Optimal Dynamic Referrals in Peer-to-Peer Content Distribution Peng Han+, Kartik Hosanagar*, Yong Tan+

Contact: Kartik Hosanagar The Wharton School of Business, University of Pennsylvania, 3730 Walnut Street, Suite 500 Philadelphia, PA 19104-6340 [email protected] Phone: 215 563 6848 Fax: 215 898 3664

Acknowledgements: The authors would like to thank Christophe Van den Bulte for helpful comments and suggestions.

+ *

University of Washington Business School, Box 353200, Seattle, Washington 98195-3200 The Wharton School, University of Pennsylvania, Philadelphia, PA 19104-6340

Optimal Dynamic Referrals in Peer-to-Peer Content Distribution Abstract Peer-to-Peer (P2P) based content distribution is fast gaining popularity for delivery of information products such as music and videos. A widely documented problem in P2P networks has been the large number of free riders – users who consume content from others in the network without redistributing it to other users. Academics and practitioners have proposed offering distribution referrals – payments to users who distribute content to others – to provide incentives for users to distribute content. This research represents one of the first studies of a P2P firm’s optimal referral strategy. We begin with a simple diffusion model in P2P networks and consider a policy in which a constant referral is provided throughout the content diffusion. Our results indicate that the optimal referral decreases with demand for content and number of unselfish users in the network. Motivated by the intuition that a referral early on plays a more significant role in content diffusion than a referral in later periods, we present a dynamic referral strategy where the firm continuously adjusts the referral. For a short-term marketing campaign, the optimal strategy involves a constant high referral at the beginning, followed by a convex decreasing trajectory that may reach zero. Interestingly, the referral may start to increase in the final few periods due to an end-of-period effect. In the infinite time horizon case, after the constant referral period, the optimal referral trajectory always decreases and converges to a stationary value. Our research represents a first step towards studying marketing issues in this emerging distribution format for entertainment and information products. Keywords: Peer-to-Peer content distribution; dynamic referrals; product diffusion in networks; networks and marketing; Internet marketing

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1. Introduction Peer-to-Peer (P2P) networks are distributed computer networks where users not only consume content, but also provide content to others. In P2P networks, most of the content resides on the users’ computers rather than on a centralized server. Thus, when a new user is interested in downloading content such as a music file, it is distributed from the computer of another user that already has the file. The content exchanged in P2P networks can be any information product, although music and movies tend to be most popular. The past few years have witnessed tremendous growth in P2P networks. By the end of 2004, there were at least 10 million people actively using P2P file-sharing technology at any time (ITIC 2004), gaining access to more than a billion music, video, software and image files (Redshift Research 2005), and generating more than half of the Internet traffic (Ross and Rubenstein 2003). P2P networks are increasingly being used for legitimate content distribution including music, video and software distribution. Compared to the high cost of setting up a centralized infrastructure, P2P allows a firm to efficiently distribute content at a relatively low cost. Further, the distribution infrastructure automatically scales as new users are added. For example, Altnet, PeerImpact, iMesh and Mashboxx are firms that use P2P networks to distribute music to users. The music is licensed from the music majors and other independent labels, the files are distributed from users’ machines and the P2P firms provide software and billing support. In addition, several media firms including AOL and NBC Universal have announced plans to launch Internet TV services using P2P technologies to allow users to download videos on demand. Similarly, P2P is being used for online radio services (e.g., Radio Free Virgin) and software delivery (e.g., distribution of the Linux RedHat operating system using BitTorrent P2P technology). A number of technologies have also emerged to prevent piracy in P2P networks. As a result, distribution and sale of information products through P2P networks are likely to become more prevalent. We consider the case of a firm that sells an information product, such as a music track, through a P2P network. Nodes or users in the P2P network are potential consumers of the product and, by virtue of the design of P2P networks, can redistribute the product. Once a few users in the network have a file, they can distribute the file to other interested users. However, a well documented problem in P2P networks is that of free riders – users that download content from other users in the network without allowing others to locate and download content from

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their machines (Adar and Huberman 2000, Asvanund et al 2004). Free riding can significantly reduce the value of P2P content distribution because the success of P2P distribution relies on file availability in the network, which in turn depends on the willingness of users to disclose and distribute content stored on their computers. A few approaches have been proposed to alleviate this problem, for example, offering higher Quality of Service (QoS) to users that share their resources (Cohen 2003, Kamvar et al. 2003). In commercial P2P networks, where users pay for content, firms may give some form of payment to users who distribute the content to other users in the network. Golle et al (2001) discuss the use of micropayments to reward peers every time they distribute content. Arora et al (2003) also examine a payment program wherein a user pays for downloading content and gets paid a small amount for distributing the content to another user. We will refer to such a payment as a referral-per-download or simply a referral. It is useful to distinguish the referrals in our paper from traditional referrals. Traditional referral fees are paid to existing customers for bringing a new customer to the firm. In contrast, the referral in the P2P setting encourages existing customers to share their content so another user can easily locate and download the content. This increases the file availability in the network which in turn increases sales. Thus, the impact is indirect. It has also been argued that the use of such payments encourages participation in legitimate P2P networks (DCIA 2005, Lang and Vragov 2005). Indeed, a number of commercial P2P systems use payments to provide incentives for sharing and distributing content to other users. For example, Altnet, a P2P search monetizing firm, pays users on the Kazaa P2P network who agree to join Altnet as distribution points. Whenever a file is distributed from a user’s computer, the user receives payments such as frequent-flier miles, free hotel stays or other similar items (CNet News 2003). Another firm, Shared Media Licensing Inc., uses a similar payment mechanism. The primary objective of this paper is to study the impact of these referrals on content diffusion in a P2P network and determine the optimal referral trajectory. The referral payment offered to users for distributing content plays a key role in determining the rate of content diffusion. Intuitively, offering a higher referral fee serves to increase content availability and consequently the rate of content diffusion. But at the same time, it reduces the margins per sale. We analytically study this trade-off and determine the optimal referral strategy. Our key finding is that the optimal referral trajectory involves three stages – the first stage entails offering a high

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aggressive referral to encourage sharing and distribution; the referral gradually decreases in stage two and then converges to a stable value in stage three. When the planning horizon is short, the third stage may not exist and further the referrals may start to rise in the final few periods due to an end-of-period effect. P2P is an emerging sales format for entertainment and other information products. Our research represents a first step towards studying marketing issues in this new sales and distribution format. The rest of the paper is organized as follows. In Section 2, we discuss the relevant literature and position our work relative to work on new product diffusion and viral marketing. In Section 3, starting with the model in Han et al. (2004), a simplified diffusion model is presented to capture the dynamics of P2P content distribution. A simple, easy-toimplement, constant referral strategy is also presented. In section 4, we formulate the dynamic referral problem as an optimal control problem, and determine the optimal referral trajectory. We discuss our results in Section 5. Finally, section 6 concludes this study and discusses extensions.

