Price Discrimination via Versioning with Limited
Quantity and Time: The Case of Special Edition Video
Games∗
Masakazu Ishihara
Stern School of Business
New York University
Joost Rietveld
Rotterdam School of Manageent
Erasmus University
Yuzhou Liu
Man Numeric
December 15, 2016
Very Preliminary and Incomplete
∗
We have benefited from discussions with Bryan Bollinger, Ron Borkovsky, Preyas Desai, Jihye Jeon,
Vineet Kumar, Carl Mela, Debu Purohit, Jiwoong Shin, S. Sriram, and Kosuke Uetake. We also thank
seminar and conference participants at Yale University, Temple University, Duke University, 2015 Marketing
Science Conference, 2016 IIOC, and 2016 Marketing Dynamics Conference for their helpful comments. All
remaining errors are ours.
Abstract
This paper investigates the role of product versioning with quantity limit as a pricediscrimination tool when it is jointly used with intertemporal price discrimination. Such
a strategy has been popular in many durable goods markets such as cars, video games,
and books. However, little empirical work has been conducted to document how these two
types of price discrimination interact and improve profits. Using panel data on sales, prices,
versioning choice, quantity limit, and product- and firm-level characteristics from the US
video game market, we first test a hypothesis that releasing a limited edition together with a
standard edition of a game could improve profitability via reducing the Coase conjecture for
the standard edition. We find that when a game offers both standard and limited editions,
its standard edition exhibits more frontloaded sales, a higher first-month price, and a lower
price cut over time than when a game offers only a standard edition, which is consistent with
softening the Coase conjecture. We then propose an empirical framework for quantifying the
benefit from offering a limited edition, by developing a dynamic equilibrium model that (1)
endogenizes firms’ versioning and quantity limit decisions and (2) consumers’ learning about
quantity limit. Our identification relies on variation in versioning choices for multi-platform
games. The proposed framework can be used to examine the impact of publicly announcing
the quantity limit on profits, and the welfare impact of different commitment mechanisms.
Keywords: Durable Goods Versioning, Quantity Limit, Intertemporal Price Discrimination,
Video Game Industry
1
Introduction
Product versioning strategy (e.g. standard versus limited editions) has been used as an
effective segmentation/price-discrimination tool in many durable goods industries such as
computer software, video games, books, and cars. Despite the increasingly popularity and
importance of this phenomenon (e.g. sales of limited edition video games can comprise up
to 15% of total product sales), little empirical work has been done in this area to document
how price discrimination via versioning interacts with intertemporal price discrimination
and improves profits. This paper studies the antecedents and consequences of product versioning strategy in the U.S. video game industry between 2005 and 2012. Specifically, we
focus on simultaneous release of multiple versions, and examine (1) when firms simultaneously introduce multiple versions of a durable good, (2) how versioning strategy helps firms
to better segment price sensitive consumers when they are also able to price-discriminate
intertemporally, and (3) welfare impact of quantity limit as a commitment device.
To answer these questions, we collect rich product- and firm-level panel data on sales,
prices, product characteristics (genre, critic/user ratings, award, series, version and associated special features/collector items, quantity limit for limited editions, etc.), and firm
characteristics (product portfolio, reputation, etc.) from the U.S. video game market. Using this data set, we first run a series of regression analyses on versioning decisions, sales
patterns, and price patterns, and find that (1) firms are more likely to introduce multiple
versions for high quality games and games that are sequels to successful earlier installments,
games that are released later in the platform’s product lifecycle, and in less competitive
1
markets, (2) publishers command higher initial selling prices for standard edition games if
a limited edition is simultaneously introduced with the standard edition and this markup
remains in tact in the six months period after a game’s release, and (3) sales of a standard
edition are more frontloaded (i.e., concentrated in the period following market launch) when
a limited edition is simultaneously introduced with the standard edition. These preliminary
analyses suggest a possibility that adding a limited edition helps reduce the demand elasticity for the standard edition by drawing in some of price sensitive consumers to the limited
edition, which allows firms to soften the Coase conjecture for the standard edition.
We then develop and estimate a dynamic equilibrium model where a monopolist firm
makes a versioning decision prior to product release. If the firm chooses to introduce a limited
edition along with the standard edition, then the firm makes the quantity limit decision.
Once the product is released, the firm makes a sequence of pricing decisions to maximize
the present discounted value of current and future profits. Consumers are forward-looking
and make an adoption decision by accounting for the future price cut and the availability of
the limited edition (if it is released). The model allows us to recover consumer preference
parameters as well as cost parameters. We estimate the model by extending the Bayesian
Markov chain Monte Carlo algorithm by Ishihara and Ching (2016).
Our preference parameter estimates suggest important heterogeneity in valuation for
standard versus limited editions across different segments of consumers: the willingnessto-pay (WTP) for the two versions are negatively correlated across some of the segments.
Consumers who have (relatively) low WTP for standard editions but (relatively) high WTP
2
for limited editions purchase limited editions upfront if limited editions are available. As
a result, firms can remove this segment of consumers from the demand for standard editions, and thus, can maintain a high standard-edition price over time (softening the Coase
conjecture). The cost parameter estimates suggest that the overall price-cost margin from
limited editions is lower than that of standard editions. This puzzling result is in line with
our interviews with video game publishers who also claim that they earn a lower price-cost
margin from limited editions. Together, our analysis suggests an important role of limited
editions that previous studies have not identified: while firms might earn a lower price-cost
margin from limited editions than from standard editions, limited editions could allow firms
to reduce the demand elasticity for standard editions and thus improve the overall price-cost
margin from standard editions and the overall profitability of multiple-edition games. Using
the parameter estimates, we plan to conduct welfare analysis and evaluate the welfare impact
of different commitment mechanisms (e.g., price commitment for standard editions).
The rest of the paper is organized as follows. Section 2 reviews the related literature.
Section 3 discusses how the addition of a limited edition improves the overall profitability of
a product. Section 4 describes the video industry game data used in this paper and presents
some empirical regularities that are suggestive of the mechanism. Section 5 presents the
proposed dynamic equilibrium model. Section 6 explains the estimation strategy. In Section
7, we discuss the estimation results. Section 8 concludes.
2
Literature Review
To be added.
3
3
An Illustration
In the dynamic structural model we propose in Section 5, monopolist firms choose product
versioning and pricing strategies so as to optimally price-segment consumers via both product
versions and time. Product versioning choice includes either standard edition only release or
simultaneous standard and limited edition release. In this section, we use a simple illustrative
example and provide an intuition behind the mechanism through which the addition of a
limited edition improves the overall profitability of a product. Moreover, in the next section,
we use the data and empirically test the predictions derived from the example.
We start with the baseline situation where a monopolist firm releases only the standard
edition of its product. In this case, the firm can potentially improve the profitability by intertemporally price-discriminating consumers with different willingness-to-pay (WTP) (cite
papers). If consumers are not perfectly patient (i.e., the discount factor is less than one),
then the firm can set the initial price high and attract a segment of consumers who have
the highest WTP for the standard edition. Once this segment of consumers have purchased
the product at the initial price, the firm lowers the price to attract consumers who have the
second highest WTP. By continuing this process, the firm can effectively set different prices
to different segments of consumers.
The top panel in Figure 1 considers an example of three segments of consumers purchasing
the standard edition at different points in time. Segment 1 has the highest WTP, Segment 2
has the second highest WTP, and so on. If we assume that all segments have a high enough
WTP to purchase the product, then the firm will set a high price in period t = 1 and sell to
4
Segment 1. In period t = 2, the firm will lower the price and sell to Segment 2, and so on.
Here, unless all consumers are myopic (i.e., the discount factor is zero), the firm cannot set
the price equal to the WTP of the target consumers in each period (the Coase conjecture,
Coase 1972).
How would the simultaneous release of a limited edition with the standard edition affect
the profitability of this product, if it is optimal to release both editions instead of only the
standard edition (baseline)? The limited edition should attract some of these three segments
and/or a segment who did not buy the standard edition previously (call them segment 4 who
have zero valuation for the standard edition, but positive valuation for the limited edition).
Which segment(s) will purchase the limited edition depends on their WTP for the standard
edition and limited edition, and the prices of these two editions in an equilibrium. It may
first appear that Segment 1 should always be the one most attracted to the limited edition
because they are most excited about the product (highest WTP for the standard edition).
However, this may not necessarily be the case. For example, consider a Batman video game
called “Batman: Arkham Knight.” It was released in June 2015 on PlayStation 4, Xbox
One, and Microsoft Windows, and came with the standard edition (game disc only) and
the limited edition that contains the following items: game disc, a steelbook case, an art
book, a comic book, in-game skin pack, and a Batman statue. These items included in
the limited edition may attract both video gamers who like Batman games, and fans of the
Batman franchise in general who like to play video games. The latter segment may not be
very interested in the standard edition (either Segment 2 or 3), but if the limited edition
5
includes special bonus items such as unique Batman figurines, their WTP for the limited
edition can be very high. As a result, the firm might set the prices of the standard and
limited editions so as to attract the latter segment to the limited edition. As we will argue
below, such a negative relationship between the WTP for the standard edition and that for
the limited edition (i.e., those who have a low WTP for the standard edition have a higher
incremental WTP for the limited edition than those who have a high WTP for the standard
edition) might play an important role in enabling the monopolist firm to increase the profits
by adding a limited edition.
Let us first consider a situation where it is optimal for the firm to attract Segment 1 to
the limited edition, as illustrated in the second panel of Figure 1.1 Segment 2 is now the
purchaser of the standard edition in period t = 1, and Segment 3 purchases the standard
edition in period t = 2. As compared to the baseline situation (standard edition only and
segment r buys it at time t = r), the firm can increase the profit only if Segment 1’s WTP
for the limited edition is very high such that the firm can earn larger profit margin from
selling the limited edition to Segment 1 than selling the standard edition to Segment 1. One
important observation here is that the price of the standard edition in period t = 1 in this
situation will be lower than that in the baseline situation, in order to sell the standard edition
to Segment 2. This lower initial price of the standard edition might put a downward pressure
on the price of the limited edition, in order to satisfy the incentive compatibility constraint
for Segment 1. Thus, the extent to which the firm can extract the very high incremental
1
See Appendix A.1 for the underlying model. To keep our illustration simple, we restrict our attention to
cases in which it is optimal to sell the limited edition to only one segment of consumers (and to all consumers
in the segment).
