The Interaction of Positive Externalities and
Congestion Effects
Ramesh Johari
Department of Management Science and Engineering, Stanford University, Stanford, CA 94305, [email protected]
Sunil Kumar
Graduate School of Business, Stanford University, Stanford, CA 94305, Kumar [email protected]
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. Next, 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
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. Negative 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
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
Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects
4
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 Nash 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 Nash 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 Nash equilibrium and arrive at the following non-obvious
result. The efficiency ratio, defined as the ratio of the total Nash welfare to the total welfare
achievable by a social planner who regulates usage, is maximized at M ∗ —the same size at which
individual Nash 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 Nash welfare; it merely maximizes the
efficiency ratio. We show that there exists an N ∗ at which total Nash welfare is maximized, and that
N ∗ ≥ 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 N ∗ and thus form
a club of size N ∗ . 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 N ∗ . We show that
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 N ∗ and show that N ∗ 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
Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects
6
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 N 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:
ui (ρ) = αρi + βρi f
X
j6=i
ρj
!
− ρi ℓ
X
j
ρj
!
− d(ρi ),
(1)
where ρj 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, αρi ; (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 N 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.
Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects
7
to N 2 , since each individual can connect to N − 1 other individuals (Metcalfe 1995). Note 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 N individuals grows as N log N Odlyzko and Tilly (2005); in our model, this
suggests each user experiences a strictly concave positive externality of the form f (x) = log x.
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
Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects
8
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 ρi , and pays a per unit price given by ℓ(
P
i ρi ).
We conclude with some basic notation that will be required throughout the paper. We define
the total social welfare as
W (ρ) =
N
X
ui (ρ).
(2)
i=1
Note 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
Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects
9
that N individuals each exert individual effort ρ/N , so the total effort is ρ. With a slight abuse of
notation, we also define:
W (ρ, N ) = ρh(ρ, N )
(3)
where h(ρ, N ) is defined by (4):
h(ρ, N ) = α + βf ((1 − 1/N )ρ) − ℓ(ρ) −
d(ρ/N )
.
ρ/N
(4)
Thus W (ρ, N ) gives the total welfare when N individuals exert total effort ρ.
We interpret W (ρ, N ) more generally as a function defined for all real N ≥ 1, not just integral
N . Indeed, in our subsequent development we make a relaxation of our model: we will assume that
N 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 N . This relationship allows us to express total welfare as a function of
real-valued N .
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.
3.1.
Self-interested usage at a fixed club size
We begin by studying the existence and uniqueness of Nash equilibria for the usage game when
N players are present. Each player i has the utility function (1), and the strategy space of each
player i is given by ρi ≥ 0. That is, each user chooses her usage level in her self-interest. A Nash
equilibrium of this game is a vector ρ such that for all i:
ui (ρ) ≥ ui (ρi , ρ−i ),
for all ρi ≥ 0.
(5)
Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects
10
(Here ρ−i = (ρ1 , . . . , ρi−1 , ρi+1 , . . . , ρN ).)
We have the following proposition characterizing the unique Nash equilibrium of the game.
Proposition 1. Suppose Assumptions 1-4 hold. Then there exists a unique Nash equilibrium
E
= ρN E /N for all i, where ρN E > 0 is the (necessarily unique) solution to the
ρN E . Further, ρN
i
following equation:
α + βf
1
1−
N
ρ
NE
NE ρN E ′ N E
ρ
NE
′
− ℓ(ρ ) +
= 0.
ℓ (ρ ) − d
N
N
(6)
Finally, ui (ρN E ) > 0 for all i.
Note that the Nash equilibrium is symmetric, and thus the total welfare is given by
W (ρN E (N ), N ). Further, via (6), we study Nash welfare under the relaxation that N is real-valued
(rather than integral).
3.2.
Self-interested choice of club size
In this section we study the behavior of the welfare of a single player at the Nash 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(N ) as follows:
I(N ) = ui
ρN E (N )
ρN E (N )
,...,
N
N
=
W (ρN E (N ), N )
.
