The Interaction of Positive Externalities and Congestion Effects

Transcription

1 The Interaction of Positive Externalities and Congestion Effects Ramesh Johari Department of Management Science and Engineering, Stanford University, Stanford, CA 94305, Sunil Kumar Graduate School of Business, Stanford University, Stanford, CA 94305, Kumar We study a system where a service is shared by many identical customers; the service is provided by a single resource. As expected each customer experiences congestion, a negative externality, from the others usage of the shared resource in our model. In addition, we assume each customer experiences a positive externality from others usage; this is in contrast to prior literature that assumes a positive externality that depends only on the mere presence of other users. We consider two points of view in studying this model: the behavior of self-interested users who autonomously form a club, and the behavior of a service manager. We first characterize the usage patterns of self-interested users, as well as the size of the club that self-interested users would form autonomously. We find that this club size is always smaller than that chosen by a service manager; however, somewhat surprisingly, usage in the autonomous club is always efficient. ext, we carry out an asymptotic analysis in the regime where the positive externality is increased without bound. We find that in this regime, the asymptotic behavior of the autonomous club can be quite different from that formed by a service manager: for example, the autonomous club may remain of finite size, even if the club formed by a service manager has infinitely many members. Key words: Externalities; network effects; congestion; club theory; services 1. Introduction Informally, an externality is any effect that a user of a system has on others, that the user does not account for; for example, the larger the collection of users accessing an online video game, the lower the quality of service delivered to each of the users. The preceding congestion effect is an example of a negative externality. To a large extent, the performance engineering of network services (such as wireless service provision, content delivery, etc.) is driven by the goal of mitigating 1

2 2 Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects the negative externality. However network services can also create positive externalities for their users; the utility derived from the online gaming system by a user, as measured by the quality of the game, conceivably would increase as overall usage of the system increases. Of course these two externalities act against each other. Several authors have recently noted this phenomenon in the evolution of peer-to-peer filesharing services. An empirical analysis suggested that negative externalities from congestion imposed a natural bound on the growth of filesharing services (Asvanund et al. 2004). A two-sided market approach is taken by Gu et al. (2007) to empirically study peer-to-peer filesharing; they treat sharers and downloaders as the two sides of the market. Their analysis suggests positive externalities between sharers and downloaders, and negative externalities on the same side of the market, again limiting system size. Recent press has highlighted this externality tradeoff in social networking services as well: the initial growth of sites such as Facebook and MySpace is driven by a large positive externality, but the utility of these sites may suffer if they grow too large (see, e.g., Economist 2007, Jesdanun 2008). One of the goals of this paper is to provide a framework for comparing these two externalities. In this paper we are interested in a particular form of negative externality, congestion. Congestion effects in resource sharing models typically depend on the usage of the resources and not on the number of users per se. For example, this is the case in the literature on congestion in telecommunication networks and interference in wireless systems. egative externalities that depend on the total usage of resources have been well understood (Kelly 1991, Bertsekas and Gallager 1992, Beckmann et al. 1956, Roughgarden 2002, Tse and Viswanath 2005), and at this point the effect that an additional user has on system performance can be well quantified for a variety of service models. We adopt this mode of analysis and incorporate positive externalities. In much of the literature on positive externalities, also called positive network effects, the mere presence of an individual generates a benefit to other users. For example, consider adoption of a common standard, e.g., a common telecommunications protocol. In this case, the adoption of the same standard by another party can only generate a positive benefit to existing users of that standard. Further, in such a setting there is not a congestion effect: more users of the standard only

3 Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects 3 improve the utility of existing users of the standard. The same reasoning applies in many other settings, including product compatibility and technology adoption. Since users participation alone generates a positive externality, in these systems the positive externality is typically a function of the count of the number of other users participating. For examples of this approach, see Buchanan and Stubblebine (1962), Oren and Smith (1981), Katz and Shapiro (1985), Farrell and Saloner (1985), Liebowitz and Margolis (1994), Farrell and Klemperer (2005), Sundarajan (2003), Economides (1996), Odlyzko and Tilly (2005), Metcalfe (1995); there are also some examples where users are modeled as a continuum, and the externality depends on the total mass of users (Laffont and Tirole 2001, Laffont et al. 2003). By contrast, in our work we are interested in situations where the quality of the system is influenced by the usage of players in a system, rather than just their mere presence. For example, in an online gaming service, a user benefits from other subscribers only if they are actually playing online. In such settings, the usage of the service often leads to a congestion effect or negative externality such as those detailed above; this typically arises due to some form of resource sharing among the users. While usage-based models are justifiable in settings such as games, there is also a second reason we take this approach. As mentioned above, in most cases where positive externalities are generated by mere presence of users, such as standards, the notion of any negative externality (let alone congestion) is moot. More always means more in these settings; we don t consider these in our work because we are interested in studying the trade-off between the two externalities. Therefore we study a usage-based externality model, as this allows us to more easily focus on the trade-off between congestion effects and network effects. The goal of this paper is to study the interaction between positive and negative externalities using a general, albeit stylized, model that could represent many applications. We focus on systems where multiple users share a common resource, and derive a benefit from each other s usage. Our model assumes the users experience a payoff that is separable in the positive and negative externality. We