6
WTP by Segment 1 will depend on Segment 2’s WTP for the standard edition.
Next, suppose that it is optimal for the firm to attract Segment 2 to the limited edition,
as in the third panel of Figure 1. This is the case when the incremental WTP by Segment
2 is very high (e.g., Batman game fans who value the bonus items included in the limited
edition). In this situation, two important mechanisms are at work. First, the profit margin
from Segment 2 can improve as compared to the baseline situation. This may be likely if
Segment 2’s WTP for the standard edition is low. Second, the removal of Segment 2 gives
Segment 1 a larger option value of waiting. This is because Segment 1 knows that once they
purchase the product, the firm will have an incentive to cut the price to attract Segment
3, and Segment 3’s WTP for the standard edition is lower than Segment 2’s. As a result,
in order to sell the standard edition to Segment 1 in period t = 1, the firm will have to
lower the initial price of the standard edition to match the larger option value of waiting as
compared to the baseline situation. This in turn reduces the profit margin from Segment 1.
Thus, in order for this situation to be optimal, the increase in profit margin from Segment
2 must be at least as great as the decrease in profit margin from Segment 1.
In the fourth panel of Figure 1, we consider a situation where it is optimal for the firm to
attract Segment 3 to the limited edition. In this situation, we expect two potential benefits
relative to the baseline situation. First, as before, the profit margin from Segment 3 can
improve. Second, because Segment 3 is removed, Segment 2 will have no incentive to wait
for a price cut. As a result, the firm can set the price to Segment 2’s WTP in period t = 2.
This will have two effects. First, it improves the profit margin from Segment 2. But it also
7
improves the profit margin from Segment 1 as well, because Segment 1 would now pay a
higher period 2 price if they chose to wait. This will allow the firm to set a higher initial
price for the standard edition. In other words, if the limited edition appeals to Segment 3,
then the firm is able to soften the Coase conjecture for the standard edition and improve the
profitability from those who purchase the standard edition.
Finally, if it is optimal to attract the segment of consumers who do not buy the standard
edition in the baseline situation, the overall profit might simply increase by the profit earned
from the limited edition and there should not be any change in the pricing and profit of the
standard edition.
Before summarizing our above arguments, we note that the commitment to the limited
quantity might play a crucial role in the price-segmentation. If instead the firm releases a
non-limited “special” edition in addition to the standard edition, then the firm could also
suffer from the Coase conjecture for the non-limited special edition. By limiting the quantity,
the firm is able to pre-commit to capturing only the segment of consumers that it wants to
remove from the standard edition demand.
The actual situation may be more complicated as there are likely a continuum of consumer
segments with different WTPs for the standard and the limited edition. However, these four
examples cover fundamental mechanisms behind the dual second-degree price discrimination.
In particular, if the limited edition attracts a new segment of consumers (as in the last
situation), the benefit from adding a limited edition can at most be the new profit from the
limited edition. On the other hand, if the limited edition creates cannibalization, pricing
8
and profitability of the standard edition will change as a result of the addition of a limited
edition. How they change will depend on which segment switches to the limited edition.
As long as it is profitable to introduce a limited edition, all three possibilities can occur in
reality. However, these three possibilities generate different testable empirical implications
for the sales and price path of the standard edition. In Table 1, we summarize the predicted
changes in the sales and price path of the standard edition, as compared to the baseline
situation. For a more formal derivation, please refer to Appendix A.1.
First, we expect that the addition of the limited edition can change the frontloadedness
of the sales of the standard edition. Suppose that we measure the frontloadedness as the
ratio of period-1 sales to total sales. Then, if either Segment 2 or 3 purchases the limited
edition, we expect that the ratio becomes higher, because the period 1 sales remains the
same, but the total sales for the standard edition declines. However, the ratio can go up or
down if Segment 1 purchases the limited edition. If Segment 4 purchases the limited edition,
it remains unchanged.
Second, as we explained above, the initial price of the standard edition will be lower if
Segment 1 or 2 purchases the limited edition, but can be higher if Segment 3 purchases the
limited edition. Finally, the price cut over time can be larger or smaller if Segment 1 or 2
purchases the limited edition, but will be smaller if Segment 3 purchases the limited edition.
As we noted, as long as it is profitable to have a limited edition, any of these situations
will improve the overall profiability of the product. However, it is worthwhile to examine
which mechanisms are at work. For example, in our interview with a major video game
9
publisher, we learned that the profit margin from a limited edition is typically lower than
that from a standard edition at the initial price level. If this is indeed the case, we expect
that the first situation (Segment 1 buys the limited edition) is less likely to be happening.
Also, if the third situation is likely, then the benefit from the addition of a limited edition
is not simply because of a higher profit margin, but also because it helps reduce the Coase
conjecture for the standard edition. In the next section, we use the data and empirically
examine these theoretical implications.
4
Data
We situate our study in the console video game industry. Video game consoles are a canonical
example of a two-sided market (Clements & Ohashi, 2005; Dubé, Hitsch & Chintagunta,
2010, Zhao, 2016). The industry comprises a hardware, or console, side and a software, or
game, side. Video game platforms are brought to market approximately every eight years and
are sold to consumers at or below costs. Platform manufacturers including SONY, Microsoft,
and Nintendo subsidize the consumer side in order to quickly ramp up a platform’s installed
base which makes for a more appealing value proposition to third-party game publishers.
Platform manufacturers earn royalty fees from every third-party video game that is sold on
their platform and occasionally develop and publish their own games (Lee, 2013). Because of
this price structure where income from hardware is subsidized in lieu of greater participation
and income from software developers, video game platforms are often seen as a necessary
evil for video games. In 2015, the US video game console market was estimated to generate
over 10 billion USD with over 50% of the revenue coming from software sales (IDG, 2016).
10
The vast majority of the papers on the video game industry looks at issues relating to
demand for the hardware side. Here we add to a small but burgeoning body of work that
estimates demand for the software side (Gil & Warzynski, 2015; Ishihara & Ching, 2016;
Nair, 2007; Rietveld & Eggers, 2016). We are specifically interested in how the addition of
a limited edition video game affects demand for the standard edition. A small section (<
10%) of the video games are released with a limited addition alongside the standard edition.
Special edition games have identical content as their standard edition counterparts, but add
special features and bonus items such as figurines and art books (discussed in more detail
below). The release date for the two versions coincide, and their hardware compatibility and
requirements also are the same. There are no changes in the royalty fee (which is based on
an agreed price per disc) charged by the platform owner across versions of a game. While
special editions are prevalent across geographical markets (i.e. North America, Europe, and
Japan), here we focus on the US market which is the largest both in terms of demand for
and supply of video games.
4.1
U.S. Video Game Data
Our estimation sample comes from two home video game consoles: Xbox 360 by Microsoft
(released in November 2005) and PlayStation 3 by SONY (released in November 2006). Our
sample starts at the inception of each console and ends on February 2012. We compiled
a data set for our study from four sources. First, we collected data on monthly sales and
prices of all games as well as part of product characteristics (publisher, genre, release pattern) from the NPD group. NPD sales data are collected from retail scanners that cover
11
over 80% of the consumer-electronics retail AVC in the US (Nair, 2007). Second, we supplemented the product characteristics by collecting additional characteristics (i.e. whether
a game is a sequel, licensed from external media (e.g. movie, TV show), and/or features
a star actor) from multiple online video game websites, games box covers, and Wikipedia.
Third, we collected data on product quality proxies from expert review aggregation website
Metacritic.com. Metacritic tracks over 300 publications from which it collects expert review
scores that are then transformed into a weighted Metascore ranging from 0 to 100.2 Finally,
for games with special editions, we collected all items included in the limited edition from
CollectorsEdition.org. CollectorsEdition keeps track of special edition video games and their
contents within specific geographical regions.
In our sample, a “special” edition may be named as “gold edition,” “platinum edition,”
or “limited edition.” We use these terms interchangeably, but necessary conditions for these
special editions are that (1) they add bonus items (either physical or digital) to the standard
edition, and that (2) their quantity is limited (even if the edition name does not say limited).
Table 2 presents summary statistics about the observed versioning strategies. The majority
of special editions are accompanied by physical items such as art books, figurines, and posters
(colloquially referred to as “swag”). The most popular item in our sample are bonus discs,
which mostly contain additional content that is not part of the game (e.g., making of the
game, animated movie). Art books that contain concept art of game characters and setpieces
2
We believe Metascores are a valid proxy for quality as many publishers use Metascores as an indicator
of quality, as is reflected in their contracts with independent game developers (see for example Activision’s
multi-game contract with developer Bungie for the hit series Destiny: http://documents.latimes.com/bungieactivision-contract/). Moreover, the majority of the review reports that Metascores are based on are published prior to the launch of a video game, minimizing any concerns of reversed causality with respect to our
dependent variables.
12
are also popular items. Some limited edition games come with special packaging such as a
steelbook instead of a plastic case. Publishers can include more than one item in their special
edition games (e.g. steelbook packaging, figurine and poster). On average, limited edition
games include 3.44 special items with a maximum of 6 special items for one game. While
some special edition games only include additional in-game content such as a special battle
arena or additional characters, the majority of the observations in our sample (86%) have
at least one physical item.
Table 3 shows descriptive statistics for our estimation sample. The total number of games
in our sample is 1168. Of those, 658 are Xbox 360 games and 510 are PlayStation 3 games.
About 9% (n = 108) of the games in our sample have a special edition. Special edition games
are divided roughly equally across platforms with 58 games on PlayStation 3 and 50 games
on Xbox 360. Sequel games and games of high quality (i.e. Metascore ≥ 75) are more likely
to have limited editions, while games that are based on media tie-ins and those featuring
famous persons are less likely to have limited editions. Also, we see variation across genres:
Survival horror games such as Dead Space and role playing games such as Mass Effect are
more likely to have limited editions while family games such as Game Party: In Motion and
simulation games such as MySims Sky Heroes are less likely to have limited edition versions.
