A Comparison of Bundling and Sequential Pricing in Competitive Markets:
Experiential Evidence
John Aloysius
University of Arkansas
Cary Deck
University of Arkansas
Amy Farmer
University of Arkansas
Department of Information Systems
1 University of Arkansas
Fayetteville, AR 72701
[email protected]
Department of Economics
1 University of Arkansas
Fayetteville, AR 72701
[email protected]
Department of Economics
1 University of Arkansas
Fayetteville, AR 72701
[email protected]
Technological advances enable sellers to identify relationships among offered goods. Sellers can leverage
this information through pricing strategies such as bundling and sequential pricing. While these strategies
have primarily been studied under monopoly assumptions, the strategies are available to competitive
firms as well. This paper reports on a series of laboratory experiments comparing bundling and
sequential pricing while varying underlying relationship between the goods in markets where a fraction of
buyers comparison shop. The results indicate that sequential pricing is generally as profitable to the
seller; however, there is evidence that sequential pricing may be more harmful to consumers than
bundling when the goods have complementary values or the buyer’s values are positively correlated.
JEL codes: C9, D4, L1
Keywords: Pricing Strategy, Bundling, Sequential Pricing, Experimental Methods
* The authors with to thank Scott Burton, Yongmin Chen, Robert Letzler, Donald Lichtenstein, John List, Bart Wilson and participants at the FTC Conference on Microeconomics, the Southern Economic Association Annual Meetings, and a seminar at Chapman University for helpful comments and suggestions on an earlier draft of this paper. Financial support from the Center for Retailing Excellence and the Information Technology Research Institute, both of the Sam M. Walton College of Business, is gratefully acknowledged. A Comparison of Bundling and Sequential Pricing in Competitive Markets:
Experiential Evidence
Technological advances enable sellers to identify relationships among offered goods. Sellers can leverage
this information through pricing strategies such as bundling and sequential pricing. While these strategies
have primarily been studied under monopoly assumptions, the strategies are available to competitive
firms as well. This paper reports on a series of laboratory experiments comparing bundling and
sequential pricing while varying underlying relationship between the goods in markets where a fraction of
buyers comparison shop. The results indicate that sequential pricing is generally as profitable to the
seller; however, there is evidence that sequential pricing may be more harmful to consumers than
bundling when the goods have complementary values or the buyer’s values are positively correlated.
JEL codes: C9, D4, L1
Keywords: Pricing Strategy, Bundling, Sequential Pricing, Experimental Methods
Introduction
Each basket of goods purchased at many major retailers is recorded.1 This vast volume
of data enables retailers to indentify the underlying distribution of buyer values and the
relationships between those values through data mining techniques.
Some goods are
complements (customer values for the bundle are greater than the sum of the values for the
individual products) and in other cases the goods are substitutes (customer values for the bundle
are less than the sum of the valuations for the individual products). For example a computer and
a printer are complements since customers have added utility for a printer when they have
computer and vice versa.
Separately, individual buyers may have positively or negatively
correlated values for two goods. For example, someone interested in a given topic is likely to
have a high value for books on that topic, while an uninterested person is likely to have low
values for the books. Additivity between values for two goods and the correlation of values for
two goods are distinct concepts. Again referring to the book example, two books covering the
1
Sellers can also connect different baskets to the same customer through “rewards” or “preferred customer”
programs or through credit card information. Online retailers have additional methods including the use of cookies
to track customers. Cookies also enable sellers to identify comparison shoppers (see Deck and Wilson 2006 for a
discussion of the pricing implications for this type of tracking).
1 same content may be substitutes whereas two books presenting different perspectives on the
topic may be complements.
With an understanding of buyers’ values, multi-good sellers choose a portfolio of prices
in order to maximize profit. For example, a seller can engage in pure components pricing by
setting a single price for each good. Alternatively, a seller could engage in mixed bundling by
offering standalone prices for the individual goods as well as a price for a bundle of goods.2
Bundling has a long history in the economics literature, going back to Stigler (1968). Adams and
Yellen (1976) develop the now common modeling framework and demonstrate how mixed
bundling could increase monopoly profits even with independent goods.
In their
interdisciplinary survey of bundling, Stremersch and Tellis (2002) make two pertinent
observations. First, the literature has focused primarily on profit maximization by a monopolist
(p. 60).3 Second, one of the most promising areas of further research appears to be the impact of
competition on the impact of bundling (p. 71). They also highlight the distribution of customer
conditional reservation prices as the likely predominant condition for optimality of bundling.
While, academic analysis has primarily focused on monopoly sellers, competitive firms
commonly use price bundling as a strategic marketing tool. Be it consumer durables (e.g.,
computers and printers), consumer packaged goods (e.g., a shrink wrapped package of shampoo
and conditioner), services (e.g., cable TV and internet), firms use bundling as a means to
improve measures of business performance including profits and market share.
As an alternative strategy to bundling, Aloysius, Deck, and Farmer (2009) introduce the
notion of sequential pricing with discrimination, where sellers observe the order in which buyers
are quoted prices and condition subsequent prices on previous purchase decisions.
New
technologies such as item level Radio Frequency Identification (RFID) tags allow a seller to
track which items a customer intends to purchase (and thus which ones they do not) and adjust
prices accordingly as a shopper moves around a store.4 Online retailers can use electronic
2
The term pure bundling refers to the case where the seller only offers the goods in a bundle. McAfee, McMillan, and Whinston (1989) extend their monopoly results to a duopoly. Chen (1997) analyzes a
situation in which firms compete in a duopoly for a single product and the firms also produce other products under
conditions of perfect competition. Bakos and Brynjolfsson (2000) show that bundling low marginal cost goods in
competitive environments can produce economies of aggregation, even without economies of scale or network
externalities.
4
RFID technology is currently being used to improve the efficiency of the supply chain by tracking palates as they
move through the distribution process. Retailers including Sam’s Club are pursuing individual item RFID tagging
3
2 “shopping carts” in a similar manner. The same technology that enables Amazon to make a
recommendation based upon search history also enables it to charge a premium (or offer a
smaller discount) to buyers who are more likely to have a high value for the item. Aloysius, et al
(2009) report that sequential pricing generates greater monopoly profit than mixed bundling
when the goods are either close substitutes or have highly positively correlated values.
The theoretical complexity of monopoly bundling or sequential pricing, as discussed in
the next section, is exacerbated under competition where some customers comparison shop. This
intractability does not preclude retailers from engaging in these practices nor should it
discourage researchers from studying the practices’ implications. As described by Smith (1994),
laboratory experiments provide a means for establishing empirical regularities and comparing
institutions and environments.