4 4 Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects emphasize that our analysis is descriptive: we provide structural insight into the tradeoffs between positive and negative externalities. We begin by studying systems with a fixed number of users with identical utilities. We start by assuming that usage of the service cannot be explicitly regulated. Rather it is implicitly regulated by the externalities, with each user choosing her usage selfishly to optimize her utility. We show that there is a unique ash equilibrium to this usage game. We use this to analyze the behavior of the system as the number of users changes. We show that there exists a finite M where the individual ash utility is maximized. That is, we show that there is an optimal size from the perspective of an individual user. This is not particularly surprising since one expects that when the club size is small, the inclusion of an additional user increases the positive externality more than the negative externality and that this effect is reversed one the club gets big enough. We also study the total welfare at the ash equilibrium and arrive at the following non-obvious result. The efficiency ratio, defined as the ratio of the total ash welfare to the total welfare achievable by a social planner who regulates usage, is maximized at M the same size at which individual ash utility is maximized. Moreover the maximal efficiency ratio is unity. In other words, selfish users who are free to form their own club will form a club of size M, and will proceed to use it in an efficient manner. Of course, M is not the size that maximizes the total ash welfare; it merely maximizes the efficiency ratio. We show that there exists an at which total ash welfare is maximized, and that M. That is, users left to themselves may form a club that is too small when total welfare is considered. Although we don t explicitly consider pricing in this paper, a monopolist wanting to charge a fixed access price would choose to charge the individual utility at and thus form a club of size. This result differs from previous equilibrium size results in the literature (Oren and Smith 1981, Katz and Shapiro 1985), in that it does not require heterogeneity among users for a finite club to be formed by a monopolist. We then turn our attention to asymptotic behavior as the effect of the positive externality increases without bound. In particular we study the asymptotics of M and. We show that

5 Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects 5 it is possible that the limit of M s be finite even in this asymptotic regime. We also study the asymptotics of and show that could grow to infinity even when M remains finite. This suggests that the scale of the clubs where admission is self-governed versus those where it is externally controlled can be dramatically different, despite having the same operational structure. In the literature relevant to service management, there are few papers that combine both positive and negative externalities in the same analysis. Even where appropriate, like communication systems (Oren and Smith 1981), the positive externality literature typically does not consider congestion and other negative externalities. More prominently, nearly all the work on congestion pricing for communication networks and other service systems typically focuses entirely on the negative externality, and pays little attention to the positive externality; for an overview, see, e.g., Falkner et al. (2000), Briscoe et al. (2003), Kelly (1997), Srikant (2004), He and Walrand (2003), Allon and Federgruen (2005). The same is typically the case in the literature on transportation systems (Beckmann et al. 1956). Our work here can be viewed as an extension of this congestion model literature to include positive externalities. The most closely connected models to our own work are those of club theory; this literature contains results analogous to our observation that there exists a finite M that maximizes individual welfare. Typically, clubs are excludable, partially nonrival goods; a simple example is a swimming pool, where congestion sets in only when the capacity limit for the pool is reached. In a typical club model, individuals feel a negative externality from congestion effects, but feel a positive benefit from the presence of others if the cost of building the club is shared among the members; see, e.g., Buchanan (1965), Berglas (1976), Sandler and Tschirhart (1980), Cornes and Sandler (1996), Scotchmer (2002) for more details. Indeed, there is a close connection between transportation models and club goods, in those cases where the cost of road construction is shared among the members of the public through tolls (Berglas and Pines 1981); a similar approach has recently been taken to study peer-to-peer filesharing Courcoubetis and Weber (2006). In such models, the typical notion of optimal club size is the one which maximizes an individual s welfare; a typical result is that the tradeoff between the congestion effect and the benefit of cost sharing leads to a finite

6 6 Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects optimal club size M, a similar result to that obtained in our analysis. However, our subsequent analysis follows a different path from the usual analyses in club theory: in particular, our analysis of the efficiency properties of M is distinct from the club theory literature. 2. Model Let denote the number of customers of a single service provider; we refer to these customers as users or players throughout the paper. Each user i has a utility given by: ( ) ( ) u i (ρ) = αρ i +βρ i f ρ j ρ i l ρ j d(ρ i ), (1) j i j where ρ j is the usage level of user j. We interpret usage as the extent to which this user participates in the system; for example, in an online messaging or social networking service, usage is proportional to the time spent in the system by the user. The user s utility is composed of four terms: (1) the utility directly received from his usage, αρ i ; (2) the positive externality, denoted by the function f; (3) the negative externality, i.e., congestion, denoted by the function l; and (4) a personal cost of effort, denoted by the function d. The positive externality depends on the aggregate usage of all other players, because we assume the payoff of a single player increases when all other players increase their effort. On the other hand, the negative externality depends on the total usage in the system, including all players; this reflects the fact that even if no other players exert effort, player i would still experience a congestion effect due to his own usage (e.g., loss or delay in queueing systems). We make the following assumptions. 1 Assumption 1. The function f(x) is nonnegative, strictly increasing, concave, and continuously differentiable over x 0. The preceding assumption is satisfied by common positive externality models. For example, Metcalfe s Law states that the aggregate value of a network of individuals grows proportional 1 Throughout the paper, given a function g(x) defined on x 0, we interpret the derivative g (0) as the right directional derivative at zero.

