The population dynamics of websites

Transcription

1 The population dynamis of websites [Online Report] Kartik Ahuja Eletrial Engineering UCLA Simpson Zhang Eonomis Department UCLA Mihaela van der Shaar Eletrial Engineering UCLA ABSTRACT Websites derive revenue by advertising or harging fees for servies and so their profit depends on their user base the number of users visiting the website. But how should websites ontrol their user base? This paper is the first to address and answer this question. It builds a model in whih, starting from an initial user base, the website ontrols the growth of the population by hoosing the intensity of referrals and targeted ads to potential users. A larger population provides more profit to the website, but building a larger population through referrals and targeted ads is ostly; the optimal poliy must therefore balane the marginal benefit of adding users against the marginal ost of referrals and targeted ads. The nature of the optimal poliy depends on a number of fators. Most obvious is the initial user base; websites starting with a small initial population should offer many referrals and targeted ads at the beginning, but then derease referrals and targeted ads over time. Less obvious fators are the type of website and the typial length of time users remain on the site: the optimal poliy for a website that generates most of its revenue from a ore group of users who remain on the site for a long time e.g., mobile and online gaming sites should be more aggressive and protetive of its user base than that of a website whose revenue is more uniformly distributed aross users who remain on the site only briefly. When arrivals and exits are stohasti, the optimal poliy is more aggressive offering more referrals and targeted ads. Categories and Subjet Desriptors J.4 [Soial and Behavioral Sienes] Keywords Websites, referrals, population dynamis. INTRODUCTION Many popular websites suh as Faebook, Google, and Netflix derive a signifiant portion of their revenue through advertising or by harging subsription fees to their users 2. Given suh a revenue model it is ritial for the websites to obtain and maintain a healthy user base. Hene, a ritial question that needs to be answered by suh websites is: how an the websites ontrol their user base? The user traffi on a website omes from two hannels: sponsored and non-sponsored. Sponsored traffi is steered to the website through targeted ads, referrals, paid keywords, disounts et. Hene, the website needs to deide how aggressively it should advertise as well as send referrals to ontrol the user base given the inurred osts. This paper aims to study and design poliies whih an be deployed by websites to ontrol their user base through referrals and targeted ads in order to maximize their profits. Among the works on user base dynamis in the eonomis literature, the ones that relate the most to this work are [5] and [3]. In [5] the authors analyze the effets of searh fritions in building a ustomer base on the firm s profits, investment, sales, et. In [3] the authors study informative advertising and analyze the effet of a deline in the ost of information dissemination on the firm and ustomer dynamis. The key results in [5][3] are derived by alibrating models based on data, while this work instead provides a theoretial foundation for understanding user base dynamis. Also, the fous of these works [5][3] are on a firm selling a produt to homogeneous users; in ontrast the users on the website in our model an be heterogeneous and generate varying revenue depending on the number of advertisements they lik. We propose a dynami ontinuous time model for the population of users present on a website. We assume that the website starts with a small initial user base and at every moment in time, the website an reah out to potential users via referrals and targeted ads. The website does this by paying a ost to inentivize its urrent users to send referrals to friends or by paying for targeted ads on other platforms. Thus, the website must adopt a poliy that balanes the marginal benefit of inreasing its user base versus the marginal ost of providing the referrals and targeted ads. Our model aounts for the fat that users are heterogeneous, and therefore provide different benefits to the website 3. We model this by making the natural assumption that the benefit funtion of the website is onave in the population level: the value from adding new users dereases with an inreasing population level. If the website starts with a low initial population, we show that the optimal poliy will be to give many referrals per unit time initially and then derease the referrals over time. This See and enue oming from only.5% of the user base See 3 For instane, in mobile gaming apps half of the rev- soial-networking-popular-aross-globe/. 2 See