2. Prior Work Study of diffusion in a network setting is a relatively new area. Han et al. (2004) analytically model the content diffusion process in P2P networks. In addition to demonstrating that the diffusion process follows a classic S-shape curve, their model also shows that the referrals have a significant impact on the diffusion speed. Izal et al (2003) present an empirical study of product diffusion in the BitTorrent P2P network. Asvanund et al (2004) empirically examine the impact of network externalities on value and optimal size of P2P music sharing networks. Lang and Vragov (2005) propose a dynamic pricing model for distributing content in P2P networks and compare the result with a centralized client-server distribution system. While their work reveals many economic insights, they do not model the P2P diffusion process or consider dynamic payments. Direct research on P2P content distribution in the marketing community is limited. However, there are several streams of work that are very relevant to the study of content distribution in P2P networks. We discuss some of these streams below. New product diffusion models: There is a vast body of literature on new product diffusion, dating back to 1960s. Building on the Bass model (Bass 1969), work has also been done to incorporate effects of advertising and promotion (Horsky and Simon 1983; Lillien et al 1981), competition (Krishnan et al. 2000), and pricing (Bass 1980) to the diffusion model. There has

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also been considerable work done on estimation procedures for the model (Bemmaor 1994; Schmittlein and Mahajan 1982). Usually, the marketing mix has been modeled using a multiplicative term (Robinson and Lakhani 1975; Bass 1980) or incorporated in the market potential (Kalish 1983) or the coefficient of innovation parameters (Horsky and Simon 1983). As we will see later in Section 3, our referral fee encourages current adopters to distribute content and hence directly appears in a viral or imitation term of the diffusion model. Thus, the optimal referral trajectory computed from our model is fundamentally different from, say, the optimal price path (Dolan and Jeuland 1981; Kalish 1983) that has been analyzed in the literature. Viral Marketing: Research work on viral marketing or word of mouth (WOM) effects has explored how adopters can influence other consumers to also adopt. The phenomenon has been modeled by Bannerjee (1992), who presents a model of consumer herd behavior. The importance of social contagion on product sales and diffusion has been widely recognized. In fact, the original Bass model for new product diffusion (1969) also incorporated a coefficient of imitation to model the influence of current adopters on future adoption. Kalish (1983) and Horsky (1990) discuss the impact of WOM effects on the price path chosen by a monopolist. Van den Bulte and Lillien (2003) find evidence of social contagion in diffusion of antibiotics. Godes and Mayzlin (2004) propose measures for word-of-mouth effects in online settings such as discussion forums. The referral fees in P2P networks seek to encourage users who currently own a file to share and distribute the file. This increases availability of the file and thus sales. Thus, the referral fee increases contagion within the P2P network. In this sense, P2P referrals are similar to viral marketing initiatives such as the use of referral rewards to enhance WOM effects (see for e.g., Biyalogorsky et al 2001) However, there are several differences between traditional viral marketing and P2P referrals. Traditional viral marketing is a marketing initiative and does not require setting up a distribution infrastructure to facilitate the same. Further, viral marketing has focused on increasing product interest or awareness through social contagion. However in the P2P setting, the idea is to use current adopters as distribution points and hence reduce the distribution costs. Thus, a separate P2P distribution infrastructure is also needed. Further, the primary goal of the referral in P2P is not necessarily to increase awareness but to ensure that users interested in purchasing a copy of the file are able to locate a distribution point.

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3. The Diffusion Process We begin by discussing our setting and listing our primary assumptions. We consider a firm that sells an information product such as a music track on a P2P network. The firm can be the copyright holder of the content or the online retailer (i.e., the P2P firm) or both.1 Also, we limit our focus to the simplest P2P network configuration, namely a completely flat P2P network. In a flat P2P network, all nodes are symmetric in terms of their roles and connectivity in the network. Figure 1 provides an example of a flat P2P networks. Although many P2P applications adopt hierarchical structures, the dynamics of the diffusion process are fundamentally similar in flat and hierarchical P2P networks (Han et al. 2004), but the diffusion model for flat networks is more tractable.

Figure 1. A Flat P2P Network Suppose there are N nodes in the network and there is no intermediary, i.e., a node that buys and sells the content while not needing it. This implies that N is also the market potential, or the total number of nodes that may eventually buy the content. The nodes seeking the content are called seeking nodes, while the nodes that have the content already are called satisfied nodes. The nodes that are satisfied and willing to share and distribute the content with others are called seeds. Note that a satisfied node does not have to be a seed, but a seed must be a satisfied node first. The number of satisfied nodes and seeds at time t are denoted by Q(t) and S(t) respectively. The average number of requests for a file per unit time per seeking node, β, reflects the demand 1

For example, cooperative promotions between retailers and labels (copyright holders) are also common in the music industry.

