Sequential selling and information dissemination in the presence of
network effects∗
Junjie Zhou†
Ying-Ju Chen‡
October 10, 2014
Abstract
In this paper, we examine how a seller sells a product/service with a positive consumption
externality, and customers are uncertain about the product’s/service’s value. Because early
adopters learn this value, we consider the customers’ intrinsic signaling incentives and positive
feedback effects. Anticipating this, the seller commits to provide price discounts to the followers, and charges the leader a high price. Thus, the profit-maximizing pricing features the cream
skimming strategy. We also show that the lack of seller’s commitment is detrimental to the
social welfare; nonetheless, the sequential selling still boosts up the seller’s profit compared with
the simultaneous-selling case. Embedding a physical network with arbitrary payoff externality
among customers, we investigate the optimal targeting strategy in the presence of information
asymmetry. We provide precise indices for this leader selection problem. For undirected graphs,
we should simply choose the player with the highest degree, irrespective of the seller’s commitment power. Going beyond this family of networks, in general the seller’s commitment power
affects the optimal targeting strategy.
Keywords: revenue management, signaling, information transmission, social networks
JEL classifications: D82, L14, L15
∗
We are grateful for the financial support provided by NET Institute (www.NETinst.org). All remaining errors
are our own.
†
School of International Business Administration, Shanghai University of Finance and Economics, 777 Guoding
Road, Shanghai, 200433, China; e-mail: [email protected].
‡
School of Business and Management & School of Engineering, The Hong Kong University of Science and Technology, Clear Water Bay, Kowloon, Hong Kong; e-mail: [email protected].
1
1
Introduction
Commercial social networks share many things in common. First, there are strong payoff externalities, i.e., a user’s consumption utility depends crucially on the behaviors of other users (e.g., time
and effort spent on the network); see Farrell and Saloner (1986) and Rochet and Stole (2002) for
some classical discussions on these network effects. Second, as new games, items, virtual goods,
and other novel services are launched to the social networks on a daily (if not hourly) basis, users
may have limited knowledge about the level of satisfaction while making their purchasing decisions
(Jackson (2008)). In accordance, there might be a large variation among the users in terms of their
a priori knowledge. This creates heterogeneity among users (on top of their endowed preferences),
and information dissemination, if possible, plays a pivotal role in promoting new experiments or
services (Acemoglu et al. (2011), Acemoglu et al. (2014), and Bala and Goyal (1998)).
Being aware of the above features of social networks, service providers’ advance knowledge
about the underlying relationships and individual networks creates abundant opportunities for
social coordination and profit maximization. This paper tackles the revenue management problem
in the social network environment from a novel angle which, to the best of our knowledge, has
never been explored in the literature. Our model setup features the positive externalities among
the customers, the seller’s price discrimination power, and the inherent information transmissions
among them. The product quality is uncertain, and only one customer has access to this valuable
information. The prior knowledge of this customer, labeled as the “maven,” may result from her
unique expertise, and the seller is aware of her existence and identity.
We consider the two-stage selling scheme in which the maven chooses her quantity first as the
leader, and other customers (followers) choose their quantities simultaneously in the second stage.
We show that the leader strategically uses her decision to convey the information to the followers.
This signaling effect influences the followers’ consumptions and subsequently creates a positive
feedback that reinforces the leader’s own incentive. Anticipating this, the seller commits to provide
price discounts to the followers, and charges the leader a high price. Thus, the profit-maximizing
pricing features the cream skimming strategy – a decreasing pricing pattern along the sequence.
Our analysis also reveals that the optimal pricing conveys no information regarding the product
quality, thereby eliminating the possibility of price signaling. This suggests that all the results are
2
robust against any specification of the seller’s prior knowledge.
Additionally, we observe that the sequential selling scheme is not only profitable for the seller
but also beneficial to the society. This is because the sequential selling gives the leader a chance
to credibly signal her information to the followers. This information channel partially mitigates
the customers’ ignorance of their consumptions on others’ payoffs, and implicitly enforces the coordination among customers without direct communication. When the seller cannot commit to the
prices, the aforementioned followers’ compensation becomes incredible. This subsequently limits
the seller’s ability to extract revenue from the leader, and the equilibrium quantity of each customer
returns to the level in the simultaneous-move game with complete information. Consequently, the
lack of seller’s commitment is detrimental to the social welfare as well as the consumer surplus.1
Nevertheless, the sequential selling still boosts up the seller’s profit because of the leader’s strong
signaling incentive. While the resulting consumptions are identical to the simultaneous-move outcomes, the leader bears the responsibility (and the cost) of information transmissions.
While the majority of analysis is conducted in a simplified setup with homogeneous utilities, we
can easily accommodate the possibility of heterogeneous payoff structures. This general framework
then embeds a physical network with arbitrary payoff externalities among customers. With this
physical network, a new question arises: who should be chosen as the leader in the first stage?
This corresponds to the optimal targeting strategy and is one of the central topics in the literature
on social networks. We provide precise indices for the leader selection in the two scenarios: with
and without commitment. For undirected graphs, i.e., all the social influences are two-way and
symmetric, we should simply choose the player with the highest degree. This gives rise to a
natural rule of thumb and it is robust against the seller’s commitment power. Going beyond this
family of networks, in general the seller’s commitment power affects the optimal targeting strategy.
We also observe that the profit-maximizing pricing is again cream skimming irrespective of the
physical network configuration. Finally, we examine the optimal selling schemes when customers
are endowed with other functional forms of utilities, and the general messages continue to prevail.
The rest of this paper is organized as follows. Section 2 reviews some relevant literature.
1
This stands in strict contrast with the implications from the intertemporal pricing literature. As argued by Coase
(1972) and Besanko and Winston (1990), the seller’s commitment issue can create fierce competition between his