N
(7)
Then I(N ) is utility of a single player at the unique Nash equilibrium when N players are present.
We start with the following key lemma that characterizes the behavior of ρN E (N ) as N → ∞.
Lemma 1. Suppose Assumptions 1-4 hold. Then ρN E (N ) is a strictly increasing function of
N ≥ 1, with supN ≥1 ρN E (N ) < ∞.
Next, we show that I(N ) is quasiconcave; further, it is maximized at N ∈ [m∗ , M ∗ ], for some
choice of m∗ , M ∗ .
Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects
11
Proposition 2. Suppose Assumptions 1-4 hold. Then I(N ) is quasiconcave as a function of
N . Further, there exist m∗ , M ∗ such that 1 ≤ m∗ ≤ M ∗ , and N ∈ arg maxN >1 I(N ) if and only if
m∗ ≤ N ≤ 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 Nash 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−
N
ρ
NE
(N ) = ℓ′ (ρN E (N )).
(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, ρN 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
Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects
12
“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 ∗ .
4.1.
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 N 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. Note
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 N ≥ 1. Let W S denote the optimal
objective function value in (9)-(10). If W S > 0, then the unique optimal solution ρS to (9)-(10) is
given by ρSi = ρS /N for all i, where ρS > 0 is the unique solution to the following problem:
ρ 1
.
max αρ + βρf
ρ − ρℓ(ρ) − N d
1−
ρ≥0
N
N
(11)
Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects
13
The preceding result characterizes the efficient usage vector, for a fixed N : 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 N ≥ 1. Let W S denote the optimal
objective function value in (9)-(10). Then W S > 0, and there exists a unique solution ρS to (9)-(10).
This solution is given by ρSi = ρS /N , where ρS solves (11), and 0 < ρS < ∞.
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 N ≥ 1, let ρS (N ) denote the unique solution to (11). Note that as in our analysis of the
Nash equilibrium, we can use (11) to view ρS (N ) as a function on real-valued N , rather than just
integral N .
Corollary 2. Suppose that Assumptions 1-4 hold. Then W (ρS (N ), N ) is non-decreasing in N .
Moreover, supN W (ρS (N ), N ) < ∞.
4.2.
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 N people enter is W (ρN E (N ), N ); the service manager’s goal is
to choose a value of N that maximizes total welfare.
We have the following result.
Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects
14
Proposition 4. Suppose Assumptions 1-4 hold. Then W (ρN E (N ), N ) → 0 as N → ∞. Thus
there exists N ∗ such that
W (ρN E (N ∗ ), N ∗ ) = sup W (ρN E (N ), N ).
(12)
N ≥1
That is, the total Nash welfare is maximized at N ∗ . Further, for any such N ∗ , there holds M ∗ <
N ∗ < ∞.
Note that although N ∗ 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 N ∗ , we will carry out such a comparison in an asymptotic setting in Section 6.
5.
Efficiency at the autonomous club size M ∗
In this section we study the ratio of the total welfare achieved at the Nash 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 N , we consider the
efficiency ratio of the Nash equilibrium to the social optimum, i.e., the ratio of their respective
aggregate utilities:
Γ(N ) =
W (ρN E (N ), N )
,
W (ρS (N ), N )
(13)
where ρN E (N ) is the unique solution to (6) with N players; and ρS (N ) is the unique solution to
(11) with N players. (This ratio is also known as the price of anarchy in the literature; see, e.g.,
Papadimitriou 2001.) Obviously 0 ≤ Γ(N ) ≤ 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) = x2 , d(x) = 0, α = 1, and β > 1.
Figure 1 plots the ratio Γ(N ) against N . As expected, when N = 1, the ratio is one: social welfare
maximization coincides with the individual optimization of the single individual in the system. As
Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects
15
1
0.9
0.8
Γ(N )
0.7
0.6
0.5 1
2
3
4
5
6
7
8
9
10
11 12
N
Figure 1
Efficiency ratio.