That is, through experiments we can identify how market
outcomes change as a function of changes to the value structure and the pricing strategy of the
sellers. Further, this type of research does not rely upon the auxiliary assumption of fully
rational agents, the appropriateness of which has been called into question by recent research
(e.g. Ellison 2006).
Our paper uses controlled laboratory experiments to explore the implication of bundling
and sequential pricing in competitive markets. The remainder of the paper is organized as
follows. The next section reviews the relevant literature providing mathematical detail where
useful for our experimental study. The third section extends the theoretical development to the
case where some customers comparison shop a la Varian (1980). The fourth section describes
the experimental design and the fifth section presents the findings. As a prelude to the results,
we find that sequential pricing is generally as profitable to sellers as bundling, but sequential
pricing can be more harmful than bundling in some environments. In all cases, the symmetric
Nash equilibrium bidding strategy does not accurately describe aggregate behavior. A final
section contains concluding remarks.
Bundling and Sequential Pricing
(Weier 2008). The technology is touted as a means of avoiding stock outs and reducing theft. This technology, in
conjunction with smart shopping carts, will also enable bricks and mortar stores to offer the same types of
recommendations and detailed product information that is available online. United States Patent 5729697 is for just
such a smart shopping cart. This technology could move prices from the physical shelf to the shopper’s private
monitor.
3 The core theoretical work on product bundling is due to Stigler (1968), Adams and
Yellen (1976), Schmalensee (1984) and McAfee, McMillan and Whinston (1989).5 The general
result is that customers with a high degree of asymmetry in product valuations will buy an
individual product that they favor, while customers with less extreme product valuations will buy
the bundle. This allows a seller to raise single item prices without losing all of the displaced
customers (as some buy the bundle). The literature on bundling is beginning to clarify optimal
pricing strategies, issues related to effective deployment of those strategies, and the environment
in which these strategies may be deployed (Guiltinan 1987, Simon and Dolan 1998, Estelami
1999, Noble and Gruca 1999). From a marketing perspective, Mulhern and Leone (1991)
analyze retail pricing and promotion strategies while Venkatesh and Mahajan (1997) examine
strategies for marketing products using their branded components. From an information systems
perspective, Bakos and Brynjolfsson (1999) and Hitt and Chen (2005) study strategies for pricing
a large number of information goods. Nettesine, Savin, and Xiao (2006) present a stochastic
dynamic program that analyzes the problem of dynamically selecting complementary products,
as well as the problem of pricing products to maximize profits.
Venkatesh and Kamakura (2003) present an analytical model of contingent valuations
and find that the degree of complementarity or substitutability determines whether products
should be sold as pure components, pure bundles, or mixed bundles. They also find that
typically, complements and substitutes should be priced higher than independently valued
products. In their model, a customer has a value for good A,
, and for good B,
. The customer’s value for buying both goods depends upon the degree of substitutability
or complementarity, θ, so that
.
With mixed bundling, the customer
simultaneously observes the prices of PA , PB , and PAB . The customer will purchase the bundle
if vAB - PAB > max (0, vA - PA , vB - PB ) with similar conditions holding for purchasing items A
5
There are three main arguments as to why firms would bundle, relying on exploitation of market power. One
reason is as a form of price discrimination. Another is as an entry-deterrent strategy (see Nalebuff 2004). A third is
that a firm leverages market power in one market to capture a greater share of a more competitive market (see
Caliskan et al. 2007 for a discussion of this literature as well as an experimental test of such models). In this paper
we are focusing exclusively on the first reason.
4 and B separately. A buyer who cannot obtain a non-negative surplus from making a purchase
will not buy anything and earn 0.
The monopolist produces each item at constant marginal costs of cA and cB respectively.
Given the framework above, now consider a seller’s price setting problem.
We introduce the
i
notation RAi , RBi , and R AB
to denote the region of possible buyer values from which a buyer
i
. Of course, these
would choose to purchase A, B, and the bundle AB given PAi , PBi , and PAB
regions depend upon the joint distribution of preferences. Therefore monopoly profit is given by
Πm = ( PA ‐ cA)
m
∫∫ f (v
A
) f (vB ) + ( PBm ‐ cB) RAm
∫∫ f (v
A
m
) f (vB ) + ( PAB
‐ cA‐ cB) RBm
∫∫ f (v
A
) f (vB ) . m
RAB
The optimization problem is conceptually straight forward. Differentiating the profit function
with respect to PA , PB , and PAB .yields three first order conditions which must be satisfied
simultaneously. While a general solution is unattainable, Venkatesh and Kamakura (2003) use a
uniform distribution and numerically solve for the solution. In contrast to bundling where prices are set simultaneously, Aloysious, et al. (2009)
introduce sequential pricing where customers observe one price at a time. Without loss of
generality, one can assume that the good A purchase decision is made first. They consider two
cases: one in which the seller sets a single price for the second good and one in which the seller
can condition the price of the second good on the purchase of the first good.
In the case where the seller prices good B without information regarding the purchase of
A, the consumer will buy A iff PA ≤ vA. A consumer who chooses not to purchase A will buy B
iff
whereas if A was purchased, then the consumer will buy B iff
, which can be expressed as
. When the seller sets PB,
he or she knows the probability that A will have been purchased, conditional on the price of A
and the distribution of preferences. Thus, the seller sets the price of A knowing the probability A
will be bought and therefore, how that probability will affect the subsequent purchase of B.
Solving the problem in reverse, the seller sets the price of B conditional on the chosen price of A
and the corresponding probability that A was purchased. This yields the optimal price of B as a
function of PA. Given that response function, the seller maximizes expected total profit as a
function of the prices, where PB=f(PA).
5 Aloysious et al. (2009) work through the exercise in the case of uniform distributions.
and
Specifically, when
, the two period expected profit
maximization problem written as a function of PA is
=
where the terms are profit from customers who buy only A, only B and both goods, respectively. Aloysious, et al. (2009) present numerical solutions for the optimal prices as a function of θ. If instead the seller can condition the good B price on whether or not the consumer has
purchased A, then there will be two separate optimal response functions for setting the price of
good B depending on whether good A was purchased or not. Each case yields an optimal
expected profit from the sale of B, ant the seller will take these optimal responses and
corresponding profits into account when setting PA. Using the uniform distribution and the
implied probability of a purchase of subsequently purchasing good B depending upon PA,
Aloysious, et al. (2009) show that the profit maximization problem in terms of PA becomes
The novelty of sequential pricing as described by Aloysious, et al. (2009) is that the
information regarding the intent to purchase one item is used to price another item on the same
shopping experience without having to identify the customer. Other studies of dynamically
priced goods have been confined to single goods and do not consider cross category effects on
other goods. Cope (2006) presents dynamic strategies for maximizing revenue in internet retail
by actively learning customers’ demand responses to price. Zhang and Krishnamurthi (2004)
provide a decision support system of micro-level promotions in an internet shopping
environment, that provides recommendations as to when, how much, and to whom to give price
promotions. The system derives the optimal price promotion for each household, on each
shopping trip by taking into account the time-varying pattern of purchase behavior and the
impact of the promotion on future purchases.