These structural differences across product characteristics suggest that the decision to release
a limited edition video game is endogenous and may be strategic.
Figures 2 and 3 show typical sales and price patterns for games with both standard and
special editions. In the video game industry, sales peak in the first month, and then quickly
13
drop over time following an exponential decline. Selling prices typically stay at their initial
level for a few months before they start declining. The timing and speed of price drops vary
across games. The game in Figure 2 (Little Big Planet 3 - PS3) shows that both standard
and special editions had a sharp decline in sales. The special edition was sold out in the
third month. The price of the special edition stayed almost constant for the three months.
The game in Figure 3 (Fallout 3 - PS3) shows different patterns. Both sales of standard and
special editions declines, but the special edition sales for Fallout 3 lasted much longer than
for Little Big Planet 3 (the sales spike of the standard edition in the third month is due to
the December holiday season effect). The special edition price quickly drops in the first few
months, and became comparable to that of the standard edition (sometimes lower).3 The
figures further show that sales of limited edition games are a fraction of the sales for the
standard edition, while the initial selling price for limited edition games is higher than that
of standard edition games. Standard edition games that receive a limited edition sell 315,216
units on average in their first month, while limited edition games sell 47,594 units. Similarly,
the average first month selling price for standard edition games that receive a limited edition
is 58.82 USD, while this is 78.91 USD for limited editions. These observations suggest
that although a special edition offers more than what the standard edition offers (as special
editions always include the standard edition), for some consumers, the willingness-to-pay for
a special edition could be lower than that for the corresponding standard edition.
3
The steep drop in price for the special edition of Fallout 3 may be driven by the retailer. There are
considerable costs associated with keeping limited edition games on shelves and in stock. Special editions
often do not fit on regular shelves for video games and when they do they take up more space than standard
edition games due to their bulky packaging and added bonus items. Retailers look to supplant these items
with newer products when they sit idle on their shelves for too long.
14
In Table 4 we list descriptive statistics of interest for all the games in our sample as well
broken down by subsamples of standard edition games that did and standard edition games
that did not receive a limited edition. As mentioned before, standard edition games with a
limited edition have average first month sales of 315,216 units. Standard edition-only games
yield significantly lower unit sales, 67,527 units on average. This pattern is consistent when
we look at cumulative unit sales after six months: 636,027 units for standard edition games
with limited edition, and 161,509 units for standard edition-only games. We also find that
unit sales are more frontloaded for standard edition games that are accompanied by a limited
edition: while 35% of total unit sales after six months for standard edition-only games occurs
in the first month, this is 50% for standard edition video games that did receive a limited
edition. When we look at games’ selling prices a similar picture emerges. Standard edition
games with limited editions have higher initial selling prices than standard edition-only
games: 58.82 USD versus 53.82 USD. Moreover, this gap in selling prices widens over games’
lifecycles as sixth month selling prices are 46.98 USD for games that receive a standard
edition where they are 39.91 USD for standard edition-only games. All aforementioned
differences are significant (p < 0.01) when we subject them to a two-tailed t-test. We
also find that standard edition games with limited editions have greater staying power than
standard edition-only games: despite their higher initial selling prices we find that limited
edition paired standard editions drop their prices by 11.84 USD over a six month period
whereas standard edition-only games drop their prices by 13.91 USD. This difference in
price drop is marginally significant (p < 0.10). Heterogeneity in product characteristics
15
notwithstanding, we note that these patterns are consistent with our theory of how limited
editions soften the Coase conjecture. We will investigate this further in the next subsection.
4.2
Reduced-form analysis
To further unpack the dynamics of versioning strategy, we run a series of regressions and
investigate (1) the factors that influence firms’ versioning decisions, and (2) whether the
data show a pattern consistent with our theory of softening the Coase conjecture. For the
latter, if an introduction of a special edition helps remove price-sensitive, yet special-editionprone consumers from the demand for a standard edition, we should expect to observe that
standard edition video games have (1) higher initial selling prices, (2) lower price drops over
time, and (3) more frontloaded sales.
First, we attempt to further unpack firms’ versioning strategy. We estimate a probit
model and examine the impact of product characteristics and platform-characteristics (e.g.
market size, competition) on a game’s probability of receiving a limited edition. We control
for platform fixed effects, genre fixed effects, month of release fixed effects, year of release
fixed effects, and publisher fixed effects.4 We report heteroscedasticity robust standard
errors clustered at the firm level. Table 5 presents the estimation results. While our results
are mostly consistent across model specifications, we focus on Model 4, which we consider
the most robust, for interpretation of our findings. We find that high quality games and
sequel games are more likely to receive a limited edition. Games with Metascores equal
to, or greater than, 75 are 13% more likely to receive a limited edition than games with
4
Note that there is one genre (i.e. family/party) and 33 firms without any limited edition games in our
data. Observations in these categories (n = 178 games) are dropped from the analyses.
16
Metacores below 75 (p < 0.01). Games that are based on a successful earlier instalment
are 6% more likely to receive a limited edition compared to completely new, or innovative,
games (p < 0.01). These findings suggest that publishers select games for limited editions
that have a proven track record in the market and games for which they know there exists
a loyal following. We also find that the market structure in which a game is launched
affects publishers’ versioning strategy. Games that launch exclusively on one platform are
marginally less likely to receive a limited edition version. Platform exclusive games have
a 5% lower probability of receiving a limited edition compared with platform agnostic, or
“multihoming” (Landsman & Stremersch, 2011), games (p < 0.10). Furthermore, we find
that the platform’s installed base (computed as the number of console owners at the time
of the game’s release, based on monthly hardware sales data) has a positive and significant
effect on the likelihood of having a limited edition (p < 0.01). We also find that a game
released in a less competitive market, as measured by platform-genre HHI, is more likely to
have a special edition (p < 0.01). Taken together, these results suggest that publishers are
more likely to release limited edition video games in markets that are larger in size and lower
in competition.
Second, in order to assess if the presence of a limited edition product softens the Coase
conjecture, we first estimate the effect of a limited edition video game on the initial (i.e. first
month) selling price of the standard edition. We estimate an OLS regression that includes
a dummy variable indicating if a standard edition video game received a limited edition
or not. Furthermore, to control for publishers’ non-random versioning decisions (as per the
17
discussion in the previous section), we manually compute a selection correction term (λ) that
is akin to Heckman’s (1979) Inverse Mills Ratio and include this as an additional regressor
in Model 5 of Table 6 (Shaver, 1998). We focus on this model for the interpretation of our
findings. We find that having a limited edition video game allows publishers to command
a 3.10 USD higher initial selling price compared to the baseline situation of not having a
limited edition (p < 0.01).
We next estimate the effect of having a limited edition on games’ absolute price drop (in
USD) after six months. We add first month price as an additional control variable and we
drop the market variables (platform installed base, platform-genre HHI ) since these change
over time and our dependent variable is measured over a six month period (adding these
variables does not change the reported results). Results are reported in Table 7. We again
focus on Model 5 for the interpretation of our findings. We find that, despite their higher
initial selling price, standard edition games that receive a limited edition have a marginally
lower price drop than standard edition-only games. The price drop for games that receive a
limited edition is 1.55 USD less than for standard edition-only games (p < 0.10). This result
is robust to using a relative measure of price drop. We further find that a 1 USD increase in
first month price correlates with a 0.36 USD increase in price drop (p < 0.01); high quality
games drop their prices by 3.93 USD less than low quality games (p < 0.01); licensed games
also have marginally greater staying power as these games drop their price by 2.11 USD less
compared to non-licensed games (p < 0.10); the same applies to platform exclusive games
that enjoy a 2.57 USD less price drop than platform agnostic video games (p < 0.05); and,
18
finally, games that feature famous persons have 2.16 USD greater price drops than games
that do not feature any stars (p < 0.05).
Lastly, we estimate the effect of receiving a limited edition video game on the frontloadedness of standard edition video games. We replicate our earlier analysis and add games’
price drop after six months as an additional control variable. Results are reported in Table
8 and we again focus on Model 5 for the interpretation of our findings. Consistent with our
conjecture, we find that standard edition video games that receive a limited edition have
more frontloaded sales than standard edition-only video games. Video game sales are 5.8
percentage points more frontloaded (i.e. the share of first month to sixth month unit sales)
when accompanied by a limited edition (p < 0.01). We also find that a 1 USD increase
in first month price leads to a 0.6 percentage point increase in frontloadedness; sales for
sequels are 2.6 percentage points less frontloaded than sales for new releases (p < 0.05);
video games based on licenses have 3.2 percentage points less frontloaded sales than games
based on original intellectual property (p < 0.05); and, sales for platform exclusive games
are 6.4 percentage points more frontloaded than sales for multi-platform games (p < 0.01).
The reported findings for our price drop and frontloaded sales measures are robust to reestimations that take three and twelve month timeframes instead of six months as currently
reported.
Taken together, these findings - games that receive a limited edition have higher initial
selling prices, enjoy marginally lower price drops, and have more frontloaded sales – are
largely consistent with our conjecture that if firms are able to remove price-sensitive segments
19
from the demand for the standard edition, they can maintain a higher price, which reduces
an incentive for consumers to delay purchase, resulting in more frontloaded sales.
In summary, we find supporting evidence for our theory of special editions as a device to
soften the Coase conjecture for their standard editions. In other words, even if the profitability of a limited edition is lower than that of its standard edition (e.g., price-cost margin),
it could still be optimal for a firm to introduce the limited edition in order to improve the
profitability of the standard edition. Another important aspect of this observation is that not
all games seem to benefit from this mechanism. Our data show that less than 10% of games
have a special edition. Thus, there are games that do not potentially benefit from introducing a special edition. One possibility is a cost of introducing a special edition. However, it
seems difficult to justify the decision is purely based on the cost, because the cost of items
included in limited editions should not depend on games themselves. Thus, we conjecture
that the decision may be driven by demand-side elements, in particular, the willingness-topay by different segments of consumers and how it differs across games, and within games
but across editions (standard and special editions). In a separate theoretical note, we find
that if consumers’ willingness-to-pay for different versions are negatively correlated across
different consumer segments, then it could be optimal to introduce a limited edition, even
if it earns a lower price-cost margin than its standard edition. However, if the versioning
decision is endogenous to unobserved heterogeneity in consumer preferences, then estimating
consumer preferences parameters with the observed versioning decisions might lead to biases
in consumer preference parameters. In order to deal with this issue, we propose a dynamic
20
equilibrium model that endogenizes firms’ versioning decision, and explicitly compute the
counterfactual profits under different possible versioning decisions.
