Title Sequential pricing for social networks with multi-state diffusion

Author(s) Niu, G; Li, VOK; Long, Y

CitationThe 2013 IEEE Global Communications Conference (GLOBECOM2013), Atlanta, GA., USA, 9-13 December 2013. In Globecom.IEEE Conference and Exhibition, 2013, p. 3176-3181

Issued Date 2013

URL http://hdl.handle.net/10722/191600

Rights Globecom. IEEE Conference and Exhibition. Copyright © IEEE.

Globecom 2013 - Symposium on Selected Areas in Communications

Sequential Pricing For Social Networks With Multi-State Diffusion

Guolin Niu, Victor O.K. Li, Yi Long Department of Electrical and Electronic Engineering

The University of Hong Kong Pokfulam Road, Hong Kong, China

Email: {glniu.vli.yilong}@eee.hku.hk

Ahstract-The rapid development of online social networks (OSN) makes viral marketing through the word-of-mouth effect possible. Designing effective marketing strategy is critical in monetizing the social networks. However, most existing studies focus on conducting effective influence maximization analysis to propagate information widely instead of explicitly incorporating the pricing factor to design intelligent marketing strategies. In this paper, we study the sequential pricing models in which a monopoly seller iteratively posts a sequence of public unique prices for a product in stages, and any interested buyer can buy the product at the posted price at a particular stage if his valuation is above the posted price. Specifically, our model is built on the multi-state diffusion scheme where an active user may be in the "AWARE" and "ADOPT" states. Users in "ADOPT" state could influence their neighbors' valuation for the given product. To realize the goal of revenue maximization, we develop a Dynamic Programming Based Heuristic (DPBH) to obtain the optimal pricing sequence. Application of the DPBH in the revenue maximization problem shows that it performs well in both the expected revenue achieved and in running time. This leads to fundamental ramifications to many related OSN marketing applications.

Keywords-Pricing, Revenue Maximization, Marketing, Mone­tizing Online Social Network.

I. INTRODUCTION

With the rapid development of online social networks (OSN) such as Facebook, Twitter and Google+, large-scale, in­stantaneous information dissemination becomes possible. This offers the potential for a surge of innovations and opportunities in social advertising and viral marketing [1]. However, most of the existing studies focus on conducting effective influ­ence maximization analysis to propagate information widely instead of explicitly incorporating the pricing factor to design intelligent marketing strategies. In other words, extracting the economic value of social networks is still lacking [2]. Suppose a seller wants to sell a digital product via a social network, with intelligent selling strategies, i.e., pricing schemes, the revenue could be maximized. This is one way to monetize the social network, and is the research problem we will address in this paper.

There is a series of studies on viral marketing in the context of social networks. Domingos [3] and Richardson [4] are the first to formulate this as an algorithmic problem. They studied how to effectively promote a new technology or a new idea in an OSN by targeting a group of influential users. The seminal paper by Kempe et al. [5] then reformulated the problem further as "influence maximization" (INFMAX)

978-1-4799-1353-4/13/$31.00 ©2013 IEEE 3176

utilizing discrete optimization theory. Given a social graph, and a positive number k, INFMAX is defined as finding a seed-set of k nodes such that by targeting them initially, the viral spread is maximized.

While the INFMAX problem has been studied intensively, especially in the spreading of information and free products, it does not discuss the dependence of adoption on price. In eco­nomics, the pricing factor has been considered fully. However, the network effects are always addressed in a global or macro view. A recent stream of literature in computer science starts to study the "Revenue Maximization" (REVMAX) problem over social networks. The pioneering work by Hartline et al. [6] and Dismitris et al. [7] studied optimal marketing for digital goods in OSNs and proposed the "Influence and Exploit" (IE) strategy. In IE, seed nodes are offered products free of charge, and then the seller could approach other users individually to offer specific prices. However, price discrimination, although useful for revenue maximization in some settings, may result in negative reactions from buyers. For example Oliver and Shor [8] [9] suggest that price discrimination could reduce the likelihood of purchasing. Moreover, it is impractical for sellers to adjust prices too frequently. Hence, we will refor­mulate the REVMAX problem such that price discrimination is prohibited.

