas the x-axis is the number of weeks since
the movie’s release.
model based on individual-level “rational” behavior
to explain the pattern shown in Figure 1. Instead of
using an a priori functional form to “fit” the sales
curve, we begin by specifying a utility function for
an individual consumer. We then derive the utility-
maximizing decision rule for each consumer (i.e., her
optimal purchase timing), specify the degree of het-
erogeneity across consumers, and aggregate across
their individual decisions to derive the aggregate
sales pattern. Thus, the functional form that describes
the aggregate sales pattern is an outcome of our model,
rather than an a priori assumption. In this respect,
our model is similar in spirit to recent NEIO mod-
els in marketing (e.g., Chintagunta et al. 2006), where
empirical patterns are often explained as outcomes of
consumer utility-maximizing decisions.
At the heart of our model is a forward-looking con-
sumer (e.g., Song and Chintagunta 2003, Sun et al.
2003) interested in a DVD title, who visits a DVD
ordering Web page at random time intervals, and
who does not remember the DVD release date. On
each visit, she decides whether to preorder/purchase
the DVD instantly, or waits until her next visit to
reconsider. Under any general interarrival distribu-
tion, the consumer’s decision then corresponds to
an optimal stopping problem (Chow et al. 1971),
and the optimal solution is to follow a “threshold”
rule (Ferguson 2000), i.e., to preorder the DVD if
the consumer arrives within a certain time from the
release date. Based on the optimal stopping rule, we
obtain the distribution of an individual consumer’s
purchase timing. Allowing heterogeneity across con-
sumers, we obtain the aggregate temporal sales pat-
tern that recovers the qualitative characteristics of the
observed pattern in Figure 1. In addition to this basic
model, we further extend our framework to allow for
a segment of consumers who do remember the DVD
release date and buy directly at that time.
We calibrate our basic (DVD-I) and extended
(DVD-II) models to a data set containing weekly
DVD preorder/sales information provided by a major
Internet retailer. Both models are seen to substan-
tially outperform benchmark models such as Moe and
Fader (2002) and the Weibull-Gamma model (Jaggia
and Thosar 1995). We further demonstrate some
potential managerial implications from our study. We
investigate the role of preorder timing through a pol-
icy experiment, estimate residual sales, and forecast
post-release sales based only on preorder information.
The remainder of this paper is organized as fol-
lows. In §2 we develop our model of DVD purchase
timing. Section 3 provides an overview of our data
set, along with key summary statistics. In §4, we
compare our model to other benchmark models to
assess model performance, then interpret the param-
eters estimates. In §5, we demonstrate some potential
managerial implications stemming from our model.
Finally, §6 concludes and outlines directions for future
research.
2. Model
In this section, we describe our model of DVD pur-
chase timing in detail. For clarity of exposition, we
first focus on an individual consumer in §§2.1–2.4.
We describe our model setup and notations in §2.1,
and specify the consumer’s utility function in §2.2.
In §2.3, we show how the consumer’s purchase tim-
ing decision can be viewed as an optimal stopping
problem in which the solution takes the form of a
“threshold” rule (Ferguson 2000). In §2.4, the distri-
bution of an individual consumer’s purchase timing
is derived under the assumption of exponentially dis-
tributed interarrival times. In §2.5, we account for
consumer heterogeneity and derive the sales distri-
bution by aggregating across consumers. This gives
us the DVD-I model and its associated likelihood
function. In §2.6, we extend the DVD-I model to the
DVD-II model which allows for a second segment of
consumers who remember the release date and come
back in that week to purchase the DVD.
2.1. Model Setup
Figure 2 shows the timing of the important events in
our model corresponding to a specific movie. Time
t = 0 represents the theatrical release date. At time
t = l, the Internet retailer allows consumers to begin
preordering the DVD. The DVD is released (and
shipped to fulfill preorders) at time t = k, which is
called the “window” in the movie industry.
1
1
Note that we make the assumption of instantaneous order fulfill-
ment. That is, we assume that if a DVD is ordered after its release,
it will be immediately shipped and the consumer will receive it
instantaneously. Similarly, if a DVD is ordered before its release,
the order will be shipped and will arrive on the release date. This
assumption is made for analytical convenience; similar results hold
Hui, Eliashberg, and George: Modeling DVD Preorder and Sales: An Optimal Stopping Approach
Marketing Science 27(6), pp. 1097–1110, © 2008 INFORMS 1099
Figure 2 Timeline of the Major Events in Our Model
Theatre release Preorder starts DVD release
DVD release window
lk
t
Time (t)
0
Next, we specify the arrival process by which a con-
sumer visits the DVD ordering Web page. On each
visit, the consumer decides whether to purchase the
DVD, or to wait till her next visit to reconsider. As
shown in Figure 3, we index the consumer’s visits
by n = 0 1 2, where the nth visit takes place at
time t
n
. The interarrival time between the nth and the
(n + 1)th visit is denoted by w
n
(i.e., w
n
= t
n+1
− t
n
),
which are assumed to be drawn i.i.d. from a general
distribution (known both to the consumer and the
researcher). To denote this distribution, we use Gw
and gw for cumulative distribution function and the
probability distribution function, respectively.
2.2. Specification of Consumer Utility
We use the following notation to specify the utility of
the DVD to the focal consumer. Let
˜
us be the instan-
taneous utility from the DVD at time s after obtaining
it. Let y be the price
2
of the DVD, and let
˜
x
y
s be the
instantaneous utility from an outside option (obtain-
able at price y) at time s after obtaining it. This speci-
fication allows us to compare the utility of the DVD to
the utility of other outside options that the consumer
would forego by purchasing the DVD. For instance,
the consumer could have used that money to buy a
CD or a new book.
We assume that consumers discount future utili-
ties by a constant discounting factor, so that (relative
to the time point t = 0) utilities obtained at time t
are discounted by a factor e
−t
, where >0. This
assumption is widely used in intertemporal choice
models (e.g., Frederick et al. 2002, Lowenstein and
Prelec 1992). The discount rate is taken as a constant
and is assumed to be fixed across all titles and all con-
sumers. We further assume that Internet search cost
if we allow for a constant shipping delay. (In the actual data, DVDs
are typically shipped within 1–3 days, which is very short com-
pared to the time unit of the data set which is weeks.) Further, we
assume that there is no scarcity for the DVD, an assumption which
we have verified with our data provider.
2
Throughout this paper, we assume that the price of a DVD
remains constant over the preorder period and afterward. We have
verified this assumption empirically with our data provider. We
further assume that consumers’ expectation of prices are rational;
i.e., they also expect prices to remain constant during their plan-
ning horizon.
Figure 3 Arrival Process of a Consumer to the DVD Ordering
Webpage
t
klt
3
t
2
t
1
t
0
w
0
w
1
w
2
Time (t)
Visit (nth)
0
01 23
is zero (e.g., Bakos 1997, Wu et al. 2004); i.e., the con-
sumer does not incur any cost in visiting the website.