5
Model
Our proposed equilibrium model consists of (1) forward-looking consumers’ adoption decision, and (2) monopoly firm’s decision on versioning (standard edition only versus standard
and limited editions), quantity limit (if a limited edition is released), and post-release pricing
strategy. We will describe the model in the context of video games, but it can be extended
to other durable goods markets with appropriate modifications.
5.1
Demand-side model
In order to focus on firms’ versioning strategy and consumers purchase decisions on versions,
we assume that each game g forms an independent market (no competition across games).
Let t index time, and t = 1 be the release period. In each period, forward-looking consumers
decide whether to buy or not. If they choose to buy and both editions (standard and limited)
are available in that period, then they decide which edition to buy. If the limited edition is
not released for game g or is sold out, they will buy the standard edition. Once consumers
buy the game, they exit the market. Let β be the consumers’ discount factor. Since each
game eventually becomes outdated, we assume that there is a terminal period T after which
games become unavailable for purchase.
We describe the choice set conditional on the firm’s decision on versioning. Let Jtg be the
set of options available for game g at time t. Let j = 0, 1, 2 be no purchase, standard edition
21
purchase, and limited edition purchase, respectively. We assume that {0, 1} ⊂ Jtg ∀g, t, but
{2} ⊂ Jtg only for those games with a limited edition and for time periods where the limited
edition has not been sold out. Consumers form expectation about (1) future prices (standard
and limited editions differently), and (2) availability of the limited edition. For the latter,
we assume that consumers do not observe the initial production level of the limited edition,
but observe when it is soldout. Instead of assuming fully rational consumers who form belief
about the future states based on firms’ optimal strategy, we assume that (1) consumers’ price
expectation processes can be approximated by a Markov process, and that (2) consumers
form belief about future soldout states for the limited edition based on (a) time since release,
and (b) type of bonus items included in the limited edition.5
5.1.1
Single-period utility function
Consumer i derives the following utility from purchasing edition j of game g at time t:
g
g
ugijt = γjr
+ µr t − αr pgjt + ξjt
+ gijt ,
where r indexes consumer segment (r = 1, . . . , R), and consumer i belongs to one of R
g
segments; γjr
is segment-r consumers’ initial valuation for edition j of game g; µr is segment-r
consumers’ sensitivity to newness of game g; αr is segment-r consumers’ price sensitivity, and
g
pgjt is the price of game g’s edition j at time t; ξjt
is an i.i.d. unobserved aggregate shock for
game g’s edition j at time t, and we assume that ξjt ∼ N (0, σξ2j ); gijt is an idiosyncratic error,
which we assume is extreme-value distributed. We model the decision in a nested framework,
and estimate the within-group correlation. We assume that the mean flow utility from no
5
We are currently estimating a model with rational expectations derived from firms’ optimal policies.
22
purchase is zero.
g
We model γjr
, segment-r consumers’ initial valuation for game g’s edition j as follows.
For the standard edition of game g (j = 1), we assume it is distributed as:
g
2
),
γ1r
∼ N (γ̄1r , σ1r
r = 1, . . . , R.
For the limited edition (j = 2), we assume that the initial valuation is given by
0
0
g
g
+ (X g , Z g )0 δr ,
= γ1r
γ2r
where X g is a vector of observed characteristics of game g (e.g., sequels), and Z g is a vector
of observed characteristics of game g’s limited edition (e.g., items included in the limited
edition). These specifications can be restrictive, but allow us to evaluate counterfactual
profitability of introducing a limited edition for our sample of games without a limited
edition.
5.1.2
Value functions
To simplify the notation, we drop g superscript. Given our assumption on consumers’ expectation processes, the state includes (1) set of available editions, (2) price for each available
edition, (3) unobserved demand shock for each available edition, and (4) time since release.
Let st = (Jt , {pkt , ξkt }k∈Jt \{0} , t) be the state vector at time t, where Jt = {0, 1, 2} or {0, 1}.
Let Vr (st ) be the integrated value function for segment-r consumers at state st :
Vr (st ) = E max{Vjr (st ) + ijt }
j∈Jt
where Vjr (st ) is the choice-specific value function for segment r and is given by
(
βEst+1 [Vr (st+1 )|st ]
if j = 0
Vjr (st ) =
γjr + µjr t − αr pjt + ξjt if j > 0.
23
The expectation operator for the future value function is taken with respect to (1) future
prices, (2) future demand shocks, and (3) set of available editions. As we described, we
assume a Markov process for prices: pjt+1 ∼ N (ρj0 + ρj1 pjt , σj2 ). For the set of available
editions, we assume that consumers’ belief is approximated as
0
Pr(Jt+1 = {0, 1}|Jt = {0, 1, 2}) =
exp((1, t, Z g )0 δ)
,
1 + exp((1, t, Z g0 )0 δ)
and Pr(Jt+1 = {0, 1}|Jt = {0, 1}) = 1, Pr(Jt+1 = {0, 1, 2}|Jt = {0, 1}) = 0, and Pr(Jt+1 =
{0, 1, 2}|Jt = {0, 1, 2}) = 1 − Pr(Jt+1 = {0, 1}|Jt = {0, 1, 2}).
5.2
Supply-side model
Our supply-side model starts with a monopolist publisher who has developed a standard
edition of game g with characteristics X g (e.g., sequel). These are exogenous to our model.
Given the standard edition of game g, prior to release, the publisher decides (1) whether
or not to release a limited edition as well, and if it does, (2) what type of bonus items to
include and how many units to produce. The choice on the inclusion of a bonus item is
assumed to be a discrete choice problem, and we also assume that the production level of
the limited edition has to be fixed prior to release. This assumption is motivated by the
observation that publishers typically outsource the production of bonus items included in
limited editions well before the release (about six months), and it is too costly to re-negotiate
additional production post-release.
Once the game is released, the publisher chooses a sequence of price(s) so as to maximize
the present discounted value of current and future profits. We describe the publisher’s
problem below.
24
5.2.1
Versioning decision
The publisher who has developed the standard edition of game g decides whether or not
to introduce a limited edition (in addition to the already developed standard edition). Let
cm (Z g ) be the average total cost of producing one unit of a limited edition with characteristics
Z g . Since we do not observe the development cost and the manufacturing cost of bonus items
included in a limited edition separately, we will consider the average total cost of producing
one unit of the limited edition. Let
0
g
cm (Z g ) = Z g λm + νm
,
g
νm
∼ N (0, σν2m ).
Finally, let Q2t be the unit of the limited edition remaining at the beginning of time t.
Let Wag be the value function for game g from choosing a versioning action a = 1, 2,
where a = 1 indexes the standard edition only, and a = 2 indexes the standard and limited
editions. For a = 1, the value function is given by
" T
#
X
g
W1g = max
E
β t−1 π1t (pg1t ; f1t
) ,
g
{p1t }T
t=1
t=1
g
g
where π1t (pg1t ; f1t
) is the per-period profit function at time t at state f1t
from choosing the
price of the standard edition, which is given by
g
g
g
g
),
) = (pg1t − c1 )D1t
(pg1t ; f1t
π1t
(pg1t ; f1t
and c1 is the marginal cost (e.g., printing DVDs, packaging, licensing fee paid to the platg
g
g
= (ξ1t
, {Mrtg }R
form), which we assume is fixed over time, and f1t
r=1 ) and Mrt is the number
of segment-r potential buyers at time t for game g. The demand function is given by
g
g
D1t
(pg1t ; f1t
)
=
R
X
Mrtg Pr(j = 1|sgt ),
r=1
25
where st = (p1t , ξ1t , t) and the choice probability Pr(j = 1|sgt ) is derived by a logit choice
probability.
For a = 2, the value function is given by
" T
#
!
X
g
g
W2g =
max
E
β t−1 π2t
(pg1t , pg2t ; f2t
, Q2t , Z g ) Qg21 − cm (Z g )Qg21 ,
g
g
T
g
Z ,Q21 ,{pjt j=1,2}t=1
t=1
g
g
where π2t (pg1t , pg2t ; f2t
, Q2t , Z g ) is the per-period profit function at time t at state f2t
from
choosing the prices of the standard and limited editions, which is given by
g
g
g
g
g
π2t (pg1t , pg2t ; f2t
, Q2t , Z g ) = (pg1t − c1 )D1t
(pg1t , pg2t , f2t
) + (pg2t − c2 ) min{D2t
(pg1t , pg2t , f2t
), Q2t },
and c2 is the marginal cost (e.g., licensing fee, as the manufacturing cost is included in cm ),
g
g 2
g
and f2t
= ({ξjt
}j=1 , {Mrtg }R
r=1 , Q2t ). The demand function for Djt is given by
g
g
Djt
(pg1t , pg2t ; f2t
)=
R
X
Mrtg Pr(j|sgt ),
r=1
where the choice probability Pr(j|sgt ) is derived from a nested logit choice probability with
the within-group correlation η.
The publisher introduces only the standard edition if and only if
W1g ≥ W2g .
5.2.2
Pricing decision
After the publisher makes the versioning decision, it maximizes the present discounted value
g
of future profits by making a sequence of pricing decisions. Let Watg (fat
) be the publisher’s
g
value function at time t given state f1t
and versioning decision a. Below we drop g superscript
to simplify the notation. For a = 1, the value function is given by
W1t (f1t ) = max π1t (p1t ; f1t ) + βEf1t+1 [W1t+1 (f1t+1 )|f1t , p1t ].
p1t
26
The first-order condition with respect to p1t is
D1t (p1t , f1t ) + (p1t − c1 )
∂Ef1t+1 [W1t+1 (f1t+1 )|f1t , p1t ]
∂D1t (p1t , f1t )
+β
= 0.