While many studies have been conducted on REVMAX, most existing work does not distinguish the different types of active node states by simply assuming that active nodes will definitely adopt the given product. Researchers from different disciplines have demonstrated the differences between awareness and adoption. In sociology, Rogers (1962) [10] proposed five stages of product adoption including knowledge, persuasion, decision, implementation, and confirmation. Be­sides, in management science, Kalish [II] also characterized the adoption of a new product as consisting of two stages, namely, awareness and adoption. In computer science, simu­lations using real world data sets [12] also show that diffusion models which utilize "influenced state" and "adoption state" to depict "active" could reflect reality well. In particular, [13] proposed a profit maximization algorithm under multi­state diffusion settings. However, they still follow the "IE" framework, i.e., price discrimination still exists. Moreover, in their work sequential pricing scheme is not addressed.

To address the aforementioned limitations, we propose to study the problem of revenue maximization over online social networks, by incorporating a multi-state diffusion scheme. In particular, we investigate marketing strategies to maximize

Globecom 2013 - Symposium on Selected Areas in Communications

TABLE I. NOTATIONS

Notation Description

N'i X,(t) X,(t) Y,(t) Ye( t) Z, (t) Ze(t) A(t)

Set of u/ s friends.

Initial set of users in ADOPT state at the beginning of pricing stage t Final set of users in ADOPT state at the end of pricing stage t Initial set of users in AWARE state at the beginning of pricing stage t Final set of users in AWARE state at the end of pricing stage t Initial set of users in INACTIVE state at the beginning of pricing stage t Final set of users in INACTIVE state at the end of pricing stage t Likelihood that any user is exposed to the external advertising

p, K

at pricing stage t Posted price of the given product at pricing stage t Total number of pricing stages

Wi. (Xi. (t)) User u;,' s valuation of the product, and it depends on the users

who already own the product, IX,(t)1 :S IX (t)1 :S IXe(t)1

revenue of durable digital goods, where the cost of producing a copy of the goods is negligible and a customer is only interested in buying a single copy, By designing an optimal sequential pricing policy based on dynamic programming, the maximum revenue could be achieved within reasonable time. Our contributions could be summarized as follows.

• We propose a multi-state diffusion scheme by in­corporating both the awareness propagation and the adoption diffusion. Here the awareness information is determined by social network effects and external influence (like mass media effects), while the actual adoption depends on the product price information and buyer's valuation of the product.

• We formally formulate the sequential pricing problem over OSN with a multi-state diffusion scheme. Given K possible stages, we determine a price vector p with K sequential prices to maximize the revenue. Recently, Akhlaghpour et al. [14] also studied the iterative pricing problem, but they did not consider the comprehensive diffusion process.

• We design an optimal pricing algorithm for any given K via dynamic programming.

• We conduct experiments using two real world datasets to evaluate our proposed algorithm. Experimental re­sults show that the algorithm performs well in the expected revenue achieved and in running time.

The rest of the paper is organized as follows. Section II discusses the proposed framework including the multi-state diffusion scheme and the REVMAX problem formulation. Section III presents our pricing strategies. Experiments are discussed in Section IV. Section V provides conclusions and future work.

II. PROPOSED FRAMEWORK

In this section we will firstly introduce the proposed multi­state diffusion model (MSDM) and then formally formulate the REVMAX problem. The notations are shown in Table I.

A. Diffusion Model

In this part, we introduce our multi-state diffusion scheme. The social graph can be represented in the form of a directed graph G = (V, E) , where V = {Ul' U2, ... , UN} is the set of nodes (users), and Ui is the unique ID of User i, E =

3177

I W,(x,) < p

I B,(X, Y,Z) I I W,(X,) ? p INACTIVE ADOPT

II-B,(X, Y,Z) I

Fig. I. The State Transition Diagram

0 Potential adopter

e External influence

• Acti ve neighbors

@ Aware neighbors

(3 Inactive neighbors

Fig. 2. The Awareness Diffusion

{ {U, V} Iu, V E V} is the set of edges. Each user belongs to one of three states: INACTIVE (User is not aware of the product), AWARE (User is aware or develops some interest about the product), and ADOPT (User decides to adopt the product). The state transition diagram is shown in Fig. I. Here, 9i(X, Y, Z) is the influence function for Ui, and it dictates u/ s awareness decision. Wi (Xi (t))) is Ui I S valuation function, and it depends on the set of u/ s friends who has already bought the product. Here, we assume that the users in our system are fully aware of their friends' state information. More details will be specified in the following subsections.