7 Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects 7 to 2, since each individual can connect to 1 other individuals (Metcalfe 1995). ote that in this case the positive externality is count-based, i.e., it depends only on the total number of other individuals. In our setting, the analogous externality model is one where the positive externality scales in proportion to the total effort of other individuals, i.e., f(x) = x. However, our assumptions on f are weaker than this, and in particular allow for externality models that exhibit diminishing marginal returns in the effect other individuals have on a single individual (i.e., concavity of f). In a setting where externalities are count-based, it has been alternatively proposed that the aggregate value of a network of individuals grows as log Odlyzko and Tilly (2005); in our model, this suggests each user experiences a strictly concave positive externality of the form f(x) = logx. Assumption 2. The function l(x) is nonnegative, strictly increasing, convex, and continuously differentiable over x 0. Further, the function l(x)+xl (x) is convex over x 0. The preceding assumption is satisfied by a broad range of models. For example, to ensure that l(x) + xl (x) is convex, it suffices to assume that l (x) is convex (in addition to l(x) having the assumed properties). Although the preceding assumption implicitly assumes that l(x) < for all x, we emphasize that the results of this paper continue to hold even if l(x) as x C, where C < ; i.e., if l has a finite asymptote. In this case the analysis can proceed via the use of extended real-valued functions (Rockafellar 1970). Thus our analysis includes standard congestion models, particularly delay functions in queueing systems. For example, the previous condition is satisfied by queueing delay functions of the form l(x) = 1/(C x); in this case we define l(x) = for x C. Assumption 3. The function d(x) is nonnegative, nondecreasing, convex, and continuously differentiable over x 0, with d(0) = 0. Further, the functions d(x)/x and d (x) are convex over x 0. Assumption 3 is also not very strong: it will be satisfied, for example, by queueing delay functions of the form d(x) = x/(c x). Indeed, one interpretation of the function d(x) is that it captures a constraint on the local resources available to an individual, e.g., communication or processing

8 8 Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects resources; while the function l captures globally shared resources. We require d(0) = 0 merely as a normalization. Analysis of our model is most interesting in those scenarios where the positive externality is strong enough to ensure that positive participation is efficient, but not so strong that usage approaches infinity in equilibrium. In other words, we need to ensure the positive and negative externalities are both active in our model. We make the following assumption. Assumption 4. (1) α +βf(0) > l(0)+d (0); and (2) α +βf(ρ) l(ρ) as ρ. The first of these conditions ensures that all users exert positive effort at any efficient outcome. The second condition ensures that eventually, the negative externality dominates the positive externality; without this condition, it is possible that the social welfare may be optimized at infinite total effort. Conceptually, our composite model interpolates between well studied classes of games. For example, when l 0, there is no negative externality; in this case we obtain a standard supermodular game used to model network effects (akin to Diamond s search model, cf. Diamond 1982, Milgrom and Roberts 1990). On the other hand, when d 0 and f 0, we obtain a Cournot oligopsony with market price function given by l(x) (Tirole 1988); in other words, the game can be interpreted as one where each player chooses a quantity ρ i, and pays a per unit price given by l( ρ i i). We conclude with some basic notation that will be required throughout the paper. We define the total social welfare as W(ρ) = u i (ρ). (2) i=1 ote that since all users share the same utility function, we are effectively measuring all players utilities in the same units. Because all users in our model share the same utility function, we will find that both social optima and equilibria are symmetric under our assumptions. For this reason, we also define a version of the total social welfare that assumes a symmetric usage allocation. Formally, suppose

9 Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects 9 that individuals each exert individual effort ρ/, so the total effort is ρ. With a slight abuse of notation, we also define: where h(ρ,) is defined by (4): W(ρ,) = ρh(ρ,) (3) h(ρ,) = α +βf((1 1/)ρ) l(ρ) d(ρ/) ρ/. (4) Thus W(ρ,) gives the total welfare when individuals exert total effort ρ. We interpret W(ρ,) more generally as a function defined for all real 1, not just integral. Indeed, in our subsequent development we make a relaxation of our model: we will assume that is no longer restricted to be an integer and is real valued. While (1) is no longer directly usable, we will show that both social optima and equilibria are characterized by determining the total effort ρ at a fixed value of. This relationship allows us to express total welfare as a function of real-valued. Proofs of all results developed in the paper can be found in the e-companion to the paper. 3. Self-interested behavior This section studies our model from the vantage point of the individual acting in her own selfinterest. We assume first that the number of players is exogenously fixed, and then study the usage decisions that would be made by individuals. We then consider the behavior of the system when the number of players is allowed to vary. We identify the system size that the users would choose in their self-interest in Subsection 3.2; we refer to this as the autonomous club size, as it is the system size that is chosen by users acting autonomously Self-interested usage at a fixed club size We begin by studying the existence and uniqueness of ash equilibria for the usage game when players are present. Each player i has the utility function (1), and the strategy space of each player i is given by ρ i 0. That is, each user chooses her usage level in her self-interest. A ash equilibrium of this game is a vector ρ such that for all i: u i (ρ) u i (ρ i,ρ i ), for all ρ i 0. (5)