To simplify notation, we let
u =
0
e
−s
˜
us ds (1)
x =
0
e
−s
˜
x
y
sds (2)
which correspond, respectively, to the “net present
utility” of the DVD and of the outside option (rela-
tive to utility at time t = 0). We further assume that
u>xfor our focal consumer, given his/her interest in
the DVD title. Similar to the “participation constraint”
in economics (Mas-Colell et al. 1995), this assump-
tion ensures that the DVD is sufficiently attractive
that the consumer will buy it—the question is when.
Later in §2.5, we describe how we can estimate mar-
ket potential, i.e., the number of consumers who have
u>x, for each DVD in our data set.
We now specify the net utility (relative to time
t = 0)
3
that the consumer receives if she preorders/
purchases the DVD on her nth visit to the DVD Web
page. The consumer may only preorder/purchase the
DVD if t
n
≥ l. Thus, if t
n
<l, no decision needs to be
made. In the discussion below, we separately consider
the two remaining cases: (i) l ≤ t
n
<k(prerelease), and
(ii) t
n
>k (post-release).
First, consider l ≤ t
n
<k. In this case, as shown in
Figure 4, the consumer preorders and pays for the
DVD at time t
n
, and receives the DVD at time t = k.
Her net utility (relative to time t = 0), Y
n
, is a sum
of two parts: the forgone (discounted) utility of the
outside option, and the (discounted) net utility from
the DVD which will arrive at time t = k. Formally,
Y
n
=−
t
n
e
−t
˜
x
y
t − t
n
dt+
k
e
−t
˜
ut − k dt (3)
3
Note that to ensure that the utilities mentioned are in the same
units and thus directly comparable, all the utilities in our model are
discounted relative to the time t = 0, the theatrical release time. This
specification is done without loss of generality, because discounting
is a monotonic transformation and thus the relative order of utilities
is preserved. As a concrete example, assume that at time t
1
, a con-
sumer is deciding between options A and B, with (instantaneous)
utilities U
A
and U
B
, respectively. Because U
A
>U
B
⇔ U
A
e
−t
1
>
U
B
e
−t
1
, the consumer’s decision is invariant to the time point rel-
ative to which utilities are discounted.
Hui, Eliashberg, and George: Modeling DVD Preorder and Sales: An Optimal Stopping Approach
1100 Marketing Science 27(6), pp. 1097–1110, © 2008 INFORMS
Figure 4 Consumer’s Net Utility for Preordering the DVD
t
t
n
k
–x +u
Time
Utility
(Net present utility of
outside option)
(Net present utility
from DVD)
(DVD release)
Changing variables in the integral (with z = t − t
n
and
v = t − k), we obtain
Y
n
=−e
−t
n
x + e
−k
u (4)
where u and x are defined in Equations (1) and (2).
In the other case, when t
n
>k (as shown in Fig-
ure 5), the consumer pays for and receives the DVD
at the same time t
n
. In this case, we have
Y
n
=−
t
n
e
−t
˜
x
y
t − t
n
dt+
t
n
e
−t
˜
ut − t
n
dt
Changing variables in the integral (with z = t −t
n
), we
have
Y
n
=−e
−t
n
x + e
−t
n
u = e
−t
n
u − x (5)
where, again, u and x are defined in Equations (1)
and (2).
Together, Equations (4) and (5) give us a complete
specification for the consumer’s discounted net utility
Y
n
(with respect to t = 0):
Y
n
=
−e
−t
n
x + e
−k
u for l ≤ t
n
<k
−e
−t
n
x − u for t
n
≥ k
(6)
Note that our utility specification above is similar to
other specifications based on discount utilities and
has been employed to derive optimal stopping rules
(Allaart 2004, Dubins and Teicher 1967). From the pre-
vious literature on optimal stopping theory, the exis-
tence of an optimal solution to our problem has been
proven. A formal proof is available upon request.
2.3. Consumer Preorder/Purchase
Timing Decisions
Under our model setup, the consumer faces an opti-
mal stopping problem (Chow et al. 1971) when decid-
ing when to place an order for a DVD. Intuitively,
Figure 5 Consumer’s Net Utility for Purchasing the DVD After Its
Release
t
t
n
k
–x
(Net present utility from outside option)
+u (Net present utility from DVD)
Time
Utility
(Release date)
the consumer prefers to (pre-) order the DVD as close
to the actual release date as possible. She derives the
most utility if she purchases the DVD at the release
date; i.e., there is no advantage for her to preorder
a DVD because she has to pay in advance while
only receiving the DVD on the release date. However,
assuming her time of visit to the website is stochas-
tic, if she decides to wait, she may “miss” the release
date and have to purchase afterward (therefore receiv-
ing a lower utility due to discounting). As we show
in the following derivation, the consumer’s optimal
decision rule is characterized by a “threshold rule”
(e.g., Ferguson 2000); i.e., she will preorder the DVD
if she happens to visit the website at time t
n
, and the
gap between time t
n
and the release date k is smaller
than a predetermined threshold.
We solve for the optimal stopping visit n
∗
by con-
sidering three cases: (i) t
n
<l, (ii) t
n
≥ k, and (iii) l ≤
t
n
<k. In the first case (t
n
<l), no decision needs to
be made because preorder is not yet allowed. In the
second case (t
n
≥ k), there is no advantage for the con-
sumer to wait any longer, and she should purchase
the DVD immediately. Formally, for any t
n
≥ t
n
≥ k,
Y
n
=−e
−t
n
x − u ≤−e
−t
n
x − u = Y
n
(7)
The third case (l ≤ t
n
<k) is most interesting
because the consumer has to trade off between the
opportunity cost of ordering now (and paying in
advance) versus the risk of waiting longer than nec-
essary to enjoy the DVD. This is similar to the idea
of a “delay premium” (e.g., Lowenstein 1988); i.e.,
some consumers are willing to pay a premium to
ensure that their enjoyment from the DVD will not be
delayed. According to a manager at our data provider,
our specification is consistent with their understand-
ing of why consumers preorder DVDs in the absence
of preorder discounts and stock-out possibilities, the
environment under which our data set was collected.
Thus, the consumer makes a (pre-)order/wait deci-
sion each time she visits the DVD Web page. She
buys the DVD at time t
n
if her utility of buying
exceeds the expected utility of “waiting.” Because our
model satisfies the property of “monotonicity” (see
Appendix I), the 1-step look-ahead rule is equivalent
to the globally optimal solution.
4
More precisely, the
consumer buys the DVD at visit n (i.e., time t
n
)if
her discounted net utility of buying now exceeds the
expected utility of buying at visit n + 1, i.e., if Y
n
≥
EY
n+1
n
, where
n
denotes the probability measure
with respect to the information available up to the
nth visit.