2 is appliable for most websites that are new to the market and are unlikely to be able to aquire a high initial population. We also analyze how the website s optimal poliy depends on the average time that a user stays on the website before exiting. We show that this depends ritially on the revenue distribution of the website aross its users. For instane, some websites rely heavily on a small set of users for revenue, e.g. mobile and online gaming websites, whih rely on a small group of players who spend a lot on games. For suh websites the optimal poliy will inrease the referrals and targeted ads if the average stay time of the users dereases. On the other hand, some websites obtain revenue from its users more uniformly, e.g. subsription based websites suh as Netflix. For suh websites the optimal poliy will beome more aggressive to send more referrals and ads if the average time that a user stays on the website inreases. We also extend our model to allow for stohasti user arrivals/exits, and we show that the above results are robust: if there is high unertainty in user arrivals, we show that the website s poliy will be to send more referrals and ads on average. 2. MODEL We assume that there is a ontinuum of potential users and the firm an attrat these users by posting ads on searh engines (Google) and other websites (Faebook) or by giving referrals. New users an visit the website either through suh sponsored media or arrive diretly through exogenous methods, suh as arriving upon the website through a nonsponsored link. We assume that the rate of suh arrivals is a onstant θ. This is a simplifying assumption made due to page limitations and the key results of this work ontinue to hold even under more general arrival rates that depend on the population levels suh as θ.p s with s. The users are heterogeneous, i.e. the revenue that the users generate for the website (e.g., the number of ads a user liks) varies aross the users. Every user who visits the website stops using the website after a random time. This time is a random variable drawn from an exponential distribution and the average time a user stays is. Denote the total mass of the users using the website at time t as p(t), where p : [, ) [, ). The initial user base is denoted as p(). This total mass of the users represents the unique user statisti, whih is often used as a metri to evaluate a website s popularity 4. The revenue that the website generates per unit time inreases with the mass of the users. It is denoted as b(p), where b : R [, ) is a ontinuously differentiable inreasing funtion. Moreover, as mentioned in the introdution b(p) is assumed to be a onave funtion in p. The long-term benefit of the website onsidering a disount rate of an be omputed as B(p(.)) = b(p(t))e t. The website hooses the intensity of advertisements (measured in terms of number of sponsored links, referrals, targeted ads) to be posted on other platforms at eah time t and as a result it ontrols the rate of sponsored arrivals λ(t), where λ : [, ) R + is a ontinuous and bounded funtion. The rate of hange of population on the website is = θ + λ(t) p. The first two terms in the differential equation represent the rate of diret and sponsored arrivals respetively, while the third term represents the users ex- 4 iting the website. The firm bears a ost per unit time for the advertisements, whih inreases with the sponsored user arrivals λ(t). It is given by (λ), where : R [, ) is a ontinuously differentiable inreasing funtion. Given the heterogeneous pool of potential users it is harder to inrease the intensity of arrivals when λ is large. Hene, we assume that the ost (λ) is stritly onvex in λ (as in [5]). We also assume that there is no ost when there are no sponsored arrivals () =, and that the ost beomes unbounded as the sponsored arrivals approah, i.e. lim λ (λ) =. The long-term disounted average ost to the website is C(λ(.)) = (λ(t))e t. The firm desires to maximize its total disounted profit, i.e. B(p(.)) C(λ(.)). The ontinuous-time optimization problem of the firm is stated as follows max B(p(.)) C(λ(.)) λ(t) R +, t subjet to = θ + λ(t) p(t), p() is given Next, we analyze the behavior of the optimal poliy. 3. RESULTS We an show that the optimal poliy exists (See [] for details) and is denoted by λ p() (t). Denote the orresponding population dynami as p p() (t). If the optimal poliy and the orresponding population dynami onverge, the steady state is ahieved. The next theorem provides onditions for the existene of the steady state and also establishes that the steady state is unique. This theorem uses the the steady state population level ˆp, whih is given by with b (p) = db(p) X(ˆp) = b (ˆp) + and (λ) = d(λ) dλ. ( ˆp θ) = () Theorem. Steady state: Existene and Uniqueness i) If < b () < and lim p X(p) < then there exists a unique solution ˆp to the steady state equation (). ii) If p() < ˆp then the optimal poliy λ p() (t) dereases with time and onverges to ˆλ = ˆp θ and the orresponding population dynami p p() (t) inreases and onverges to ˆp. For the proofs of all the theorems refer to the online version []. Theorem proves that if there is a positive marginal benefit from inreasing the user base at very low population levels and if there is a negative marginal benefit from inreasing the user base at very high populations then there exists a population level (between very low and very high population levels) where the marginal benefit is zero, whih orresponds to the unique steady state. In addition we see that if the initial population level is low, whih is true for most of the websites when they are launhed, then the firm is more aggressive with advertising in the initial stages (loser to the launh) in omparison to the later stages (loser to the steady state). For the rest of the paper it is assumed that the firm starts with a low initial population, i.e. p() < ˆp. Also, we will use the terms advertisements and referrals, firm and website interhangeably heneforth. 3. Poliy for Different User Behaviors There are several interesting questions that one an ask about how the optimal poliy depends on the user s behavior: What happens to the poliy if the average time that a