N increases, the ratio initially falls. However, the figure reveals a interesting insight: for N > 1,
there exists a unique value of N at which the ratio Γ(N ) is maximized. More surprisingly, the
maximum value achieved by this ratio is exactly one (if integrality constraints on N are ignored).
That is, at this value of N the Nash welfare is identical to the maximal welfare. Below this level of
N , the usage level of the system does not generate the efficient level of positive externality (unless
N = 1); and above this level of N , the usage level of the system creates an excessive negative
externality. In this section we evaluate efficiency of the Nash equilibrium, and formalize this insight.
That is we show that there is always a range of club sizes at which Γ(N ) = 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
N in the system approaches infinity. Observe from Proposition 4 that the Nash equilibrium welfare
approaches zero as N → ∞. 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(N ) approaches zero as N → ∞.
We can be more precise about this convergence. We have the following theorem.
Proposition 5. Suppose Assumptions 1-4 hold. Then ρN E (N )h(ρN E (N ), N ) → 0 as N → ∞;
16
Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects
further, there exist constants K1 , K2 > 0 such that for all N ≥ 1:
K1
K2
≤ Γ(N ) ≤
.
N
N
Thus Γ(N ) ∼ Θ(1/N ) as N → ∞.
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 N . As in Figure 1, when N = 1, then clearly the Nash 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 N > 1.
Theorem 1. Suppose Assumptions 1-4 hold. Then Γ(N ) is a continuous function of N if N ≥ 1,
with Γ(1) = 1. Further, with m∗ and M ∗ defined as in (EC.5) and (EC.6) we have:
1. If 1 < N < m∗ , then Γ(N ) < 1, and ρN E (N ) < ρS (N ).
2. If N > 1 and m∗ ≤ N ≤ M ∗ , then Γ(N ) = 1, and ρN E (N ) = ρS (N ).
3. If N > M ∗ , then Γ(N ) < 1, and ρN E (N ) > ρS (N ).
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 (ρN E (N ), N ) is a “moving target”—it is strictly increasing in N ; thus achieving full
efficiency at the Nash 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 Nash 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.)
Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects
17
When N < m∗ , an efficiency loss arises because individuals do not sufficiently internalize the
positive externality; in this regime, the total Nash effort ρN E (N ) falls short of the socially optimal
level ρS (N ). When N > M ∗ , an efficiency loss arises because individuals do not sufficiently internalize the negative externality; in this regime, the total Nash effort ρN E (N ) exceeds the socially
optimal level ρS (N ). When N ∈ [m∗ , M ∗ ], the two effects exactly balance; this is clearly shown by
the definition of J(N ) in (EC.4). When N ∈ [m∗ , M ∗ ], we have J(N ) = 0, or from (8):
βf
′
1
1−
N
ρ
NE
(N ) = ℓ′ (ρN E (N )).
In this case the Nash 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, N ∗ of Section 4.2, the usage will be necessarily inefficient since
N ∗ > M ∗ . In other words, although the total Nash welfare is higher at N ∗ than at the autonomous
club sizes, the Nash equilibrium usage at N ∗ 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 N ∗ 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).
Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects
18
We characterize the asymptotic behavior of M ∗ and N ∗ 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.
6.1.
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:
xℓ′ (x)
;
x→∞ ℓ(x)
ǫ = lim
(14)
we assume the preceding limit exists. For each c ≥ 1, define φ(c) as:
d′ (x/c)
;
x→∞ xℓ′ (x)
φ(c) = lim
(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
1
M = inf M ≥ 1 :
+ φ(M ) ≤ 1 −
.
M
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
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:
1
2/M + φ(M ) = 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 N players, and
the payoff to each player is βρi (C − ρi ). If one player is asked how many other players she would
like to play with, then the player will solve:
C
max
N ≥1
N
C
C−
N
.
Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects
20
It is straightforward to check that the optimal solution to this problem is N = 2: one additional
player brings a large positive externality, but additional players past N = 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.
6.2.