10 10 Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects (Here ρ i = (ρ 1,...,ρ i 1,ρ i+1,...,ρ ).) We have the following proposition characterizing the unique ash equilibrium of the game. Proposition 1. Suppose Assumptions 1-4 hold. Then there exists a unique ash equilibrium ρ E. Further, ρ E i following equation: α +βf Finally, u i (ρ E ) > 0 for all i. = ρ E / for all i, where ρ E > 0 is the (necessarily unique) solution to the (( 1 1 ) )ρ E (l(ρ E )+ ρe l (ρ E ) ) d ( ρ E ) = 0. (6) ote that the ash equilibrium is symmetric, and thus the total welfare is given by W(ρ E (),). Further, via (6), we study ash welfare under the relaxation that is real-valued (rather than integral) Self-interested choice of club size In this section we study the behavior of the welfare of a single player at the ash equilibrium, as the number of players increases. We show that this function is quasiconcave, and we characterize the optima of the individual welfare. We interpret these optima as the club size that would be preferred by self-interested individuals who autonomously form a club; for this reason we refer to the optima as the autonomous club sizes. Formally, define I() as follows: ( ) ρ E () I() = u i,..., ρe () = W(ρE (),). (7) Then I() is utility of a single player at the unique ash equilibrium when players are present. We start with the following key lemma that characterizes the behavior of ρ E () as. Lemma 1. Suppose Assumptions 1-4 hold. Then ρ E () is a strictly increasing function of 1, with sup 1 ρ E () <. ext, we show that I() is quasiconcave; further, it is maximized at [m,m ], for some choice of m,m.

11 Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects 11 Proposition 2. Suppose Assumptions 1-4 hold. Then I() is quasiconcave as a function of. Further, there exist m,m such that 1 m M, and argmax >1 I() if and only if m M, where m and M are given by (EC.5) and (EC.6) respectively. Further, m = M if at least one of the following holds: (1) f is strictly concave; or (2) l is strictly convex. To interpret this result, consider a scenario where a club forms autonomously, and usage is determined according to the ash equilibrium. For members in the club, the individual benefit is maximal when the club size lies in [m,m ], and thus an autonomous club would admit members at most up to M, and no further. We refer to any club size in [m,m ] as an autonomous club size; when m = M, the autonomous club size is unique. The preceding result is merely descriptive: it allows us to explicitly characterize the autonomous club sizes. From the proof in the online companion, we observe that the autonomous club sizes are exactly those where the following condition holds: (( βf 1 1 ) ) ρ E () = l (ρ E ()). (8) Thus at the autonomous club sizes, there is a balancing between marginal positive and negative externality effects according to the preceding equation. As we will see in our subsequent development, this balancing is closely linked to a surprising welfare optimality property of the autonomous club sizes. In principle, even at the autonomous club size, individual users could utilize the system in an inefficient manner. That is, ρ E (m ) could be quite different from the ρ that a social planner would choose. In Section 5, we will show that this does not happen: the usage of the system is efficient at the autonomous club sizes. 4. The service manager s viewpoint The preceding section studied our model from the point of view of self-interested utility maximization on the part of the individual. In this section, we consider instead scenarios where a service manager is able to control usage and/or access to the service. We assume the service manager takes decisions to maximize welfare. In one interpretation of this assumption, the service manager is a

12 12 Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects social optimizer, acting on behalf of the entire community. An alternate interpretation views the service manager as a pure monopolist: in this case the service manager s objective becomes welfare maximization, on the presumption that all surplus can be extracted through a nondiscriminatory, static pricing scheme. For example, the service manager might charge a fixed fee for the service, and then choose the fee so as to extract full surplus from the system. We consider two distinct settings. First, we formulate a setting where the number of players is exogenously fixed, and the service manager can control the usage of the players. This setting also allows us to show that if the service manager were to control both usage and the number of players, then under our assumptions the manager would choose to admit infinitely many players to the service. In the second setting we consider, the service manager can control access, but not usage; this is arguably more realistic for many online services where, e.g., a monthly fee might be charged, but usage is not monitored. In this setting we show that finite optimal club sizes exist, and that any optimal club size is at least as large as M Usage control We start by considering the optimal solution to the following problem that a service manager will solve, if she is able to dictate the vector ρ of the usage of all the users in the system: maximize W(ρ) (9) subject to ρ 0. (10) We refer to a solution to this problem as a social optimum, following standard terminology. ote that we are allowing the service manager to potentially choose asymmetric usage allocations as well; however, as the following result demonstrates, the service manager s optimal solution is symmetric. Proposition 3. Suppose Assumptions 1-3 hold, and fix 1. Let W S denote the optimal objective function value in (9)-(10). If W S > 0, then the unique optimal solution ρ S to (9)-(10) is given by ρ S i = ρ S / for all i, where ρ S > 0 is the unique solution to the following problem: [ (( max αρ+βρf 1 1 ) ) ( ρ ) ] ρ ρl(ρ) d. ρ 0 (11)