3 user stays on the website dereases? Or if the diret arrivals to the website inrease? How does the new poliy ompare with the old poliy both in the steady state (i.e. t ) and at a finite time t? We first address these questions at time t, i.e. in the steady state. 3.. Comparison of the poliy in the steady state If the average stay time of the users dereases then the firm is faed with the following question: Is spending more on advertisements worth it? The answer is not straightforward beause of the following dilemma. The average stay time of the users redues whih disourages the firm, but the users leaving at a faster rate will also redue the population and the total benefit may thus fall sharply. We answer this question next. Define an operator Φ θ,, (b(p), p) = d2 b(p) 2 p + db(p) + The next theorem shows that if dereases and Φ θ,, (b(ˆp), ˆp) < then the intensity of ads inreases, otherwise the intensity dereases. Theorem 2. Poliy omparison with hange in in steady state t : Loal Behavior i) If the average time that a user stays on the website is dereased by ɛ > and if Φ θ,, (b(ˆp), ˆp) < then the advertisements in the steady state inrease. ii) If the average time that a user stays on the website is dereased by ɛ > and if Φ θ,, (b(ˆp), ˆp) > then the advertisements in the steady state derease. We assume that the hange in, i.e. ɛ to be small (See [] for details). Theorem 2 is interpreted as follows. If Φ θ,, (b(ˆp), ˆp) < then the perentage inrease in the marginal benefit resulting from a derease in the user population is high, i.e. d 2 b(p) 2 p= ˆp db(p) p= ˆp < (+ )ˆp. Therefore the website should send out more ads if the average stay time redues. Conversely, if Φ θ,, (b(ˆp), ˆp) > then the perentage inrease in marginal benefit resulting from a derease in population is not suffiient to ompensate for the marginal ost of additional ads. Therefore, the firm redues the ads. The above theorem haraterizes the loal behavior of the firm s poliy under the impat of small hanges in the average stay time. Next, we haraterize how the firm s global behavior under the impat of arbitrary hanges in average stay time. We fous on a speifi benefit funtion for better exposition, while the results presented extend to a larger lass of funtions. Consider a benefit funtion defined for p, as b(p) = p a with < a <. For a fixed a, we define a ( a) a threshold = behavior of the firm.. Then the following is true for the Result. Poliy omparison with hange in in steady state t : Global Behavior If the average stay time of the users dereases upto then the firm should inrease the intensity of advertisements in the steady state, while if the average stay time falls below then the firm should derease the advertisements. Next, we analyze the firm s behavior depending on its revenue distribution aross the users, whih is refleted by a. If a is small then the firm s benefit saturate very fast. This ours if the firm relies heavily on a set of ore users for revenue. Example of suh websites are online and mobile. Intensity of referrals in steady state a < a 2 = a ( a) Firm whih relies uniformly on users for revenue Firm whih relies on a set of ore users for revenue 2 = a2 ( a2) Average stay time of the users Figure : Impat of average stay time on firm s behavior in steady state gaming, sine these websites rely heavily on their ore users. Observe that a small value of a implies that the threshold is very small as well. Hene, for suh firms a redution in the average stay time leads to an inrease in the advertisements in the steady state. On the other hand, onsider the ase when a is large and lose to. If a is lose to then the firm s benefit saturates slowly. This ours if the firm relies uniformly on all the users for revenue. Examples inlude subsription based websites suh as Netflix. Observe that a large value of a implies a high threshold. Hene, for suh firms an inrease (derease) in the average stay time leads to a inrease (derease) in the advertisements in the steady state. As an example, the inrease in the ontent quality and thus user stay time on Netflix led Netflix spending on expanding its user base 5. Fig. shows the example of the above two types of websites. Next, we understand the effet of an inrease in the diret arrivals on the intensity of advertisements in steady state. Theorem 3. Poliy omparison with hange in θ in steady state t : Global Behavior If the intensity of the diret arrivals θ is inreased then the intensity of advertisements in the steady state dereases. The intuition behind Theorem 3 is as follows. An inrease in the diret arrivals makes the firm derease the advertisements in a ontrolled manner, suh that the total population on the website for the new higher level of diret arrivals is higher than the ase with original lower level. Hene, the firm an redue its ost while simultaneously inreasing the total population. We have analyzed the impat of the revenue distribution of the firm and the average stay time of the users on the firm s poliy in steady state. Now we want to understand the impat on the poliy while it is on the path to steady state. For this, we will assume quadrati benefit and ost funtions from now on due to spae limitations. However, the results obtained extend to a larger lass of benefit and ost funtions Comparison of the poliy on the path to steady state Websites are hosted on servers and the traffi that an be handled by a server is limited. Hene, we assume that the website has a maximum apaity, whih orresponds to the maximum mass of users that the website an support with 5 reasons-to-invest-in-netflix/