current and future selves and ultimately benefits the consumers.
3
Section 3 introduces the model setting, and Section 4 derives the optimal pricing strategies. In
Section 5, we extend our model setup to incorporate some alternative scenarios and discuss possible
variants of our model characteristics. Section 6 concludes. Proofs of all results are provided in the
appendix.
2
Literature review
Our paper is related to the vast literature on network externalities, where an individual’s utility
generated from a product or service is enhanced by others’ use. The original theory developed by
Rohlfs (1974) captures the self-enhancing and self-fulfilling characteristics of network effect, and
addresses simply networked goods such as telephone and fax machine. Later, Katz and Shapiro
(1985) introduce the cross-sided network effect, and deal with the network effect between complementary goods, such as razor set and the blades. Since then the discussion has been focusing
on the multiplicity of equilibria and fulfilling expectation, the efficiency of technology adoption
(Farrell and Saloner (1986)), market dominance, and the inter-connection pricing to induce/reduce
compatibility (Rochet and Stole (2002)). See Economides (1996) for an extensive survey of this
literature.
Recent papers (such as Ballester et al. (2006) and Candogan et al. (2012)) acknowledge the
local network effects, i.e., customers (agents) pay more attention to their close friends than to
others who are far away from their life circles. Nevertheless, in these papers the customers choose
the consumption levels simultaneously, whereas we allow for sequential selling and incorporate
information asymmetry. Some researchers examine how in a social network, players’ adoption
decisions are affected by the influencers or early adopters (e.g., Katona et al. (2011) and Katona
(2013)). Unlike our paper, these influences are mainly driven by the network effects rather than
the signaling effect. Our sequential selling schemes are related to the interactive pricing strategy
proposed by Akhlaghpour et al. (2010) and Hartline et al. (2008). They focus on the algorithmic
procedures for the optimal pricing problem, and do not incorporate the information asymmetry
among customers.
Our paper also adds to the growing literature on social learning (Banerjee (1992) and Bikhchandani et al. (1992)), in which effective information aggregation requires all the players to voluntarily
4
disclose their information (presumably through some observable actions). In our context, however,
information transmissions are in a top-down manner – the leader conveys information to the followers who have no private information ex ante. Some papers study the dynamics of belief formation
and information aggregation through communication and choices (such as Acemoglu et al. (2014),
Bala and Goyal (1998), and references in (Jackson 2008: Chapter 7)). Past literature either takes
as given the network structure or studies other forms of information transmissions through cheap
talk, verifiable disclosure, or direct communication. In contrast, in our paper the information transmissions are facilitated through costly signaling instruments – the quantity decisions. In Campbell
et al. (2013), players intend to exchange information to signal their social status. This alternative
signaling effect leads to a fundamentally different targeting strategy from us. In addition, the signaling device in Campbell et al. (2013) is the communication itself, whereas in our paper the leader
has to signal via her consumption choice.
Two recent papers consider the possibility of sequential-move network games. Zhou and Chen
(2014a) extend the framework of Ballester et al. (2006) and allow the network designer to determine
the sequence of moves. They show that sequential-moving gives rise to a second-order enhancement
of aggregate contribution, and in the two-stage game the optimal selection of leaders (targeting
problem) is isomorphic to a weighted maximum-cut problem. Zhou and Chen (2014b) incorporate
incomplete information into the same network game. They prove that the signaling incentives can
lead to a first-order materialistic effect on the equilibrium outcomes, and the resulting targeting
strategy can be fundamentally different. The current paper differs from Zhou and Chen (2014a)
and Zhou and Chen (2014b) in that we eliminate strategic complementarity; thus, the sole driver
for the benefit of sequential moves comes from informational externality. Specifically, in this paper
if information is symmetric, the equilibrium outcomes will be the same irrespective of the sequence
of players’ moves. This allows us to highlight the information dissemination channel. Moreover, we
study the sequential-selling (pricing) problem and the impact of seller’s commitment power on the
customers and social welfare. These discussions have no counterparts in Zhou and Chen (2014a)
and Zhou and Chen (2014b).
5
3
Model
We consider a stylized setting wherein a monopoly seller sells a product or service to a group of
N ≥ 2 customers.
Customers. Let p = (p1 , · · · , pN ) denote the price vector set by the seller and x = (x1 , · · · , xN )
the profile of consumption levels selected by these customers. Given these vectors, a customer
i ∈ {1, ..., N } obtains a payoff:
N
X
√
ui (x, p) = θ
2 xj − p i xi ,
(1)
j=1
where θ measures the intrinsic product quality. From (1), a priori all customers are endowed with the
same utility functions, and each customer’s consumption has a positive effect on all other customers’
payoffs. The identical utility assumption is made to crystallize the mis-coordination problem in
this network game, because it largely simplifies the derivations and exposition. In Section 5,
we introduce a more general framework in which the payoff externality also exhibits a (physical)
network structure. Abstracting from the physical network structure facilitates a treatment of the
structure generated by the information flows. Incidentally, the functional form in (1) is also made
for ease of exposition; we will examine other functional forms of the customers’ utilities in Section
5 as well.
Seller’s profit and social welfare. Let c denote the seller’s constant marginal cost of
providing the product. Thus, the seller’s profit is
R(x, p) =
N
X
(pi − c)xi .
i=1
Define the social welfare as the sum of customers’ utilities and the seller’s profit:
W (x) =
N
X
ui (x, p) + R(x, p).
i=1
From expression (1), we obtain:
W (x) = N θ
N
N
N
X
X
X
√
√
xi =
(2N θ xi − cxi ) .
2 xi − c
i=1
i=1
i=1
Our focus is on the optimal (profit-maximizing) selling scheme in this context when θ is privately
observed by only some customers. Before we proceed, we first introduce two benchmark cases.
6
The welfare-maximizing outcomes. Suppose a social planner observes the realization of
nP
o
√
N
2N
θ
θ and dictates the quantities {xi }’s to solve maxxi
x
−
cx
. It is readily seen that
i
i
i=1
the solution is:
xFi B
=
Nθ
c
2
,
where the superscript indicates the first-best quantities. The corresponding maximum welfare is
WFB =
N 3 θ2
c .
The complete-information outcomes. We now consider the second benchmark: each
customer knows the realization of θ, but freely chooses her own quantity, xi . The seller moves first
and announces prices to maximize his profit. Afterwards, suppose that customers simultaneously
make their quantity decisions. Customer i chooses quantity xi to maximize the following:


 √

X √
max θ(2 xi +
2 xj ) − p i xi .
xi 

j6=i
This gives rise to a dominant strategy:
1
= pi , or xN
θq
i =
xN
i
θ
pi
2
,
which is independent of {xj , j 6= i}, the choices of other customers.
The above independence also implies that when θ is commonly known, if customers make
decisions sequentially, the resulting choices will be the same irrespective of the sequence of moves.
Note that this is substantially different from Zhou and Chen (2014a) and Zhou and Chen (2014b), in
which strategic complementarity embedded in customer utilities generates the benefit of sequential
moves. Viewed in this way, our setting is specifically tailored to single out the salient feature of
information transmissions. In Section 4, we will illustrate how informational externality alone can
generate the momentum amongst customers.
In the first stage, the seller chooses the price vector that maximizes his profit:
(N
)
(N
2 )
X
X
θ
max
(pi − c)xN
= max
(pi − c)
.
i
N
N
pi
p∈R
p∈R
i=1
i=1
The optimal price pi follows directly from the first-order condition:
2
θ
−2θ2
N
−
c)
+
(p
= 0 ⇐⇒ pN
i
i = 2c,
N )3
pN
(p
i
i
7
which is independent of θ. Observe that the seller need not know the true realization of θ when
choosing his profit-maximizing price vector. The equilibrium quantity under this optimal price is
θ 2
N
xN
i = ( 2c ) , the seller’s profit is R =
N θ2
4c ,
and the social welfare is W N =
(4N −1)N θ2
.
4c
Comparison between two benchmark scenarios. We observe that the equilibrium quanF B for two reasons. First, for the profit-maximization purpose, the seller
tity xN
i is lower than xi
charges a price that is higher than the marginal cost c; this subsequently reduces the quantities
selected by the customers. Second, when making their quantity decisions, customers neglect the
positive externality their consumptions impose on others. The lack of coordination gives rise to
a lower consumption level for each customer. Since the welfare function W (x) is increasing in xi
whenever xi ≤ xFi B , we have W N < W F B . Incidentally, to implement first-best outcomes when
the customers can simultaneously choose their quantities, prices need to be set as follows:
pi =
c
, ∀i = 1, ..., N.
N
This price, however, is lower than the seller’s marginal cost c; thus, the seller in this case bears a
negative payoff and has no incentive to implement this price vector.
Information structure. In what follows, we examine the optimal selling scheme from the
seller’s perspective. Ex ante, θ is drawn from the distribution F : [0, θ̄] → [0, 1], and for simplicity we
assume that F has full support on Θ = [0, θ̄]. One customer, called the maven, knows the realization
of θ. The presence of this maven is common knowledge, and we will also refer to this customer as
the leader L in various places. The maven’s knowledge may result from her unique expertise of
accessing the intrinsic value of the product. If no such maven exists, the game degenerates to the
second benchmark except that now every customer uses the expected value E[θ].
We will not explicitly specify the seller’s prior knowledge regarding θ, because it turns out to
be irrelevant for our analysis. For the concreteness of our setup, it is convenient to think of the
seller as having no direct access to the realization of θ. In fact, our analysis reveals that even if the
seller knows θ ex ante, the equilibrium outcomes remain the same so long as he cannot credibly
disclose this information to the customers. In other words, there is no room for price signaling in our
context. This is because the optimal pricing does not convey any information about θ (as we will
demonstrate later). In addition, exactly the same equilibrium outcome arises for any intermediate
knowledge in between these two extreme cases (for example, ex ante the seller observes a signal
8
that is correlated with θ with an arbitrary precision).
The seller’s goal is to find the selling scheme to maximize his (expected) payoff, taking into
account the interactions among the customers. We will explore the possibility of sequential selling.
As argued by Akhlaghpour et al. (2010) and Hartline et al. (2008), this “interactive pricing” is
a promising marketing strategy that best explores the business potential of social networks. In
addition, we will also discuss the role of seller’s commitment in this pricing context. Since the game
involves multiple rounds of strategic interactions between the seller and the customers, we adopt
the perfect Bayesian equilibrium (PBE) as our solution concept (Fudenberg and Tirole (1991)).
4
Optimal pricing strategies
In this section, we investigate the optimal pricing problem when the seller sells to the maven in
the first stage, and then other customers (followers) choose their quantities simultaneously in the
second stage. We will first allow the seller to announce the entire price vector and commit to it.
In the end we comment on the alternative scenario wherein the seller has no commitment power
on setting the prices for the followers.
4.1
Quantity decisions
By backward induction, we start with the customers’ quantity decisions. We will focus on a
particular form of equilibrium in which the maven, acting as the leader, chooses her quantity
according to the following:
x∗L (θ) = (kL θ)2 ,
where kL is a constant yet to be determined. This particular form indicates a one-to-one correspondence between the value of θ and the quantity x∗L (θ). Consequently, upon observing the
leader’s quantity choice, each customer perfectly infers the realization of θ. This suggests that the
equilibrium is fully separating.
In this two-stage selling scheme, if other customers observe the leader’s choice xL , they hold
9
√
xL
kL .
the same posterior belief about the state: θ̂ =
Thus, each customer i intends to maximize:

 √x


X
√
√
L √
max
(2 xi + 2 xL +
2 xj ) − pi xi ,
xi  kL

j6=i,L
√
x
and the optimal quantity is x∗i = ( pθ̂j )2 = ( kL pLj )2 . Observe that the quantity x∗i is monotonic in
xL . This creates a positive feedback effect to the leader, as other consumers react positively to the
leader’s consumption, and their consumptions directly influence the leader’s payoff. Anticipating
other customers’ responses, the leader will choose xL to maximize
 




X √xL
√
 − pL xL .
max θ 2 xL + 2
xL 

kL pj
j6=L
The first-order condition yields:

θ 1 +

X
j6=L
1 
1
p ∗
− pL = 0.
kL pj
xL (θ)
Recall that x∗L (θ) = (kL θ)2 . The consistency requirement of perfect Bayesian equilibrium
(PBE) leads to:

θ 1 +

X
j6=L

1  1
− pL = 0 ⇐⇒ 1 +
kL pj kL θ

X
j6=L
1 
= pL kL .
kL pj
Solving this quadratic equation gives rise to the following expression for kL :
q
q
P
P
1
1
+
+ 4 p1L j6=L p1j
1 + 1 + 4pL j6=L p1j
pL
p2L
1
=
,
kL =
≥
2pL
2
pL
P
where the inequality follows from the non-negativity of {pj }’s. Thus, the term 4pL j6=L
(2)
1
pj
captures
the positive feedback effect mentioned in the introduction. We summarize our finding as the next
proposition.
Proposition 1. For fixed price vector p, there exists a separating perfect Bayesian equilibrium in
which on the equilibrium path:
1. xSL (p) = (kL θ)2 , where kL is given in (2).
√ S
x (p)
2. xSi (p) = ( kLLpi )2 = ( pθi )2 , ∀i 6= L.
10
Proposition 1 provides a full characterization of the equilibrium quantities. First, we observe
that kL ≥
1
pL ,
implying that the leader is willing to consume more than what she would do
in the simultaneous-move game. This is because she intends to convey the information of θ to
the remaining customers. The leader knows that her action shapes each follower’s belief about
the product quality, and a higher consumption level indicates a better quality. Therefore, this
signaling effect subsequently boosts the leader’s equilibrium consumption. On the other hand,
all the remaining customers make the correct inference about the true state on the equilibrium
path. Consequently, each customer consumes ( pθi )2 , which is exactly the quantity she chooses in a
simultaneous-move game. It is noteworthy that in the information network literature, information
transmissions are typically assumed to be truthful (e.g., (Jackson 2008: Chapter 7) and Acemoglu
et al. (2014)). In contrast, the credible disclosure channel is absent in our model; thus, the costly
signaling also takes into account the leader’s incentive problem for truthful revelation.
4.2
Pricing
Given the perfect Bayesian equilibrium characterized in Proposition 1, we can formulate the seller’s
profit for each price vector p as follows:

RS (p) := R(xS (p), p) = (pL − c) 
1+
q
P
1 + 4pL j
2
1
pj
θ +
2pL
X
(pj − c)(
j6=L
θ 2
) ,
pj
and the corresponding social welfare is:
W S (p) := W (xS (p)) = [N θ2kL θ − c(kL θ)2 ] +
X
[N θ2
j6=L
θ
θ
− c( )2 ].
pj
pj
Profit maximization. A natural objective for the seller is profit maximization. In this case,
we let RS := maxp RS (p) be the maximum profit from the seller’s perspective, and let pSR :=
arg maxp RS (p) be the profit-maximizing price vector. The following theorem characterizes the
optimal pricing in this two-stage selling scheme.
SR = c to
Theorem 1. The seller will optimally charge price pSR
L = (4N − 2)c to the leader and pj
all the remaining customers (followers). Accordingly,
θ 2
) and
1. The optimal pricing p is independent of the state θ. The leader chooses quantity ( 2c
θ 2
each follower chooses quantity ( c ) .
11
2. The corresponding seller’s profit is
RS =
4N − 3 θ2
.
4
c
3. The corresponding welfare is
S
W =
2
1
θ
N − + (N − 1)(2N − 1)
.
4
c
Theorem 1 leads to a number of observations, and we highlight them as corollaries. First, the
price vector contains no information regarding the realization of θ.
Corollary 1. The optimal pricing p is independent of the state θ.
Corollary 1 suggests that if a customer is uninformed, by looking at the price vector she cannot
infer anything from the seller. This remains true if the seller either knows the true product quality
ex ante, has learned it from interacting with the maven (leader), or is completely ignorant about
this information. This irrelevance result eliminates the possibility of price signaling in our social
network context. Given this property, the only source of information is the leader’s consumption
level. Thus, in our setup information transmissions are confined within the network, which allows
us to isolate the effects of the maven’s signaling incentive and learning from the leader. On the flip
side, the irrelevance result also facilitates the implementation of optimal pricing. Since the optimal
pricing strategy does not require the knowledge of the true state, the seller can attain his optimal
profit even if he has no access to the product quality (as described in Section 3).
Next, we compare the sequential selling scheme with the simultaneous one to identify the new
economic forces.
Corollary 2. Compared with the complete-information selling scheme:
1. The seller charges the leader a higher price, but compensates the remaining customers (followers) using a lower price.
2. The leader consumes exactly the same quantity, but the followers consume strictly more.
3. The seller obtains a higher expected payoff (i.e., RS > RN ).
4. The social welfare is higher (i.e., W S > W N ).
12
The first two parts of Corollary 2 show that the seller can capitalize on the leader’s intrinsic
signaling incentive. In accordance with this strategy, the seller should commit to boost the followers’
quantities by setting a low price. This then leads to a strong positive feedback effect, and allows
the seller to charge the leader a high price. Thus, Corollary 2 suggests that with information
transmissions, the profit-maximizing pricing features the cream skimming strategy – decreasing
prices along the sequence. This property turns out to be a general principle (as elaborated in
Section 5).
The last two parts of Corollary 2 suggest that the sequential selling scheme is not only profitable
for the seller but also beneficial to the society. This is because the sequential selling gives the leader
a chance to credibly signal her information to the followers. This information channel partially
mitigates the customers’ ignorance of their consumptions on others’ payoffs. Corollary 2 also
leads to a non-trivial recommendation for information provision. Since the complete-information
benchmark can be regarded as the situation in which the seller simply discloses the product quality
to all customers, our results show that this pervasive provision is suboptimal for both profit and
welfare maximization. Rather, some form of exclusivity is desired (this phenomenon is also noted
in Zhou and Chen (2014b)).
To better understand these results, let us now explore the rationale for why the price to the followers shall be set lower than 2c, which is the optimal price with complete information.
Recall that
r
1
1 P
1
1
+4
+
j6=L pj
pL
pL
p2
P
L
1
s
2
2
R (p) = θ (pL − c)kL + j6=i (pj − c) p2 . From (2), we obtain that kL (p) =
,
2
j
which is strictly decreasing in pj , j 6= i, i.e.,
∂kL
∂pj
< 0, j 6= i. For any t 6= i, for pt ≥ 2c, we obtain
that
1
∂(pt − c) p2
∂kL
∂Rs (p)
t
= θ2 (pL − c)2kL
+ θ2
< 0.
∂pt
∂pt
∂pt
|
{z
} |
{z
}
<0
≤0 as pt ≥2c
Therefore, charing any pt greater or equal to 2c is never optimal for the seller. This works for any
follower t. Moreover, the seller will optimally charge price pL ≥ 2c for the leader. Therefore, cream
skimming emerges as the optimal pricing strategy.
Welfare maximization. We can also discuss what the seller can do if he were to maximize
the social welfare. We note that even though the seller has no direct control over the social welfare,
he can use prices as the instruments to induce appropriate quantity choices by the customers.
13
Proposition 2. There exists a price vector
pL = c, and pj =
c
, ∀j 6= L,
N
that implements the first-best quantities.
Proposition 2 suggests that the implementation is feasible, even though the customers still
ignore the externality they impose on others while choosing their quantities. This is because the
prices endogenize this externality, and the seller again utilizes the leader’s information transmission
and signaling incentive to charge a high price. However, the seller’s net profit is negative because he
has to provide a deep discount to the followers (pj =
c
N ).
Welfare maximization goes fundamentally
against the seller’s own profit concern.
4.3
Commitment issue
Insofar we assume that the seller can credibly commit to the price vector in this two-stage sequential
selling scheme. This assumption is reasonable if it can be enforced by either an authoritative third
party or by the seller’s own reputation concern. However, in certain situations it appears to be a
bit strong. In this subsection, we investigate the impact of seller’s commitment problem.
To this end, we allow the seller to optimally adjust the prices for the followers after observing
the leader’s quantity choice. In this case, we again assume that the leader’s quantity choice takes
the same form:
x∗L (θ) = (kLnc θ)2 ,
but now the coefficient kLnc may differ as it reflects the leader’s anticipation of the seller’s pricing
decisions. With this new strategy in mind, the followers update the belief θ̂ about the state
after observing the leader’s choice. We can then characterize the optimal pricing in the following
proposition.
Proposition 3. Suppose that the seller cannot commit to the second-stage prices. At optimality:
θ 2
1. The seller will charge pnc
L = 2N c for the leader, and the leader will choose xL = ( 2c ) .
2. The seller will charge prices pnc
j = 2c to the followers, and each follower will choose xj (xL ) =
θ 2
xL = ( 2c ) .
3. The seller’s profit is Rnc =
3N −2 θ2
4
c ,
and RN < Rnc < RS .
14
4. The social welfare is W nc =
(4N −1)N θ2
4
c ,
which is the same as W N .
In the absence of commitment power, the seller will charge the followers the profit-maximizing
P
price 2c. This is because in the second stage, the seller tries to maximize j6=i (pj − c) p12 . This
j
subsequently limits the seller’s ability to extract profit from the leader, because she now anticipates
that the followers’ aggregate consumptions will not live up to the same level. It is interesting to
point out that the followers just mimic the leader in terms of quantity decisions (xj (xL ) = xL ). On
the equilibrium path, each customer chooses the same quantity as that in the simultaneous-move
game. Consequently, in the absence of seller’s commitment power, the social welfare returns to the
original level.
Nevertheless, the sequential selling still boosts up the seller’s profit even if she cannot commit
to the prices (Rnc > RN ). This is because while the resulting consumptions are identical, the leader
bears the responsibility (and the cost) of information transmissions. Therefore, the seller can again
capitalize on the leader’s signaling incentive, and the commitment problem only partially weakens
the benefit of price discrimination. The customers become the unfortunate victims of the seller’s
lack of commitment. Note that the social welfare is the same as that in the simultaneous-move
game, but the sequential selling increases the seller’s profit and therefore hurts the customers. In
closing, Table 1 summarizes the results in this section.
Prices
Quantities
Profit
Welfare
First-best
Simultaneous
n/a
pi = 2c
xi = ( Ncθ )2
θ 2
xi = ( 2c
)
n/a
N θ2
4 c
(4N −1)N θ2
4
c
N 3 θ2
c
Commitment
pL = (4N − 2)c
pi = c, ∀i 6= L
θ 2
xL = ( 2c
)
xi = ( θc )2 , ∀i 6= L
4N −3 θ2
4
c
N−
1
4
2
+ (N − 1)(2N − 1) θc
No commitment
pL = 2N c
pi = 2c, ∀i 6= L
θ 2
xi = ( 2c
) , ∀i
3N −2 θ2
4
c
(4N −1)N θ2
4
c
Table 1: Equilibrium outcomes in different scenarios.
5
Extensions and discussions
In this section, we consider some variants of our model characteristics and examine the robustness
of our results.
15
5.1
Physical network structure
In our basic framework, customers are endowed with homogeneous payoff structures, and the externality one customer imposes on others is the same across different customers. This is certainly
a simplification. In this section, we extend our setting to incorporate asymmetric payoffs. Given
that there are N customers, in general a customer i’s payoff shall be expressed as follows:


N
 √

X
√
ui (x) = θ 2 xi +
2gij xj − pi xi ,


(3)
j=1
where gii = 0 and gij ≥ 0. With these heterogeneous {gij }’s, a network structure is embedded in
the relationships among the customers.
In the network economics literature, matrix G ≡ [gij ] is called the adjacency or externality
matrix (see, e.g., Candogan et al. (2012)). Our basic framework is a special case in which gij = 1,
for all i 6= j. Thus, it can be regarded as an unweighted (complete) graph wherein each customer is
equally important to all others. An interpretation of payoff functions in 3 is that when gij is high,
customers i and j are close neighbors in this social network; thus, their utilities largely depend on
the consumption levels of each other. On the other hand, a small gij implies that these customers
are far apart. Collectively, this adjacency matrix captures the detailed local network effects beyond
the first-order measures (such as “degrees”) of the social network. As in Candogan et al. (2012),
we assume that the seller knows the underlying network configuration G. This assumption is
appropriate for Facebook and Twitter business models because the entire relationship building
processes are recorded.
Unlike the complete graph case, the identity of the first-mover, i.e., the leader, matters for
general G. The natural question to ask is: from the seller’s perspective, who should be chosen as
the leader in the first stage? This corresponds to the optimal targeting strategy and is one of the
central topics in the literature on social networks. In a different context, Zhou and Chen (2014a)
examine this problem with complete information regarding the product quality and strategic complementarity among players.
To determine the appropriate leader, we should first solve the subgame perfect equilibrium
when each of these players is selected as the leader. We can then compare the seller’s maximal
16
profits across these scenarios. To this end, suppose that player i is the leader; accordingly, players
in N \{i} are the followers in the second stage. We have the following proposition.
Proposition 4. With the physical network structure, suppose that player i is the leader.
1. For fixed price vector p, there exists a separating perfect Bayesian equilibrium. On the equilibrium path, for leader i, xi (p) = (ki θ)2 , where
q
P
1
1
+
+ 4 p1i j6=i gij p1j
pi
p2i
ki =
.
2
For each follower j, xj (p) = ( pθj )2 , ∀j 6= i.
2. With commitment, the maximal profit for the seller is


2
X
θ 
Ri∗ =
1+
(1 + gij )2  .
4c
j6=i
Moreover, the leader is charged a higher price than the followers.
3. Without commitment, the maximal profit for the seller is


2
X
θ 
1+
(1 + 2gij ) .
Rinc∗ =
4c
j6=i
Proposition 4 provides very precise indices for the leader selection in the two scenarios: with
and without commitment. The first part of Proposition 4 is similar to Proposition 1, with the
generalization that now the customers have preferential treatments amongst others. The latter
two parts, however, lead to precise criteria based on which we can select the leader. Namely,
P
with commitment, the seller should simply select the customer arg maxi=1,...,N j6=i (1 + gij )2 . By
P
contrast, without commitment, the seller chooses arg maxi=1,...,N j6=i (1 + 2gij ). We also observe
that the profit-maximizing pricing is decreasing, which echoes the cream skimming strategy in
Corollary 2.
Below, we present some examples to illustrate our strategy recommendations. The first example considers undirect graphs, i.e,. all the social influences are two-way and symmetric.
Example 1. If G is the adjacent matrix of an undirected graph, i.e. gij = gji ∈ {0, 1}. Then
Ri∗ =
θ2
θ2
(1 + 4di + (N − 1 − di )) = (N + 3di ),
4c
4c
Rinc∗ =
θ2
(N + 2di ).
4c
P
Here di = N
k=1 gik is the degree of node i. The solutions to both problems are the same: the player
with the highest degree.
17
The above example gives rise to a natural rule of thumb. When the social influences are
mutual between each pair of players, and the intensities are the same in every relationship, we
simply choose the player that is the most connected. In this way, her signaling incentive is the
strongest. Thus, she has an intrinsic reason to convey this message through her action to others.
Next, we switch to the general weighted {gij }’s. We find that the seller’s commitment power
affects the optimal targeting strategy, i.e., arg max Ri∗ may be different from arg max Rinc∗ . See the
two examples below for illustration.
Example 2. Let us consider a six-player network with the following network specification:


0 0.8
0.1
0
0
0
0.8 0
0.1
0
0
0


0.1 0.1
0
0.7 + e 0
0
,

G=
0 0.7 + e
0
0.1 0.1

0
0
0
0
0.1
0 0.8
0
0
0
0.1
0.8 0
where e is a parameter whose value can be adjusted. In Figure 1, we present a graphic illustration.
The number on each edge is the corresponding gij , and we have ignored the edges with gij = 0.
Note that in this network, players (nodes) 1, 2, 5, and 6 are symmetric, and players 3 and 4 are
symmetric.
For any e > 0, player 3 (or equivalently player 4) has the highest Rinc∗ .
2
2
For Ri∗ , we know that R1∗ = θ4c (4 + 1.12 + 1.82 ) and R3∗ = θ4c (3 + 1.12 + 1.12 + (1.7 + e)2 ). When
0 ≤ e < 0.0406895, we have R1∗ > R3∗ ; when e > 0.0406895, the opposite holds, i.e., R1∗ < R3∗ .
6
1
0.1
0.1
0.8
0.7+e
3
0.1
0.8
4
0.1
2
5
Figure 1: An example in which the seller’s commitment affects the optimal targeting strategy.
The above example assumes a symmetric network (G is symmetric). However, the discrepancy between the targeting choices with and without commitment does not rely on this symmetry
assumption. In the next example, we provide another instance for which this is true.
18
Example 3. Consider a three-player network with the network specification:


0
0.95 0.05
0
0.56 .
G = 0.56
0.7 0.4
0
It is easy to see that R1∗ > R3∗ > R2∗ , whereas R1nc∗ < R3nc∗ < R2nc∗ .
5.2
Power utility functions
Insofar we restrict our attention to the square-root functional form for the customers’ utilities. A
legitimate question is whether the results hinge on this specific feature. To this end, we consider a
broader family of utility functions:
ui (x) = θ(
a
1 + a X 1+a
) − p i xi ,
xi
a
(4)
i∈N
where the parameter a ∈ (0, ∞) determines the skewness of the power utilities. When a = 1, we
return to our square-root case in the basic framework. With this alternative utility functional form,
the simultaneous-move game yields the following equilibrium outcomes:
−1/(1+a)
θxi
= pi , or xi = (
and it is verifiable that the optimal pricing is pN
i =
θ 1+a
) ,
pi
(1+a)c
a .
For this alternative utility function, we have the following parallel results.
Proposition 5. With power utilities described in (4) and player i as the maven/leader:
1. For fixed price vector p, there is a separating equilibrium in which the leader/maven’s strategy
is xi (θ) = (ki θ)1+a where ki is the unique solution to
X 1
1+
(
)a = p i k i .
pj ki
j6=i
√
The follower’s belief is θ̂ =
1+a
xi
ki
, and chooses quantity xj = ( pθ̂j )1+a , j 6= i.
2. The maximum seller’s profit is
R∗ =
θ1+a
aa
·
· (N − 1)21+a + 1 .
a
1+a
c
(1 + a)
The corresponding optimal prices are
P
(ki∗ )a + j6=i (kj∗ )a
∗
pi =
,
(ki∗ )1+a
where ki∗ =
1
c
·
a
1+a
and kj∗ =
2
c
·
a
1+a , j
6= i.
19
and
p∗j =
1
, j 6= i.
kj∗
Proposition 5 suggests that the main messages in this paper do not hinge on the specific
functional form. Rather, it is the information dissemination that shapes the optimal pricing pattern
as well as the seller’s profitability. Naturally, we can also work out the case without commitment,
but the results are parallel to those already obtained in Section 4. Thus, we omit them for brevity.
6
Conclusions
In this paper, we examine how a seller sells a product/service with a positive consumption externality, and customers are uncertain about the product’s/service’s value. In the leader-follower
setup, we identify the leader’s intrinsic signaling incentive and the positive feedback effect among
the customers. The profit-maximizing pricing turns out to be the cream skimming strategy – a decreasing pricing pattern along the sequence. Our analysis reveals that the optimal pricing conveys
no information regarding the product quality, thereby eliminating the possibility of price signaling.
The sequential selling scheme is not only profitable for the seller but also beneficial to the society.
This is because the sequential selling gives rise to an information channel, and it partially mitigates
the customers’ ignorance of their consumptions on others’ payoffs. We also show that the lack of
seller’s commitment is detrimental to the social welfare, but the sequential selling nonetheless still
boosts up the seller’s profit.
We then accommodate a physical network with arbitrary payoff externality among customers
and examine the optimal targeting strategy of selecting the leader. We provide precise indices for
the leader selection in the two scenarios: with and without commitment. For undirected graphs,
the player with the highest degree should be selected irrespective of the seller’s commitment power.
Going beyond this family of networks, in general the optimal targeting strategy crucially depends
on whether the seller can commit to the pricing pattern. We also observe that the profit-maximizing
pricing is again cream skimming irrespective of the physical network structure. Finally, we examine
the optimal selling schemes when customers are endowed with other functional forms of utilities,
and the general messages continue to prevail. Overall, our results speak to the sophisticated social
interactions in the presence of information asymmetry, payoff externality, and social learning.
Our paper can be extended along several directions. For example, in our model only one
customer has prior knowledge about the product quality. Extending to the multi-maven setting is
20
definitely interesting, because a natural question is then how many mavens are needed in order to
maximize the seller’s profit as well as the social welfare. More generally, we may allow the seller
to strategically decide the sequence of customers to offer the products/services to. In such a scenario, the sequence itself forms a new network configuration that facilitates endogenous information
transmissions. On the methodological side, this creates a non-trivial challenge. Allowing the seller
to choose the sequence implies that signaling takes place in multiple rounds; this dynamic signaling
problem may require new techniques to handle. It is also intriguing to characterize the optimal
sequence of such information transmissions, and we leave it for future work.
A
Appendix. Proofs
In this appendix, we provide the detailed proofs of our main results.
Proof of Proposition 1. The equilibrium quantities follow directly from the arguments in
the main text.
Proof of Theorem 1. We can solve for the optimal price pSR explicitly here by changing
variables. Instead of maximizing over the price vector p, we can choose (kL , tj =
1
pj , j
6= L) as the
new variables. Given this substitution, we can rewrite profit Rs (p) as follows:




X
Rs (p) = θ2 (pL − c)kL2 +
(tj − ct2j ) .


j6=L
Using (2), we know that:
pL =
1+
tj
j6=L kL
P
kL
=
kL +
P
j6=L tj
.
2
kL
(5)
Plugging in (5) and simplifying the notation, we obtain the following expression:








X
X
X
Rs (p) = θ2 (kL +
tj ) − ckL2 +
(tj − ct2j ) = θ2 (kL − ckL2 ) +
(2tj − ct2j ) .