1) Awareness Diffusion: Inactive nodes become AWARE due to word-of-mouth effect or advertising. Thus the likeli­hood of becoming AWARE is proportional to the number of "transmitters" in the social network (i.e. X(t), Y (t), Z(t) and the exogenous advertising effects (i.e. A( t) . See Table I for the notations. Fig. 2 shows how different types of users influence the target user.

For a given user Ui, the awareness diffusion probability equation is shown as follows.

9.(X Y Z) = A(t) +aIXi(t)1 +f3IYi(t)1 +)Zi(t)1 (1) , , , . INil INil I INil

IXi(t)1 = IX(t) n Nil (2)

IYi(t)1 = IY (t) n Nil (3)

IZi(t)1 = IZ(t) n Nil (4)

This awareness equation says that, an inactive user ui will become AWARE with probability 9i(X, Y, Z), and remains INACTIVE with probability 1 - 9i(X, Y, Z). A(t) is the external influence, and other terms represent the influence exerted by the neighbors of Ui. Here, we make a close-world assumption that adopters transmit information more effectively than AWARE individuals, i.e., a > (3 > 0, and INACTIVE nodes do not have influential power, i.e., I = O.

Globecom 2013 - Symposium on Selected Areas in Communications

2) Adoption Diffusion: This section introduces the adoption diffusion scheme. Due to personal differences or human diversities, different people will value the same product differently. For example, the income, personal interests and actual needs will affect the adoption process. Existing literature [15] uses the term "reservation prices" to represent user valuation. A reservation price is the maximum amount a customer would be willing to pay given current available information. If the actual price is below the individual's reservation price, he will buy; otherwise, he will not. As more users adopt the product, the valuation of the product increases. At stage t, Wi (Xi (t)) represents U/ o5 valuation of the product.

(5)

Note that Ii represents U/ o5 intrinsic value in buying certain products and \If i (Xi (t)) is purely detennined by the set Xi (t). So we have \lfi(0) = 0, Wi(0) = h It means that even if no friends of Ui buy the given product, U/ o5 valuation is still not zero, since a customer may have certain intrinsic value for a given product.

If the current announced price is Pt, Ui will adopt the given product under the condition that Wi(Xi(t))�Pt. Here, we make the assumption that the seller does not know the buyer's exact valuation, but it does know the distribution F(W) from which the valuation is drawn. Moreover, we assume that users have positive externalities on each other, i.e., for two subsets, S, T � V, if S � T � V, then Wi(S) :::; Wi(T).

3) Diffusion Dynamics with Sequential Pricing: Instead of just publishing one unique price at all times, we allow the seller to post K public prices sequentially to promote the given product. Each pricing stage ends when no more buyers are willing to buy the product at the current price. Then the seller has to lower the price to the current maximum valuation among the AWARE nodes, and start the next pricing stage. The K-stage sequential pricing process ends if either of the following two conditions occurs: 1) The price has been posted for K stages; 2) The number of nodes in AWARE state is O. Obviously, following the proposed diffusion framework, the sequence of prices should be decreasing. The diffusion dynamics within a given pricing stage t is as follows.

• Find Ui, the node with maximum valuation within the AWARE set. Set the public posted price for the current stage Pt = Wi and set U/ o5 state to ADOPT.

• Update the states for U/ o5 existing neighbours using Algorithm 1 (Node Decision Algorithm).

• If newly ADOPT nodes or AWARE nodes are found, continue the diffusion process.

• The current pricing stage t ends if no more state transition is possible.

B. Problem Formulation

In this part we formulate the Revenue Maximization Prob­lem. Suppose there is a seller and a set V of potential buyers connected as in graph G = (V, E). The seller could post a public price which is visible to all nodes in an iterative way.

3178

Algorithm 1 Node Decision Algorithm

INPUT ui 8 current state o5i, current price Pt if o5i is INACTIVE then

Ui becomes AWARE with probability gi if SUCCESS then

if Wi > Pt then set o5i to ADOPT

else set o5i to AWARE

end if end if

else if o5i is AWARE then if Wi > Pt then

set o5i to ADOPT end if

else add Ui to ADOPT nodes queue

end if

When the buyer accepts the offered price, i.e., the price of the product is smaller than the buyer's valuation, the seller earns the price of the product as the revenue. Suppose the price at stage t is Pt, and the corresponding number of buyers is Bt, then the system revenue at pricing stage t, i.e., Rt, will be:

R(t) = Pt x Bt (6)

Consider that in the whole marketing period, the sys­tem could post at most K different prices, i.e., there are K pricing stages. Then our goal is to find a price vector

P = {PI, P2, ... , P K} of K prices to maximize the cumulative revenue 2:0<t<::K R(pt), then the optimal pricing sequence is:

p = argmax "'"' R(t) = argmax "'"' PtBt (7) �O<t<::K �O<t<::K This is defined as the REVMAX(K) problem.