4
The proof one-step look-ahead rule is the optimal solution to the
monotonic optimal stopping problem and can be found in Ferguson
(2000).
Hui, Eliashberg, and George: Modeling DVD Preorder and Sales: An Optimal Stopping Approach
Marketing Science 27(6), pp. 1097–1110, © 2008 INFORMS 1101
Letting
n
= k − t
n
(the gap between the current
time and the release date), we have
Y
n+1
buy at visit n
=
e
−t
n
e
−
n
u − e
−w
n
x for w
n
≤
n
e
−t
n
e
−w
n
u − e
−w
n
x for w
n
>
n
(8)
⇒ EY
n+1
n
= e
−t
n
n
0
e
−
n
u−e
−w
n
xgw
n
dw
n
+
n
e
−w
n
u − xgw
n
dw
n
(9)
where g is the probability density function of the
interarrival time defined in §2.2. Equations (8) and (9)
precisely describe the consumer’s trade-off discussed
above. Note that two implicit assumptions are made
here. First, we assume that consumers know k. This
assumption is generally valid given that the release
date is displayed prominently on the DVD page.
The second assumption is that the consumers do not
remember the release date. We later relax this assump-
tion in the DVD-II model of §2.6, where we allow for
a segment of consumers who do remember the DVD
release date, and thus purchase during the week of
the release.
After some algebraic manipulations shown in
Appendix II, we obtain
Y
n
≥EY
n+1
n
⇔
x
u
1−Ee
−w
n
≤ e
−
n
1−G
n
−
n
e
−w
n
gw
n
dw
n
≡H
n
(10)
where the second term of the inequality (10) is
denoted by H
n
. Because the left side of inequality
(10) is independent of
n
, we only need to show that
H
n
is a decreasing function of
n
to prove that the
optimal solution takes the form of a threshold rule.
Differentiating H
n
with respect to
n
,
dH
n
d
n
= e
−
n
−g
n
+ 1 − G
n
e
−
n
−+ e
−
n
g
n
=−e
−
n
1 − G
n
< 0 (11)
Thus, the optimal stopping rule is to buy the DVD
whenever
n
= k − t
n
is below a threshold that
depends on the ratio (u/x), , and the interarrival
time distribution g. We write
n
∗
= min"n
n
≤ Cu/x g $ (12)
where Cu/x g captures the dependence of the
threshold on the ratio (u/x) and on the interarrival
Figure 6 Illustration of the Threshold Rule
Possible DVD purchase time
C(u/x, g
γ)
k
l
t
0
Note. The shaded area indicates the possible time of DVD preorder/purchase
for a specific consumer.
time distribution g.
5
We denote the corresponding
purchase time as t
n
∗
.
Combining the above three cases, the consumer
preorders a DVD if she visits the DVD Web page at
time t ≥ d, where d = max"l k − Cu/x g $ is
the larger of the time when preorder is available and
the threshold in Equation (12). This result is depicted
graphically in Figure 6; the shaded area represents the
possible DVD purchase time for our focal consumer.
2.4. Individual-Level Distribution of
Purchase Timing Under Exponential
Interarrival Times
Following previous literature on Internet visit behav-
ior (Moe and Fader 2004, Park and Fader 2004), we
restrict our attention, in this subsection, to the case
where the consumer’s interarrival times follow an
exponential distribution with parameter %, and derive
the corresponding distribution of her purchase time.
The assumption of exponentially distributed interar-
rival times has been widely used in marketing and
operations research in models of consumers’ interar-
rival/interpurchase times (e.g., Morrison and Schmit-
tlein 1988) and other duration models (e.g., Morrison
and Schmittlein 1980).
Substituting the exponential density, gw
n
= %e
−%w
n
into Equation (10) and performing the algebraic sim-
plifications outlined in Appendix III, the following
threshold decision rule is obtained. The consumer will
preorder the DVD if
n
≤
1
% +
ln
u
x
= C (13)
The form of the threshold rule above leads to two
immediate, intuitive insights. First, the threshold is an
increasing function of u/x. Thus, the more net dis-
counted utility a consumer derives from the DVD
(relative to her outside option), the more likely she
is to preorder. Second, the threshold is a decreasing
5
Dependence of the threshold on the discount rate is not denoted
explicitly because it is treated as a constant in our model, as we
discussed earlier.
Hui, Eliashberg, and George: Modeling DVD Preorder and Sales: An Optimal Stopping Approach
1102 Marketing Science 27(6), pp. 1097–1110, © 2008 INFORMS
function of %; the higher a consumer’s frequency of
visits to the DVD website, the less likely she is to pre-
order. Intuitively, because more frequent visitors have
more chances to visit the DVD Web page, they can
afford to wait a little longer in the hope of preordering
closer to the actual release date. Their risk of missing
the release date is lower.
Considering the cumulative distribution function
(CDF) of t
n
∗
(the purchase time) conditional on t
n
∗
−1
(the time of the last visit before purchase), we have
F
t
n
∗
t t
n
∗
−1
= 1 − Pt
n
∗
>t t
n
∗
−1
= 1 − Pw
n
∗
−1
>t− t
n
∗
−1
w
n
∗
−1
>d− t
n
∗
−1
= 1 − e
−%t−d
for t>d (14)
where d = maxlk − C as defined earlier. Note that
in the second step above, we utilize the “memoryless”
property of the exponential distribution. Because the
expression in Equation (14) is independent of t
n
∗
−1
,
we also have the unconditional CDF of t
n
∗
, F
t
n
∗
t =
1 − e
−%t−d
. Thus, the purchase time t
n
∗
follows a
shifted exponential distribution with parameter %;
i.e., t
n
∗
+ d follows an exponential distribution with
parameter %. The corresponding probability density
associated with Equation (14) is the (shifted) exponen-
tial density, f
t
n
∗
t = %e
−%t−d
.
2.5. Consumer Heterogeneity: DVD-I Model
We now show that under a simple and reason-
able specification of heterogeneity across consumers,
namely that u/x
i
(where i indexes consumer) fol-
lows a Pareto distribution with parameter v, i.e.,
f u/x
i
= vu/x
i
−v−1
for u/x
i
∈ 1,
6
our
model can generate the observed temporal pattern
of DVD preorder and sales. Using a change of vari-
able, lnu/x
i
follows an exponential distribution
with rate parameter v.
7
The Pareto distribution has
been found to be a good approximation in various
economic contexts involving heterogeneity between
units, e.g., wealth distributions (Wold and Whit-
tle 1957) and income distributions (Clementi and
Gallegati 2005).
Although we allow u/x
i
to vary across the popu-
lation of DVD buyers, we keep the parameter % fixed.
We make this simplifying assumption for two reasons.