6 There are several examples of sellers using customer behavior to infer preferences, and
using that information either to drive revenues or for customer relationship management.
Montgomery et al. (2004) show how clickstream data about the sequence of pages or path
navigated by web buyers can be used to infer users’ goals and future path. There is a literature
on behavior-based price discrimination (for a survey see Fudenberg and Villas-Boas 2006) in
which firms use information about consumers’ previous purchases to offer different prices and/or
products to consumers with different purchase histories. Empirical data shows that even the
information contained in observing one historic purchase occasion by a customer boosts net
target couponing revenue by 50% (Rossi et al. 1996). Better ability to predict preferences has
been shown to potentially reduce price competition (Chen et al. 2001). Acquisti and Varian
(2005) show analytically that it is optimal to price so as to distinguish between high-value and
low-value customers. There is empirical evidence that competing firms have been able to price
discriminate profitably by charging different prices across consumer segments (Basenko et al
2003).
Moon and Russell (2008) develop a product recommendation model based on the
principle that customer preference similarity stemming from prior purchase behavior is a key
element in predicting current purchase. These studies exploit customer revealed preference for a
good in order to set prices for future purchases of that good.
A Model of Competition
The previous work addresses bundling and sequential pricing for monopolies only, so in
this section we extend the analysis of Venkatesh and Kamakura (2003) and Aloysious et. al.
(2009) to examine these pricing strategies with competition. To introduce competition into
multiple product markets, we follow Varian’s (1980) model of informed and uninformed buyers.
We assume that a fraction 1- α of the customers are fully informed of all competitor prices.
These comparison shoppers can be thought of as those people that seek out price information,
say by looking at advertisements in print media or by using pricebots or other online price
comparison tools. The remaining customers visit a single seller. These loyal buyers may be
considered to have an idiosyncratic preference for a given seller, high search or travel costs, etc.
7 Each of the n firms acts as a monopolist to the α/n customers who are loyal to that seller, but the
seller is unable to determine which customers comparison shop and which do not.6
The problem faced by an uninformed potential customer with values vA, vB, vAB, who
i
observes simultaneously the prices of PAi , PBi , and PAB
from seller i, is straightforward. The
i
customer will purchase the bundle AB if vAB - PAB
> max (0, vA - PAi , vB - PBi ). Similar conditions
hold for purchasing items A and B separately. A buyer who can obtain a non-negative surplus
from making a purchase will not buy anything and earn 0.
The problem faced by an informed customer is more complex. The informed customer
i
will purchase the bundle AB from seller i if vAB - PAB
> max (0, vA - PAi , vB - PBi , vA - PAj , vB - PBj ,
vAB - PABj ) ∀j ≠ i, with similar conditions holding for purchasing item A only or item B only.
Any tie between sellers is assumed to be broken randomly. Therefore, an informed buyer makes
the best decision from the 3n+1 choices and cannot create a bundle by purchasing components
from different sellers.7,8
Let Πm denote the expected profits that a monopolist would earn from setting the
monopoly prices. Since a firm could simply act as a monopolist to its loyal, uninformed
customers, any competitor can guarantee itself a security profit of αΠm /n. We first note that
there are no pure strategy equilbria nor are there any mass points in the symmetric mixing
distribution because the probability of a tie would lead some seller to cut its price or revert to
monopoly pricing. Any strategy in the support of the mixed strategy equilibrium must generate
the same expected payoff which we denote by Π* where Π*≥ αΠm /n. If every strategy generates
Π* by definition of a mixed strategy, then the expected value of the entire game will also be Π*.
Thus, the joint distribution function g ( PA , PB , PAB ) that characterizes the equilibrium must satisfy:
∫∫∫ g ( P , P , P
A
B
AB
i
) ×E(Π| PAi , PBi , PAB
)=
∫∫∫ g ( P , P , P
A
B
AB
) ×Π* = Π*
(1)
6
See Deck and Wilson (2006) for theory and experiments regarding the ability to track customers in a single good
market where some customers comparison shop.
7
Buyer behavior must be individually rational so the buyer has the option to buy nothing.
8
Perhaps travel or time costs make visiting multiple stores prohibitive or slight differentiation makes products from
different sellers incompatible.
8 Venkatesh and Kamakura (2003) show that for general θ a closed form solution does not
exist in the relatively simple monopoly problem. We do note that if α = 0 then there is no
i
m
= PAB
∀i.
competition and the equilibrium is for PAi = PAm , PBi = PBm and PAB
Of course, the
monopoly prices are a function of f(vA)f(vB) and θ. On the other hand, if α = 1 and all customers
i
= cA+ cB ∀i.
comparison shop, the equilibrium is PAi = cA, PBi = cB and PAB
Without directly solving for g ( PA , PB , PAB ) we can identify price strategies that are not in
the support of the symmetric mixed strategy equilibrium. Since a seller could focus exclusively
on its uninformed customers, if g ( PA , PB , PAB ) > 0 then it must be that the strategy ( PA , PB , PAB )
generates a profit of at least (αΠm/n) / (1 - α+α/n) from loyal shoppers where the denominator is
the probability a firm is visited either by an informed or an uninformed shopper.
Any time a
seller chooses a price that generates less profit, it is direct evidence that the seller is not playing
the symmetric mixed strategy Nash equilibrium. As we will discuss in the experimental results,
there is compelling evidence that subjects in fact do not play the symmetric Nash equilibrium
strategy.
The competitive markets with sequential pricing are similar.