∂p1t
∂p1t
For a = 2 and Q2t > 0, the value function is given by
W2t (f2t ) = max π2t (p1t , p2t ; f1t , Q2t ) + βEf2t+1 [W2t+1 (f2t+1 )|f2t , p1t , p2t ].
p1t ,p2t
The first-order condition with respect to pjt is given by
∂Djt (p1t , p2t , f2t )
∂pjt
∂Ef2t+1 [W2t+1 (f2t+1 )|f2t , p1t , p2t ]
∂D−jt (p1t , p2t , f2t )
+ (p−jt − c−j )
+β
= 0.
∂pjt
∂pjt
Djt (p1t , p2t , f2t ) + (pjt − cj )
Let p∗jt be the optimal price of edition j predicted by the model.
6
6.1
Estimation Strategy
Estimation algorithm
We estimate the proposed model by Bayesian Markov chain Monte Carlo (MCMC) algorithm
proposed by Imai et al. (2009) (IJC) and extended by Ishihara and Ching (2016) to a
finite-horizon model. In our equilibrium model, we need to compute the value functions
for consumers and the monopolist publisher for each of the games in our sample. In each
MCMC iteration, we conduct a modified backward induction for both consumers’ and the
monopolist publisher’s value functions at a randomly chosen sequence of state points over
time, and store those partially solved value functions. When we evaluate a value function at
a given parameter vector and at a state, we use those stored partially solved value functions
and approximate the value function using the Gaussian kernel. Another benefit from using
27
g
by Hierarchical Bayes, which reduces the
a Bayesian approach is that we can estimate γ1r
computational burden.
Similar to Ching (2010), the likelihood functions are derived by assuming a measurement
error in the observed sales and prices. Let Dgjt and pgjt be the observed unit sales and price
of game g’s edition j. Then,
g
Dgjt = Djt
+ εjt
εjt ∼ N (0, σε2 ),
and
pgjt = p∗jt + ηjt
6.2
ηjt ∼ N (0, ση2 ).
Identification
Our identification of the demand and cost parameters closely follows Nair (2007), Lee (2013),
Derdenger and Kumar (2013), Gowrisankaran and Rysman (2012), and Gowrisankaran et al.
(2014). The distinct feature from these models is that the identification of the initial game
valuations, γ1r and γ2r , are not only identified by the sales of observed standard and limited
editions, but also via the optimal versioning decision, in order to control for the potential
endogeneity of versioning decision that could be driven by γ1r and γ2r , as illustrated in
Section 3. We further exploit variation in versioning choices for multi-platform games. That
is, there are games whose standard edition is available on both PlayStation 3 and Xbox 360
but whose limited edition is available only on one of the consoles. Such variation reduces
the reliance of our identification on the supply-side model structure.
28
7
Preliminary Results
In estimating the proposed dynamic equilibrium model, we assume: (1) consumers’ and
firms’ discount factors are 0.975 (according to the monthly interest rate), (2) we allow for
three discrete segments of consumers (R = 3), (3) the characteristics of game g (X g ) includes dummies for sequel games, licensed games, and platform-exclusive games, and (4) the
characteristics of limited edition bonuses for game g (Z g ) is a dummy for whether limited
edition bonuses include a physical item or not.6 This implies that in order to derive the
versioning decision, for each game, we need to compute two counterfactual value functions.
For example, if a game has no limited edition, then the value function from a single version
must be larger than either the value function with a limited edition that includes a physical
item or the value function with a limited edition that does not include a physical item. (5) as
a preliminary analysis, we randomly selected 200 games as the estimation sample (24 games
with a limited edition).
We first present the estimation results for preference parameters in Table 9. In our
estimation, we include holiday season effect (December dummy) in the utility function for
purchase as a surprise to consumers (i.e., consumers’ value functions do not account for a
dynamic effect due to the holiday effect). We find a strong positive holiday season effect,
which is reasonable given that the sales of video games increases sharply in December.
The within-group correlation parameter is close to zero and non-significant. As for the
heterogeneous parameters, we identify the following three segments. Segment 1 (about 24%
6
Allowing for other detailed characteristics is possible, but it dramatically increases the computational
burden.
29
of the population) has the lowest mean initial valuation across games (4.42), a moderate
price sensitivity (-0.137), and the lowest sensitivity to newness (-0.799). This segment, as
compared to the other two segments, generates a smaller sales, especially for limited editions.
Segment 2 (about 68%) has the highest initial valuation for both standard and limited
editions, and the highest price sensitivity. They have a moderate sensitivity to newness. This
segment is key to our analysis. We find that only this segment has a higher average valuation
for limited editions than for standard editions. A closer look at the coefficient associated
with observed game characteristics shows that the incremental valuation for limited editions
is mainly driven by the sequel dummy, a factor that is highly correlated with the decision to
introduce a limited edition. Thus, for sequel games, this segment plays an important role in
improving the profitability of the standard edition. Recall that this segment has the highest
price sensitivity and a moderate sensitivity to newness. Thus, if only a standard edition is
introduced, this segment of consumers are likely to delay purchase, reducing the profitability
of the standard edition.
Finally, segment 3 (about 8%) has the lowest price sensitivity and the highest sensitivity
to newness. They are more likely to purchase right after a game’s release. Also, they value
limited editions less than standard editions. Table 10 compares the average initial valuation
for (1) standard only edition, (2) standard edition with a limited edition, and (3) limited
edition, across the three segments.
Next, the cost estimates are reported in Table 11. We find that the marginal cost for
standard editions is about $16.74, which is close to what the industry typically reports
30
(about $12). The marginal cost for limited editions is not significantly different from zero,
which may be reasonable as most of the production cost is incurred prior to release and it is
possible that even the licensing fee is paid prior to release. Finally, as we expect, a limited
edition with a physical item is more expensive to produce than that without a physical item.
Based on the cost estimates, we compute the average price-cost margin for (1) standard
only edition, (2) standard edition with a limited edition, and (3) limited edition. Figure 4
shows the average price-cost margin over the first six months after a game release. Consistent
with our intuition about softening the Coase conjecture, we find that the price-cost margin
of standard editions with a limited edition is consistently higher than the other two. Also,
consistent with our interviews with video game publishers, we find that the price-cost margin
of limited editions is the lowest.
7.1
Counterfactuals for welfare analysis
Using the estimates from the structural model, we plan to evaluate the welfare impact of
different commitment mechanisms.
8
Conclusion
To be added.
31
References
Ching, Andrew T. 2010. Consumer Learning and Heterogeneity: Dynamics of Demand for
Prescription Drugs After Patent Expiration. International Journal of Industrial Organization 28(6) 619–638.
Coase, R.H. 1972. Durability and Monopoly. Journal of Law and Economics 15(1) 143–149.
Derdenger, Timothy, Vineet Kumar. 2013. The Dynamic Effects of Bundling as a Product
Strategy. Marketing Science 32(6) 827–859.
Gowrisankaran, Gautam, Marc Rysman. 2012. Dynamics of Consumer Demand for New
Durable Goods. Journal of Political Economy 120(6) 1173–1219.
Gowrisankaran, Gautam, Marc S. Rysman, Grace Yu. 2014. Computing Price-Cost Margins
in a Durable Goods Environment. Working Paper, Boston University.
Imai, Susumu, Neelam Jain, Andrew Ching. 2009. Bayesian Estimation of Dynamic Discrete
Choice Models. Econometrica 77(6) 1865–1899.
Ishihara, Masakazu, Andrew Ching. 2016. Bayesian Estimation of Finite-Horizon Discrete
Choice Dynamic Programming Models. Working Paper, Stern School of Business, New
York University.
Lee, Robin S. 2013. Vertical Integration and Exclusivity in Platform and Two-Sided Markets.
American Economic Review 103(7) 2960–3000.
Nair, Harikesh. 2007. Intertemporal Price Discrimination with Forward-Looking Consumers:
An Application to the US Market for Console Video-Games. Quantitative Marketing and
Economics 5(3) 239–292.
32
Table 1: Theoretical Implications
Which segment buys
Price cut in the mid-term
Frontloadedness of sales
Initial price
the limited edition?
(from t=1 to t=2)
Segment 1
Higher, Lower, or Unchanged
Lower
Larger, Smaller, or Unchanged
Segment 2
Higher
Lower
Larger, Smaller, or Unchanged
Segment 3
Higher
Higher, Lower, or Unchanged
Smaller
Segment 4
Unchanged
Lower or Unchanged
Smaller or Unchanged
Table 2: Summary Statistics: Bonus Items Included in Special Editions
Type
Special case (e.g. steelbook)
Figurine
Comic book
Artbook
Story book/journal
Strategy guide
Bonus disc
Soundtrack
Additional game contents
Card/posters
Other small items
Games % of games
39
36.11%
32
29.63%
23
21.30%
44
40.74%
9
8.33%
13
12.04%
66
61.11%
33
30.56%
43
39.81%
29
26.85%
27
25.00%
33
Table 3: Summary Statistics: Versioning Strategies
All games
High quality
Sequel
Licensed
Star
Platform exclusive
PlayStation 3
Xbox 360
Genre:
1st person shooter
Action
Adventure
Family/party
Fighting
Racing
Role playing game
Shooter
Simulation
Sports
Survival horror
All games Limited edition
1168
108
510
91
732
82
481
28
237
10
272
24
658
58
510
50
132
239
69
55
48
126
69
72
54
270
34
17
21
5
0
10
8
19
8
1
9
10
Limited Edition (%)
9.25%
17.84%
11.20%
5.82%
4.22%
8.82%
8.81%
9.80%
12.88%
8.79%
7.25%
0.00%
20.83%
6.35%
27.54%
11.11%
1.85%
3.33%
29.41%
Table 4: Summary Statistics: Key Variables
All games (N=1168)
SD
Min
Max
173342.2
119
1718307
351732.9
5937
3865565
0.1897348
0.003
0.857
8.750318
19.83
119.17
12.51289
9.21
94.67
11.29704
-13.19
52.72
Standard only (n=1060)
Limited edition received (n=108)
Variable
Mean
SD
Min
Max
Mean
SD
Min
Max
First month unit sales
67527.25 126411.10 119.00
945894.00 315215.60 336693.60 119.00 1718307.00
Sixth month unit sales
161508.70 258151.70 5937.00 2166535.00 636027.30 695330.70 9115.00 3865565.00
First-to-sixth month sales ratio
0.35
0.19
0.00
0.86
0.50
0.17
0.01
0.84
First month selling price
53.82
9.04
19.83
119.17
58.82
2.07
47.44
59.98
Sixth month selling price
39.91
12.57
9.21
94.67
46.98
9.96
16.32
59.65
First-to-sixth month price drop
13.91
11.39
-13.19
52.72
11.84
10.15
-2.98
42.95
Notes: ** p < 0.01, * p < 0.05, + p < 0.10. Mean differences and significance levels reported from a two-tailed t-test with equal variances.