First, models with a similar homogeneity assumption
have been successfully applied to model box office
6
The range for u/x
i
here is consistent with our assumption that
(among DVD buyers) the ratio of the utility from the DVD com-
pared to the outside option, u/x
i
, is larger than 1.
7
In the Technical Appendix at http://mktsci.pubs.informs.org, we
consider relaxing this assumption by allowing lnu/x
i
to follow
a gamma distribution. We find that it only provides a marginally
better fit at the cost of the analytic tractability and computational
efficiency of a closed-form solution. Thus, we focus on using the
exponential distribution assumption in this paper.
sales patterns (e.g., Sawhney and Eliashberg 1996).
Second, assuming a homogeneous % maintains ana-
lytical tractability when deriving the marginal distri-
bution of purchase times across the population.
Under these specifications,
u/x
i
∼ Paretov lnu/x
i
∼ expv (15)
the distribution of the threshold across consumers
(C
i
denotes the threshold for the ith consumer) is
C
i
=
1
% +
lnu
i
/x ⇒ C
i
∼ expv% +
⇒ C
i
∼ exp* (16)
where * = v% + . The marginal distribution of
purchase timing across the population, ft, is then
obtained by integrating the distribution function
associated with Equation (14) across all consumers
(indexed by i), under the mixture distribution speci-
fied by Equation (16):
ft =
−
f
i
t C
i
+C
i
dC
i
=
0
%e
−%t−d
1
"t>d$
*e
−*C
i
dC
i
= %*e
−%t
0
e
%d−*C
i
1
"t>d$
dC
i
(17)
From this expression we obtain the following
closed-form expressions for the marginal distribu-
tion ft and the corresponding cumulative distri-
bution function Ft of the DVD purchase time (see
Appendix IV):
ft=
%*
%+*
e
%k
1−e
−%+*k−l
+%e
%l−*k−l
e
−%t
t ≥ k
%*
%+*
e
−*k−t
+
%
2
%+*
e
−%t−l−*k−l
l ≤t<k
(18)
Ft=
%
% + *
e
−*k
e
*t
− e
*l
+ e
*+%l
e
−%l
− e
−%t
for t ≤ k
= Fk+
A
%
e
−%k
− e
−%t
for t>k (19)
where A = %*/% + *e
%k
1 − e
−%+*k−l
+ %e
%l−*k−l
.
The properties of ft are interesting. First, the
density is strictly exponentially decreasing in the
region t ≥ k. This is consistent with the observation
in Lehmann and Weinberg (2000) that post-release
revenues are exponentially decreasing. For l ≤ t<k,
ft is a sum of two terms. The first term gives us
the exponentially increasing preorder pattern that we
observe in the data. The second term acts as a “correc-
tion term” due to the fact that preorders are available
Hui, Eliashberg, and George: Modeling DVD Preorder and Sales: An Optimal Stopping Approach
Marketing Science 27(6), pp. 1097–1110, © 2008 INFORMS 1103
Figure 7 Distribution of Aggregate Sales Derived from Our Model
(Equation (18); = 03, = 04)
x(t)
k
t
only from t = l onward; its magnitude decreases if
preorders are allowed earlier, and vanishes if pre-
orders are always available (i.e., in the limit l →−.
The density ft in Equation (18) is plotted in
Figure 7. As can be seen, the pattern in Figure 7
closely resembles that of Figure 1; in particular, (pre-
order) sales increase exponentially before the release
date and decrease exponentially after it. This pro-
vides some evidence that our model, based on the
individual-level utility maximization and forward-
looking behavior, provides an explanation for the pat-
tern of the observed aggregate sales data.
By varying % and * and plotting the resulting den-
sity function, we can study our model properties
more closely and further understand the sources of
identification for our model parameters. Figures 8
and 9 plot ft as % and * are varied, respectively.
From the figures, we see that % primarily controls
the rate of decline, whereas * mainly controls the
rate of ascent. More specifically, larger values of %
and * correspond to faster rates of decline and ascent,
respectively. Thus, the rate of decline after release pro-
vides the source of identification for the parameter %,
whereas the rate of ascent prerelease provides the
source of identification for the parameter *.
Because our proposed model is defined in contin-
uous time while actual preorder/sales data are usu-
ally recorded in discrete time units (e.g., week or
day), we need to first discretize our model before
Figure 8 Plots of pdf by Varying
λ = 0.15, τ = 0.30
λ = 0.10, τ = 0.30
λ = 0.20, τ = 0.30
0.12
0.10
0.08
0.06
0.04
0.02
0.00
0 5 10 15 20 25
pdf
Figure 9 Plots of pdf by Varying
λ = 0.15, τ = 0.30
λ = 0.15, τ = 0.20
λ = 0.15, τ = 0.40
0 5 10 15 20 25
0.12
0.10
0.08
0.06
0.04
0.02
0.00
pdf
calibrating our model parameters on actual data. This
is the same issue faced by Moe and Fader (2002); to
tackle this problem, we use an approach proposed
by Schmittlein and Mahajan (1982) and applied in
the paper by Moe and Fader (2002). First, we let M
denote the market potential, i.e., the total number of
consumers who have u>x, for a DVD. We let p
j
denote the probability that an individual consumer
preorders/purchases the DVD in week j so that
p
j
= Fj+ 1 − Fj (20)
where F denotes the cumulative distribution func-
tion of the purchase timing (Equation (19)).
Finally, we derive the likelihood function of the
data given model parameters (M% *). Let
y
i
=
y
il
y
il+1
y
iT
i
be the vector of sales of a DVD,
where y
ij
denotes preorders/sales of the ith DVD at
(calendar) week j, and T
i
denotes the last week of the
data collection window for the ith DVD. We have
DVD-I l
y
i
M%*
=
M!
T
j=l
y
ij
!
M −
T
j=l
y
ij
!
T
j=l
p
j
y
ij
·
1 − FT + 1
M−
T
j=l
y
ij
(21)
The parameters M % * for each DVD can then be
estimated by maximizing Equation (21) given the
weekly preorder/sales data. We refer to Equation (21)
as the DVD-I model.
2.6. Model Extension: DVD-II
The DVD-I model makes the assumption that con-
sumers do not remember the release date of a DVD.
However, some consumers do in fact remember the
release date, perhaps from their visit to the DVD page,
or from prerelease media advertising. To account for
this segment of consumers, we extend the DVD-I
model to what we call the DVD-II model.
Similar to the specification used in the paper by
Moe and Fader (2002), we denote the proportion of
consumers who remember the DVD release date by 0.
We assume that this particular segment of consumers
will purchase the DVD during the release week; i.e.,
t = k. Under the DVD-II model, we obtain a mixture
Hui, Eliashberg, and George: Modeling DVD Preorder and Sales: An Optimal Stopping Approach
1104 Marketing Science 27(6), pp. 1097–1110, © 2008 INFORMS
specification (e.g., Muthen and Masyn 2005):
p
j
= 1 − 0F j + 1 − Fj for j = k
p
k
= 1 − 0F k + 1 − Fk+ 0
(22)
where p
j
denotes the probability that an individual
consumer preorders/purchases on week j. Finally,
we derive the likelihood function associated with the
DVD-II model:
DVD-II l
y
i
M%*
=
M!