With or without
discrimination, loyal customers visit one seller, make a good A purchase decision, and then
observe a possibly conditional price for good B. By contrast, comparison shoppers first observe
the price of good A from each seller and then visit the seller offering the largest non-negative
buyer surplus from good A (i.e. comparing vA- PA across sellers and visiting the seller offering
the best deal if it beats buying nothing and receiving a surplus of 0), with ties between sellers
again broken randomly. Comparison shoppers who purchase good A then observe the possibly
conditional good B price of that specific seller and make a purchase if it increases their buyer
surplus (i.e. (1+θ)(vA + vB) - PA - PB > vA - PA for the nondiscriminatory case and (1+θ)(vA + vB) PA - PB|A > vA - PA with discrimination). Informed customers who do not purchase good A never
visit a seller and thus never have the opportunity to purchase good B. One could assign the
informed buyer that does not purchase A to a randomly selected store, but this approach is
unappealing without a compelling reason why the buyer would observe the store’s website or go
visit the physical location in such situations. We believe that this potential loss of customers is
an important implication of using a sequential pricing strategy that was not apparent from
previous theoretical research on monopolies, but is nonetheless a real issue for competitive firms;
9 i.e. when pricing good A, the possibility of lost customers in market B factors into the pricing
decision.
Again, there is no pure strategy equilibrium nor are there mass points in the symmetric
mixed strategy equilibrium. As before, a seller could focus on its loyal customers and earn a
profit of least αΠm/n, where Πm now denotes monopoly profit under the appropriate form of
sequential pricing. The equilibrium is again characterized by an equation similar to (1), but
where g (•) is only over PA as the good B prices are determined by PA. As before, any price
strategy that does not generate an expected profit of least (αΠm/n) / (1-α+α/n) cannot be supported
in equilibrium. Further, to be consistent with equilibrium behavior it must be that the sellers are
playing the optimal good B price response(s) to the chosen PA. In the event that the firm can
condition the price of good B on the purchase of good A, the optimal prices are the same here as
under monopoly. When sellers cannot set conditional prices, the optimal good B price will differ
from the monopoly case; this situation involves a change in the probability that a customer
observing PB will have purchased good A due to the existence of comparison shoppers.
Ultimately, there is once again compelling evidence that subjects do not play the symmetric
mixed strategy Nash equilibrium.
Experimental Design
To explore the impact of bundling and sequential pricing in competitive markets, a total
of 36 laboratory sessions were conducted. The sessions consist of 4 replicates of each of the 9
treatments in a 3×3 between subjects design. The first factor of the design is the relationship
between two experimental goods, A and B. In the “independent values” condition, the buyer’s
values for the two goods are independent and the value of the bundle made from purchasing both
goods is the sum of the values of the separate items (i.e. θ = 0). Specifically, vA ~ U[0,100] and
vB ~ U[0,100] in the independent values case. For the “complements” condition, the single item
values are the same as in the previous case but the value from buying both items is 1.3(vA + vB),
that is θ = 0.3. The third condition involves positively “correlated values,” where vA and vB are
jointly distributed according to the density function f(vA,vB)=
10 for vA, vB
integers ∈ [0,100]. For this distribution the correlation is ρ = 0.5.9 Bundle values are additive in
the correlated values condition (i.e. θ = 0).
This distribution is admittedly arbitrary, as is the
uniform distribution; however, this distribution has the advantage of being easy to describe to
subject sellers.10
The second factor in the experimental design was the pricing strategy available to the
sellers. In the “Bundling” condition seller i could post PA, PB, and PAB for good A alone, good B
alone, and the bundle consisting of goods A and B, respectively. Notice that sellers could
engage in pure bundle pricing by setting PAB < PA, PB and could engage in pure components
pricing by setting PAB = PA + PB. In the “Sequential Pricing with Discrimination” condition
sellers could post PA for good A and PB|A and PB|¬A for good B depending on whether or not the
buyer purchased good A. In the “Sequential Pricing without Discrimination” condition sellers
could post PA and PB for goods A and B, respectively. Note that sequential pricing without
discrimination is not the same as pure components pricing because while the seller does not
observe behavior, the buyer will not observe the price of B at the time the decision is being made
to purchase good A. This timing difference for buyers generates different pricing strategies for
sellers.
Each session involved n = 4 seller subjects posting prices for 750 paid market periods.11
In each period a new buyer would visit the market having drawn a realized value pair (vA, vB)
from the appropriate distribution and make a purchase decision.
Since buyers make one
purchase decision, there is no incentive for them to not truthfully reveal their willingness to buy.
Therefore, the buyer role was automated, a common practice in posted offer market experiments
where demand withholding is not a critical element of the design (see Davis and Holt 1993).
With this buyer automation, each period lasted only 3 seconds. The use of nearly continuous
posted offer markets provides considerable seller experience and “improve[s] the drawing power
9
The correlations in the Baseline and Complements conditions are both ρ = 0. The distributions used in the laboratory are consistent with those used in the numerical monopoly analysis of
Aloysius, et al (2009). Chen and Rirodan (2008) use a similar construction for analyzing monopoly and oligopoly
behavior over a uniform distribution over two dimensions with correlation. In their problem each firm sells a single
good.
11
Subjects did not know the total number of periods in the experiment nor were the practice periods singled out for
the subjects. Subjects did know the total length of the session, but typically the experiments finished well before the
allotted time had expired. 10
11 of underlying equilibrium predictions.” Davis and Korenok (2009, p. 465).12 Our choice of n is
somewhat arbitrary; however, previous experimental work has suggested that 4 firms are
sufficient to avoid collusion (see Holt 1995, p. 407). Following Varian (1980) the markets consist of two types of buyers.
Loyal (or
uninformed) customers account for α = 80% of the market so each seller served as a monopolist
to α/n = 20% of the market. Comparison (or informed) shoppers account for 1 - α = 20% of the
market. A buyer’s type is independent of its (vA, vB) realization. Therefore, conditional on a
particular seller being visited, there is a 50% chance that the seller is in direct competition for a
comparison shopper. Thus, as described in the previous section, no subject seller should select a
set of prices that generate less than Πm/2. With the chosen parameters, it is straightforward
(although tedious) to calculate optimal monopoly prices and profits for each treatment and thus
the security profits. These security profits drawn from Aloysius, et al. (2009) are presented in
Table 1. While the profit differences in Table 1 are not nominally large, these profits are per
period expected profits, and our sellers are competing for 750 periods.
Subject sellers could adjust their prices at any point during the experiment and received
feedback about the prices charged and profit earned by each rival after every period. During the
experiment, subjects had an onscreen tool that would identify which potential uninformed
customers would buy each good and the expected profit based upon a subject specified pricing
strategy.13 This tool is shown in Figure 1 for the Sequential Pricing with Discrimination and
Positively Correlated Values treatment. Value combinations that would lead to purchases of
good A only were shaded in yellow while value combinations that would lead to the purchase of
good B only were in blue. Value combinations such that the person would buy both A and B
were shaded green. Combinations for which the person would not buy anything are white and
areas shaded black could not occur given the distribution of values. A subject could click on an
area in the diagram on the right and the diagram on the left would focus on the specified area
revealing the actual buyer values in that region. For simplicity, the marginal cost of producing
both types of goods was cA = cB = 0, and therefore profit and revenue are identical.