III. METHODOLOGY

In this section, we first study a restricted case of the REVMAX(K) problem, and show that we can identify the optimal pricing strategy based on a simple dynamic program­ming approach. For the general case, due to the complexity of the problem it is hard to find an optimal solution. Hence we propose a Dynamic Programming Based Heuristic (DPBH) to achieve a good approximation for REVMAX(K) within polynomial time.

A. A Special Case of REVMAX(K)

To better understand the properties of REVMAX(K), we first study a special case of the problem. We assume that a user's valuation of the given product is deterministic, i.e., for each Ui, we have Wi(X) = Wi. For simplicity, the awareness diffusion probability gi (X, Y, Z) is generated before the diffusion starts. Under this setting, we generate a pricing vector p = {PI, P2, ... , PT}, PI > P2··· > PT following the greedy algorithm (Algorithm 2). Here, T is the total number of pricing stages required to get all potential users to adopt.

The system revenue here is calculated as

revenue = PIXP1 + P2XPIP2 + ... + PTXPIP2 .. PT (8)

Globecom 2013 - Symposium on Selected Areas in Communications

Algorithm 2 Pricing Sequence Generation

Set t=I, then Pt = max(Wi), for each Ui in V Add Pt to vector p update node states in node set V according to Algorithm I t = t+I while AWARE node set is not empty do

set Pt = max(Wi), for each Ui in AWARE set add Pt to vector p update node states in node set V

end while return vector p

Here, XPIP2",PT is the number of adopters who adopt the item at Stage T when the price is Pj at stage j, j <::: T. Let XPT be the number of adopters who bought the item when only one pricing stage is allowed in which the public price is set to be PT. Thus XP1 + XP1P2 + ... + XpIP2 ... PT = XpT, which means that XpIP2 ... PT = Xp./, - XpT_1'

Then the REVMAX(K) problem is to find a decreasing sequence of K prices from the vector p to maximize the revenue. In computer science, "A problem exhibits optimal substructure if an optimal solution to the problem contains optimal solutions to the sub-problems." [16]. The optimal substructure of REVMAX(K) is as follows:

R[K,p] = max {R[K - 1,p'] + p'(Xp' - Xp)} (9) O<p'<p

where R[K,p] represents the maximum revenue achieved when the number of pricing stages is K and the maximum price is p. Then REVMAX(K) under the restricted setting may be solved using dynamic programming.

B. The General Case of REVMAX(K)

Now we consider the general REVMAX(K) problem. While it is difficult to get an optimal solution, using heuristics, we could get a polynomial time approximation regardless of the specific form of the valuation distribution. Our proposed Dynamic Programming Based Heuristics (DPBH) consists of the following four steps:

• Specify the price bounds Pmin and Pmax, ensuring that the optimal pricing sequence p = {P1, P2, ... , P K } satisfies Pi E [Pmin, Pmax], i = 1,2, ... , K.

• Construct candidate price set using Algorithm 3.

• Simulate the diffusion process for sufficient runs to get the expected buyer number for all prices within the candidate price set.

• Use dynamic programming to choose the optimal pricing sequence based on the optimal substructure:

R[K,p] = max {R[K -l,p']+p'(Xp'-Xp)} (10) O<p'<p

Let the size of the candidate price vector generated by

Algorithm 3 be Ai[. Ai[ = IIOg;;F l' and the computational

complexity of our proposed DPBH is KM = K Ilog;�'Fl where K is the number of pricing stages.

3179

Algorithm 3 Candidate Price Set Construction

set imin = llogf+�' J, imax = Ilogf+�x l for i = imin to imax do

Pi = (1 + E)(imax+imin-i) end for return p

TABLE II. STATISTICS OF THE TWO REAL-WORLD NETWORKS

Dataset arXiv HEP-PH DBLP community

Nodes 11,204 1,772 Edges 235,238 14,984

Graph density 0.0019 0.0048

Now we discuss the performance of the proposed heuris­tic. Considering that both the awareness diffusion and user valuation function are probabilistic, the adopt number Xp and valuation function Wi(X) are all random variables. Then our goal is to maximize the expected revenue E[revenue(K)] with K pricing stages. By simulating the diffusion process for a polynomial number of trials, we could use the average adoption number as an estimation of the expected adopt number.