4 Intensity of referrals at time t.4.3 th Intensity of referrals at time, t Time, t Figure 2: Impat of average stay time of the user on firm s behavior out of the steady state when th Figure 3: Impat of average stay time of the user on firm s behavior out of the steady state when th a non-negative marginal benefit. The benefit per unit time is a ontinuously differentiable onave funtion b : R + R defined as b(p) = 2 ( p) 2, where is the apaity of the website. In the previous subsetion the benefit funtion was inreasing in p, while this is no longer the ase when apaity is inorporated in the model. We will assume that the apaity is suffiiently large, whih will ensure that the results of the previous setion ontinue to hold. We disuss the impat of firm hoosing to invest in inreasing the apaity in the online version of the paper []. We also assume that the ost per unit time for sponsored arrivals is quadrati (as in [5]) and is given as (λ) =.λ 2. We will assume through out this setion that the apaity is suffiiently high, i.e. θ. This ondition ensures that if there are no advertisements given at any time then the population in the steady state is less than the apaity. Under this assumption we an show that the result in Theorem, 2 and, 3 ontinue to hold. Hene, in the optimal poliy the advertisements will derease with time and ahieve the steady state value ˆλ (from Theorem ). If the average stay time of the users were to derease then the firm will ontinue to inrease the advertisements ˆλ until the stay time falls below a threshold, beyond whih the advertisements derease. Next, we analyze the effet of a dereased stay time on the poliy outside the steady state. Theorem 4. Poliy omparison with hange in : at finite time t i) If the average stay time of the user dereases to a value above the threshold th then the intensity of advertisements inreases at all time instanes t larger than some finite t. ii) If the average stay time of the user dereases to a value below the threshold th then the intensity of advertisements at every time instane t dereases. See the expression for th and t in the online version []. This theorem generalizes the result in subsetion 2.2.., where the behavior was analyzed in the steady state. In Fig. 2 we an see that if the average stay time of the user redues but does not fall below th then the firm spends more on advertisements this would happen when the website is suffiiently old, i.e. t t. This behavior from an inumbent website ould serve as a barrier to entry to an entrant website, whose arrival possibly results in the derease in the average stay time. Fig. 3 shows the seond ase, i.e. if the average stay time of the user dereases below the threshold th the website gives out less referrals in the new optimal poliy. Next, we understand the effet of an inrease in the diret arrivals on the poliy outside the steady state. It an be shown that there will be a derease in the advertisements at all times (See []). This generalizes Theorem 3. The intuition for this result is the same as that for Theorem 3. Unertain user arrivals/exits: We an show that even under unertain arrivals/exit of the users the key results presented in the deterministi setting will ontinue to hold (See []). The population dynami with unertain user arrival/exit hanges into the following stohasti differential equation dp = (θ + λ(t) P ) + σp dwt. Here Wt is the Weiner proess/brownian motion. The firm s objetive in this ase is to maximize the expetation of the long term total disounted profit (benefit-ost). If we onsider the quadrati benefit and osts onsidered in the previous setion, Theorem 4 an be extended to the stohasti setting as well (See []). In this ase the omparison is done between the expetation of the intensity of sponsored arrivals. An interesting result emerges is when we ompare the intensity of the advertisements under varying levels of unertainty σ. It an be shown that inreasing the level of unertainty makes the optimal poliy more aggressive in giving advertisements (See []). 4. CONCLUSION This paper has been the first to systematially haraterize how a website should build its user base. We showed that the optimal poliy requires that websites starting with small initial populations send ads aggressively in the beginning and then derease them with time. For websites that derive revenue from a set of ore users, e.g. mobile and online gaming, the optimal poliy inreases ads when the average stay time of the user on the site dereases, while for websites with a more uniform revenue distribution aross users the optimal poliy dereases ads. These results extend to a stohasti setting with noise in population dynamis, and it is shown that more unertainty in user arrivals/exits leads to a more aggressive optimal poliy for ads. 5. REFERENCES [] K. Ahuja, S. Zhang, and M. van der Shaar. The population dynamis of websites. Available reportpopdynamis.pdf [online]. [2] L. M. Benveniste and J. A. Sheinkman. On the differentiability of the value funtion in dynami models of eonomis. Eonometria: Journal of the Eonometri Soiety, pages , 979.