12
See Deck and Wilson (2006) and Deck and Wilson (2008) for other studies using nearly continuous posted offer
experiments.
13
A similar tool for the comparison shoppers would require that subjects specify price vectors for each of their
rivals, a task that was viewed as being overly cumbersome. It is possible that this design tool led subjects to be
overly focused on loyal customers; however, the results suggest that this is not the case.
12 Subjects were recruited from undergraduate courses at a state university. While many of
the participants are in the business school, students from other disciplines participated as well.
The students were recruited directly from classes and through the laboratory’s subject database.
Some of the subjects had prior experience in experiments; however, none had previously
participated in any related experiments. Each laboratory session lasted 90 minutes, including
approximately 30 minutes for the self paced written directions and the completion of a
comprehension handout.14 After completing the handout, responses were checked by an
experimenter and any remaining questions were answered. Once all of the participants were
ready, the actual experiment began.
At the conclusion of the experiment, subjects were paid based upon their earned profit at
the rate $400 in profit = US $1. The average salient payment was approximately $18.00.
Participants also received a fixed payment of $7.50 for arriving on time and participating in the
study. Subjects were paid in private and were dismissed from the experiment once they had
collected their money. Multiple sessions occurred concurrently in the laboratory to prevent
subjects from being able to identify which other participants were sellers in the same market.
Experimental Results
The analyzed data consist of 72,000 market pricing decisions.15 Figure 2 shows the
average realized seller profit by session for each of the nine treatments. Figure 3 shows the
distribution of price choices for each treatment. Of course, observations from the same subject
or even from the same session are not independent. Therefore, linear mixed effects models are
utilized to control for the repeated measures present in the data at the period level. Nonparametric permutations tests are utilized for comparisons at the session level since each session
is independent.
<<Insert Figure 3 Here>>
14
Copies of the directions and the handout are available from the authors upon request.
500 periods per subject × 144 subjects = 72,000 market decision periods. The last 500 periods of each session are
used in the statistical analysis to control for learning effects.
15
13 We first note that all of our competitive markets generate greater consumer surplus and
lower profits than under monopoly. It is not surprising that by introducing competition into these
environments surplus is redistributed from sellers to buyers. However, in this setting there are
only 4 sellers and 80% of customers are loyal, leaving only 20% of the population of consumer
over which to compete. Even with such a small degree of competition, these pricing strategies
do not seem to protect sellers from competitive losses. We see this very clearly in the sequential
pricing game where sellers dramatically lower the price of good A in order to be able to compete
in the good B market. Below we analyze each pricing strategy, but it is important to note that
one of the main contributions here is to adapt these strategies to a competitive environment. In
so doing, it is clear that these seller strategies do not seem to harm consumers by protecting
sellers from the effects of even a small level of competition.
The next question is how bundling and sequential pricing compare. The results are
discussed in separate subsections for each value relationship (independently valued goods,
positively correlated goods, and complementary goods). Within each subsection we report a
series of results. First we compare the seller profit, buyer surplus and overall efficiency across
pricing strategies.16 Next we summarize the pricing decisions for each pricing strategy. Given
the structural differences between sequential pricing and bundling, we evaluate bundling
separately, but we do compare the other two strategies directly to evaluate the effect of
discrimination. We do not make comparisons across environments because the total surplus
available differs and this relationship is not a choice variable for sellers. The final subsection
discusses the consistency of behavior with the symmetric Nash equilibrium mixed strategy.
The Case of Independently Valued Goods
Based upon the permutations test, seller profit is significantly greater with bundling than
with sequential pricing without discrimination (p-values = 0.043), but does not differ between
16
Efficiency is measured as the percentage of the total possible gains from trade that were actually realized. In these
markets, a trade is fully efficient is the buyer purchases both goods A and B since the values are non-negative and
the costs of production are 0.
14 bundling and sequential pricing with discrimination (p-value = 0.429) or between sequential
pricing with and without discrimination (p-value = 0.514) despite the fact that security profits are
10% greater under bundling than either sequential pricing strategy. The results indicate that
there is no significant difference between pricing strategies with respect to buyer surplus or
efficiency. Table 2 provides the results of permutation tests for comparing the welfare measures
across pricing strategies.
Sellers in the bundling treatment could engage in mixed bundling, pure components, or
pure bundling. Overwhelmingly, they engaged in mixed bundling (83% of observations) and
rarely engaged in pure component pricing (3% of observations). Given the symmetry of the
market, it is reasonable to expect that sellers in the bundling treatment would set PA = PB, and in
fact this is what was observed.17 Therefore, we combine the single item prices in the left portion
of Figure 3 and in reporting that the average single item price was 39.9 while the average price
for a bundle was 55.3. The 31% = 1 – 55.3/(2×39.9) bundle discount is similar to the 35%
offered under monopoly.
These prices represent the typical price that a loyal uninformed
customer would observe. For comparison shoppers, the average minimum single item price fell
to 27.6 and the lowest bundle price fell to 37.0 (a bundling discount of 33%).
For both sequential pricing conditions, good A prices tend to be much lower than good B
prices, as sellers are competing for customers via good A prices (see Figure 3). There is also
evidence of subjects using a leader pricing strategy, as a substantial proportion of sellers
(regardless of whether or not they could price discriminate) priced at or close to marginal cost.18
In this environment of independently valued goods, the ability to price discriminate should not
be exercised when sellers are engaging in sequential pricing as the purchase of good A reveals no
information to the seller about the buyer’s marginal value of subsequently purchasing good B. In
fact, given the structure of the buyer values, sequential sellers should charge 50 for good B (i.e.
PB|A = PB|¬A = 50 with discrimination and PB = 50 without discrimination). Because of this result,
there is no reason for the price of good A to differ based upon whether or not sellers could
subsequently discriminate.
17
The average difference in the single item prices was less than 1 × 10-10 in all three conditions and not statistically
different from 0 in any condition.
18
See Hess and Gerstner (1987) for a discussion of this strategy.
15 The average price of good A is only 18 when sellers cannot base the price of good B on
the good A purchase decision as reported in Table 3. This amount increases to 21.6 = 18 + 3.6
when sellers do have the ability to set conditional prices for good B, but as predicted this
difference is not significant (p-value = 0.41).