Asymptotics of N ∗
We now turn our attention to the asymptotic behavior of N ∗ as β increases without bound. For
each β, let N ∗ (β) be a second best optimal choice of club size, i.e., a solution to (12). Note that
N ∗ (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 N ∗ (β) ≥ M ∗ (β) for all β, under the same assumptions we also have N ∗ (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
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)
;
x→∞ ℓ′ (x) + xℓ′′ (x)
φ = lim
we assume the preceding limit exists, and that φ = 0.
2
For each β, let N ∗ (β) be a second best optimal choice of club size, i.e., a solution to (12). Under
these assumptions, N ∗ (β) approaches a limit N (possibly infinite) as β → ∞. Further:
1. If ǫ = ∞ or ǫ = 1, then N = ∞.
2. If 1 < ǫ < ∞, then N is the unique positive solution to:
2
N −
In particular, for large ǫ, we have N ≈
√
4ǫ
N − ǫ = 0.
ǫ−1
(16)
ǫ. As ǫ approaches 1, we have N ≈ 4/(1 − 1/ǫ).
It is interesting to compare Theorem 2 with Theorem 3. In Figure 2, we plot the values of M
and N in the case where φ(c) = 0 for all c, and ǫ varies between 1 and infinity. As ǫ → 1, both M
and N grow to infinity; indeed, for small ǫ, we have N ≈ 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 N ≈
√
ǫ. In particular,
N 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, N ∗ (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
Note 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.
Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects
22
8
7
6
M,N
5
4
3
2
5
10
15
20
25
ǫ
30
35
40
M
N
Figure 2
M vs. N .
for C > 0, and d(x) ≡ 0. This is a case where ǫ = ∞, so M = 2 while N = ∞. 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:
max N ×
N ≥1
C
N
C
C−
N
.
The term in parentheses is the welfare of a single individual; maximization of that quantity alone
yields an optimal solution of N = 2. However, maximizing N times the welfare of a single individual yields N = ∞. 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 N 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
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 Nash 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 N ∗
individuals enter, and per time period revenue to the service manager approaches the total Nash
welfare at N ∗ . 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 N , 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
Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects
24
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 National Science Foundation under grant no. DMI-0620811.
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
N 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 N 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
Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects
25
on a much slower schedule than the growth of the club, and nevertheless individuals will autonomously find
it in their best interest to let the club size grow without bound.
Formally, we define ℓ(ρ; N ) = 1/(KN γ −ρ), for γ ∈ [0, 1], and ρ < KN γ ; we define ℓ(ρ; N ) = ∞ for ρ ≥ KN γ .
Thus ℓ(·; N ) is given by an M/M/1 delay function, with service capacity KN γ . For simplicity we assume
that d(x) ≡ 0 in this section, and that f (x) = x. Under these assumptions, if γ = 0, then the discussion of
the last section shows that even if β is large, the optimal club size remains small. In this section we ask: if
capacity expands, will the optimal club size grow accordingly?
For fixed N , this model satisfies Assumptions 1-3, so we know that there exists a unique total Nash
equilibrium effort ρN E (N ). Recall that one way to model club formation is that the current members continue
to admit new members as long as individual utility at the Nash equilibrium is increasing. We are thus
interested in characterizing the behavior of the individual utility at the Nash equilibrium, defined in (7)
(with ℓ(ρ) replaced by ℓ(ρ; N )). We have the following theorem.
Theorem 4. Consider a model where α, β > 0; f (x) = x; ℓ(ρ; N ) = 1/(KN γ − ρ) for γ ∈ [0, 1]; and d(x) ≡
0. Then I(N ) → ∞ if γ > 1/2, and I(N ) → 0 if γ < 1/2.
Whereas in Section 6 we studied the scaling behavior of club size as β scales, the preceding result lets us
study the scaling behavior of the club as the “capacity” (i.e., KN γ ) increases but β stays fixed. Our result
suggests that current members will continue to admit new members indefinitely as long as available capacity
√
grows faster than N , where N is the number of members in the club. Conversely, the club will only grow to
√
a finite size if capacity grows slower than N (since in this case I(N ) reaches a maximum at a finite value
of N ). Our interest in the result is not so much the specific form of the capacity scaling threshold; rather,
we place qualitative emphasis on the fact that capacity need only grow sublinearly to ensure that the club
size grows to infinity.