5 [3] E. M. Dinlersoz and M. Yorukoglu. Information and industry dynamis. Amerian Eonomi Review, 2(2):884 93, 22. [4] A. Dmitruk and N. Kuz kina. Existene theorem in the optimal ontrol problem on an infinite time interval. Mathematial Notes, 78(3-4):466 48, 25. [5] F. Gourio and L. Rudanko. Customer apital. The Review of Eonomi Studies, page rdu7, 24. [6] B. Zhao. Inhomogeneous geometri brownian motion. Available at SSRN , PROOFS Lemma. For the proposed model the optimal poliy always exists. p. We now show that the assumptions in [4] hold for our model as well. A. The funtion h θ, (λ, p) is linear in p and is independent of time t, therefore learly the funtion is ontinuous in (t, p). The funtion is also linear in λ. Let S(t) = [ δ, P ] denote the set that the population at time t is restrited to be in, where δ >. In our original problem we did not have suh a set to restrit the population trajetory. Hene, imposing suh a onstraint may not lead to an equivalent problem. Note that if U is hosen suffiiently high then the two problems are equivalent. Observe that if the intial population level is p() then any trajetory Proof. Existene of the optimal poliy We first show that the optimal poliy λ(.) exists. To be able to do so we will use the suffiient onditions arrived at in [4].The searh spae of optimal poliy λ(.) is restrited to measurable funtions and the optimal poliy is positive uniformly bounded above by M. The population mass at time t p(t) is restrited to be absolutely ontinuous funtion on interval [, T ] and this has to hold for all suh intervals. The initial ondition on p(t) is given, i.e. p(). The trajetory of p follows the following differential equation = h θ,(λ, p) = θ + λ(t) of p(t) whih follows = h θ,(λ, p) will always be bounded above by P = max{p(), (θ + U)} + δ. This an be justified as follows. Consider the ase when p() < (θ + U). Let us assume that the population p(t) in this ase does exeed P, whih is equal to (θ + U) + δ. Sine p(t) has to exeed (θ + U) it has to be true that there exists a t at whih >. This is due to the ontinuity of p(.). However, notie that h θ, (λ, (θ+u)) = θ+λ(t) p is non-positive beause λ U. This ontradits the laim. Consider the ase when p() > (θ+u). In this ase we need to show that the population trajetory p(t) always stays less than or equal to p(). Let us assume otherwise, i.e. the population did exeed p(). It an be shown that for the population to exeed and attain a level higher than p(), the rate p(t) = (θ+u) and t=t of hange of population has to be positive at a ertain time when p(t) = p(). This annot be true beause λ U whih will make the = h θ,(λ, (θ + U)) <. Note that in the above ase we arrived at a bound P, whih depends on p(). We an make this bound independent of p() by assuming that the initial population p() < P in <. Therefore, if S P then the problem with onstraint on p(t) S(t) will still be equivalent. A2. Sine S(t) is losed for all t the seond assumption in [4] is satisfied as well. A3. Sine p() is fixed then assumption 3 is automatially satisfied. Let U(t, p) denote the set in whih the λ(t) is restrited to be in. In our ase let us assume that U(t, p) = [, U]. Sine λ(.) is upper bounded by U this impliit onstraint suffies for the equivalene of the two problems if we add the onstraint that λ(t) U(t, p). A4. Sine the set valued funtion U(t, p) = [, U] is losed and bounded and takes a fixed value, we an dedue that it is a ontinuous mapping and the output of the mapping U(t, p) is onvex and ompat. A5. Sine the set U(t, p) and S(t) are uniformly bounded then we an say that the assumption 5 is satisfied. A6. The funtion β(p()) = in our ase and hene, ontinuity is obvious. A7. φ(t, p, λ) = ((λ) b(p))e t is ontinuous in (t, λ, p). This follows from the definition of the funtions b(.) and (.). Also the onvexity (.) is suffiient to show that φ(t, p, λ) is onvex in λ. A8. Consider the negative part of the funtion φ(t, p, λ) = ((λ) b(p))e t, whih is equal to b(p)e t and we need to T show that negative part of φ(t, p, λ) i.e. lim T T b(p(t))e t goes to zero. Observe that T T b(p(t))e t T b(p )e t = b(p )e T. Hene, learly the limit of the term goes to. Note that this limit will be for all population trajetories. In addition we also know that there is a feasible trajetory i.e. whih satisfies all the onditions above and gives a finite value of the objetive, onsider the ase when the λ(t) = then V (p()) = b(θ( e t ) + p()e t )e t. Sine b(θ( e t ) + p()e t ) b(p ) we know that the integral exists. Given all the onditions in [4] are satisfied the optimal poliy has to exist. Theorem. Steady state: Existene and Uniqueness i) If < b () < and lim p X(p) < then there exists a unique solution ˆp to the steady state equation (). ii) If p() < ˆp then the optimal poliy λ p() (t) dereases with time and onverges to ˆλ = ˆp θ and the orresponding population dynami p p() (t) inreases and onverges to ˆp. Proof. In the steady state = whih implies λ = p θ. In order to arrive at the steady state solution we would use the HJB equation, whih is a neessary ondition whih the optimal value funtion needs to satisfy. It is given as follows. V (p) = max λ (b(p) (λ) + V (p)(θ + λ p) Note that the above equation assumes that the value funtion is differentiable. Next, we show that the optimal value funtion is differentiable in our problem. In [2] the authors showed suffiient onditions on the problem that are required for a value funtion to be differentiable at a point. Define the ) are onstrained to be in. p(t) is onstrained to be in the set S and is onstrained to be in [θ P δ, θ+u +δ]. The upper bound on omes from the onstraint that λ U and the population is lower bounded by. While the lower bound follows from the fat that p(t) P. Denote the set in whih (p(t), ) are onstrained to be in as T. Hene T is given as T = S (, θ + U] set in whih the (p(t),