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for that content in the network. Note that we assume that consumers are myopic and do not consider the (expected) future revenue from referrals when adopting. In Section 6, we discuss an extension on diffusion and dynamic referrals with forward looking buyers. Since the network is flat, whenever a seeking node seeks content, a request is randomly sent to another node in the network. If the request hits a seed, the file will be transferred to the seeking node, which then becomes a satisfied node. Otherwise, the request fails and the status of the seeking node remains unchanged. While we assume a seeking node forwards only one request, the diffusion process is similar even when multiple requests are sent out as long as the number of nodes queried is small relative to the size of the network (Han et al 2004). In addition, we ignore the detailed connection patterns in the network. Thus, the dynamics of the diffusion depends only on the number of seeds, satisfied nodes and seeking nodes in the network rather than which particular nodes are the seeds. This is reasonable given the random formation of such decentralized P2P networks and their large sizes. When a seed distributes a copy of the content to a seeking node, it gets a referral payment of r(t) from the firm. Without loss of generality, we assume that, within the total population, a fraction α are always willing to be seeds (given that they have the content) even if there is no compensation for doing so. These nodes are called unselfish nodes. The rest of the population are assumed to be “rational,” i.e. they will not be a seed unless the benefit is greater than or equal to the cost of being a seed. The node’s cost of providing a download is denoted by c, which is assumed to be uniformly distributed across the population from 0 to C. A satisfied rational node will be a seed if and only if c ≤ r(t). Thus, the fraction of satisfied rational nodes that will be seeds at time t is given by r (t ) / C . The assumption of unselfish nodes is not critical as α may be set equal to zero. However, there is empirical evidence for the presence of both unselfish nodes and rational nodes in other settings where free-riding incentives exist, such as P2P technical forums (Gu and Jarvenpaa 2003). Table 1. Glossary of Terms t

Time

N

Total number of nodes in the network

Q(t)

Number of satisfied nodes at time t, Q(0) > 0; q = Q / N

S(t)

Number of seeds at time t, S(0) > 0; s = S / N

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r(t)

Referral per sale offered at time t by firm to the distributing node

α

Fraction of nodes that are unselfish

β

Average number of requests per unit time per seeking node

c

A node’s cost of providing a download, Uniformly distributed in [0,C]

C

The upper bound of the cost of providing a download

We now proceed to the model. The notation is summarized in Table 1. Given the system configuration described above, the number of seeds in the network at any time is, ⎧⎛ r (t ) ⎞ ⎪⎜ α + (1 − α ) ⎟Q(t ), if r (t ) ≤ C ; S (t ) = ⎨⎝ C ⎠ ⎪⎩Q(t ), otherwise.

(1)

Within S, the first part, αQ represents the number of unselfish satisfied nodes, while the second part refers to the number of rational nodes that will agree to be seeds. It is easy to see that when there is no referral, i.e. r = 0, the diffusion will solely rely on the goodwill of the unselfish nodes. Considering the large size of the population, it is reasonable to assume that the upper bound of sharing cost, C, is high enough so that r ≤ C always holds. That is, there is at least one individual who would rationally not distribute content. Thus, we focus on the more interesting case where S < Q in following analysis. The average rate at which seeking nodes become satisfied nodes, i.e. the time dynamic of Q is,

dQ S = β ( N − Q) = β (1 − q) S , dt N

(2)

Given that there are ( N − Q) seeking nodes at some time, β ⋅ ( N − Q) is the number of requests in that period. The probability that a request is satisfied is simply S / N , because every node has an equal probability of being queried.2 Plugging (1) into (2) and with r (t ) = r , we get dQ r dq r ⎛ ⎞ ⎛ ⎞ = β ⎜ α + (1 − α ) ⎟(1 − q)Q , or = β ⎜ α + (1 − α ) ⎟(1 − q )q . dt C dt C ⎝ ⎠ ⎝ ⎠

(3)

Given that the node will not query itself, the probability of a request being satisfied is actually S/(N-1). For a large network, we approximate this to S/N

2

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Solving the differential equation with the initial condition of Q(0) = 1 (or, q(0) = 1/N), we obtain the following function describing the diffusion process. 1 , 1 + ( N − 1)e − A( r )t

(4)

⎞ q 1 ⎛ ⎜⎜ ln + ln( N − 1) ⎟⎟ A(r ) ⎝ 1 − q ⎠

(5)

q (t , r ) = or t ( q, r ) = where

A(r ) = β (α + r (1 − α ) / C )

Based on (4), a typical diffusion curve is plotted in figure 2 for N = 1000, Q(0) = 1, C = 1, α = 0.05, β = 1, and r = 0.4 and 0.7. As can be seen, higher referrals serve to accelerate the diffusion process. An interesting problem that the firm faces is how to determine the referral to offer. While offering a large amount of referral can reduce the time to achieve a certain level of market penetration, it also reduces the margin per sale at the same time. The marginal reduction in the time to achieve a certain level of market penetration q is ⎛ ⎞ ∂t ∂t q = − = (1 − α )⎜⎜ ln( ) + ln( N − 1) ⎟⎟ ∂r ∂r ⎝ 1− q ⎠

⎛ r (1 − α ) ⎞ C⎜ +α ⎟ β . ⎝ C ⎠ 2

(6)

It is easy to see that ∂t / ∂r , the marginal benefit of referral, is decreasing in r. Figure 3 shows the relationship between t(q = 0.8), the time to achieve 80% market penetration, and r. The decreasing returns from referrals imply that there must be an optimal referral that the firm must offer to balance the benefit from faster diffusion with the cost of the referral. In the following sections, we analyze the referral optimization problem for two cases, namely a constant referral strategy, where the firm applies a constant referral throughout the diffusion process, and a dynamic referral strategy, where the firm adjusts the referral over time to maximize profit.

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1

q

0.8 0.6 r = 0.7

0.4

r = 0.4

0.2

t

0 0

10

20

30

Figure 2. A typical diffusion curve 140

t (q = 0.8)

120 100 80 60 40 r

20 0 0

0.2

0.4

0.6

0.8

1

Figure 3. Impact of r on the time to achieve 80% penetration

3.1. Constant Referral Strategy We begin by analyzing a simple referral scheme which entails providing a fixed referral throughout the product diffusion. Such a referral scheme is easy to implement. It is also easy to evaluate from the consumers’ standpoint. We will also analyze a dynamic referral policy in the next Section. Assume that the unit price of the product is constant and normalized to 1. Therefore, the profit from each sale is (1 – r). Considering the time discount effect, the present value of selling a copy at time t is (1 – r)e–ρt, where ρ is a discount factor. The optimization problem, assuming at the beginning there is only one seed in the network, is max π = N (1 − r ) ∫ r

1

1/ N

e − ρt ( q ,r ) dq ,

where t(q, r) is given by (5). When N is large, 1/N → 0. Therefore,

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(7)

1

π ≈ N (1 − r ) ∫ e − ρt ( q ,r ) dq = N 0

ρ ⎞ ⎞ − A ln( N −1) A ⎛ C ⎛A ⎜1 − ⎟ α e ⎜ − ⎟ . ⎟⎟ A + ρ ⎜⎝ (1 − α ) ⎜⎝ β ⎠⎠

(8)