13 Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects 13 The preceding result characterizes the efficient usage vector, for a fixed : we first solve the scalar optimization problem (11), then symmetrically divide the total effort among the players. Adding Assumption 4 yields the following corollary. Corollary 1. Suppose that Assumptions 1-4 hold, and fix 1. Let W S denote the optimal objective function value in (9)-(10). Then W S > 0, and there exists a unique solution ρ S to (9)-(10). This solution is given by ρ S i = ρ S /, where ρ S solves (11), and 0 < ρ S <. The second part of Assumption 4 is actually somewhat stronger than needed to establish the preceding corollary, since the personal effort cost d also acts to mitigate the effect of the positive externality; however, we retain Assumption 4 as it is technically simpler than including any assumptions over d as well. We conclude this section by considering the possibility that the system manager controls both usage and access, i.e., the number of players in the system. The following result tells us that in this case, she will choose to have a club of infinite size. For 1, let ρ S () denote the unique solution to (11). ote that as in our analysis of the ash equilibrium, we can use (11) to view ρ S () as a function on real-valued, rather than just integral. Corollary 2. Suppose that Assumptions 1-4 hold. Then W(ρ S (),) is non-decreasing in. Moreover, sup W(ρ S (),) < Access control without usage control In this section we consider a form of second best control for the service manager, where access to the service can be controlled, but usage cannot. In particular, we assume that once users enter the system, they act in their own self interest, in accordance with the description in Subsection 3.1. Thus the total welfare when people enter is W(ρ E (),); the service manager s goal is to choose a value of that maximizes total welfare. We have the following result.

14 14 Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects Proposition 4. Suppose Assumptions 1-4 hold. Then W(ρ E (),) 0 as. Thus there exists such that W(ρ E ( ), ) = sup W(ρ E (),). (12) 1 That is, the total ash welfare is maximized at. Further, for any such, there holds M < <. ote that although always exists, in general there may be multiple solutions. Proposition 4 tell us that a service manager who can control the size of the club will choose a system of finite size. However, this choice is no smaller than the autonomous club size M, i.e., the club size the users would pick for themselves. While it is not always tractable to directly compare M and, we will carry out such a comparison in an asymptotic setting in Section Efficiency at the autonomous club size M In this section we study the ratio of the total welfare achieved at the ash equilibrium, cf. Section 3.1, to the maximum possible welfare, cf. Section 4.1. We establish the result that the usage of the club is indeed efficient at the autonomous club size M. Given fixed choices of α, β, f, l, and d, as well as a fixed number of player, we consider the efficiency ratio of the ash equilibrium to the social optimum, i.e., the ratio of their respective aggregate utilities: Γ() = W(ρE (),) W(ρ S (),), (13) where ρ E () is the unique solution to (6) with players; and ρ S () is the unique solution to (11) with players. (This ratio is also known as the price of anarchy in the literature; see, e.g., Papadimitriou 2001.) Obviously 0 Γ() 1, and larger efficiency ratios imply selfish behavior is increasingly efficient. Throughout this section we will assume that Assumptions 1-4 hold. We begin with a numerical special case, where f(x) = x, l(x) = x 2, d(x) = 0, α = 1, and β > 1. Figure 1 plots the ratio Γ() against. As expected, when = 1, the ratio is one: social welfare maximization coincides with the individual optimization of the single individual in the system. As

15 Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects Γ() Figure 1 Efficiency ratio. increases, the ratio initially falls. However, the figure reveals a interesting insight: for > 1, there exists a unique value of at which the ratio Γ() is maximized. More surprisingly, the maximum value achieved by this ratio is exactly one (if integrality constraints on are ignored). That is, at this value of the ash welfare is identical to the maximal welfare. Below this level of, the usage level of the system does not generate the efficient level of positive externality (unless = 1); and above this level of, the usage level of the system creates an excessive negative externality. In this section we evaluate efficiency of the ash equilibrium, and formalize this insight. That is we show that there is always a range of club sizes at which Γ() = 1 and this coincides exactly with the autonomous club size range [m,m ]. Before we turn our attention to the efficiency ratio at the autonomous club size, we take a detour to study the efficiency ratio in the common asymptotic regime where the number of users in the system approaches infinity. Observe from Proposition 4 that the ash equilibrium welfare approaches zero as. On the other hand, the social optimum must have social welfare that is nondecreasing in the number of users, and positive under our assumptions. We conclude that the efficiency ratio Γ() approaches zero as. We can be more precise about this convergence. We have the following theorem. Proposition 5. Suppose Assumptions 1-4 hold. Then ρ E ()h(ρ E (),) 0 as ;