6 A. The first assumption requires the set T to be onvex and have a non-empty interior. Clearly the set T is onvex and has a non-empty interior. A2. Let the flow benefit funtion be given as u(p,, t) = (b(p) (λ))e t = (b(p) ( + p θ))e t. Note that b(p) is onave and (p + ) is onave in (p, ). Hene, the sum of both the funitons has to be onave in (p, ). Also, b(p)e t is ontinuously differentiable. This is beause we assume that b(p) is ontinuously differentiable and e t is ontinuosly differentiable in t. Similar argument holds for the other part of the funtion ( + p θ)e t. Hene, the flow benefit funtion is ontinuously differentiable. Therefore, the seond assumption is satisfied. A3. The optimal poliy p(.), λ(.) exists starting from p(). This follows from the Lemma, where we showed that the optimal solution exists. The previous lemma holds for any value of p() <, hene the optimal poliy exists for all p() < and therefore V (p) is well defined. Note that V (p) = (b(p(t)) (λ(t)))e t exists and is bounded beause p() and λ are bounded. A4. In this assumption it is required to show that the solution starting at p(), i.e. p p() (t) and the orresponding p() are interior in the sense stated in [2]. Note that by the onstrution of the set S the entire trajetory of p p() (t) is in the interior, beause the p(t) > δ and p(t) < P. Hene, at no point does the trajetory hit the boundary, thereby showing that this assumption is true as well. Similar argument holds for the p() as well. Given the above four above assumptions, the value funtion is differentiable at p(). The above argument an be extended to any p() provided we assume that p() < P in. This will ensure that the bound P an be now modified to max{p in, (θ + U)} + δ. The optimal solution to the HJB equation λ = (V (p)). This follows from FOC and the fat that is invertible and (x) is positive when x >. This an be justified as follows. is stritly onvex, therefore is stritly inreasing, whih implies that is invertible. It is assumed that () = and lim x (x) =, using this and strily inreasing nature of, it is lear that (x) > for x >. However, we also need to show that V (p) is positive. This means that the value funtion is inreasing with an inrease in the initial population base, this seems intuitive owing to the inreasing nature of the benefit funtion as a funtion of population. We show this below. Let p < p 2 and let s assume V (p ) > V (p 2). Let the optimal poliy starting at an initial population of p be λ p (t). Suppose we followed the same poliy λ p (t) at the initial population p 2 the population levels ahieved by the trajetory starting at p 2 will always be greater than or equal to the trajetory at p. The trajetory starting at p 2 has to be stritly greater for a finite duration, before whih it intersets with the trajetory of p. After the time the two trajetories interset the two trajetories follow the same path. It may also happen that the trajetory of p 2 always stays higher. Sine the population levels are always higher and the osts for the referrals is the same in the two ases, the value funtion at p 2 is bound to be higher. This ontradits the assumption. Hene, p < p 2 implies V (p ) < V (p 2). Hene, we know that in steady state the following has to hold. λ = ˆp θ ( ˆp θ) = V (ˆp) Sine V (ˆp) is not known the above equations are not suffiient to derive ˆp. Sine we know that in steady state ˆp b(ˆp) ( =, therefore we know that V (ˆp) = θ). Define another funtion W (p) = b(p) ( p θ). Observe that if the poliy was λ = p θ then the orresponding (b(p(t)) (λ(t)))e t = (b(p) ( p θ)) = W (p). It is lear that sine V (p) is the superemum over all the poliies, the following has to hold V (p) W (p). And the equality holds at ˆp, i.e. V (ˆp) = W (ˆp). Sine b(p) is onave and ( p θ) is also onave in p. Therefore, W (p) is onave and ontinuously differentiable in p. We an onlude that V (p) = W (p) = b (p) ( p θ). Therefore, the ondition ( ˆp θ) = V (ˆp) simplifies to ( ˆp θ) = b (ˆp) ( ˆp θ). This is equivalent to X(ˆp) = b (ˆp) + ( ˆp θ) = (2) Next, we see that given the onditions in the theorem there will be a unique solution to the above equation. If < b () < then X() > beause (x) for x < is zero ((x) is zero for all x < ) and sine lim p X(p) <. From the ontinuity of X(p) it follows that there exists a steady state ˆp. Observe that X(p) is dereasing in p, whih follows from onavity of b(p) and ( p θ). Hene, we an see that the steady state will be unique. This shows the first part of the theorem. For the seond part, we need to show that if p() < ˆp then the λ p() (t) will derease with time and settle to ˆλ, while the orresponding population p p() (t) will inrease and onverge to ˆp. = v(p) = θ + (V (p)) p. From the analysis presented above we know that v(p) is a dereasing funtion whih starts at a positive value and then dereases to a negative value and intersets at the steady state value of the population. Note that if p() < ˆp then the v(p()) > is positive initially. Consider the population level v(ˆp ɛ) where ˆp p() > ɛ > then it an be seen easily that the population will grow as fast as v(ˆp ɛ). Hene, this an be done for any ɛ, whih proves onvergene. λ = (V (p)), note that V (p) is a dereasing funtion in p. For this we need to show that V is onave, whih follows from [2]. We also give the formal argument here. Consider two initial population levels p, p 2 and the orresponding optimal referral poliy and population trajetories are given as (λ (t), λ 2(t)) and (p (t), p 2(t)) respetively. Let us onsider onvex ombination of p and p 2, p 3 = κp + ( κ)p 2. We know that the value funtion V (p 3) e t (b(p 3(t)) (λ 3(t))), where p 3(t) is the population trajetory when λ 3(t) = κλ (t) + ( κ)λ 2(t). From the linearity of the differential equation for p, we an see that p 3(t) will be κp (t) + ( κ)p 2(t). V (p 3) e t (b(κp (t) + ( κ)p 2(t)) (λ 3(t) = κλ (t) + ( κ)λ 2(t))) and from the onavity of b(p) and (λ) we an say that V (p 3) κv (p ) + ( κ)v (p 2). Hene V is onave. Hene, as pop-