Note that A is a linear function of r as specified in equation 5. Finding the optimal r is equivalent to finding the optimal value of A. Taking the first order derivative of (8) with respect to A, we have, −

ρ

dπ N ( N − 1) A =− A 3 C + A 2 X + AY + Z , 2 dA A(1 − α ) β ( A + ρ )

(

)

(9)

where X = (2 + ln( N − 1))Cρ , Y = Cρ 2 ln( N − 1) − (1 − α + Cα )(1 + ln( N − 1)) βρ , and Z = −(1 − α + Cα ) ρ 2 β ln( N − 1) . Setting dπ / dA = 0, the optimal value of A is, A* = −

X 2(3CY − X 2 )21 / 3 W 1/ 3 − + , 3C 3CW 1 / 3 3C 21 / 3

(10)

where W = 9CXY − 2 X 3 − 27C 2 Z + 4(3CY − X 2 ) 3 + (9CXY − 2 X 3 − 27C 2 Z ) 2 . Therefore, ⎞ C ⎛ A* ⎜⎜ − α ⎟⎟ . r = (1 − α ) ⎝ β ⎠ *

1

(11)

r*

0.8 0.6 0.4 0.2

β

0 0

2

4

6

8

Figure 4. Optimal referral r* vs. request rate β

12

0.8

r*

0.6

0.4

0.2

ρ

0 0

0.1

0.2

0.3

0.4

0.5

Figure 5. Optimal referral r* vs. discount factor ρ

In Figures 4, 5, and 6, we plot r* against β, ρ, and α for N = 1000, Q(0) = 1, C = 1, α = 0.05,

β = 1, and ρ = 0.12. As can be seen, r* decreases with the average request rate, β. High request rate, i.e. strong demand, helps speed up the diffusion process and reduces the need for referrals. Thus, referrals can set at a lower level for content expected to be bestsellers, such as popular shows or music by popular artists. If future revenue is discounted considerably (a large value of

ρ), a higher referral is needed to ensure more sales are achieved early on. Hence, r* is increasing in ρ. Finally, if the fraction of unselfish nodes, α, increases then less referral is needed to drive the diffusion process. Interestingly, at a certain point, the optimal referral goes down to 0 even though several selfish nodes exist in the network. This is because there are sufficient unselfish nodes in the network to ensure that product diffusion proceeds relatively smoothly without referrals. 0.6

r* 0.4

0.2

α

0 0

0.2

0.4

0.6

Figure 6. Optimal referral r* vs. fraction of unselfish nodes α

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4. Dynamic Referral Strategy Since P2P content distribution is a continuous process, a referral early on in the diffusion process will have an impact on all subsequent sales, and therefore should have more significant impact on firm profit than a referral in later periods. If we view the referral, r, as the control variable, and market penetration level, q, as the state variable, then the profit maximization problem can be formulated as an optimal control problem, and the optimal referral strategy will be the resulting optimal trajectory of the control variable. Clearly, this referral trajectory will maximize overall profit and perform better than the constant referral strategy proposed in the last section. In this section, we first study the problem in a finite time horizon with no time discount effect. This scenario is applicable to short-term marketing campaigns or situations where the content is quickly perishable and thus has to be sold in a short period. For example, content such as software patches or holiday music may have significantly different short-term demand characteristics which may warrant a short-term planning horizon. In Section 4.2, we will consider an infinite time horizon and the time discount effect to obtain the long-term optimal referral strategy. 4.1. Finite Time Horizon

For the optimal control problem, we make the same assumptions as in the constant referral strategy except that the referral, r(t), is now a function of time. In addition, since the time horizon is short, it is reasonable to set the discount factor to zero, i.e. ρ = 0. Therefore, the profit generated from each sale is (1 – r). As before, we assume that there is only one seed in the network at the very beginning. Letting T be the market planning duration, the optimal control problem can be formulated as follows:

{

}

T

{

T

max π = ∫ N (1 − r )q&dt = max ∫ (1 − r )q&dt r

0

r

0

}

(12)

subject to

r (1 − α ) ⎞ 1 ⎛ q& = β ⎜ α + ⎟(1 − q)q , q (0) = , N C ⎠ ⎝

(13)

C ≥ r ≥0.

(14)

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Note that the feasible range of r is limited by 0 and C. As shown in (1), a referral equal to C ensures that every satisfied node is a seed. Therefore any referral greater than C is suboptimal.3 Also note that there is no restriction on the final value of q, i.e. it is a free end problem. As long as the total profit is maximized, the level of penetration at time T does not matter. To solve the problem, we apply standard optimal control theory (Sethi and Thompson 2000) and define the Hamiltonian function as r (1 − α ) ⎞ ⎛ H (r , q, λ ) = (1 − r )q& + λq& = β (1 − r + λ )⎜ α + ⎟(1 − q )q , C ⎠ ⎝

(15)

where λ is the adjoint variable. In the optimal solution, it satisfies the following differential equation. ⎛ ⎝

λ& = − H q = − β (1 − r + λ )⎜ α +

r (1 − α ) ⎞ ⎟(1 − 2q) . C ⎠

(16)

Since it is a free end problem, λ has a boundary condition of λ(T) = 0 (Sethi and Thompson 2000). (15) is a quadratic function of r. By the Hamiltonian maximizing condition, we have, ⎛ (1 + λ )(1 − α ) 2r (1 − α ) ⎞ − −α ⎟ = 0. H r = β (1 − q)q⎜ C C ⎝ ⎠

(17)

Considering the feasible range of r, ⎧C , if λ > Cα /(1 − α ) + 2C − 1, ⎪ ⎪ (1 + λ )(1 − α ) − Cα r* = ⎨ , if Cα /(1 − α ) − 1 ≤ λ < Cα /(1 − α ) + 2C − 1, α 2 ( 1 − ) ⎪ ⎪⎩0, if λ < Cα /(1 − α ) − 1.

(18)

Depending on the value of λ, the optimal referral trajectory can be conceptually divided into three stages. In stage 1, the firm will offer the maximum referral, C, to accelerate the diffusion process as much as possible. Once a certain penetration level is achieved, the referral should be gradually reduced based on the value of λ. Finally, in stage 3, the firm will offer no referral since the penetration level is already high enough to sustain further diffusion. This referral strategy further confirms our intuition that referrals early on play a more important role than referrals in

3

This is related to the assumption that consumers are myopic and do not consider the (expected) referral trajectory when adopting. See Section 5 for a discussion on diffusion and referral with forward looking buyers.