Comparison shoppers (panel 3) on average
observed a good A price of 6.1 without discrimination and 11.7 = 6.1+ 5.6 with discrimination, a
difference that is not significant (p-value = 0.22 in third panel of Table 3).
The average price for good B was 33.2 when sellers could not discriminate (second panel
of Table 3). This is significantly, less than the hypothesized value of 50 (p-value <0.01).19 The
average price of 33.2 is statistically similar to the average good B price of 39.5 = 33.2 + 6.3
observed by buyers known to have bought good A (p-value = 0.27 in second panel of Table 2)
and the average good B price of 39.3 = 33.2 + 6.1 observed by those known not to have
purchased good A (p-value = 0.28 in second panel of Table 2). Comparison shoppers are quoted
an average good B price of 32 in the absence of discrimination and 37.6 = 32 + 5.6 with
discrimination, which is not a significant increase (p-value = 0.32 in fourth panel of Table 2).
The comparison of the second and fourth panels of Table 3 suggests that the low price seller is
not pricing good B differently from its rivals. In general, it seems that sellers are competing in
the market for A, and charge higher prices for B, although they fall short of the monopoly price.
As predicted, however, pricing behavior does not change when sellers can or cannot
discriminate. Across pricing strategies, there is no difference in terms of buyer surplus or overall
efficiency, but bundling produces weakly higher profits for sellers.
The Case of Positively Correlated Goods
In this environment, all three pricing strategies are observed to generate the same profit
and buyer surplus (see first two columns of Table 4). For comparison, all three practices
generate similar security profits as well (see Table 1). However, unlike what was observed with
independent values, in this environment sequential pricing without discrimination reduces
overall efficiency (p-value = 0.086).
19
With independent goods, distribution of the marginal values of good B is the same regardless of whether or not
the seller attracts comparison shoppers. For this special case PB is the same under monopoly and competition.
16 The distribution of prices by pricing strategy is shown in the middle panel of Figure 3 for
this environment. Again, sellers overwhelming choose to engage in mixed bundling (88% of
observations) and rarely engaged in pure component pricing (4% of observations). As in the
independent values environment, sellers set PA = PB. The average single item price was 31.1 and
the average price for a bundle was 45.9. This bundle discount 26% = 1 – 45.9/(2×31.1) is much
larger than the 15% offered under monopoly. For comparison shoppers, the average minimum
single item price was 18.0 and the lowest bundle price was 28.4 (a bundling discount of 21%).
For the sequential pricing treatment, sellers are again engaged in fierce competition
setting average prices for good A of 19.7 with and 19.4 without discrimination, a difference that
is not significant according to the estimation reported in Table 5 (p-value = 0.96). Without the
ability to price discriminate, sellers should charge at least 50 for good B, but they do not (p-value
< 0.01).20 With discrimination, sellers should charge a lower good B price to those who have not
purchased good A than to those who have. There is some evidence for this behavior as the price
for those known to have not purchased good A received a statically lower price for good B than
those who could not be observed (p-value = 0.08 in the one sided test).21 Comparison shoppers
observed an average good A price of 11.1 with and 8.6 without discrimination, an insignificant
difference (p-value = 0.61). For good B, comparison shoppers observed statistically similar
average prices of 38.4 with and 33.3 without discrimination (p-value = 0.38). The similarity of
these prices with the good B prices observed by loyal customers again suggests that the low good
A price sellers were not behaving differently from their rivals. Generally it seems that firms
once again compete strongly for good A, but then lower priced sellers do not use that to
capitalize further in the good B market. However, there is some evidence that sellers do raise
prices when they can observe a purchase of good A, of course, given the low price of good A
most purchase good A.
Overall, both buyers and sellers are equally as well off across
environments, but efficiency is reduced when discrimination is not possible.
20
Given that comparison shoppers only observe good B prices if they purchase good A, the distribution of good B
values that a seller faces depends on the posted good A price. The reported p-value is for testing PB = 50. 21
The p-values reported in Table 5 are for the two sided test on inequality, but here we have a one sided alternative
hypothesis. While the buyers who were identified as having purchased good A did not receive a statistically
different price from buyers in the no discrimination treatment, their prices were nominally lower making them
similar to those who were identified as not having purchased good A.
17 The Case of Complementary Goods
As before, there is evidence of a negative social outcome from engaging in sequential
pricing, in this case with discrimination, as compared to bundling (p-value = 0.071).22 The first
column of Table 6 indicates that there is no difference in profits across the three pricing
strategies despite the fact that bundling generates a greater security profit (see Table 1). At least
nominally, sequential pricing with discrimination is harming consumers, although the effect is
not significant in the two sided test reported in Table 6.
The price distributions for this environment are shown in the bottom panel of Figure 3.
As in the other treatments, sellers predominantly engage in mixed bundling (84% of
observations) and rarely engaged in pure component pricing (4% of observations). As in the
other environment, sellers set PA = PB. The average single item price was 38.8 and the average
price for a bundle was 56.3. This bundle discount of 28% is not too dissimilar to the 35%
offered under monopoly. For comparison shoppers, the average minimum single item price was
25.5 and the lowest bundle price was 35.4 (a bundling discount of 31%).
The ability to price discriminate did not significantly change the mean or minimum good
A prices for the sequential treatments as reported in Table 7 (p-values = 0.98 and 0.27
respectively). However, buyers who purchased good A paid significantly more for good B than
those who did not buy good A when sellers could price discriminate (p-value <0.01). This result
is intuitive because purchasing good A increases the marginal value of good B. Interestingly, the
mean price of good B when sellers cannot conditionally price is similar to the distribution of
prices for customers who could be identified as having purchased good A (p-value = 0.89). That
is, rather than the practice of price discrimination leading to higher prices for the high valued
buyers, it actually leads to a price discount for the low value segment. Again this pattern is
reasonable given that the good A prices are set so low that most buyer purchase good A. A
similar pattern in observed for comparison shoppers in the bottom two panels of Figure 7.
22
The comparison between bundling and sequential pricing without discrimination would be significant in the onesided test that sequential pricing is harmful.
18 Generally, with complementary goods, buyers are harmed by sequential pricing with
discrimination while seller profits are similar across pricing mechanisms. Sellers do seem to be
discriminating by charging higher prices to consumers known to have purchased good A.
Consistency of Behavior with Symmetric Nash Equilibrium
Although we do not have an explicit solution for the symmetric Nash equilibrium mixing
distribution for these markets, its support is well defined. As described previously, a firm should
never choose a pricing strategy that generates less profit than αΠm/(1‐nα+α) = .5Πm. Therefore
any observed behavior that violates this condition casts doubt on predictive power of that
equilibrium concept. Before proceeding, it is important to note that these restrictions have very
different implications for bundling and sequential pricing.