16 16 Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects further, there exist constants K 1,K 2 > 0 such that for all 1: Thus Γ() Θ(1/) as. K 1 Γ() K 2. The preceding asymptotic result tells us that at large club sizes, the system is used in an increasing inefficient manner by selfish users. So the key to understanding the behavior of this system is studying Γ for finite values of. As in Figure 1, when = 1, then clearly the ash equilibrium and social optimum coincide: the positive externality plays no role in the strategic optimization of the single player. Thus we always have Γ(1) = 1. For this reason, for the remainder of the section we will focus on the case where > 1. Theorem 1. Suppose Assumptions 1-4 hold. Then Γ() is a continuous function of if 1, with Γ(1) = 1. Further, with m and M defined as in (EC.5) and (EC.6) we have: 1. If 1 < < m, then Γ() < 1, and ρ E () < ρ S (). 2. If > 1 and m M, then Γ() = 1, and ρ E () = ρ S (). 3. If > M, then Γ() < 1, and ρ E () > ρ S (). The preceding theorem reveals several interesting features of the tradeoff between positive and negative externalities. First, it shows that self-interested optimization exactly achieves the socially optimal welfare in the interval [m,m ]. It is particularly noteworthy that the maximal social welfare W(ρ E (),) is a moving target it is strictly increasing in ; thus achieving full efficiency at the ash equilibrium is somewhat unexpected. But what is even more surprising that the range of club sizes over which efficiency is achieved coincides exactly with the range where individual ash utility is maximized. So the autonomous club size takes on additional meaning: left to themselves selfish users would form a club which will then be used efficiently without any regulation. (We note that here we mean efficiency conditional on the number of players in the system; as already noted in Section 4.1, if the number of players is controlled as well, the efficient outcome is a system of infinite size.)

17 Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects 17 When < m, an efficiency loss arises because individuals do not sufficiently internalize the positive externality; in this regime, the total ash effort ρ E () falls short of the socially optimal level ρ S (). When > M, an efficiency loss arises because individuals do not sufficiently internalize the negative externality; in this regime, the total ash effort ρ E () exceeds the socially optimal level ρ S (). When [m,m ], the two effects exactly balance; this is clearly shown by the definition of J() in (EC.4). When [m,m ], we have J() = 0, or from (8): (( βf 1 1 ) ) ρ E () = l (ρ E ()). In this case the ash equilibrium correctly balances positive and negative externalities and exactly achieves the socially optimal welfare. Another consequence of the preceding theorem is that at the size of the club chosen by a service manager who cannot control usage, of Section 4.2, the usage will be necessarily inefficient since > M. In other words, although the total ash welfare is higher at than at the autonomous club sizes, the ash equilibrium usage at is inefficient relative to the socially optimal usage levels. 6. Managed clubs vs. autonomous clubs: an asymptotic analysis In this section our primary goal is to compare the autonomous club size M with the managed club size obtained when the service manager can exercise access control, but not usage control. Thus we compare access decisions made by self-interested individuals with those made by the service manager. We carry this analysis out by considering an asymptotic regime that makes the comparison tractable. The preceding sections have characterized the notion of an optimal club size for fixed parameters in our model. In this section, we restrict attention to a setting where f(x) = x, and consider the asymptotic behavior of the system as the positive externality coefficient β increases to infinity. For technical simplicity, we assume that l is strictly convex, so that M is uniquely defined. Furthermore, we assume l(x) < for all x. The results do hold if l(x) approaches infinity at a finite value of x, as is the case for queueing delay functions such as l(x) = 1/(C x).

18 18 Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects We characterize the asymptotic behavior of M and as β ; these asymptotic results are interesting in their own right, but in particular allow us to directly compare the two system sizes. It turns out the behavior can be dramatically different as will be evident from the sequel Asymptotics of M Our assumptions on l guarantee that for a fixed β, the optimal club size M of Theorem 1 is uniquely defined; we denote this value by M (β). We begin by investigating the behavior of M (β) as β approaches infinity. Informally, this limit allows us to gauge the maximum possible autonomous club size. Since f is concave in general, it is always upper bounded by a linear positive externality; and as β, we are steadily increasing the effect of the positive externality. Our key result is the following theorem, giving the precise scaling behavior of M (β). Theorem 2. Suppose f(x) = x, and that Assumptions 2-4 hold. Assume further that l(x) is strictly convex, and that l (x) is an invertible function on [0, ). Define ǫ as: we assume the preceding limit exists. For each c 1, define φ(c) as: ǫ = lim x xl (x) l(x) ; (14) φ(c) = lim x d (x/c) xl (x) ; (15) again, we assume the preceding limit exists. Then ǫ 1, and φ(c) is monotonically nonincreasing in c. Under these assumptions, M (β) approaches a limit M (possibly infinite) as β. Further: 1. If ǫ = 1, or φ(c) = for all c, then M =. 2. If ǫ > 1, then: { 2 M = inf M 1 : M +φ(m) 1 1 }. ǫ The theorem can be used to characterize several interesting special cases. Assume that ǫ > 1. First, if d(x) 0, i.e., if no personal effort cost exists, then φ(c) is identically zero; in this case M is given by 2/(1 1/ǫ). In particular, notice that in the absence of a personal effort cost, the optimal