T
j=l
y
ij
!
M −
T
j=l
y
ij
!
T
j=l
p
j
y
ij
·
1 − FT + 1
M−
T
j=l
y
ij
(23)
The DVD-II model parameters M 0 % * for each
DVD can then be estimated by maximizing Equa-
tion (23) given the preorder/sales data.
3. Data
We obtained our data set from a major Internet retailer
for books, DVDs, music CDs, electronics, and other
household products. Our data set contains weekly pre-
order and sales figures, along with the release date,
for a sample of movie DVDs from February 2002 to
November 2004. For each DVD, we obtained the com-
plete preorder record from the week when preorder
first became available, together with weekly post-
release sales up to the 15th week after release. Similar
to the data-preparation procedure described in Moe
and Fader (2002), we eliminated DVDs with incom-
plete preorder information or very sparse sales (i.e.,
less than 100 units sold during the release week, the
criterion used in Moe and Fader 2002). Our final data
set contains a total of 251 titles, which we use in all
our subsequent analyses.
Table 1 presents key summary statistics of our data
set. The median total DVD sales (from the start of the
preorder period until the 15th week after release) is
3,084 units. Because our data set contains DVDs of
blockbuster movies as well as those from indepen-
dent distributors, total sales vary significantly across
different titles, ranging from just 526 units to over
200,000 units sold. Averaged across titles, 15.7% of the
preorders/sales occurred during the week of release
throughout our data collection window.
In our data set, a DVD is available for preorder,
on average, about 11.6 weeks before its release. The
median window between theatrical and DVD release
is about 22 weeks. This is roughly consistent with
MPAA (2006), which reported that the average win-
dow between the theatrical and DVD release is about
five months across all studios.
To study the empirical temporal pattern of pre-
orders/sales in more detail, we plot the density of
Table 1 Summary Statistics from Our Data Set
Mean Median St. dev. Min Max
Total sales
a
10,952.7 3,084.0 27,435.5 5260 221,229.0
Number of preorder 1161102660 250
weeks
% of total sales during 1571514335381
the release week
Window between 277220 30950 2890
theatrical and
DVD release (weeks)
a
Total sales represent the total number of units sold within the time period
considered.
preorders/sales versus the number of weeks since a
title’s release date, aggregated across all titles. More
specifically, we define y
is
as the number of sales for
the ith DVD during the sth week after its release.
Thus, s = 1 denotes one week after the ith DVD’s
release, s = 0 denotes the week of release, and s =
−1 denotes one week before the release. We define
r
is
as the proportion of sales for the ith DVD that
occurred during the sth week after release; i.e., r
is
=
y
is
/
s
y
is
. We denote r
s
=
i
r
is
/N as the mean of r
is
across all DVDs. A plot of r
s
versus s is shown in Fig-
ure 10. From the figure, we see that across all DVDs,
preorders/sales are generally exponentially increas-
ing before release, and exponentially decreasing after
release. This roughly replicates the preorder/sales
pattern of the title 24 Hour Party People, shown in
Figure 1.
4. Results
In this section, we describe the results of applying
our two models to our data set. In §4.1, we demon-
strate the fit of both DVD-I and DVD-II models and
compare them against two benchmarks—Moe and
Fader (2002) and the Weibull-Gamma model. After
Figure 10 Temporal Density of Preorders/Sales Across All DVDs
0.15
0.10
0.05
0.00
–10 –5 5 100
Notes. The x-axis denotes the number of weeks before/after release (the
dotted line at t = 0 indicates the week of the DVD’s release); the y -axis plots
the density of preorders/sales.
Hui, Eliashberg, and George: Modeling DVD Preorder and Sales: An Optimal Stopping Approach
Marketing Science 27(6), pp. 1097–1110, © 2008 INFORMS 1105
validating the model’s performance, we proceed to
discuss parameter estimates in §4.2.
4.1. Model Comparison
As discussed in §2, the Moe and Fader (2002) model
is a six-parameter model based on a mixture of two
Weibull distributions: one representing the behavior
of innovators, and the other representing the behav-
ior of followers. The cumulative distribution function
that it yields can be written as
Ft=
01 − e
−%
1
t
c
1
for t<k
01 − e
−%
1
t
c
1
+ 1 − 01 − e
−%
2
t−k
c
2
for t ≥ k
(24)
where 0 is the fraction of buyers associated with the
innovator segment; %
1
c
1
are the Weibull parameters
associated with the innovators segment; and %
2
c
2
are
the Weibull parameters associated with the followers
segment.
We also considered the benchmark model used by
Moe and Fader (2002), namely the Weibull-Gamma
model. This model is based on the assumption that
consumers preorder/purchase based on a Weibull dis-
tribution with rate parameters that vary according to
a Gamma distribution. The cumulative distribution
can be derived as (see Moe and Fader 2002):
Ft= 1 −
a
a + t
c
r
for t ≥ l (25)
We compare the four candidate models (DVD-I,
DVD-II, Moe and Fader 2002, Weibull-Gamma)
with three common model comparison criteria, log-
likelihood (LL), Bayesian Information Criteria (BIC)
(Schwarz 1978), mean absolute percentage error
(MAPE), and a fourth criterion which we call mean
relative absolute error (MRAE). We define MRAE to
be the average over all DVDs of
RAE
i
=
t
y
it
−
y
it
t
y
it
(26)
where y
it
denotes the actual sales for the ith DVD
during week t, and
y
it
denotes the corresponding pre-
dicted value. As opposed to MAPE, which is very
sensitive to weeks with very few sales,
8
each RAE
i
measures total absolute deviations as a percentage of
total sales. Thus MRAE, together with MAPE, allows
us to obtain a more complete assessment and compar-
ison of model fit.
The values of the four model comparison criteria,
LL, BIC, MAPE, and MRAE, are presented in Table 2
for the four models DVD-I, DVD-II, Moe and Fader
(2002), and the Weibull-Gamma. We find that the
8
For example, a prediction of 10 units of sales on a week with
actual sales of 5 units would generate an absolute percentage error
of 10 − 5/5 = 100%.