References
Allon, G., A. Federgruen. 2005. Competition in service industries. To appear, Operations Research.
Asvanund, Atip, Karen Clay, Ramayya Krishnan, Michael D. Smith. 2004. An empirical analysis of network
externalities in peer-to-peer music sharing networks. Information Systems Research 15(2) 155–174.
Beckmann, Martin, C. B. McGuire, Christopher B. Winsten. 1956. Studies in the Economics of Transportation. Yale University Press, New Haven, Connecticut.
Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects
26
Berglas, E. 1976. On the theory of clubs. The American Economic Review 66(2) 116–121.
Berglas, E., D. Pines. 1981. Clubs, local public goods, and transportation models: A synthesis. Journal of
Public Economics 15 141–162.
Bertsekas, Dimitri P., Robert G. Gallager. 1992. Data Networks. 2nd ed. Prentice Hall, Upper Saddle River,
New Jersey.
Briscoe, Bob, Vasilios Darlagiannis, Oliver Heckman, Huw Oliver, Vasilios Siris, David Songhurst, Burkhard
Stiller. 2003. A market managed multiservice Internet (M3I). Computer Communications 26(4) 404–
414.
Buchanan, J. M. 1965. An economic theory of clubs. Economica 32(125) 1–14.
Buchanan, J. M., W. C. Stubblebine. 1962. Externality. Economica 29(116) 371–384.
Cornes, R., T. Sandler. 1996. The Theory of Externalities, Public Goods, and Club Goods. Cambridge
University Press, Cambridge, United Kingdom.
Courcoubetis, C., R.R. Weber. 2006. Incentives for large peer-to-peer systems. IEEE Journal on Selected
Areas in Communications 24 1034–1049.
Diamond, P. 1982. Aggregate demand management in search equilibrium. Journal of Political Economy 90
881–894.
Economides, Nicholas. 1996. The economics of networks. International Journal of Industrial Organization
16(4) 673–699.
Economist. 2007. Social graph-iti. October 18, 2007.
Falkner, Matthias, Michael Devetsikiotis, Ioannis Lambadaris. 2000. An overview of pricing concepts for
broadband IP networks. IEEE Communications Surveys 3(2).
Farrell, J., P. Klemperer. 2005. Coordination and lock-in: Competition with switching costs and network
effects: Section iii (Network effects). Preliminary draft chapter for Handbook of Industrial Organization,
Vol 3.
Farrell, J., G. Saloner. 1985. Standardization, compatibility and innovation. The RAND Journal of Economics 16(1) 70–83.
Gu, Bin, Yun Huang, Wenjing Duan, Andrew B. Whinston. 2007. Online peer-to-peer communities: An
Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects
27
empirical investigation of a music sharing network as a dynamic two-sided network. Working paper
07-42, NET Institute.
He, L., J. Walrand. 2003. Pricing internet services with multiple providers. Proceedings of Allerton Conference.
Jesdanun, Anick. 2008. Enthusiasts find big is not always best with social networks. Associated Press, April
16, 2008.
Katz, M.I., C. Shapiro. 1985. Network externalities, competition and compatibility. The American Economic
Review 75(3) 424–440.
Kelly, Frank P. 1991. Network routing. Philosophical Transactions: Physical Sciences and Engineering
337(1647) 343–367.
Kelly, Frank P. 1997. Charging and rate control for elastic traffic. European Transactions on Telecommunications 8 33–37.
Laffont, J.-J., S. Marcus, P. Rey, J. Tirole. 2003. Internet interconnection and the off-net-cost pricing
principle. The RAND Journal of Economics 34(2) 370–390.
Laffont, J.-J., J. Tirole. 2001. Competition in Telecommunications. MIT Press, Cambridge, Massachusetts.
Liebowitz, S. J., S. E. Margolis. 1994. Network externality: An uncommon tragedy. J. Economic Perspectives
8(2) 133–150.