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later stage. Note that stages 1 and 3 may not always exist depending on the other parameter values as described below. Let subscripts 1, 2 and 3 represent the three stages, and T1, T2 and T3 denote the corresponding duration of these stages. When exactly three stages exist, T1, T2 and T3 satisfy following conditions.

λ (T1 ) = Cα /(1 − α ) + 2C − 1 , λ (T1 + T2 ) = Cα /(1 − α ) − 1 , and λ (T1 + T2 + T3 ) = λ (T ) = 0 . Also denote τ1 and τ2 as the market penetration levels at T1 and (T1 + T2), respectively. i.e. q (T1 ) = τ 1 , and q (T1 + T2 ) = τ 2 .

We further assume that λ(0) = λ*. To obtain the optimal trajectory of r, we need to solve λ in each stage and the boundary points of each stage, i.e. T1, T2, T3 and τ1, τ2. The solution details are provided below. Stage 1

If λ* > Cα /(1 − α ) + 2C − 1 , stage 1 exists and r* = C from 0 to T1. Therefore the dynamics of q and λ are captured by following differential equations, ⎧q&1 = β (1 − q1 )q1 , q1 (0) = 1 / N , ⎨& * ⎩λ1 = − β (1 − C + λ1 )(1 − 2q1 ), λ1 (0) = λ ,

(19)

where q1 and λ1 are the penetration level and adjoint variable in stage 1, respectively. Solving (19), we get the functions for q and λ in stage 1.

1 ⎧ , ⎪q1 (t ) = 1 + ( N − 1)e − βt ⎨ ⎪⎩λ1 (t ) = C − 1 + e ( 2 q1 −1) βt (λ* − C + 1) .

(20)

Stage 2

From T1 to (T1 + T2), r * =

(1 + λ )(1 − α ) − Cα . The dynamics of q and λ are as follows, 2(1 − α )

⎧ ⎛ r * (1 − α ) ⎞ & ⎜ ⎟⎟(1 − q 2 )q 2 , q 2 (0) = τ 1 , = + q β α ⎪ 2 ⎜ C ⎪ ⎝ ⎠ ⎨ * ⎪λ& = − β (1 − r * + λ )⎛⎜ α + r (1 − α ) ⎞⎟(1 − 2q ), λ (0) = Cα + 2C − 1. 2 ⎜ 2 2 ⎟ ⎪⎩ 2 1−α C ⎝ ⎠

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(21)

where q2 and λ2 are the penetration level and adjoint variable in stage 2. Note that, since q and λ are continuous, the initial value of q2 and λ2 are the ending value of q1 and λ1 respectively. Solving (21) (see appendix A for details), we have the functions for q and λ in stage 2. ⎧ β ⎞ 2⎛ ⎪q 2 (t ) = sin ⎜ arcsin τ 1 + t 2 τ 1 (1 − τ 1 ) ⎟, ⎝ ⎠ ⎪⎪ 4C τ 1 (1 − τ 1 ) ⎛ Cα ⎞ ⎨ −⎜ + 1⎟. ⎪λ 2 (t ) = β ⎛ ⎞ 1−α ⎠ ⎪ τ 1 (1 − τ 1 ) ⎟ ⎝ (1 − α ) sin 2⎜ arcsin τ 1 + t ⎪⎩ 2 ⎝ ⎠

(22)

It can be easily shown that λ 2′′(t ) ≥ 0 . Therefore λ2(t) is convex, and so is r in stage 2. Stage 3

If λ (t ) falls below Cα /(1 − α ) − 1 , then there will be a stage where the referral is set to zero. Further, if Cα /(1 − α ) − 1 > 0 = λ (T ) , then the diffusion will terminate with zero referral (i.e., in stage 3). The dynamics of q and λ in stage 3 are as follows, ⎧q& 3 = βα (1 − q3 )q3 , q3 (0) = τ 2 , ⎪ ⎨λ& = − βα (1 + λ )(1 − 2q ), λ (0) = Cα − 1. 3 3 3 ⎪⎩ 3 1−α

(23)

Solving it, we get the expressions for q and λ in stage 3, 1 ⎧ ⎪⎪q3 (t ) = 1 + (1 / τ − 1)e −αβt , 2 ⎨ C α ⎪λ3 (t ) = −1 + e ( 2 q3 −1)αβt . 1−α ⎩⎪

(24)

Thus far, we have analytically solved the optimal trajectories of q(t) and λ(t) in each of the three stages, while the values of T1, T2, T3, τ1, τ2 and λ* remain unknown. Since both q(t) and λ(t) are continuous, we have the following simultaneous equations that must be satisfied when exactly three stages exist in the diffusion process, ⎧q1 (T1 ) = τ 1 , ⎪q (T ) = τ , 2 ⎪ 2 2 ⎪λ (T ) = Cα + 2C − 1, ⎪ 1 1 1−α ⎨ Cα ⎪λ 2 (T2 ) = − 1, 1−α ⎪ ⎪λ3 (T3 ) = 0, ⎪T + T + T = T . 2 3 ⎩ 1

17

(25)

Although the above equations cannot be analytically solved due to the complexity of the functional forms of q(t) and λ(t), they can be numerically evaluated. Once these boundary points are obtained, the optimal referral trajectory, r*(t), and the resulting diffusion process, q*(t), can be easily obtained from equations 18 and 13 respectively. Figure 7 shows trajectories of r*(t) and q*(t), when N = 1000, Q(0) = 1, C = 5, α = 0.2, β = 1 and T = 30. The firm offers the highest possible referral during the first stage. Note that the referral can be higher than the unit price of the information product. Then in stage 2, the optimal referral convexly decreases until it reaches zero. Thereafter, no referral is offered in stage 3. Note that stage 3 need not necessarily exist. For example, Figure 8 shows the diffusion curve and referral trajectory when N = 1000, Q(0) = 1, C = 1, α = 0.2, β = 1 and T = 20. r* decreases first and then increases without touching zero, though it is still always convex in stage 2. In Figure 9, we have a case where the referral trajectory enters stage 3 (i.e., zero rebate) and re-emerges into the convex trajectory of stage 2 in the final few periods (N = 1000, Q(0) = 1, C = 1, α = 0.4, β = 1 and T = 20). In all these cases, the referral is high initially because there are very few seeds in the network early on and it is important to ensure every satisfied node shares the content. Once there are enough satisfied nodes, the referral starts to drop because there are enough seeds in the network. However, based on the parameters, two other effects may be observed subsequently – i) sometimes referrals may reach a zero value and ii) sometimes referrals may rise in the final few periods. We begin by investigating what happens to the referral in the last few periods and why referrals may rise in the final few periods (Figures 8 and 9). As we approach the end of the planning horizon, there may still be a large number of seeking nodes and the firm has very few periods in which to achieve additional sales. Thus, it may be willing to sacrifice margins in order to ensure higher sales in the limited time available. Hence, the firm raises the referral due to this end-of-period effect. This is also partly illustrated by plugging the transversality condition