Table 2 Comparison with Benchmark Models
Log-likelihood BIC MAPE (%) MRAE (%)
DVD-II −1842731 3796413347198
DVD-I −2240034 4563281374265
Moe and Fader (2002) −3578647 7323721640 295
Weibull-Gamma −4999076101091041012437
DVD-II model performs better on all measures than
the DVD-I model, which in turn outperforms both
the Moe and Fader (2002) and the Weibull-Gamma
model. The DVD-II model obtains the highest log-
likelihood, the lowest MAPE, and the lowest MRAE
in comparison to all models. Compared to DVD-I, it
has a lower BIC, which indicates that the two-segment
model captures the actual data more closely than the
single-segment model assumed in DVD-I. The overall
superiority of DVD-II is further revealed by boxplots
of the RAE for each DVD across the four models,
shown in Figure 11. In particular, DVD-II appears to
yield uniformly lower RAE quantiles than the other
three models.
The DVD-II model (or even the single segment
DVD-I model) performs considerably better than Moe
and Fader’s (2002) model, even with a smaller num-
ber of parameters per DVD (for each title, DVD-
I uses three parameters, DVD-II has four parame-
ters, whereas the Moe and Fader 2002 model uses
six parameters). We believe that our models perform
better because the behavioral premise behind them
is more consistent with the DVD context. Specifi-
cally, Moe and Fader’s (2002) model is based on
the assumption that there are two segments of con-
sumers: innovators who try new CDs, and follow-
ers who imitate innovators (presumably because of
positive word-of-mouth). This dimension of word-
of-mouth effects is generally less relevant for the
Figure 11 Boxplot of RAE Across DVDs for Each of the Four Models
(DVD-I, DVD-II, Moe and Fader 2002, Weibull-Gamma)
DVD-I
DVD-II
Moe and Fader
Weibull-Gamma
1.5
1.0
0.5
Hui, Eliashberg, and George: Modeling DVD Preorder and Sales: An Optimal Stopping Approach
1106 Marketing Science 27(6), pp. 1097–1110, © 2008 INFORMS
Table 3 Summary of Parameter Estimates
Mean St. dev. Median
M 11,643.6 28,488.1 3,499.4
0062 0036 0057
0165 0063 0158
0273 0094 0262
DVD market, where sequential release is more preva-
lent across distribution windows. Before a DVD is
released, the movie has already been shown in a the-
atre (typically for 5 months), and extensive word-
of-mouth information, either from other consumers
who watched the movie or from movie critics, has
already been widely available long before the DVD is
released. Thus, it is reasonable to expect that imitation
effects, the key mechanism behind the model of Moe
and Fader (2002), is less relevant in the DVD setting
than in the music CD context that they focused on.
Finally, the Weibull-Gamma model performed worst,
which is not surprising because it does not take into
account differences between the preorder and post-
release periods.
4.2. Parameter Estimation and Interpretation
We now turn to interpreting the parameter esti-
mates for the DVD-II model which are summarized
in Table 3. The estimated market potential (ulti-
mate market size) M has a mean of 11,643.6 across
DVDs. It exhibits huge variation across titles that, to
a large extent, tracks the number of units already
sold. Indeed, across titles M shows a strong posi-
tive correlation with the cumulative number of sales
observed from the start of the preorder period up
until the 15th week after release (r = 09986, p<
0001). This provides some support for the valid-
ity of our model. Overall, the ratio of mean total
observed sales (i.e., cumulative market penetration)
from Table 2 to the mean market potential is about
91%. However, despite the strong correlation of total
sales with market potential, the variation across titles
of the ratio of total observed sales to market potential
leads to some interesting managerial implications. We
return to this issue in §5.
We find that the mean proportion of consumers
who “remember” the release date of the DVD is about
6.2%. Because about 15.7% of total sales occurred dur-
ing the week of release (see Table 1), this suggests that
roughly one-half to one-third of release-week sales
are accounted for by this segment of consumers. This
information may be relevant to home video distrib-
utors as they make prerelease advertising decisions.
Finally, the mean rate of visiting the DVD website is
about 0.165, which suggests that consumers’ median
interarrival time (except for the segment of consumers
who remember the release date) is about four weeks.
5. Managerial Implications
Three managerial issues, which are of particular
interest to the retailer that provided our data, are
(i) whether/how the length of the preorder period
affects preorder/sales patterns; (ii) estimation of the
residual sales for each DVD after the 15th week; and
(iii) forecasts of post-release sales based only on pre-
orders. In §§5.1–5.3 below we discuss how implications
from our model can be used to address these issues. We
should add that such implications are only a first step
towardgenerating managerial insights from our model.
As we work more closely with the Internet retailer and
obtain additional data, more implications may be feasi-
ble. We discuss this again in §6.
5.1. Policy Experiment: Preorder Timing
Because l (the calendar time when preorder is first
available) is a retailer’s strategic variable, our model
allows retailers to adjust l, using a policy experi-
ment, to examine how preorder/sales patterns will be
affected.
We study the effect of changing the length of the
preorder period by differentiating the sales pattern
ft (Equation (18)) with respect to l. After algebraic
manipulations, we have
5ft
5l
=
%% + 2*e
−%t−*k+%+*l
> 0 t ≥ k
%
2
e
−%t−*k+%+*l
> 0 l ≤ t<k
(27)
Equation (27) indicates that the partial derivative
5f t/5l is always positive in both the region t ≥ k
and the region l ≤ t<k. Thus, for a fixed k (release
date), as l increases (i.e., the preorder period is short-
ened), the region l ≤ t<kwill become smaller, while
preorders and sales of the other periods will increase,
with the magnitude of the predicted changes gov-
erned jointly by the other model parameters %, *, and
the window k.
As a concrete numerical example, we take the title
24 Hour Party People (considered in §1) to illustrate our
approach. Originally, the title was available for pre-
order 12 weeks before its release date. We analyze the
preorder/sales patterns of the DVD under two differ-
ent scenarios: (i) if preorder is allowed only 3 weeks
before the release date, and (ii) if the preorder period
is extended to 24 weeks before the release date. The
patterns under the two scenarios are shown in Figure
12. The solid line is the preorder/sales pattern under
l = 12 (the original value); the dashed line shows the
predicted preorder/sales pattern under l = 3, whereas
the dotted line shows the predicted pattern under
l = 24.
As can be seen, adjusting the number of preorder
weeks has only a modest effect on the preorder/sales
pattern. Shortening the number of preorder weeks to
three weeks pushes some of the sales to the other
preorder post-release periods; as a result, the dashed
Hui, Eliashberg, and George: Modeling DVD Preorder and Sales: An Optimal Stopping Approach
Marketing Science 27(6), pp. 1097–1110, © 2008 INFORMS 1107
Figure 12 Policy Experiment: Preorder/Sales Pattern with Different
Number of Preorder Weeks
250
200
150
100
50
0
15 20 25 30 35
Note. Solid line: 12 weeks of preorder period; dashed line: 3 weeks of pre-
order period; dotted line: 24 weeks of preorder period.
line is always above the solid line, in line with the
analysis provided by Equation (27). On the other
hand, doubling the number of preorder weeks from
12 to 24 has a negligible effect on the preorder/sales
pattern (as shown by the dotted line). This is because
very few consumers are willing to preorder a DVD
more than three months before the release date, as is
suggested by our model estimates.