Under sequential pricing the
equilibrium condition requires that sellers choose the optimal price(s) for good B given PA. This
means the equilibrium support is a surface in the price space for sequential pricing.23 However,
since prices are sequential, it is possible for a firm to earn more than the security profit even
when pricing good A at cost. With bundling, the bundle price cannot be greater than the sum of
the individual prices nor can the bundle price be below the price of an individual item. This
means that supportable prices cannot be too low. However, with bundling the equilibrium
support is a solid in the three dimensional price space. Therefore, it is not appropriate to make
comparisons regarding consistency with the theoretical predictions across institutions
The observed behavior indicates that subjects are not pricing according to the predictions
of the theoretical model.
Subjects in the bundling treatment were not consistent with the
symmetric Nash equilibrium mixing distribution in 35% of the baseline independent value
observations, 53% of the observations in the positively correlated values treatments, and 42% of
the complementary goods treatment.
When sellers could sequentially price, but could not
discriminate 93% of observations were not consistent with the equilibrium predictions in the
baseline case while 83% and 63% were inconsistent in the complements and correlated goods
23
In the cases of positively correlated goods and complements, sequential pricing without discrimination under
competition results in a different PB than under monopoly. The optimal price depends upon the probability of the
seller setting the lowest PA and thus being visited by the high good A valued comparison shoppers. In both cases
the optimal PB is weakly higher than under monopoly since the distribution of VBs stochastically dominates the
distribution faced by a monopolist. Therefore, the monopoly PB serves as lower bound for PBs that could be in the
mixing distribution and thus in these two cases the observations that cannot be considered inconsistent with the
symmetric Nash equilibrium mixing distribution form an area in the two dimensional price space.
19 treatments, respectively. When sellers could price discriminate while sequentially pricing more
than 99% of the observations were inconsistent with the equilibrium strategy in each of the three
conditions. The reader should bear in mind that the reported percentages are a lower bound on
the number of inconsistent choices as there may be some strategies that generate half of security
profit and are still not part of the equilibrium support.
Conclusions
In this paper we extend the analysis of both bundling and sequential pricing to
competitive markets where a fraction of customers comparison shop. We characterize the
equilibrium, and although it is intractable to solve explicitly, we are able to identify lower
bounds on profits that should be generated, and where discrimination is possible we can identify
the optimal price of good B conditional on the price of A that was chosen. Through experiments
we can observe behaviors in a market where the explicit solution cannot be found.
The results of those experiments indicate first, that subjects are not playing according to
the predicted Nash equilibrium mixing distribution. Beyond that, we gain some insight into the
functioning of these alternative pricing mechanisms. Our results indicate that bundling produces
weakly higher profits when goods are independent, but otherwise, sellers obtain the same profits
regardless of the mechanisms. Further, sequential pricing adversely affects buyers when
discrimination is possible and generates an overall efficiency loss when it is not. In all three
cases we find that sequential pricing with or without discrimination leads sellers to drastically
discount the price of good A in an effort to not lose that customer in the good B market. By
adding competition to the analysis we are able to see that potential losses in a subsequent round
create significant price discounts in the initial round. Our research also demonstrates the degree
to which a small amount of competition can retard the harmful effects (for consumers) of
sophisticated pricing strategies.
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23 Table 1. Security Profits by Treatment
Pricing Strategy
Value Relationship
ρ
θ
Security Profit =Πm/2
Bundling
Independent Values
Positive Correlation
Complements
0
0.5
0
0
0
0.3
27.6
26.2
35.4
Independent Values
Positive Correlation
0
0.5
Complements
Independent Values
Positive Correlation
Complements
0
0
0.5
0
0
0
0.3
25.1
25.6
29.5
0
0
0.3
25.1
25.9
30.2
Sequential
Pricing without
Discrimination
Sequential
Pricing with
Discrimination
Table 2. Comparison of Welfare Effects for Goods with Independent Values
p-values for Permutations Tests for Differences between Pricing Strategies
Comparison
Seller
Profit
Buyer
Surplus
Efficiency
Bundling vs
Sequential Pricing with
Discrimination
0.429
0.957
0.249
0.043
0.214
0.657
0.514
0.386
0.257
Bundling vs
Sequential Pricing
without Discrimination
Sequential Pricing with
Discrimination vs
Sequential Pricing
without Discrimination
24 Table 3. Linear Mixed Effects Estimation of
Sequential Pricing for Goods with Independent Values
Estimate
t-statistic
p-value
Average Good A Pricei
Model: PAijt = α0 + α1PDj+ εi + εj + εijt Intercept
PD
18.0
3.6
6.23
0.88
<0.01
0.41
Average Good B Pricei
Model: PBijt = β0 + β1PDi×BoughtA + β2PDj×(1‐BoughtA) + εi + εj + εijt Intercept
PD×BoughtA
PD×(1-BoughtA)
Estimate
33.2
6.3
6.1
t-statistic
8.28
1.11
1.09
p-value
<0.01
0.27
0.28
Estimate
t-statistic
p-value
Minimum Price of Good Aii Model: MinPAjt = γ0 + γ1PDj + εj + εjt Intercept
PD
6.1
5.6
1.91
1.24
0.06
0.22
Low A Price Seller’s Good B Priceii Model: PB|MinPAjt = ω0 + ω1PDj×BoughtA + ω2PDj×(1‐BoughtA) + εj + εjt Intercept
PD×BoughtA
PD×(1-BoughtA)
Estimate
32.0
5.6
4.8
t-statistic
8.05
0.99
0.86
p-value
<0.01
0.32
0.39
PD is a dummy variable that equals 1 if the seller was operating in a market in which
price discrimination was possible and 0 otherwise. BoughtA is a dummy variable that
equaled 1 if the price was targeted to people who had purchased good A and 0 otherwise.
i
indicates that the unit of observation is at the individual level each period. Each session
and period is modeled as having a random effect while the treatments are modeled as a
ii
fixed effect. indicates that the unit of observation is at the market level each period.
Each session and period is modeled as having a random effect, while the treatments are
modeled as fixed effects.