Variable
First month unit sales
Sixth month unit sales
First-to-sixth month sales ratio
First month selling price
Sixth month selling price
First-to-sixth month price drop
Mean
90429.94
205385.4
0.3600753
54.28196
40.56561
13.71635
34
Mean
difference
-247688.40**
-474518.60**
-0.16**
-5.00**
-7.07**
2.07+
Table 5: Probit Estimation on Versioning Decision
Variable
High quality
Sequel
Licensed
Star
Platform exclusive
ln(Platform IB)
Platform-genre HHI
Constant
Model 1
est
se
0.926**
0.180
0.289
0.178
-0.020
0.162
-0.776*
0.308
-4.26E-04 0.153
0.209**
0.055
1.850**
0.512
-6.580**
1.027
Model 2
Model 3
Model 4ᵃ
est
se
est
se
est
se
1.128**
0.190
1.228**
0.197
1.154**
0.224
0.421*
0.170
0.451**
0.157
0.571**
0.162
0.193
0.235
0.277
0.229
0.360
0.281
-0.181
0.437
-0.189
0.443
-0.135
0.541
0.013
0.146
-0.055
0.143
-0.476+
0.252
0.235**
0.067
0.696**
0.263
0.792**
0.292
2.026**
0.367
2.109**
0.341
2.301**
0.412
-7.228** 1.120798 -13.583** 3.548212 -15.079** 3.841726
Platform FE
YES
YES
YES
YES
Genre FE
NO
YES
YES
YES
Month FE
NO
NO
YES
YES
Year FE
NO
NO
YES
YES
Publisher FE
NO
NO
NO
YES
Observations
1168
1113
1113
990
Pseudo R2
0.188
0.268
0.333
0.399
Notes: ** p < 0.01, * p < 0.05, + p < 0.10. Heteroskedasticity robust standard errors clustered at the publisher
level. ᵃ Model 4 used for computing selection correction (λ) term.
Table 6: Regression on the First-Month Price
Variable
Limited edition received
High quality
Sequel
Licensed
Star
Platform exclusive
ln(Platform IB)
Platform-genre HHI
Selection correction (λ)
Constant
Model 1
est
se
4.416**
0.970
1.654*
0.785
0.806
0.538
-1.989**
0.633
3.496*
1.304
-2.457**
0.658
-1.680**
0.346
0.994
2.071
Model 2
est
se
3.009**
0.911
1.704*
0.690
0.5222
0.458
-0.705
0.753
5.253*
2.040
-1.946**
0.648
-1.325**
0.309
2.981
2.202
Model 3
est
se
2.920**
0.929
1.812*
0.721
0.524
0.473
-0.909
0.783
5.278*
2.091
-1.669**
0.612
-0.809
0.759
3.763+
2.181
Model 4
est
se
2.648**
0.842
1.507+
0.779
-0.034
0.501
-0.905
0.787
4.403*
2.194
-2.474**
0.769
-0.772
0.717
1.524
1.976
79.461**
75.411**
68.667**
65.410**
5.368
5.417
10.429
9.793
Model 5
est
se
3.101**
0.929
-0.797
2.628
-0.984
1.439
-1.476
1.243
4.591
2.735
-1.143
1.156
-2.147
1.650
-1.784
5.601
-2.409
2.507
96.838* 34.595
Platform FE
YES
YES
YES
YES
YES
Genre FE
NO
YES
YES
YES
YES
Month FE
NO
NO
YES
YES
YES
Year FE
NO
NO
YES
YES
YES
Publisher FE
NO
NO
NO
YES
YES
Observations
1168
1168
1168
1168
990
R-squared
0.123
0.223
0.257
0.366
0.345
Notes: ** p < 0.01, * p < 0.05, + p < 0.10. Heteroskedasticity robust standard errors clustered at the publisher level. Reported results robust to
the exclusion of three outlier games with high prices.
35
Table 7: Regression on the Price Drop in the First 6 Months
Variable
Limited edition received
First month price
High quality
Sequel
Licensed
Star
Platform exclusive
Selection correction (λ)
Constant
Model 1
est
se
-1.028
1.259
0.376**
0.051
-5.907**
0.882
-3.564**
1.186
-2.199+
1.213
1.233
1.338
-4.534**
1.235
0.360
2.523
Model 2
est
se
-0.953
1.153
0.383**
0.058
-5.716**
0.828
-3.088**
1.128
-2.321+
1.293
2.043*
0.924
-4.291**
1.262
1.421
2.727
Model 3
est
se
-0.575
1.056
0.376**
0.056
-5.643**
0.783
-2.873*
1.144
-2.445*
1.192
2.292*
0.926
-3.185*
1.337
-3.051
3.447
Model 4
est
se
-2.012*
0.996
0.415**
0.077
-5.478**
0.815
-2.124+
1.144
-2.228
1.371
2.112*
0.837
-2.152*
0.967
2.058
Model 5
est
se
-1.547+
0.852
0.358**
0.086
-3.928**
1.232
-1.123
1.263
-2.114+
1.118
2.155*
0.915
-2.573*
1.190
1.815
1.470
-15.632**
8.398
4.541
Platform FE
YES
YES
YES
YES
YES
Genre FE
NO
YES
YES
YES
YES
Month FE
NO
NO
YES
YES
YES
Year FE
NO
NO
YES
YES
YES
Publisher FE
NO
NO
NO
YES
YES
Observations
1168
1168
1168
1168
990
R-squared
0.185
0.205
0.267
0.376
0.357
Notes: ** p < 0.01, * p < 0.05, + p < 0.10. Heteroskedasticity robust standard errors clustered at the publisher level. Reported results robust to
the exclusion of three outlier games with high prices. Reported results further robust to calculating the price drop measure using three and
twelve month time frames.
Table 8: Regression on the Ratio of First-Month to Total Sales
Variable
Limited edition received
First month price
Price drop
High quality
Sequel
Licensed
Star
Platform exclusive
Selection correction (λ)
Constant
Model 1
est
se
0.095**
0.022
0.007**
0.001
0.001
0.001
0.043**
0.014
0.01
0.014
-0.004
0.022
-0.065**
0.016
0.015
0.015
0.060+
0.033
Model 2
est
se
0.068**
0.023
0.006**
0.001
0.001
0.001
0.057**
0.014
0.015
0.012
0.006
0.018
-0.029
0.019
0.016
0.014
0.009
0.036
Model 3
est
se
0.070**
0.016
0.007**
0.001
-0.0005
0.001
0.059**
0.010
0.017+
0.009
0.007
0.011
-0.026
0.019
0.029**
0.011
0.03
0.066
Model 4
est
se
0.064**
0.018
0.006**
0.001
-0.001
0.001
0.045**
0.011
0.008
0.009
-0.006
0.012
-0.029
0.022
0.027*
0.013
-0.010
0.068
Model 5
est
se
0.058**
0.019
0.006**
0.001
0.0000661
0.001
-0.023
0.019
-0.026**
0.009
-0.032*
0.015
-0.029
0.023
0.064**
0.011
-0.071**
0.014
0.351**
0.076
Platform FE
YES
YES
YES
YES
YES
Genre FE
NO
YES
YES
YES
YES
Month FE
NO
NO
YES
YES
YES
Year FE
NO
NO
YES
YES
YES
Publisher FE
NO
NO
NO
YES
YES
Observations
1168
1168
1168
1168
1168
R-squared
0.200
0.245
0.451
0.501
0.530
Notes: ** p < 0.01, * p < 0.05, + p < 0.10. Heteroskedasticity robust standard errors clustered at the publisher level. Reported results robust to
the exclusion of three outlier games with high prices. Reported results further robust to calculating the price drop and sales ratio measures
using three and twelve month time frames.
36
Table 9: Demand Estimates
Variable
holiday effect
within-group correlation
heterogeneity
proportion
mean
sd
0.570
1.49
0.002
0.003
segment 1
0.059
0.238**
mean valuation for standard edition (γ 1)
4.42
mean
sd
mean
segment 2
0.066
0.675**
segment 3
0.007
0.086**
0.755
9.53**
1.72
8.69**
1.68
**
2.76
0.167
**
1.15
**
5.39
1.04
-1.59**
0.074
-0.238**
0.027
-1.10**
0.100
**
**
sd valuation for standard edition (σ γ1)
incremental valuation (δ)
constant
*
sequel dummy
-0.250
licensed dummy
**
0.024
1.18
**
0.096
exclusive dummy -0.348
physical item dummy 0.146+
0.078
**
price sensitivity (α)
0.002
0.137
**
newness sensitivity (µ)
0.021
-0.799
8.41
0.892
0.027
0.058
**
0.027
0.049
-0.683
**
0.047
**
0.008
0.076
-0.009
0.044
0.074
0.138
0.445
0.029
**
0.284
**
-4.93
0.027
0.737
1 ver: standard
$30.97
$31.71
$109.71
2 ver: standard
$40.52
$48.49
$142.71
0.005
**
0.262
0.077
-11.1
2 ver: limited
$30.64
$50.43
$122.74
Table 11: Cost Estimates
Variable
standard edition
mean
sd
marginal cost
16.74+
8.60
manufacturing cost (const)
37.04*
17.33
manufacturing cost (physical item)
**
special edition
marginal cost
23.31
-14.18
5.57
14.09
Notes: +, *, and ** indicate that zero is not contained in the 10%, 5%, and 1% credible interval, respectively.