5.2. Estimating Residual Sales
Because our model estimates the overall market
potential of each DVD, retailers may also use our
model to estimate the amount of residual sales for
each DVD beyond a certain date. By comparing mar-
ket potential with total sales reached by the end of
the 15th week (the horizon that our retailer is inter-
ested in), managers can determine which DVDs have
exhausted their market potential, and which ones still
have “legs” to continue. This may have important
implications for inventory management and/or the
allocation of promotion efforts.
The quantity of interest to retailers, therefore, is the
ratio +
i
=
t
y
it
/M
i
, i.e., the market penetration that
has already been realized by the end of the data col-
lection window (i.e., by the 15th week). A histogram
of +
i
is shown in Figure 13. The mean percentage pen-
etration by week 15 is about 91%, with a fair amount
of variation among the different titles. The 10 titles
with highest +
i
are shown in the top panel of Table 4;
among these, more than 99.5% of the total market
potential has already been realized. In contrast, the 10
titles with lowest +
i
are shown in the bottom panel
of Table 4; the percentage penetration among these
titles varies from 50% to 75%, indicating that substan-
tial residual sales potential is still available for these
titles.
Figure 13 Histogram of the Proportion of Market Potential Realized
by the End of the 15th Week Since Release Across the 223
Titles in Our Data Set
100
80
60
40
20
0
0.5 0.6 0.7 0.8 0.9 1.0
5.3. Forecasting Post-Release Sales from Preorders
To demonstrate the out-of-sample forecasting capa-
bility of our model, we consider predictive perfor-
mance on a holdout sample. More precisely, by ran-
domly dividing our data into a training set of 201
DVDs and a holdout sample of 50 DVDs, we con-
sider how well a model, calibrated on the 201 DVDs,
can predict post-release sales of the 50 DVDs using
only preorder information (up to one week prior to
release date); this is the same forecasting problem con-
sidered by Moe and Fader (2002). In the discussion
below, we compare our DVD-II model, which demon-
strated the best in-sample fit, against the state-of-the-
art model, i.e., the model developed by Moe and
Fader (2002).
In order to use preorder information to forecast
post-release sales, we treat the parameters for each
DVD as realizations from a hyper-distribution, in
effect embedding our model within a hierarchical
model; this is also the approach taken by Moe
and Fader (2002). Specifically, we assume that the
(transformed) parameters for each DVD are drawn
from a common multivariate normal distribution with
mean 6 and covariance matrix 7, as follows:
logM
i
logit0
i
log%
i
log*
i
∼MVN67 (28)
where log- and logit-transformations are taken to
ensure that the transformed parameters can take on
any real values. The parameter correlation structure
implicit in the covariance matrix 7 allows us to use
a new DVD’s preorder information to infer the poste-
rior distribution of all its parameters.
We use a similar specification for the Moe and
Fader (2002) model. In particular, the (transformed)
parameters in Equation (24) are assumed to be
Hui, Eliashberg, and George: Modeling DVD Preorder and Sales: An Optimal Stopping Approach
1108 Marketing Science 27(6), pp. 1097–1110, © 2008 INFORMS
Table 4 Top Panel: 10 Titles with Highest % of Market Potential
Realized by the 15th Week (Since the Week of Release);
Bottom Panel: 10 Titles with Lowest % of Market Potential
Realized by the 15th Week (Since the Week of Release)
Market % reached by 15th week
Title potential after release
Ten titles with highest cumulative % penetration by the
15th week after release
Seabiscuit 35147999
Rugrats Go Wild 14203999
The Powerpuff Girls—The Movie 26334999
Spirit—Stallion of the Cimarron 170190 998
Ice Age 533483998
Dumb and Dumberer: When 12780 998
Harry Met Lloyd
Like Mike 37111998
The Crocodile Hunter—Collision 20191998
Course
The Country Bears 11600 997
Pirates of the Caribbean—The 2220453996
Curse of the Black Pearl
Ten titles with lowest cumulative % penetration by the
15th week after release
Swimming Pool 235938721
Anger Management 113593674
The Italian Job 287481634
Daddy Day Care 66861614
The Big Lebowski 59643612
From Justin to Kelly 15732596
Shaolin Soccer 38148596
The Count of Monte Cristo 167839570
40 Days and 40 Nights 51617556
The Trip 82629526
drawn from a common multivariate normal distribu-
tion with mean 6
MF
and covariance matrix 7
MF
, the
same hyperprior specification used in Moe and Fader
(2002):
logM
i
logit0
i
log%
1i
logc
1i
log%
2i
logc
2i
∼ MVN6
MF
7
MF
(29)
We proceed as follows for both of the above proce-
dures. Using an MCMC procedure, we simulate from
the posterior distribution of the parameters for any
DVD in the holdout sample, conditional only on its
preorders up to one week before the release date.
We then compute the posterior mean of its param-
eters, and use these parameter estimates to forecast
the remaining sales for that DVD. These estimates are
then compared against the actual sales in the holdout
sample to compute MAPE and MRAE metrics.
The MAPE for the DVD-II model is 39.1%, and
MRAE is 35.1%, whereas the MAPE for Moe and
Fader’s (2002) model is 72.3% and MRAE is 49.9%.
Thus, the predictive performance of our DVD-II
model is substantially better than the model of Moe
and Fader’s (2002). This improvement, which again
may be due to the behavioral premise underlying the
DVD-II model, may have important managerial impli-
cations for improving inventory control and promo-
tion decisions.
6. Conclusion and Future Research
In this paper, we developed a behaviorally motivated
model of aggregate DVD preorder/sales patterns. We
modeled the purchase timing decision of an indi-
vidual consumer using an optimal stopping frame-
work that explicitly captures her forward-looking
behavior. We allowed consumer utilities to be hetero-
geneous, and derived the aggregate preorder/sales
curve (DVD-I model). We then extended our model
to handle a segment of consumers who remember
the release date, resulting in the DVD-II model. We
calibrated both models on data and showed that
they outperformed state-of-the-art benchmark mod-
els, such as the Moe and Fader (2002) model and
the Weibull-Gamma model. Finally, we demonstrated
how our model can generate managerial insights
through policy experiment, estimation of residual
sales, and forecasting post-release sales based only on
preorders.
To the best of our knowledge, the model proposed
here is the first attempt to explain the temporal pre-
order/sales pattern of DVDs using an individual-
level modeling framework. While our model is a first
step toward understanding the home video market,
it can be further extended in a number of different
directions. We briefly note some of these possibilities
below.
(i) Incorporating nonstationarity: We assumed that
the distribution of consumer visits to the DVD web-
site is stationary, and that the market potential is
fixed. In reality, these can change over time due to
advertising activities, seasonality (Einav 2007, Radas
and Shugan 1998) and other reasons. To allow for
such nonstationarities, one can specify our parame-
ters as a function of time that depends, for example,
on advertising intensity; e.g., %t = %
0
+ 9At, where
At denotes the intensity of advertising over time.