Given the estimated optimal sequence obtained from our proposed DPBH as Pdp = [Pdp1,Pdp2, ""PdpK], then the corresponding estimated achievable maximum revenue is

revenue(K) = Pdp1XPdPl +

Pdp2(XPdr>2 - X;:;:) + ... +

PdpK(XpdPK - XPdP(K_l»)

Meanwhile, the theoretical optimal revenue is:

E[revenue(K)] = PoptlE[XpoPtl]+

(11)

Popt2(E[XpoPt2] - E[XpoPtl]) + ... + (12)

PoptK(E[XpOPtK] - E[XpOP1(K_l)])

Under certain conditions, in which the random variable Xp is i.i.d, then the maximum revenue generated by the DPBH, revenue(K), can provide a performance guarantee within a factor of (1

1;00)2 of the optimal revenue,

E[revenue(K)]. The sketch of the proof is as follows. Since Xp is i.i.d and 0 <::: Xp <::: lVI, using Chernoff­Hoeffding theorem, we can show that with high probability (1 - E)E[Xp] <::: Xp <::: (1 + E)E[Xp]. Suppose Ui buys the product at price (1 + Ey < Popt < (1 + Ey+1 in the optimal solution, then Ui will buy the product when the price is (1 + EY in our solution, and we have (1 + EY > �. Combining the above approximations of the adopt number and the candidate price set construction, we show that the maximum revenue achieved using our heuristic is within

1 0 f h . (1;0 )2 0 t e optImum.

IV. EXPERIMENTS

A. Experimental Setup

Dataset. We use two real-world networks of different graph densities in our experiments. The first dataset, arXiv HEP­PH (High Energy Physics - Phenomenology) collaboration

Globecom 2013 - Symposium on Selected Areas in Communications

10000 9500 ,. [-+- DPBH Gaussian

--e-- Random Gaussian -+- DPBH Gaussian --e- Random Gaussian

:;; 9000 .0

g 1000 § z 8500 D- 15. 0

"0 8000 « 500

7500

�L---�1�0 --�20�--�30�--4�0----� Pricing Stages

\L---�2�0--�4�0--�6�0--�8�0--�100 Pricing Stages

70000L---�20�--4� 0--�6�0--�8C-0 --�1 00 Pricing Stages

(a) DPBH with 50 pricing stages (b) Revenue vs number of pricing stages with the Gaussian distribution

(c) Adopt number vs number of pricing with the Gaussian distribution

stages

g 1000 D-

500

600° r:-�=._iiiMi!iiiiiiiiiiiiiiiiliiiiiiliiiliiliiiiiiiliiiiii� r r I -+- DPBH Uniform 5500 �t --e-- Random Uniform

25000 E :i 4500 15. o

� 4000 3500

�L---�2�0 --�40�--�60�-- 8�0--�W � � � � 100 30000·L---�2�0 --�40�--�60�-- �80�-----"1 00 Pricing Stages Pricing Stages Pricing Stages

(d) DPBH with 100 pricing stages (el Revenue vs number of pricing stages with the Uniform distribution

(f) Adopt number vs number of pricing with the Uniform distribution

stages

Fig. 3. Experimental Results

network, is from the e-print arXiv and covers scientific collab­orations submitted to HEP-PH category. The edges represent the collaboration relationships between authors. Here we use the largest connected component of the arXiv HEP-PH dataset. The second graph, the DBLP computer science bibliography provides a comprehensive list of research papers in computer science. Here we just focus on one comparatively denser community of the DBLP graph. The statistics for both graphs are shown in Table II. Here, the edges are directed, so the density metric is calculated as IEI/(IVI(IVI - 1)).

Algorithms. Here we compare our proposed DPBH with the baseline random algorithm which chooses the pricing sequence randomly. For each given input, 1000 simulation runs are used to obtain an accurate estimate of the adoption spread. For all buyers, we adopt Gaussian distribution and Uniform distribution to model user's valuation behaviour. The Gaussian distribution has been justified using real datasets and utilized in existing papers [13] [17]. For unfamiliar products, the Uniform distribution is used to account for lack of knowledge [18] [19].

B. Experimental Results

In this section we show the experimental results of four metrics. To save space, we use Fig 3 to illustrate the first three metrics.