37
0.024
**
Table 10: Willingness-to-pay by segment
proportion
24%
68%
8%
0.047
**
Notes: +, *, and ** indicate that zero is not contained in the 10%, 5%, and 1% credible interval, respectively.
segment
1
2
3
sd
**
Figure 1: Illustrative Examples of Price-Segmentation
t=1
t=2
t=3
Seg. 1
Seg. 2
Seg. 3
Multi Ver.
Standard
edition
Seg. 2
Seg. 3
Limited
edition
Seg. 1
Single Ver.
Standard
edition
Multi Ver.
Standard
edition
Limited
edition
Multi Ver.
Standard
edition
Limited
edition
Seg. 1
Seg. 3
Seg. 2
Seg. 1
Seg. 2
Seg. 3
38
time
Figure 2: PS3: Little Big Planet 2
PS3 Little Big Planet 2
90
300000
80
250000
70
Unit Sales
50
150000
40
Price ($)
60
200000
30
100000
20
50000
10
0
0
0
2
4
6
8
10
12
14
16
Months in Release
Sales: Standard
Sales: Limited
Price: Standard
Price: Limited
Figure 3: PS3: Fallout 3
180000
90
160000
80
140000
70
120000
60
100000
50
80000
40
60000
30
40000
20
20000
10
0
0
0
5
10
15
20
25
30
35
40
45
Months in Release
Sales: Standard
Sales: Limited
39
Price: Standard
Price: Limited
Price ($)
Unit Sales
PS3 Fallout 3
Figure 4: Average Price-Cost Margins
45
Price-cost margin (in $)
40
35
30
25
20
15
10
5
0
0
1
2
3
4
5
6
Months in release
1 ver: standard
2 ver: standard
40
2 ver: limited
7
A
A.1
Appendix
Derivation of the Hypotheses in Table
In this section, we use a simple model and derive the empirical implications presented in
Table 1. Our goal is to examine how (1) frontloadedness of sales, (2) initial price, and (3)
price cut for the standard edition change as a result of adding a limited edition, assuming
each of the four cases in Table 1 is optimal for the monopolist. Here, we do not examine
what is optimal.
We first describe the model. There are four segments of consumers who differ in their
valuation of the product. Let qr be the willingness-to-pay for the standard edition by segment
r (r = 1, 2, 3, 4). As we considered in Figure 1, let us assume q1 > q2 > q3 > 0. Segment 4
does not buy the standard edition, so we assume q4 = 0. Let ∆r be segment-r consumers’
(additive) incremental valuation for the limited edition (i.e., the WTP for the limited edition
is qr + ∆r ). Since the limited edition “contains” the standard edition, we assume ∆r ≥ 0 ∀r.
Let pst and plt be the prices of the standard and limited editions, respectively. Consumers
thus receive the following utility from purchasing the standard/limited edition at time t:
(
qr − pst
if buying the standard edition,
urt =
qr + ∆r − plt if buying the limited edition.
We normalize the utility for no purchase to zero. In the example 1, only one segment buys
the limited edition at time t = 1, we only need to consider pP
l1 . We assume that the size of
each segment (mr ) is strictly positive and sums up to one, r mr = 1. Let β ∈ [0, 1) be
the discount factor common across all consumers and the firm. Finally, the marginal cost is
assumed to be zero for both standard and limited edition. We assume that the limited edition
has to be produced prior to the release of the product, and thus, the additional production
cost for limited edition’s bonus items is sunk at the time of making the pricing decisions. The
additional product cost will be important if we are interested in optimal versioning strategy.
But for the present analysis, we assume a certain optimal versioning strategy and derive the
optimal pricing path. Thus, the additional production cost is irrelevant.
A.1.1
The Baseline Case: Standard Edition Only
To examine how adding a limited edition might change the variables of our interest, we first
consider a baseline case where the monopolist sells only the standard edition.
We assumed that segment-r consumers buy at time t = r, and segment-4 consumers do
not buy. In order for this to be the optimal pricing strategy, the price of the standard edition
at time t, pst , have to satisfy the following incentive compatibility and individual rationality
constraints:
q1 − ps1 ≥ max{β(q1 − ps2 ), β 2 (q1 − ps3 ), 0}
β(q2 − ps2 ) ≥ max{q2 − ps1 , β 2 (q2 − ps3 ), 0}
β 2 (q3 − ps3 ) ≥ max{q3 − ps1 , β(q3 − ps2 ), 0}
41
Here we omit the constraint for segment-4 consumers. Based on these constraints, we can
solve the optimal pst by backward induction.
pbs1 = (1 − β)q1 + (1 − β)βq2 + β 2 q3
pbs2 = (1 − β)q2 + βq3
pbs3 = q3 ,
where the superscript b denotes that these are the optimal prices for the baseline case.
We can now derive (1) the frontloadedness of sales, (2) initial price, and (3) price cut for
the standard edition. First, the frontloadedness is defined as the ratio of the first-period
m1
sales to the total sales, which is m1 +m
in the baseline case. The initial price is pbs1 =
2 +m3
(1 − β)q1 + (1 − β)βq2 + β 2 q3 . The price cut, which is defined as the absolute difference
between pbs1 and pbs2 is (1 − β)(q1 − (1 − β)q2 − βq3 ).
A.1.2
When Segment-1 Consumers Buy the Limited Edition
Consider a situation where segment-1 consumers buy the limited edition at t = 1, segment-2
consumers buy the standard edition at t = 1, and segment-3 consumers buy the standard
edition at t = 2. Note that in order to support this path, the role of quantity limit is
important. The firm can credibly commit to selling the limited edition to only segment1 consumers. In this case, the following incentive compatibility and individual rationality
constraints need to be satisfied:
q1 + ∆1 − pl1 ≥ max{q1 − ps1 , β(q1 − ps2 ), 0}
q2 − ps1 ≥ max{q2 + ∆2 − pl1 , β(q2 − ps2 ), 0}
β(q3 − ps2 ) ≥ max{q3 + ∆3 − pl1 , q3 − ps1 , 0}
0 ≥ max{∆4 − pl1 , −ps1 , −βps2 }
Based on these constrains, we solve the optimal pricing by backward induction. First, at
t = 2, only segment-3 and segment-4 consumers remain. Since the firm will not sell to
segment-4, the optimal price is ps2 = q3 . At t = 1, ps1 and pl1 have to satisfy
ps1 ∈ [q3 , (1 − β)q2 + βq3 ] ≡ A
pl1 ∈ [max{∆4 , q3 + ∆3 , ∆2 + ps1 }, min{∆1 + (1 − β)q1 + βq3 , ∆1 + ps1 }] ≡ B
It is easy to see that A is non-empty. For B, we first notice that min{∆1 + (1 − β)q1 +
βq3 , ∆1 + ps1 } = ∆1 + ps1 , ∀ps1 ∈ A. Thus in order for B to be non-empty for some ps1 ,
we need ∆1 + ps1 > max{∆4 , q3 + ∆3 , ∆2 + ps1 } for some ps1 , which results in ∆1 > ∆2 and
∆1 + (1 − β)q2 + βq3 > max{∆4 , q3 + ∆3 }. Intuitively, this condition guarantees that ∆1 is
large enough such that it is optimal for the firm to sell the limited edition only to segment-1
consumers. It is then immediate that the firm can maximize the profits by setting ps1 and
pl1 at their maximum values: ps1 = (1 − β)q2 + βq3 and pl1 = ∆1 + (1 − β)q2 + βq3 .
42
In summary, we have
pl1 = ∆1 + (1 − β)q2 + βq3
ps1 = (1 − β)q2 + βq3
ps2 = q3
Now we examine (1) frontloadedness of sales, (2) initial price, and (3) price cut for
2
. Without any further
the standard edition. First, the frontloadedness of sales is m2m+m
3
assumptions, we cannot determine whether this is higher or lower than its baseline size
m1
. Next, we consider the initial price:
m1 +m2 +m3
pbs1 − ps1 = (1 − β)[q1 − (1 − β)q2 − βq3 ] > 0.
Thus, the initial price in this scenario is lower than that in the baseline case. Finally, the
change in the price cut is given by
(pbs1 − pbs2 ) − (ps1 − ps2 ) = (1 − β)[(q1 − q2 ) − (1 − β)(q2 − q3 )],
which can be positive or negative. Thus, the change is indeterminate.
A.1.3
When Segment-2 Consumers Buy the Limited Edition
Consider a situation where segment-2 consumers buy the limited edition at t = 1, segment-1
consumers buy the standard edition at t = 1, and segment-3 consumers buy the standard
edition at t = 2. In this case, the following incentive compatibility and individual rationality
constraints need to be satisfied:
q1 − ps1 ≥ max{q1 + ∆1 − pl1 , β(q1 − ps2 ), 0}
q2 + ∆2 − pl1 ≥ max{q2 − ps1 , β(q2 − ps2 ), 0}
β(q3 − ps2 ) ≥ max{q3 + ∆3 − pl1 , q3 − ps1 , 0}
0 ≥ max{∆4 − pl1 , −ps1 , −βps2 }
Based on these constrains, we solve the optimal pricing by backward induction. First, as
before, at t = 2, the optimal price is ps2 = q3 . Now at t = 1, ps1 and pl1 have to satisfy
ps1 ∈ [q3 , (1 − β)q1 + βq3 ] ≡ A
pl1 ∈ [max{∆4 , q3 + ∆3 , ∆1 + ps1 }, min{∆2 + (1 − β)q2 + βq3 , ∆2 + ps1 }] ≡ B
It is easy to see that A is non-empty. For B, we first notice that
(
∆2 + ps1
if ps1 ∈ [q3 , (1 − β)q2 + βq3 ],
min{∆2 +(1−β)q2 +βq3 , ∆1 +ps1 } =
∆2 + (1 − β)q2 + βq3 if ps1 ∈ [(1 − β)q2 + βq3 , (1 − β)q1 + βq3 ].