Similarly, the market potential M can also be modeled
as a function of advertising intensity. Further, season-
ality effects can be handled using the methodology
developed in Radas and Shugan (1998).
(ii) Dynamic pricing: We assume that the price of a
DVD is constant over time. Although this assump-
tion may be reasonable for movie DVDs (as verified
empirically with our data provider), it may not hold
for other categories, for example, books, video games,
music CDs, where promotions tend to be offered
early after their release. Researchers may extend our
optimal stopping framework to handle dynamic pric-
ing, by allowing consumers to act strategically based
Hui, Eliashberg, and George: Modeling DVD Preorder and Sales: An Optimal Stopping Approach
Marketing Science 27(6), pp. 1097–1110, © 2008 INFORMS 1109
on their expectation of future price changes (e.g., Nair
2007, Sun 2006).
(iii) Movie characteristics and cross-category analysis:
It may be interesting to investigate how characteris-
tics of a movie such as genre, story (Eliashberg et al.
2007), star involvement (e.g., Wei 2006) are related to
the sales pattern of its DVD. In addition, preorder-
ing has become prevalent in many different categories
(e.g., books, video games, music CDs). Variants of our
model can be applied to each category to study how
category characteristics affect model parameters. In
other categories, the preorder/sales pattern may be
different from the pattern in Figure 1; a concave-up
pattern may also be possible, in which case the model
extension discussed in Appendix V may be useful.
(iv) Joint modeling of box office and DVD revenue:
Our framework can be potentially extended to
an integrated model of a consumer’s decision of
when/whether to watch a movie in a theatre and
whether/when to purchase the corresponding DVD.
This extension allows one to understand not only the
cannibalization/synergy effects between movie and
DVD, but also the role of the “window,” in a more
structural manner. Currently, our policy experiment
only addresses changes in sales when l (the length of
the preorder period) is varied; with the above exten-
sion, one can then conduct policy experiments by
varying k (the window) as well. The optimal window
can then be derived, a topic of interest to many other
marketing researchers (e.g., Eliashberg et al. 2006,
Lehmann and Weinberg 2000, Prasad et al. 2004).
Already, some progress has been made in this area
(e.g., Luan 2005); we believe that more managerial
insights can be gained with more research attention.
Appendix
I. Proof of Monotonicity of Our Model Setup
A monotone optimal stopping problem is defined as follows
(Ferguson 2000):
Let A
n
be the event "Y
n
≥ EY
n+1
n
$. A problem is mono-
tone if A
0
⊂ A
1
⊂ A
2
···almost surely. From Equations (9)
and (10), we show that Y
n
≥ EY
n+1
n
⇔
n
≤ C for some
constant C. Consider Y
n+1
.Ift
n+1
>k, then the consumer
will buy at time t
n+1
based on Equation (7). If t
n+1
≤ k, then
we have
n+1
= k − t
n+1
≤ k − t
n
≤ C ⇔ Y
n+1
≥ EY
n+2
n+1
Thus, the condition of monotonicity is satisfied.
II. Derivation of the Threshold Rule for
General Interarrival Time
EY
n+1
n
= e
−t
n
n
0
e
−
n
u − e
−w
n
xgw
n
dw
n
+ u − x
n
e
−w
n
gw
n
dw
n
= e
−t
n
e
−
n
uG
n
− x
n
0
e
−w
n
gw
n
dw
n
+ u − x
n
e
−w
n
gw
n
dw
n
= e
−t
n
e
−
n
uG
n
+ u
n
e
−w
n
gw
n
dw
n
− xEe
−w
n
Thus,
Y
n
≥ EY
n+1
n
⇔−x + e
−
n
u ≥ e
−
n
uG
n
+ u
n
e
−w
n
fw
n
dw
n
− xEe
−w
n
⇔ e
−
n
1−G
n
−
n
e
−w
n
gw
n
dw
n
≥
x
u
1−Ee
−w
n
III. Derivation of the Threshold Rule for Exponentially
Distributed Interarrival Time
Start from Equation (10),
x
u
1 − Ee
−w
n
≤ e
−
n
1 − G
n
−
n
e
−w
n
gw
n
dw
n
Because w
n
∼ exp%, we have gw
n
= %e
−%w
n
and G
n
=
1 − e
−%w
n
.
Ee
−w
n
=
0
e
−w
n
%e
−%w
n
dw
n
= %
0
e
−%+w
n
dw
n
=
%
% +
n
e
−w
n
gw
n
dw
n
=
n
e
−w
n
%e
−%w
n
dw
n
=
%
% +
e
−%+
n
Thus, we have
x
u
1 −
%
% +
≤ e
−
n
1 − 1 − e
−%
n
−
%
% +
e
−%+
n
⇔
x
u
% +
≤ e
−%+
n
−
%
% +
e
−%+
n
⇔
x
u
≤ e
−%+
n
⇔
n
≤
1
% +
ln
u
x
IV. Distribution of Purchase Timing with
Consumer Heterogeneity
First, consider the case t ≥ k. Clearly, it is also true that t>d
and hence 1
"t>d$
= 1. Thus,
ftt ≥k = %*e
−%t
0
e
%d−*C
i
dC
i
= %*e
−%t
k−l
0
e
%k−C
i
−*C
i
dC
i
+
k−l
e
%l−*C
i
dC
i
= %*e
−%t
e
%k
k−l
0
e
−%+*C
i
dC
i
+e
%l
k−l
e
−*C
i
dC
i
Hui, Eliashberg, and George: Modeling DVD Preorder and Sales: An Optimal Stopping Approach
1110 Marketing Science 27(6), pp. 1097–1110, © 2008 INFORMS
= %*e
−%t
e
%k
e
−%+kk−l
−%+*
+
1
%+*
+e
%l
e
−*k−l
*
=
%*
%+*
e
%k
1−e
−%+*k−l
+%e
%l−*k−l
e
−%t
Second, for the case l ≤ t<k,
ft l ≤ t<k
= %* e
−%t
k−l
k−t
e
%k−C
i
−*C
i
dC
i
+
k−l
e
%l−*C
i
dC
i
= %* e
−%t
e
%k
k−l
k−t
e
−%+*C
i
dC
i
+ e
%l
k−l
e
−*C
i
dC
i
= %* e
−%t
1
%+*
e
%k
e
−%+*k−t
−e
−%+*k−l
+
1
*
e
%l
e
−*k−l
=
%*
% + *
e
−*k+*t
− e
−%t+%l−*k+*l
+ %e
−%t+%l−*k+*l
=
%*
% + *
e
−*k−t
+
%
2
% + *
e
−%t−l−*k−l
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