The Optimal Pricing Sequence. Fig. 3(a) and Fig. 3(d) show the optimal sequence for K = 50 and K = 100, respectively. As shown in both figures, the decreasing trend of the price sequence will be affected by the valuation function. For the Uniform distribution, it drops linearly. For the Gaussian distribution, the trend is comparatively less steep.

3180

Revenue vs Number of Pricing Stages. Fig. 3(b) and Fig. 3(e) study the increase of maximized revenue as the number of pricing stages increases, under the Gaussian and Uniform distributions, respectively. In both cases, the maximum achiev­able revenue increases with the increase of pricing stages. However, there is an obvious diminishing returns phenomenon, i.e., within a limited number of pricing stages (around 20 for DPBH, and 30 for the random algorithm), the seller could extract almost the same revenue that is achieved by more pricing stages.

Adopt Number vs Number of Pricing Stages. Fig. 3(c) and Fig. 3(t) study the increase of adopt number as the number of pricing stages increases under the Gaussian and Uniform distributions, respectively. The results show that maximized influence (adopt number) does not necessarily mean maxi­mized revenue. Under the Gaussian distribution, the random algorithm performs equally well as DPBH for the adopt number metric. Under the Uniform distribution, while DPBH performs much better than the random algorithm in terms of the revenue, the random algorithm outerperforms DPBH in terms of the adopt number.

Graph Density vs Pricing Strategy. Finally, we study how the network topology may influence the pricing strategy. Here we use two real world networks with different graph densities. Fig. 4 and Fig. 5 show how the pricing strategy changes under different network structures, for the Gaussian and the Uniform distributions, respectively. Here, the DBLP network is almost three times denser than the arXiv network. Although the magnitudes of the maximum achievable revenue are different due to the differences in the network scale, the diminishing returns phenomenon still holds for both networks, for both the Gaussian and the Uniform distributions.

Globecom 2013 - Symposium on Selected Areas in Communications

Fig. 4.

X 108 x 10e 11 r--':diiiiiill�IIIIIIIIIIIlIlllllIIIIIIIlllllIlllllIIIIII,1 .25

C", 10.5 r l --+- arXiV Gaussian 1.2 C "Ci5 10 --e--DBLP Gaussian 1.15·� � � (B 9.5 P 1.1 (9 > 0.. � 1.05 ffi "'" e Q) 8.5 (j) ::J ::J " (j) &

" 0.95 � 7.5 0.9 0::::

70"---'=0------::40=-------=6:- 0 ---8= 0--�10g·85 Pricing Stages

Graph Density vs Pricing Strategy with the Gaussian Distribution

�10e x10� 8��--�----�--�==========�5 l -+-arXiV Uniform -e-DBLP Uniform

�------""" I r 4 5

p i (j)

3 �

, "---�--�--�--�--� , o 20 40 60 80 100 Pricing Stages

(j) &

Fig. 5. Graph Density vs Pricing Strategy with the Uniform Distribution

V. CONCLUSION

We study the problem of revenue maximization over online social networks, by incorporating the multi-state diffusion scheme. In particular, we investigate marketing strategies to maximize the revenue of durable digital goods. By designing a sequential optimal pricing algorithm using dynamic pro­gramming, the maximum achievable revenue could be obtained within polynomial time. Moreover, our algorithm could pro­vide performance guarantee under specific cases. Experimental results using real-world datasets demonstrate the effectiveness of our proposed heuristic.

In our current studies, the cost of the product and repeat purchase effect are not considered. We will consider these issues in our future work.

REFERENCES

[1] J. Leskovec, L.A. Adamic, and B.A. Huberman, "The dynamics of viral marketing;' ACM Trans. Web, vol. I, no. I, May 2007. [Online]. Available: hup://doi.acm.org/IO.1145/1232722.1232727

[2] J. Kleinberg, "Cascading behavior in networks: Algorithmic and eco­nomic issues;' in Algorithmic Game Theory, Cambridge University Press, 2007, pp. 613-632.

[3] P. Domingos and M. Richardson, "Mining the network value of customers," in Proceedings of the seventh ACM SIGKDD international conference on Knowledge discovery and data mining, ser. KDD '01. New York, NY, USA: ACM, 2001, pp. 57-66. [Online]. Available: hUp://doi.acm.org/l0.1145/502512.502525

[4] M. Richardson and P. Domingos, "Mining knowledge-sharing sites for viral marketing," in Proceedings of the eighth ACM STGKDD international conference on Knowledge discovery and data mining, ser.