Thus in order for B to be non-empty for some ps1 , we need (1) ∆2 > ∆1 , and (2) ∆2 + (1 −
β)q2 + βq3 > max{∆4 , q3 + ∆3 }. Now the optimal price of the limited edition can be set to
43
pl1 = ∆2 + (1 − β)q2 + βq3 , but the optimal price of the standard edition depends on the
parameters:
(
(1 − β)q1 + βq3
if (1 − β)q1 + ∆1 ≤ (1 − β)q2 + ∆2 ,
ps1 =
(1 − β)q2 + βq3 + ∆2 − ∆1 if (1 − β)q1 + ∆1 > (1 − β)q2 + ∆2 .
In summary, we have
pl1 = ∆2 + (1 − β)q2 + βq3
(
(1 − β)q1 + βq3
ps1 =
(1 − β)q2 + βq3 + ∆2 − ∆1
if (1 − β)q1 + ∆1 ≤ (1 − β)q2 + ∆2 ,
if (1 − β)q1 + ∆1 > (1 − β)q2 + ∆2 .
ps2 = q3
Now we examine (1) frontloadedness of sales, (2) initial price, and (3) price cut for
1
, which is greater than its
the standard edition. First, the frontloadedness of sales is m1m+m
3
m1
baseline size m1 +m2 +m3 . Next, we consider the initial price. For (1−β)q1 +∆1 ≤ (1−β)q2 +∆2 ,
we have
pbs1 − ps1 = (1 − β)β(q2 − q3 ) > 0.
For (1 − β)q1 + ∆1 > (1 − β)q2 + ∆2 , we have
pbs1 − ps1 = (1 − β)(q1 − q2 ) + (1 − β)β(q2 − q3 ) − (∆2 − ∆1 )
> (1 − β)β(q2 − q3 ) > 0,
where the second inequality comes from the condition (1 − β)q1 + ∆1 > (1 − β)q2 + ∆2 .
Thus, in either case, the initial price in this scenario is lower than that in the baseline case.
Finally, the price cut for (1 − β)q1 + ∆1 ≤ (1 − β)q2 + ∆2 is
(pbs1 − pbs2 ) − (ps1 − ps2 ) = (1 − β)2 (q3 − q2 ) < 0,
meaning that the price cut is larger. When (1 − β)q1 + ∆1 > (1 − β)q2 + ∆2 , the price cut is
(pbs1 − pbs2 ) − (ps1 − ps2 ) = (1 − β)(q1 − q2 ) − (∆2 − ∆1 ) − (1 − β)2 (q2 − q3 ),
which can be positive or negative. Thus, the change is indeterminate.
A.1.4
When Segment-3 Consumers Buy the Limited Edition
Consider a situation where segment-3 consumers buy the limited edition at t = 1, segment-1
consumers buy the standard edition at t = 1, and segment-2 consumers buy the standard
edition at t = 2. In this case, the following incentive compatibility and individual rationality
constraints need to be satisfied:
q1 − ps1 ≥ max{q1 + ∆1 − pl1 , β(q1 − ps2 ), 0}
β(q2 − ps2 ) ≥ max{q2 + ∆2 − pl1 , q2 − ps1 , 0}
q3 + ∆3 − pl1 ≥ max{q3 − ps1 , β(q3 − ps2 ), 0}
0 ≥ max{∆4 − pl1 , −ps1 , −βps2 }
44
Based on these constrains, we solve the optimal pricing by backward induction. First, at
t = 2, segment-2 consumers remain. Thus, the optimal price is ps2 = q2 . Now at t = 1, ps1
and pl1 have to satisfy
ps1 ∈ [q2 , (1 − β)q1 + βq2 ] ≡ A
pl1 ∈ [max{∆4 , q2 + ∆2 , ∆1 + ps1 }, min{q3 + ∆3 , ∆3 + ps1 }] ≡ B
It is easy to see that A is non-empty. For B, we first notice that q3 + ∆3 < ∆3 + ps1 ∀ps1 ∈
[q2 , (1 − β)q1 + βq2 ]. Thus in order for B to be non-empty for some ps1 , we need q3 + ∆3 >
max{∆4 , q2 + ∆2 , q2 + ∆1 }. Now the optimal price of the limited edition can be set to
pl1 = q3 + ∆3 , but the optimal price of the standard edition depends on the parameters:
(
(1 − β)q1 + βq2 if (1 − β)q1 + βq2 + ∆1 ≤ q3 + ∆3 ,
ps1 =
q3 + ∆3 − ∆1
if (1 − β)q1 + βq2 + ∆1 > q3 + ∆3 .
In summary, we have
pl1 = βq3 + ∆3
(
(1 − β)q1 + βq2
ps1 =
q3 + ∆3 − ∆1
if (1 − β)q1 + βq2 + ∆1 ≤ q3 + ∆3 ,
if (1 − β)q1 + βq2 + ∆1 > q3 + ∆3 .
ps2 = q2
Now we examine (1) frontloadedness of sales, (2) initial price, and (3) price cut for
1
, which is greater than its
the standard edition. First, the frontloadedness of sales is m1m+m
2
m1
baseline size m1 +m2 +m3 . Next, we consider the initial price. For (1−β)q1 +βq2 +∆1 ≤ q3 +∆3 ,
we have
pbs1 − ps1 = β 2 β(q3 − q2 ) < 0.
Thus, the initial price in this scenario is higher than that in the baseline case. For (1 −
β)q1 + βq2 + ∆1 > q3 + ∆3 , we have
pbs1 − ps1 = (1 − β)q1 + βq2 + ∆1 − (q3 + ∆3 ) − β 2 (q2 − q3 ),
which can be positive or negative. Finally, the price cut for (1 − β)q1 + βq2 + ∆1 > q3 + ∆3
is
(pbs1 − pbs2 ) − (ps1 − ps2 ) = (1 − β)β(q2 − q3 ) > 0,
meaning that the price cut is smaller. When (1 − β)q1 + βq2 + ∆1 > q3 + ∆3 , the price cut is
(pbs1 − pbs2 ) − (ps1 − ps2 ) = (1 − β)q1 + βq2 + ∆1 − (q3 + ∆3 ) + (1 − β)β(q2 − q3 ) > 0
because of the condition (1 − β)q1 + βq2 + ∆1 > q3 + ∆3 . Thus the price cut is smaller
regardless of the condition.
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A.1.5
When Segment-4 Consumers Buy the Limited Edition
Consider a situation where segment-4 consumers buy the limited edition at t = 1, segment-1
consumers buy the standard edition at t = 1, segment-2 consumers buy the standard edition
at t = 2, and segment-3 consumers buy at t = 3. In this case, the following incentive
compatibility and individual rationality constraints need to be satisfied:
q1 − ps1 ≥ max{q1 + ∆1 − pl1 , β(q1 − ps2 ), β 2 (q1 − ps3 ), 0}
β(q2 − ps2 ) ≥ max{q2 + ∆2 − pl1 , q2 − ps1 , β 2 (q2 − ps3 ), 0}
β 2 (q3 − ps3 ) ≥ max{q3 + ∆3 − pl1 , q3 − ps1 , β(q3 − ps2 ), 0}
∆4 − pl1 ≥ max{−ps1 , −βps2 , −β 2 ps3 , 0}
Based on these constrains, we solve the optimal pricing by backward induction. First, notice
that at t = 2, segment-1 and segment-4 consumers have already bought, and only segment-2
and segment-3 consumers remain. Thus, the optimal prices at t = 2 and t = 3 are identical
to those of the baseline case: p2s = (1 − β)q2 + βq3 and p3s = q3 . Now at t = 1, ps1 and pl1
have to satisfy
ps1 ∈ [(1 − β 2 )q2 + β 2 q3 , (1 − β)q1 + (1 − β)βq2 + β 2 q3 ] ≡ A
pl1 ∈ [max{q3 + ∆3 , (1 − β 2 )q2 + β 2 q3 + ∆2 , ∆1 + ps1 }, ∆4 ] ≡ B
It is easy to see that A is non-empty. In order for B to be non-empty for some ps1 , we need
∆4 > max{q3 + ∆3 , (1 − β 2 )q2 + β 2 q3 + ∆2 , (1 − β 2 )q2 + β 2 q3 + ∆1 }. Now the optimal price
of the limited edition can be set to pl1 = ∆4 , but the optimal price of the standard edition
depends on the parameters:
(
(1 − β)q1 + (1 − β)βq2 + β 2 q3 if (1 − β)q1 + (1 − β)βq2 + β 2 q3 + ∆1 ≤ ∆4 ,
ps1 =
∆4 − ∆1
if (1 − β)q1 + (1 − β)βq2 + β 2 q3 + ∆1 > ∆4 .
In summary, we have
pl1 = ∆4
(
(1 − β)q1 + (1 − β)βq2 + β 2 q3
ps1 =
∆4 − ∆1
if (1 − β)q1 + (1 − β)βq2 + β 2 q3 + ∆1 ≤ ∆4 ,
if (1 − β)q1 + (1 − β)βq2 + β 2 q3 + ∆1 > ∆4 .
ps2 = (1 − β)q2 + βq3
ps3 = q3
It is immediate that when (1 − β)q1 + (1 − β)βq2 + β 2 q3 + ∆1 ≤ ∆4 , the optimal pricing for
the standard edition is identical to that in the baseline case.
Now we examine (1) frontloadedness of sales, (2) initial price, and (3) price cut for the
m1
, which is identical to its
standard edition. First, the frontloadedness of sales is m1 +m
2 +m3
m1
baseline size m1 +m2 +m3 . Next we consider the initial price and price cut. We know that when
(1 − β)q1 + (1 − β)βq2 + β 2 q3 + ∆1 ≤ ∆4 , the optimal pricing for the standard edition in
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this scenario is identical to that in the baseline case. Thus, there is no change in the initial
price and price cut. When (1 − β)q1 + (1 − β)βq2 + β 2 q3 + ∆1 > ∆4 , the initial price in this
scenario is lower than that in the baseline case. Since ps2 = pbs2 , the price cut is smaller.
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