j6=L
j6=L
j6=L
Obviously, the above expression is maximized when
kL =
1
1
, tj = , ∀j 6= L.
2c
c
21
(6)
Therefore, the corresponding prices are
pj = c, ∀j 6= L, and pL = (4N − 2)c (by equation (5)).
Given these prices, the equilibrium quantities, the seller’s profit, and the social welfare follow
immediately from straightforward algebra.
Proof of Corollary 1. This follows immediately from Theorem 1.
Proof of Corollary 2. The first two parts follow directly from Theorem 1. The third part
holds true because
RS − RN =
3(N − 1) θ2
> 0.
4
c
The last part is established by
3 θ2
W S − W N = (N − 1)(N − ) > 0. 4 c
Proof of Proposition 2. To implement the first-best outcomes, we need each customer to
choose xF B on the equilibrium path. By Theorem 1, this is the same as choosing
kL =
N
1
N
, and
= , ∀j 6= L
c
pj
c
Using (2), we obtain that pL = c, and pj =
c
N , ∀j
6= L.
Proof of Proposition 3. With the leader’s new strategy in mind, the followers update the
belief θ̂ about the state after observing the leader’s choice. In the second stage, the seller faces a
simultaneous-move game among the remaining customers, and he optimally chooses pj = pN = 2c
for the followers. This price coincides with that in the one-shot simultaneous-move game, because
the leader’s choice has been sunk. Accordingly, each follower chooses xj =
θ̂
2c ,
∀j 6= L.
We can then plug in these quantity choices and return to the leader’s decision making. The
procedure remains the same as in deriving Proposition 1. Thus, kLnc has almost the same expression
given in (2), except now that the prices for the followers are 2c due to the seller’s inability to commit.
We now characterize the optimal pricing for the leader. In (6), we are forced to choose tj =
j 6= L. Therefore, at optimality, kLnc =
1
2c ,
1
2c ,
and the corresponding price, given by (5), is 2N c. The
seller’s profit and the social welfare follow immediately. 22
Proof of Proposition 4. For a given network matrix G, for each price vector p and firstmover i, we can show that the equilibrium quantities are given by xs (θ) = (ks θ)2 , where {ks }’s are
determined by
ki = g(ti ,
X
gij kj ),
and kj = tj , j 6= i.
j6=i
√
Here ts =
1
ps , s
= 1, · · · , n and g(x, y) :=
x+
x2 +4xy
.
2
Straightforward algebra shows that
x ≤ g ≤ x + y, gxx ≤ 0, gxy > 0, and gyy < 0, g 2 = x(g + y).
Now consider the seller’s profit as a function of the price vector p:


N
X
X
Ri (p) =
(ps − c)xs (θ) = θ2 (pi − c)ki2 +
(pj − c)kj2  .
s=1
Note that ki2 = ti (ki +
j6=i
P
j6=i gij kj )
and kj = tj , j 6= i. Consequently, the seller’s profit can be
rewritten as follows:




X
X 1
X
1
Ri (p) = θ2 ( )ti (ki +
gij kj ) − cki2 +
( − c)kj2  = θ2 (ki − cki2 ) +
(1 + gij )kj − ckj2  .
ti
kj
j6=i
j6=i
j6=i
(7)
With commitment, the seller can freely pick any price ps , or equivalently kj in (7). Clearly,
1+g
1
and kj∗ = 2cij , j 6= i. The maximum profit is just
the maximal profit is achieved when ki∗ = 2c
P
2
Ri∗ = θ4c 1 + j6=i (1 + gij )2 . The corresponding prices can be computed as follows: p∗j = k1∗ =
2c
1+gij
for j 6= i and
p∗j =
2c
1+gij
p∗i
=
1
t∗i
=
ki∗ +
∗
j6=i gij kj
∗
2
(ki )
P
ki∗ + j6=i gij kj∗
∗
i
(ki∗ )2
j
P
for node i. Moreover, since gij ≥ 0, we obtain that
≤ 2c. On the other hand, p =
≥
1
ki∗
= 2c. Therefore, p∗i ≥ p∗j , ∀j 6= i.
Without commitment, the seller will charge price 2c to all of the followers in the second stage.
1
Therefore, we are forced to set kj = 2c
in (7). In other words,
 




X
1
Rinc∗ = max θ2 (ki − cki2 ) +
(1 + gij )kj − ckj2  |kj = , j 6= i .

ki 
2c
j6=i
1
The maximum is obtained by setting ki∗ = 2c
, and the maximal profit without commitment is


2
X
θ 
Rinc∗ =
1+
(1 + 2gij ) .
4c
j6=i
23
Proof of Proposition 5. We again start with the conjectured equilibrium quantities:
xs = (ks θ)1+a ,
where {ks }’s are yet to be determined. If the customer is a maven who knows the true state ex
ante, she will use the true state directly. On the other hand, an originally uninformed customer
will update her belief about θ after observing her predecessors’ decision, and then substitute θ by
her estimate θ̂. In a fully separating equilibrium, the estimate coincides with the true state.
It can be verified that in equilibrium, for the follower j 6= i in the second stage, kj =
1
pj .
On
the other hand, for the leader i, ki is the unique solution to
X kj
1+
( )a = pi ki
ki
or equivalently
kia +
j6=i
P
j6=i kj
1+a
ki
a
= pi .
(8)
The argument is similar to the proof of Proposition 1; hence, we omit the details for conciseness.
Given the above equilibrium quantity decisions, the seller’s profit can be written as:
R(p) =
X
(ps − c)xs =

X
(ps − c)(ks θ)1+a

X 1
= θ1+a pi ki1+a − cki1+a +
k 1+a − cki1+a 
kj j
j6=i


X
X
= θ1+a kia +
kj a − cki1+a +
kja − cki1+a 
j6=i
j6=i


= θ1+a kia − cki1+a +
X
1+a
2kja − cki
,
j6=i
where we use equation (8) in the third equality.
The optimal {ki }’s to (9) are given by:
ki∗ =
1
a
·
,
c 1+a
and kj∗ =
2
a
·
, j 6= i,
c 1+a
and the corresponding prices can be computed using (8):
P
(ki∗ )a + j6=i (kj∗ )a
1
∗
pi =
, and p∗j = ∗ , j 6= i.
∗
1+a
(ki )
kj
The corresponding maximum profit is
R∗ =
θ1+a
aa
·
· ((N − 1)21+a + 1). ca
(1 + a)1+a
24
(9)
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