λ (T ) = 0 into equation 17 to get Cα ⎧1 ⎪ − r (T ) = ⎨ 2 2(1 − α ) ⎪⎩ 0

if

1 Cα − >0 2 2(1 − α ) otherwise

(26)

Thus, the optimal referral will be zero in the final period if α (fraction of nodes that are unselfish) or C (upper bound of cost of providing download) are high. If there are a lot of unselfish nodes, then a referral is not needed once there are sufficient satisfied nodes and hence

18

the referral can be zero in the final few periods. Further, if the cost of providing a download is high, then the referral will also have to be very high in order to provide incentives to contribute. It is not worth providing such high referrals if there are a reasonable number of seeds, so the referral is zero when C is high. Conversely, if α or C are low, referrals may rise in the final few periods to end up in a non-zero value. Note that the referral may reach zero at an intermediate time even if r (T ) ≠ 0 (see Figure 9). In additional numerical tests, we find that the referral is more likely to reach a zero value some time during the planning horizon if α or C are high and the planning horizon, T, is long. In Figure 10, we plot the referral trajectory against T (N = 1000, Q(0) = 1, C = 1, α = 0.2, β = 1). A longer planning horizon makes it more likely that the rebate will touch zero at some point. 6

r *, q *

5 4 3

r*

2

q*

1 0

t T1

0

T2

T3

10

20

30

Figure 7. Typical trajectories of optimal r and q (α = 0.2, C = 5, T = 30) 1.2

r *, q *

1 0.8

r*

q*

0.6 0.4 0.2

t

0 0

5

10

15

20

Figure 8. A case without stage 3 (α = 0.2, C = 1, T = 20)

19

1.2

r *, q *

1 0.8

r*

q*

0.6 0.4 0.2

t

0 0

5

10

15

20

Figure 9. A case where stage 3 is followed by recurrence of the stage 2 trajectory (α = 0.4, C = 1, T = 20) r*

1.2 1 0.8 0.6

T =10

0.4

T =15

T =20 T =25 T =30

0.2

T

0 0

10

20

30

Figure 10. Impact of the length of the planning horizon on the rebate trajectory (α = 0.2, C = 1)

A variable that may be of particular interest to the firm is the duration of stage 1, T1, since it entails the highest cost for each copy sold. The primary reason for offering such an aggressive referral early on is to ensure that most early seeking nodes are able to locate the file. Once there is a reasonably large number of satisfied nodes, the firm can afford to lower the referral because there will be sufficient seeds to help sustain the diffusion process. Clearly, the duration of stage 1 will depend on the fraction of unselfish nodes (α) in the network and the average request rate (β). Figure 11 shows that T1 is concavely decreasing in α. As the fraction of unselfish nodes increases, the need for an aggressive referral is reduced. When there are a large number of unselfish nodes, the duration of stage 1 shrinks to zero. The impact of β, the demand for the product, is a little more interesting (Figure 12). As β increases, T1 increases initially but soon

20

decreases convexly until it reaches zero. When β is extremely small, the demand for the product is so weak that the market penetration will be low even with an aggressive referral. Therefore the best strategy for the firm is to reduce costs by offering a low referral. As demand increases, the cost of a high referral offered early on will be offset by increased sales in later periods. Therefore the duration of stage 1 increases. However once the demand exceeds a certain threshold value, the high demand ensures that a large number of satisfied nodes (and hence seeds) will be present early in the diffusion. Therefore T1 decreases with β after that threshold value. T1 3

2

1

α

0 0

0.2

0.4

0.6

Figure 11. Duration of stage 1, T1, as a function of fraction of unselfish nodes α T1 12

8

4

β

0 0

1

2

3

Figure 12. Duration of stage 1, T1, as a function of average request rate β

21

4.2. Infinite Time Horizon with Discount Effect

We now consider an infinite planning horizon with discounting of future profits. The present value of the profit generated from a sale at time t is (1 − r )e − ρt . Then the optimal control problem for infinite time horizon can be formulated as follows.

{

∞

max π = ∫ N (1 − r )e − ρt q&dt subject to

0

}

r (1 − α ) ⎞ 1 ⎛ q& = β ⎜ α + ⎟(1 − q)q , q (0) = , C ⎠ N ⎝

(27)

C ≥ r ≥0.

The current-value Hamiltonian function is given by r (1 − α ) ⎞ ⎛ H (r , q, λ ) = (1 − r )q& + λq& = β (1 − r + λ )⎜ α + ⎟(1 − q )q , C ⎠ ⎝

(28)

where λ is the current-value adjoint variable satisfying the following differential equation. ⎛ ⎝

λ& = ρλ − H q = ρλ − β (1 − r + λ )⎜ α +

r (1 − α ) ⎞ ⎟(1 − 2q) . C ⎠

(29)

By applying the Hamiltonian maximizing condition, we can derive the same expression for r* as (18). As in Section 4.1, the distribution process can be conceptually divided into 3 stages, and the optimal trajectory of λ, q, and optimal r can be obtained by solving the coupled differential equations, (27) and (29). Unfortunately, the time discount factor, ρ, makes the problem analytically intractable. However, if the limiting state of λ is known, the problem becomes a TPBVP (Two Point Boundary Value Problem), which can be solved (Sethi and Thompson 2000)4. Considering the infinite time horizon and the continuous nature, it is reasonable to assume that the system will eventually converge to a long-run stationary equilibrium, where q, λ and r become independent of time. Formally, the stationary equilibrium is defined by the three tuple

{q , r , λ } satisfying the following conditions (Sethi and Thompson 2000). ⎧q& (q , r ) = 0, ⎪& ⎨λ = ρλ − H q (q , r , λ ) = 0, ⎪ H (q , r , λ ) = 0, ⎩ r 4

(30)

Strictly speaking, the problem is not a real TPBVP since the end point is infinity. However, a good approximation can be obtained by setting a large enough value as the ending point in numerical analysis.