3181

KDD '02. New York, NY, USA: ACM, 2002, pp. 61-70. [Online]. Available: hup://doi.acm.org/IO.11451775047.775057

[5] D. Kempe, J. Kleinberg, and E. Tardos, "Maximizing the spread of influence through a social network," in Proceedings of the ninth ACM SIGKDD international conference on Knowledge discovery and data mining, ser. KDD '03. New York, NY, USA: ACM, 2003, pp. 137-146.

[6] J. Hartline, V. Mirrokni, and M. Sundararajan, "Optimal marketing strategies over social networks," in Proceedings of the 17th international conference on World Wide Web, ser. WWW '08. New York, NY, USA: ACM, 2008, pp. 189-198. [Online]. Available: hUp://doi.acm.org/1 0.1145/1367497.1367524

[7] D. Fotakis and P. Siminelakis, "On the efficiency of influence­and-exploit strategies for revenue maximization under positive externalities;' in Internet and Network Economics, ser. Lecture Notes in Computer Science, P. Goldberg, Ed. Springer Berlin Heidelberg, 2012, vol. 7695, pp. 270-283. [Online]. Available: hUp://dx.doi.org/l 0.1007/978-3-642-35311-6_20

[8] R.L. Oliver and M. Shor, "Digital redemption of coupons: satisfying and dissatisfying effects of promotion codes," Journal of Product & Brand Management, vol. 12, pp. 121 - 134, 2003.

[9] M. Shor and R.L. Oliver, "Price discrimination through online couponing: Impact on likelihood of purchase and profitability," Journal of Economic Psychology, vol. 27, no. 3, pp. 423 - 440, 2006. [Online]. Available: hUp://www.sciencedirect.com/science/artic1e/ pii/SO 167487005000917

[10] E. Rogers, Difji,sion of innovations. Free Press of Glencoe, 1962. [Online]. Available: hup://books.google.com.hk/books?id= zwO-AAAAIAAJ

[11] S. Kalish, "A new product adoption model with price, advertising, and uncertainty," Management Science, vol. 31, no. 12, pp. pp. 1569-1585, 1985. [Online]. Available: hUp://www.jstor.org/stable12631795

[12] S. Bhagat, A. Goyal, and L.v. Lakshmanan, "Maximizing product adoption in social networks," in Proceedings of the .fifth ACM international conference on Web search and data mining, ser. WSDM , 12. New York, NY, USA: ACM, 2012, pp. 603-612. [Online].

Available: hup://doi.acm.org/1 0.114512124295.2124368

[13] W. Lu and L. Lakshmanan, "Profit maximization over social networks," in Data Mining (ICDM), 2012 IEEE 12th International Conference on, 2012, pp. 479-488.

[14] H. Akhlaghpour, M. Ghodsi, N. Haghpanah, V.S. Mirrokni, H. Mahini, and A. Nikzad, "Optimal iterative pricing over social networks," in Proceedings of the 6th international conference on Tnternet and network economics, ser. WINE' 10. Berlin, Heidelberg: Springer­Verlag, 2010, pp. 415-423. [Online]. Available: hup://dl.acm.org/ citation.cfm?id= 1940179.1940215

[15] R. Schmalensee, "Product differentiation advantages of pioneering brands," American Economic Review, vol. 72, no. 3, pp. 349-65, June 1982. [Online]. Available: hup://ideas.repec.org/a/aea/aecrev/ v72yI982i3p349-65.html

[16] C.E. Leiserson, R.L. Rivest, C. Stein, and T.H. Cormen, Introduction to algorithms. The MIT press, 2001.

[l7] A.X. Jiang and K. Ley ton-Brown, "Estimating bidders?valuation distri­butions in online auctions," in Proceedings of l.lCAT-05 Workshop on Game Theoretic and Decision Theoretic Agents, 2005.

[18] Y. Shoham and K. Ley ton-Brown, Multiagent systems: Algorithmic, game-theoretic, and logical foundations. Cambridge University Press, 2008.

[19] F. Bloch and N. Querou, "Pricing in net-works," 2011, working Paper, Unpublished. [Online]. Available: hUp://www.ea4.ehu.es/s0031-con/es/contenidos/informacion/ 00031_seminarios/es_00031_se/adjuntoslF. %20Bloch. pdf