25 Table 4. Comparison of Welfare Effects for Goods with Positively Correlated Values
p-values for Permutations Tests for Differences between Pricing Strategies
Comparison
Seller
Profit
Buyer
Surplus
Efficiency
Bundling vs
Sequential Pricing with
Discrimination
0.643
0.757
0.314
0.371
0.214
0.086
0.500
0.471
0.586
Bundling vs
Sequential Pricing
without Discrimination
Sequential Pricing with
Discrimination vs
Sequential Pricing
without Discrimination
26 Table 5. Linear Mixed Effects Estimation of
Sequential Pricing for Goods with Positively Correlated Values
Estimate
t-statistic
p-value
Average Good A Pricei
Model: PAijt = α0 + α1PDj+ εi + εj + εijt Intercept
PD
19.7
-0.3
4.32
-0.05
<0.01
0.96
Average Good B Pricei
Model: PBijt = β0 + β1PDi×BoughtA + β2PDj×(1‐BoughtA) + εi + εj + εijt Intercept
PD×BoughtA
PD×(1-BoughtA)
Estimate
39.5
-6.4
-7.7
t-statistic
10.15
-1.17
-1.40
p-value
<0.01
0.24
0.16
Estimate
t-statistic
p-value
Minimum Price of Good Aii Model: MinPAjt = γ0 + γ1PDj + εj + εjt Intercept
PD
11.1
-2.5
3.19
-0.50
<0.01
0.61
Low A Price Seller’s Good B Priceii Model: PB|MinPAjt = ω0 + ω1PDj×BoughtA + ω2PDj×(1‐BoughtA) + εj + εjt Intercept
PD×BoughtA
PD×(1-BoughtA)
Estimate
38.4
-5.1
-8.4
t-statistic
9.32
-0.88
-1.44
p-value
<0.01
0.38
0.15
PD is a dummy variable that equals 1 if the seller was operating in a market in which
price discrimination was possible and 0 otherwise. BoughtA is a dummy variable that
equaled 1 if the price was targeted to people who had purchased good A and 0 otherwise.
i
indicates that the unit of observation is at the individual level each period. Each session
and period is modeled as having a random effect while the treatments are modeled as a
ii
fixed effect. indicates that the unit of observation is at the market level each period.
Each session and period is modeled as having a random effect, while the treatments are
modeled as fixed effects.
27 Table 6. Comparison of Welfare Effects for Goods with Complementary Values
p-values for Permutations Tests for Differences between Pricing Strategies
Comparison
Seller
Profit
Buyer
Surplus
Efficiency
Bundling vs
Sequential Pricing with
Discrimination
0.586
0.186
0.071
0.871
0.300
0.142
0.587
0.457
0.729
Bundling vs
Sequential Pricing
without Discrimination
Sequential Pricing with
Discrimination vs
Sequential Pricing
without Discrimination
28 Table 7. Linear Mixed Effects Estimation of
Sequential Pricing for Goods with Complementary Values
Estimate
t-statistic
p-value
Average Good A Pricei
Model: PAijt = α0 + α1PDj+ εi + εj + εijt Intercept
PD
13.0
3.4
5.33
0.98
<0.01
0.37
Average Good B Pricei
Model: PBijt = β0 + β1PDi×BoughtA + β2PDj×(1‐BoughtA) + εi + εj + εijt Intercept
PD×BoughtA
PD×(1-BoughtA)
Estimate
51.8
-1.0
-17.7
t-statistic
10.80
-0.14
-2.60
p-value
<0.01
0.89
<0.01
Estimate
t-statistic
p-value
Minimum Price of Good Aii Model: MinPAjt = γ0 + γ1PDj + εj + εjt Intercept
PD
5.4
3.7
2.29
1.11
0.02
0.27
Low A Price Seller’s Good B Priceii Model: PB|MinPAjt = ω0 + ω1PDj×BoughtA + ω2PDj×(1‐BoughtA) + εj + εjt Intercept
PD×BoughtA
PD×(1-BoughtA)
Estimate
53.4
-0.5
-19.6
t-statistic
8.80
-0.05
-2.28
p-value
<0.01
0.96
0.02
PD is a dummy variable that equals 1 if the seller was operating in a market in which
price discrimination was possible and 0 otherwise. BoughtA is a dummy variable that
equaled 1 if the price was targeted to people who had purchased good A and 0 otherwise.
i
indicates that the unit of observation is at the individual level each period. Each session
and period is modeled as having a random effect while the treatments are modeled as a
ii
fixed effect. indicates that the unit of observation is at the market level each period.
Each session and period is modeled as having a random effect, while the treatments are
modeled as fixed effects.
29 Figure 1. Onscreen Pricing Tool for
Sequential Pricing with Discrimination and Positively Correlated Values
Figure 2. Profit and Buyer Surplus by Session
Independent Values
Correlated Values
Complements
125
125
100
100
75
75
75
50
50
50
25
25
25
125
100
0
0
0
125
125
125
100
100
100
75
75
75
50
50
50
25
25
25
0
0
0
150
150
150
125
125
125
100
100
100
75
75
75
50
50
50
25
25
25
0
0
0
Bundling
Sequential Pricing
30 Profit
Buyer Surplus
Monopoly Profit
Sequential Pricing
with Discrimination without Discrimination
31 Figure 3. Frequency of Prices by Treatment
0.5
0.5
0.4
Price of Good A
Price of Good B without A
Price of Good B with A
0.4
Single Item Price
0.5
0.4
Values
0.2
0.2
0.2
0.1
0.1
0.1
Bundle Price
0.3
0-10
11-20 21-30 31-40 41-50 51-60 61-70 71-80 81-90 91-100 101-110 111-120 120+
0-9 10-19 20-29 30-39 40-49 50-59 60-69 70-79 80-89 90+
0.5
0.4
0.4
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
0
0
0-10
11-20 21-30 31-40 41-50 51-60 61-70 71-80 81-90 91-100 101-110 111-120 120+
0.5
0
0-9 10-19 20-29 30-39 40-49 50-59 60-69 70-79 80-89 90+
0.5
0.4
0.4
0.3
0.3
0.3
0.2
0.2
0.2
0.1
0.1
0.1
0
0
0-10
11-20 21-30 31-40 41-50 51-60 61-70 71-80 81-90 91-100 101-110 111-120 120+
Bundling
0
0-9 10-19 20-29 30-39 40-49 50-59 60-69 70-79 80-89 90+
Sequential Pricing
with Discrimination
32 0-9 10-19 20-29 30-39 40-49 50-59 60-69 70-79 80-89 90+
0.5
0.4
Complements
0-9 10-19 20-29 30-39 40-49 50-59 60-69 70-79 80-89 90+
0.5
0.3
Values
0
0.5
0.4
Correlated
0.3
0
0
Price of Good A
Price of Good B
Independent
0.3
0-9 10-19 20-29 30-39 40-49 50-59 60-69 70-79 80-89 90+
Sequential Pricing
without Discrimination