22

where q = lim t →∞ q(t ) , r = lim t →∞ r (t ) , and λ = lim t →∞ λ (t ) .

The optimal trajectory of r can be numerically computed by setting λ& =0. Figure 13 shows a typical optimal referral trajectory and the corresponding diffusion process for infinite time horizon case with N = 1000, Q(0) = 1, C = 1, α = 0.2, β = 1 and ρ = 0.12. The firm offers a referral as high as C at the beginning, and gradually reduces it to a stationary value as the diffusion proceeds. A primary difference between the infinite and finite horizon solutions is the absence of an end-of-period effect wherein referrals rise in the last few periods of the finite horizon referral. While Figure 13 shows an instance where the referral settles to a non-zero stationary value, we do find instances where the referral converges to zero (i.e., r = 0 ). 1.5

r *, q * 1

r*

q*

0.5

t 0 0

10

20

30

40

50

Figure 13. Trajectory of optimal r and q with infinite time horizon 4.3 Profit Gain from Dynamic Referral

In contrast to a constant referral strategy, the dynamic referral strategy performs better because it accounts for the fact that referrals early on will impact future diffusion and adjusts the referral trajectory appropriately. However, the constant referral strategy is easy to implement. Hence, it is useful to know when (and by how much) dynamic referrals outperform constant referrals. In Figure 14, we plot firm profit against demand ( β ) under the optimal constant and optimal dynamic referral strategies. The remaining parameters are as follows: N = 1000, Q(0) = 1, C = 1, α = 0.1, and ρ = 0.12. When β is extremely small (i.e. there is no demand for the content), the optimal profit approaches zero irrespective of the referral strategy. When β is very large (i.e., the content is very popular), there is no need to offer a referral and hence the profit under both referral strategies again converges. For intermediate values of β , dynamic rebating

23

significantly outperforms constant rebating. In Figure 15, we plot the profit against the fraction of unselfish nodes in the network ( α ). The remaining parameters are N = 1000, Q(0) = 1, C = 1,

β = 1 and ρ = 0.12. As the number of unselfish nodes in the network increases, the need to offer referrals diminishes. Hence it does not matter whether constant or dynamic referrals are offered. When α is small, dynamic referral significantly outperforms a constant referral strategy. 400

π

300

Constant Referral Strategy Dynamic Referral Strategy

200

100

β

0 0

0.5

1

1.5

2

Figure 14. Profit against product demand for constant and dynamic referrals π 400 300 200 100

Constant Referral Strategy Dynamic Referral Strategy

α

0 0

0.2

0.4

0.6

0.8

1

Figure 15. Profit versus fraction of unselfish nodes for constant and dynamic referrals

5. Discussion P2P networks are fast gaining popularity as distribution channels for information products due to their distributed nature and nearly unlimited scalability. A useful approach to reduce freeriding and speed product diffusion within the network is to offer referral payments to users who distribute content to other users. Our research represents one of the first studies on the impact of

24

these distribution referrals on the dynamics of product diffusion in P2P networks and on optimal referral strategies for P2P firms. We first study a constant referral strategy, where the firm applies a fixed referral throughout the diffusion process. Closed form solutions are derived and the impact of various parameters on the optimal referral is studied. We find that the optimal referral decreases with product demand and the fraction of unselfish nodes, but increases with the discount factor. When the fraction of nodes that are unselfish is reasonably large, the firm does not need to offer any referral. Motivated by the intuition that a referral early on should play a more important role than a referral in later periods, we extend our research to study a dynamic referral strategy, where the firm continuously adjusts the referral over time. Two versions of this dynamic strategy are discussed. The first, characterized by a finite time horizon and no time discount effect, is applicable to short-term marketing campaigns. For this case, we find that the optimal referral trajectory usually, but not always, involves three stages. During the first stage, the firm sets the referral as high as needed to ensure every satisfied node is a seed. The referral may even be greater than the price charged by the firm. This aggressive early referral serves to increase content availability and hence speed up the product diffusion. In stage 2, the optimal referral will be gradually reduced. The referral may touch zero and then stay at zero in stage 3. Alternatively, the referral may not reach zero and instead reach a minimum non-zero value. Finally, we also observe an end-of-period effect where referrals demonstrate an increasing convex trajectory in the final few periods. In the last few periods, the firm may be willing to sacrifice margins to increase sales because of the limited time in which to achieve additional sales. We also consider an infinite time horizon model, which can be used for long-term marketing decision-making. Our numerical study suggests that the firm should offer the highest possible referral at the beginning. Subsequently, the referral should decrease and converge to a stationary value eventually. We do not find any instance in the numerical analysis of infinite horizon referrals where the referral increases with time. This is consistent with our hypothesis that the referral increases in the finite horizon model because of an end-of-period effect.

6. Conclusions and Future Directions Our research opens a number of other avenues for future research. In this paper, we limit our focus to the simplest network topology, namely flat P2P network structure. While previous research has shown that the diffusion dynamics in flat P2P networks are fundamentally similar to