7 ulation inreases λ will derease with time and onverge. Theorem 2. Poliy omparison with hange in in steady state t : Loal Behavior i) If the average time that a user stays on the website is dereased by ɛ > and if Φ θ,, (b(ˆp), ˆp) < then the advertisements in the steady state inrease. ii) If the average time that a user stays on the website is dereased by ɛ > and if if Φ θ,, (b(ˆp), ˆp) > then the advertisements in the steady state derease. Proof. i) In steady state the value funtion an be omputed as follows V (ˆp) = b(ˆp) (ˆλ), where ˆλ = ˆp θ. Differentiating V (ˆp) w.r.t ˆp, we get V (ˆp) = b (ˆp) (ˆλ) (See Proof of Theorem.). We also know that V (p) = (λ), substituting V (p) we get (ˆλ)( + ) = b (ˆp). Expressing this equation in terms of only ˆλ, we get b ((θ + ˆλ)). (ˆλ). Note that g(ˆλ) = b ((θ + ˆλ)). + + = (ˆλ) is a ontinuously differentiable dereasing funtion in ˆλ. We are interested in determining how the behavior of the optimal referrals hange when the rate at whih users leave = is inreased by a small amount. Note that we will use = + (b ( θ+ˆλ in the equations that follow. Let dg (ˆλ) d = ) (θ+ˆλ) λ 2 d + b ( θ+ˆλ ) (+ ) ase when b ( θ+ˆλ ) (θ+ˆλ) + b ( θ+ˆλ λ 2 d equivalent to Φ θ,, (b(ˆp), ˆp) = d2 b(ˆp) 2 ) (+ ) ). Let us onsider the ) <, whih is ˆp + db(ˆp) s + <. In this ase dg (ˆλ) d > and we laim that the optimal ˆλ will inrease. We know that hanging will lead to a new value for ˆλ, whih will be unique. Let us assume that ˆereases, whih would mean that (ˆλ) will derease as well. Also we know that dg (ˆλ) d >, whih means that g(ˆλ) at the new value of λ will be positive. This violates the ondition for optimality, hene ˆλ inreases. Note that in the above the ritial thing to showing the result is Φ θ,, (b(ˆp), ˆp) < and we only allowed for a small hange in beause we only have loal information about the derivative at ˆp. ii) Consider the other ase Φ θ,, (b(ˆp), ˆp) = d2 b(ˆp) ˆp + 2 db(ˆp) s + >. In this ase dg (ˆλ) d <, whih means that ˆλ annot inrease beause this would make g(ˆλ) negative. Result. Poliy omparison with hange in in steady state t : Global Behavior If the average stay time of the users dereases upto then the firm should inrease the intensity of advertisements in the steady state, while if the average stay time falls below then the firm should derease the advertisements. Proof. For the above result we will onsider benefit funtion to take the form, b(p) = p a. Φ θ,, (b(p)) is given as (p a )((a ) + ). Observe that if then + Φ θ,, (b(p)) <, whih leads to an inrease in the poliy. For the other part also we an see that if a derease in the average stay time makes the firm inrease the advertisements in the poliy. Theorem 3. Poliy omparison with hange in θ in steady state t : Global Behavior If the intensity of the diret arrivals θ is inreased then the intensity of advertisements in the steady state dereases. Proof. Note that in this proof as well, we would substitute = We know that in the steady state the following + b ( θ+ˆλ ). = (ˆλ) holds. If θ is inreased then the term on the left b ( θ+ˆλ ). + dereases, this is due to the fat that b (p) <. For the steady state equation to hold, i.e. b ( θ+ˆλ ). + = (ˆλ) the term on the right an only derease, whih happens only if ˆereases. This ombined with the fat that a solution ˆλ always exists and is unique, leads to the onlusion that ˆereases. Theorem 4. Poliy omparison with hange in : at finite time t i) If the average stay time of the user dereases to a value above the threshold th then the intensity of advertisements inreases at all time instanes t larger than some finite t. ii) If the average stay time of the user dereases to a value below the threshold th then the intensity of advertisements at every time instane t dereases. Proof. Next, we study the firm s behavior when the total apaity of the system is finite. Let us arrive at the analytial form for the referrals by solving the HJB equation for the ase given as follows. V (p) = max λ ( ( p)2 λ 2 + V (p)(θ + λ p)) The solution for optimal λ is λ = max{ 2 V (p), }. It an be shown that the value funtion is onave. Therefore, we an see that λ = max{ 2 V (p), } is a dereasing funtion in p. Sine the optimal λ is different in different regimes we give a ase by ase analysis. Assume that the rate at whih users leave is high, i.e. a low average stay time θ. Reall that we are using =. Let us assume that the value funtion V (p) = Ap2 + 2Bp+C for p. Let us onsider the regime where V (p) = 2Ap + 2B and p, whih implies λ = (2Ap + 2B). 2 Substituting λ = (2Ap + 2B) and V (p) = 2 Ap2 + 2Bp + C in the HJB equation we get (Ap 2 + 2Bp + C) = ( 2A + A2 )p2 + (2 + AB + 2Aθ B )p + Bθ + B2 4 In the above equation we used the fat that λ = Ap+B and p. If Ap + B and p for a measurable set of values p then the the solution to the above is found by equating the oeffiients we get A = (2+) 2 (2 +) 2 +4, B = 2 Aθ+. Substituting A and B in the expression Ap+B + A and p, we get the following threshold on p, p th = B A = ( θ)(2 ++ (4+4λ 2 d )) + (+(+)θ). Note 2(λ 2 d ++) λ 2 d ++ that p th >. Hene, we know that there exists < p p th. For eah < p p th the equation (4) has to hold, whih is only possible if the oeffiients on left of (4) are the same as