19 Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects 19 club size remains finite regardless of the magnitude of the positive externality. Another important special case arises when φ(c) = for all c; in this case, M is infinite as well. Finally, it can be shown that if φ(c) is continuous and M is finite, then M is the unique solution to the following equation: 2/M +φ(m) = 1 1 ǫ. (For further details, please see the proof of the theorem in the e-companion.) To gain intuition behind the definition of φ(c), suppose that c individuals are in the system, and exert a total effort x, that is symmetrically divided among them. In this case d (x/c)δ is the marginal change in a player s personal effort cost if she exerts an additional δ units of effort, while xl (x)δ is the marginal change in the negative externality if other players exert a total additional effort δ. Informally, when φ(c) is large, then d (x/c) is large relative to xl (x) for sufficiently large x; and in this case, the system behaves as if the negative externality were not present at all in which case the club size always grows to infinity, since additional players are always desirable. On the other hand, if φ(c) is small, system behavior is dominated by the congestion externality, and beyond some fixed finite size additional players only congest the system. We also note that it is possible to extend the result above in the case where the limits defining either ǫ or φ( ) do not exist, by considering subsequences where necessary; for simplicity of presentation we assumed the limits exist. We conclude with an example inspired by the M/M/1 congestion function; i.e., suppose that l(x) = 1/(C x) for C > 0, and d(x) 0. In this case our result suggests that M (β) 2 as β ; i.e., in the limit the optimal club has only two members! This result obtains in large part because of the fixed capacity constraint induced by the M/M/1 queueing model. To obtain further intuition, consider a split the pie game where a pie of size C will be equally split among players, and the payoff to each player is βρ i (C ρ i ). If one player is asked how many other players she would like to play with, then the player will solve: ( )( C max C C ). 1

20 20 Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects It is straightforward to check that the optimal solution to this problem is = 2: one additional player brings a large positive externality, but additional players past = 2 only force the pie to be split further with diminishing externality gains. Effectively, when β, the M/M/1 congestion externality induces a system that behaves as if players are splitting a fixed size pie; and thus the limiting club size is small. As we will see in the next section, the conclusion is quite different when the service manager controls access to the service Asymptotics of We now turn our attention to the asymptotic behavior of as β increases without bound. For each β, let (β) be a second best optimal choice of club size, i.e., a solution to (12). ote that (β) may not be unique; our results hold for any sequence of optimal club sizes. For analytical simplicity, and in light of the discussion in the preceding section, we consider only two extremes: one where φ(c) = for all c, and one where φ(c) = 0 for all c (where φ is defined as in (15)). The first case implies d (x/c) grows faster than xl (x), so that the personal effort cost dominates congestion for large x. The second case implies d (x/c) grows slower than xl (x), so that congestion dominates the personal effort cost for large x. One of these two regimes is straightforward to analyze. If φ(c) = for all c, we have already shown in Theorem 2 that M (β) under the assumptions of that theorem. Since Proposition 4 implies (β) M (β) for all β, under the same assumptions we also have (β). Thus if the autonomous club size approaches infinity, then the service manager s second best choice of club size also approaches infinity. For this reason we focus our attention in this section on the other extreme, where φ(c) = 0 for all c. This is equivalent to assuming that φ(1) = 0, since φ( ) is a nonincreasing function (cf. Theorem 2). Our primary result is the following. Theorem 3. Suppose f(x) = x, and that Assumptions 2-3 hold; also suppose Assumption 4 holds for all β > 0. Assume further that l(x) is strictly convex, and that l (x) is an invertible

21 Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects 21 function on [0, ). Define ǫ as in (14); we assume the limit there exists (though it may be infinite). Define d (x) φ = lim x l (x)+xl (x) ; we assume the preceding limit exists, and that φ = 0. 2 For each β, let (β) be a second best optimal choice of club size, i.e., a solution to (12). Under these assumptions, (β) approaches a limit (possibly infinite) as β. Further: 1. If ǫ = or ǫ = 1, then =. 2. If 1 < ǫ <, then is the unique positive solution to: ( ) 4ǫ 2 ǫ = 0. (16) ǫ 1 In particular, for large ǫ, we have ǫ. As ǫ approaches 1, we have 4/(1 1/ǫ). It is interesting to compare Theorem 2 with Theorem 3. In Figure 2, we plot the values of M and in the case where φ(c) = 0 for all c, and ǫ varies between 1 and infinity. As ǫ 1, both M and grow to infinity; indeed, for small ǫ, we have 2M = 4/(1 1/ǫ) i.e., the limiting club size created by the service manager is approximately twice the club size created by self-interested club members. On the other hand, for large ǫ, we observe that M 2, while ǫ. In particular, becomes infinite as ǫ. In particular, consider the case where ǫ =, i.e., loosely speaking, the congestion function is very steep. In this case M (β) remains finite, and in fact approaches 2. On the other hand, (β) grows to infinity in this situation. This suggests that the scale of the clubs where admission is based on self-governance can be dramatically different from those where admission is externally controlled, despite having the same operational structure. We can gain further intuition for the case where ǫ = by once again considering an example inspired by the M/M/1 queueing model, as in the last section; i.e., suppose that l(x) = 1/(C x) 2 ote that here φ is defined in terms of second derivatives of d and l. This is slightly stronger than the definition of φ(c) in Theorem 2: an application of L Hôpital s rule immediately shows that this assumption implies φ(1) = 0, and hence φ(c) = 0 for all c.