25

those in hierarchical P2P networks, extending the results on optimal referrals to hierarchical P2P networks would prove useful. Modeling complex network structures may also allow us to model heterogeneity among the nodes in the network. Some nodes may play a more influential role in the network (for example, “supernodes” in a hierarchical P2P network). With high and low “influentials”, firms have to figure out not only the amount of referral but how to target these referrals to nodes. There exists a considerable opportunity to apply pricing and consumer behavior models from marketing to address emerging issues in online content distribution. Our paper is a first step in this direction. Below, we discuss some of these opportunities. Forward looking consumer behavior: Our diffusion model assumes that consumers are myopic

and do not account for the referral in choosing whether (and when) to buy the product. An interesting extension would be to incorporate the impact of the referral on product demand. Recently, issues surrounding the impact of forward looking consumer behavior on adoption and diffusion of products have got some attention. For example, Song and Chintagunta (2003) propose a consumer adoption model with heterogeneous forward-looking consumers and estimate the model parameters using market data on digital cameras. In a diffusion setting, Narasimhan (1989) incorporates consumer price expectations in a diffusion model and finds that the optimal price path set by the firm is cyclical. It would also be interesting to model forwardlooking behavior and study whether similar cycles are observed in the referral trajectory. The answer is not straightforward because the referral plays a different role in our diffusion model when compared to price in traditional diffusion models. As discussed earlier, the referral impacts a term in our diffusion model that is fundamentally viral in nature. Referral Vs Price Cut: The distribution referral rewards users for distributing content to other

users and thus serves to reduce free-riding, increase content availability and hence sales. As an alternative to a referral, a firm can also offer a price cut or subsidy early on to increase adoption. As long as there are a reasonably large number of unselfish nodes, an early price cut can bring a sufficient number of unselfish nodes as seeds which in turn can address issues related to content availability. Thus, an interesting extension would be study when firms may prefer price-cuts over referrals. Further, an optimal hybrid scheme that incorporates both a referral and a dynamic price would shed more light into the problem. As discussed, referrals and price impact different components of the diffusion model and will likely have dissimilar impact on rate of diffusion. A

26

similar trade-off between traditional referral rewards and price cuts has been studied by Biyalogorsky et al (2001). The authors find that consumer delight, measured by the extent to which consumer surplus exceeds a predefined threshold, is a primary factor in determining whether firms should use referrals over price cuts. Inter-firm Competition: Both the above extensions would require building a model of

consumer preference and modeling the realized demand for content as a function of referral and price. Another extension may be to model competition among firms. Generally, information products like a song by an artist are unique and enjoy considerable differentiation. However, consumers may offer compare products because of budget constraints. For example, two hip-hop albums released at nearly the same time can cannibalize each other’s sales. Similarly, two P2P firms may compete in selling music tracks. Thus, studying the impact of competition on the optimal referral trajectory will be fruitful. Prior work on the impact of competition on dynamic pricing policies is particularly relevant here (see for e.g., Rao and Bass 1985; Eliashberg and Jeuland 1986). As highlighted above, there a number of open questions related to the role of the marketing mix in P2P content distribution. We believe that this is a promising new stream of research and that our research is a crucial first step in that direction.

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Appendix A Solving the coupled differential equations for Stage 2, we obtain, from T1 to (T1 + T2), r* =

(1 + λ )(1 − α ) − Cα , 2(1 − α )

(A.1)

and the coupled different equations describing the dynamics of q and λ, ⎧ ⎛ r * (1 − α ) ⎞ & ⎜ ⎟⎟(1 − q 2 )q 2 , q 2 (0) = τ 1 , = + β α q ⎪ 2 ⎜ C ⎪ ⎝ ⎠ ⎨ * ⎪λ& = − β (1 − r * + λ )⎛⎜ α + r (1 − α ) ⎞⎟(1 − 2q ), λ (0) = Cα + 2C − 1. 2 ⎜ 2 2 ⎟ ⎪⎩ 2 1−α C ⎝ ⎠

Plugging (A.1) into (A.2), we get,

30

(A.2)

β (1 − q 2 )q 2 ⎧ ((1 + λ2 )(1 − α ) + Cα ), q 2 (0) = τ 1 , ⎪⎪q& 2 = 2C ⎨ β (1 − 2q 2 ) ⎪λ&2 = − ((1 + λ2 )(1 − α ) + Cα )2 , λ2 (0) = Cα + 2C − 1. ⎪⎩ 4C (1 − α ) 1−α

(A.3)

From (A.3) we can solve λ2 as

λ2 =

2Cq& 2 ⎛ Cα ⎞ −⎜ + 1⎟ . β (1 − α )(1 − q 2 )q 2 ⎝ 1 − α ⎠

(A.4)

Differentiating the first differential equation in (A.3) with respect to t, and letting Λ = (1 + λ 2 )(1 − α ) + Cα , we get,

q&&2 = Also, Λ =

Λβ (1 − 2q 2 )q& 2 + β (1 − α )(1 − q 2 )q 2 λ&2 . 2C

(A.5)

β (1 − 2q 2 ) 2 2Cq& 2 Λ . Plugging them into (A.5) and rearranging it, , and λ&2 = − β (1 − q 2 )q 2 4C (1 − α )

we get, ′ ⎛ ⎞ ′ &2 q (1 − 2q 2 )q& 22 ′ ⎟ =0 q&&2 = ⇒ (ln q& 2 ) − ln (1 − q ) q = 0 ⇒ ⎜ ln ⎜ 2(1 − q 2 )q 2 (1 − q ) q ⎟⎠ ⎝

(

)

(A.6)

Let q 2 = sin 2 θ , where θ ∈ (0, π / 2) . Then (1 − q 2 ) q 2 = cos 2 θ sin 2 θ , q& 2 = 2 sin θ cos θθ& . (A.6) becomes

(ln 2θ& )′ = 0 ⇒ θ&& = 0 ⇒ θ (t ) = C

1

+ tC 2 ,

(A.7)

where C1 and C2 are constants to be determined. Considering the initial condition, q2(0) = τ1, we have C1 = arcsin τ 1 .

Since λ 2 (0) =

(A.8)

Cα + 2C − 1 , we have 1−α C2 =

β 2

τ 1 (1 − τ 1 ) .

(A.9)

Therefore,

θ (t ) = arcsin τ 1 + t

β 2

31

τ 1 (1 − τ 1 ) .

(A.10)

⎧ β ⎞ 2⎛ ⎪q(t ) = sin ⎜ arcsin τ 1 + t 2 τ 1 (1 − τ 1 ) ⎟, ⎝ ⎠ ⎪⎪ C τ τ − 4 ( 1 ) ⎨ ⎛ Cα ⎞ 1 1 −⎜ + 1⎟. ⎪λ (t ) = β ⎛ ⎞ 1−α ⎠ ⎪ τ 1 (1 − τ 1 ) ⎟ ⎝ (1 − α ) sin 2⎜ arcsin τ 1 + t ⎪⎩ 2 ⎝ ⎠

32

(A.11)