8 reases, while the seond term an inrease/derease. However, note that as t inreases the effet of the hange in se- the oeffiients on the right. Hene, A = (2+) 2 (2 +) 2 +4, ond term deays and after a ertain threshold t the effet 2 B = Aθ+. Therefore, for < p p th the optimal poliy of inrease in the first term will dominate. + A will be λ = Ap+B. For p > p th the referrals will have to be beause we know that referrals are dereasing funtion of p and they annot go below zero. Having omputed optimal referral poliy is a funtion of the population level, we will now ompute the optimal referral poliy and the population levels as a funtion of time. The optimal poliy and the orresponding population dynami, when the average stay time of the user is low, i.e. or λ θ d θ and the initial population is low as well, i.e. p() < p th is omputed as follows. (B + Ap) = θ + p The steady state of the above dynami is ˆp = +( +)θ. λ 2 d ++ θ+b A = If p() ˆp then p=p() is positive. The population will inrease and saturate to attain the steady state value ˆp. Sine the population trajetory never rosses p th, λ = Ap+B. Solving the population dynami above, we get p(t) = ( + (λ.( d + )θ) 2 +4λ 2 (λ 2 d + + ) )( e d +4 +4)+ 2 t ) +.( 2 +4λ 2 d +4 +4)+ t p()e 2 Sine the population is inreasing, the optimal referrals will derease and saturate to attain λ = ˆp θ. Optimal referrals λ = Ap(t) + B are given as ( λ(t) = ( θ) λ 2 d ( p() +.( 2 +4λ 2 d ( + ( + )θ) (λ 2 d + + ) ) e +4 +4)+ t ) ( 2 + 4λ 2 d ) If th, where th = max{ θ+ 2 θ 2 +(θ+), } equivalently and if dereases then we need to th argue that the optimal referrals will derease at all times. Observe that if θ+ 2 θ 2 +(θ+) in λ(t), i.e. or then the first term ( θ) λ 2 d ++ dereases with inrease in. In addition if then (+(+)θ) (λ 2 d ++) Also, e.( 2 +4λ 2 d +4 +4)+ t ( 2 +4λ 2 d +4+4) in. Note that p() ˆp ensures that 2( p()+ (+(+)θ) ) (λ 2 d ++) e dereases as well. dereases with inrease.( 2 +4λ 2 d +4 +4)+ t 2 dereases and this is true for all times t ( 2 +4λ 2 d +4+4) Consider the ase when th, where th = this is equivalent to θ+ 2 θ 2 +(θ+) under this ondition the first term in λ(t), i.e. θ+ 2 θ 2 +(θ+). Observe that ( θ) λ 2 d ++ in- Unertain user arrivals/exits: In this setion we allow for unertainty in the arrivals and exits of users. This modifies the user population differential equation to the following stohasti differential equation, dp t = (θ + λ(t) P ) + σp dw t. (Reall that = ) The HJB equation orresponding to this ase is V (p) = max λ (2 ( p) 2 λ 2 + V (p)(θ + λ p) + 2 V (p)σ 2 ) In the above setting we allow for negative referrals for analytial tratability, but we do not allow that the population beomes negative. Hene, the optimal λ should ensure that the population is always positive. Solving the optimal λ we get λ = V (p). Let s assume that V (p) = A p 2 + B p + C 2 and substitute λ = V (p) = 2A p+b in the above HJB equation. Equating oeffiients on both the sides we get A 2 2 = 2 + (2 +) 2 +4( +σ2 ) A 2( +σ2 ) and B = 2 (A θ+) (+ A ). Note that λ d < and θ + B >, these onditions together with 2 the fat that the initial population p() ensure that the population dynami stays above zero (see [6]). The expeted population at a time t is given as E[P (t)] = e ( A 2 A ( )t ) + p()e ( A )t. The expeted number of referrals are given as E[λ(t)] = A E[P t ]+B If p() 2 A = ( +)θ+ 2. ( + A )( A ) then the expeted population E[P (t)] dynami dereases with time and onverges to attain 2 A. The orresponding trajetory of expeted number of referrals will inrease with time, this is beause the referrals derease with inrease in population. If p() 2 A then the expeted population E[P (t)] dynami inreases with time and onverges to attain 2. A The orresponding trajetory of expeted number of referrals will derease with time, this is beause the referrals derease with inrease in population Theorem 5. The average number of referrals given to a population of size P inreases with in an inrease in the variane σ 2 of the Brownian motion.. The average number of referrals given at any time t inreases with in an inrease in the variane σ 2 of the Brownian motion provided the initial population of the users is suffiiently high. We know that λ = A P + B. We know that 2 B = 2 A θ+ and from the expression observe that if A inreases B inreases as well. Next, we show that as σ in- +λ, d A reases then A inreases. Sine da dσ da dσ = 2 σ((2 +) +2( ))+2σ 3 (2λd +) 2 +4( +σ2 )( +σ2 ) 2 >, it shows the first part of the theorem.

9 We know that A and B inrease with inrease with σ. In addition if we an show that E[P (t)] inreases as well, then the E[λ(t)] will inrease as well. The expression for E[P (t)] is sum of two terms, the first term inreases with σ. 2 A 2 A learly The seond term is given by (p() )e ( A )t. If p() 2 A ( A )2 A db dσ da dσ is satisfied then the seond term dereases as well. This is a suffiient ondition and it an be heked by replaing in the derivative of the seond term. Hene, the E[λ(t)] will inrease.