22 22 Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects M, ǫ M Figure 2 M vs.. for C > 0, and d(x) 0. This is a case where ǫ =, so M = 2 while =. We again interpret this result in terms of a split-the-pie game. We again fix the total effort ρ, but we now assume that the service manager chooses the number of players so that the total welfare is maximized. In other words, the service manager acts to maximize: ( )( C max C C ). 1 The term in parentheses is the welfare of a single individual; maximization of that quantity alone yields an optimal solution of = 2. However, maximizing times the welfare of a single individual yields =. Thus, in this case the service manager chooses to create infinitely many, infinitesimally small slices of the fixed pie. 7. Extensions and Future Directions In this section we detail several extensions and open directions. First, we note that in our model, the capacity of the system does not scale as the number of users grows; in other words, we have analyzed the system assuming that the resource congestion function l( ) stays fixed as varies. In part, this constraint on resources is responsible for the fact that the autonomous club size does not grow large. However, we could consider an alternate model where capacity is allowed to scale as the number of users in the system increases. One such model is analyzed in Appendix 7, where capacity

23 Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects 23 of an M/M/1 queueing system is scaled with the number of users. It is shown there that sublinear scaling of the capacity suffices to ensure that the autonomous club size approaches infinity. Another key feature of such models that we have not studied here is the dynamics of entry. Because we study ash behavior in our model, it is as if all users are present in a static pool of potential entrants. It is possible, however, to consider dynamic pricing and entry as well. It is possible to show that with a sufficiently powerful pricing scheme, the service manager can again extract the second-best surplus (cf. Section 4.2). For example, if individuals arrive sequentially over time, and the service manager is allowed to charge each individual a nondiscriminatory service use fee per time period, then it is possible to show that the service manager can charge so that individuals enter, and per time period revenue to the service manager approaches the total ash welfare at. We omit the details of this straightforward analysis. Our paper leaves several open directions of significant interest. First, although our model has considered a macroscopic model of externality effects, there are important microscopic effects that are system-specific that we have not captured. For example, in a peer-to-peer file sharing system, users also choose to contribute their own resources to a common pool; indeed, it is users upload bandwidths that are employed to be able to share files with other peers. In effect, this setting requires that the function l again changes with the number of users, but now that the change is also controlled by users in their own self-interest. Important questions arise here regarding the design of mechanisms to avoid free-riding by system participants. In addition, our paper has considered only a single club. This approach has allowed us to focus exclusively on welfare effects of interactions among club members, and also of the incentives of a service provider interested in welfare maximization. However, a major departure from the model of this paper involves studying a model where multiple clubs can be chosen. We envision two distinct possibilities. First, a single service manager may choose to form multiple subgroups of a single club (as is the case, for example, when subcommunities form in social networking services such as Orkut and Facebook). Second, a model with multiple clubs may be used to study competition

24 24 Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects among multiple service managers to acquire users. Both these models remain open directions for future work. Finally, as discussed in the Introduction, we focus here on a usage-based externality model to explore the tradeoff between positive and negative externalities. A more general model would allow for both count-based and usage-based externalities; for example, a cell phone user both increases the utility of the cell phone to other users, through a product compatibility effect (a count-based externality), but also affects others directly through her usage. In our paper we focus attention on the usage-based externality. However, allowing both externalities in one model remains an intriguing direction for future research. Acknowledgments This work was supported by the Media X Program at Stanford University, and the ational Science Foundation under grant no. DMI Appendix. Scaling Capacity We note that in our model, the capacity of the system does not scale as the number of users grows; in other words, we have analyzed the system assuming that the resource congestion function l( ) stays fixed as varies. As discussed at the end of Section 6.1, this constraint on resources is in part responsible for the fact that the autonomous club size does not grow large in the M/M/1 queueing model. This is also why the service manager splits the capacity into infinitely many infinitesimally small slices in the M/M/1 example at the end of the previous section. In this section, we briefly consider an alternate model where capacity is allowed to scale as the number of users in the system increases, as is reasonable to expect in many settings. For example, a service provider may adhere to a provisioning rule that upgrades capacity on a schedule determined by subscription. We consider a setting where the capacity of an M/M/1 queueing system is scaled with the number of users. Our main finding is that a sublinear scaling of the capacity suffices to ensure that the autonomous club size approaches infinity. This result can be interpreted in a setting where a service manager chooses to allow a club to operate autonomously; for example, many social networking sites operate without extensive control from the service manager. In this case our result suggests that the service manager need only upgrade capacity