Metcalfe, Bob. 1995. Metcalfe’s Law: a network becomes more valuable as it reaches more users. Infoworld .
Milgrom, P., J. Roberts. 1990. Rationalizability, learning, and equilibrium in games with strategic complementarities. Econometrica 58 1255–1277.
Odlyzko, A., B. Tilly. 2005. A refutation of Metcalfe’s law and a better estimate for the value of networks
and network interconnections. Preprint.
Oren, S. S., S. A. Smith. 1981. Critical mass and tariff structure in electronic communication markets. Bell
J. Economics 12(2) 467–487.
Papadimitriou, Christos. 2001. Algorithms, games, and the Internet. Proceedings of ACM Symposium on
the Theory of Computing. 749–753.
Rockafellar, R. Tyrrell. 1970. Convex Analysis. Priceton Landmarks in Mathematics and Physics, Princeton
University Press, Princeton, New Jersey.
Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects
28
Roughgarden, Tim. 2002. Selfish routing. Ph.D. thesis, Cornell University.
Sandler, T., J.T. Tschirhart. 1980. The economic theory of clubs: an evaluative survey. Journal of Economic
Literature 18(4) 1481–1521.
Scotchmer, S. 2002. Local public goods and clubs. A.J. Auerbach, M. Feldstein, eds., Handbook of Public
Economics. Elsevier, 1997–2042.
Srikant, R. 2004. The Mathematics of Internet Congestion Control . Birkhauser, Boston, Massachusetts.
Sundarajan, A. 2003. Network effects, non-linear pricing and entry deterrence. Preprint.
Tirole, Jean. 1988. The Theory of Industrial Organization. MIT Press, Cambridge, Massachusetts.
Tse, David, Pramod Viswanath. 2005. Fundamentals of Wireless Communication. Cambridge University
Press, Cambridge, United Kingdom.
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e-companion to Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects
Proofs
Proof of Proposition 1. First observe that given Assumptions 1-3, the utility (1) of player i is a
concave and continuously differentiable function of ρi , for a fixed value of ρ−i . This follows because
the first two terms of ui (ρ) are linear functions of ρi , and the last two terms are the negation of
convex functions in ρi . We conclude that ρN E is a Nash equilibrium if and only if the following
conditions hold for all players i:
E
E
) = 0,
α + βf (ρN E − ρN
) − (ℓ(ρ) + ρi ℓ′ (ρ)) − d′ (ρN
i
i
if ρi > 0;
(EC.1)
≤ 0,
if ρi = 0.
(EC.2)
Define g(r; ρ) as:
g(r; ρ) = α + βf (ρ − r) − (ℓ(ρ) + rℓ′ (ρ)) − d′ (r).
E
Observe that the left hand side of (EC.1)-(EC.2) is equal to g(ρN
; ρN E ). Since f is strictly increasi
ing and ℓ is strictly increasing (so ℓ′ (ρN E ) > 0), observe that for a fixed value of ρ, g(r; ρ) is a strictly
decreasing function of r for 0 ≤ r ≤ ρ. Thus for a fixed value of ρ ≥ 0, there exists at most one value
of r such that g(r; ρ) = 0. We define a function R(ρ) as follows: R(ρ) = 0 if g(0; ρ) < 0; R(ρ) = ρ if
g(ρ; ρ) > 0; and otherwise, R(ρ) is the unique solution of g(r; ρ) = 0 in 0 ≤ r ≤ ρ. Comparing with
E
E
(EC.1)-(EC.2), we conclude that ρN
= R(ρN E ) for all i, which is only possible if ρN
= ρN E /N
i
i
for all i. Thus every Nash equilibrium is symmetric.
Next, we claim that ρ = 0 cannot be a Nash equilibrium. If it were, from (EC.2), we must have:
α + f (0) − ℓ(0) − d′ (0) ≤ 0.
However, this violates Condition 1 of Assumption 4, so we conclude ρ = 0 cannot be a Nash
equilibrium.
E
Since every Nash equilibrium must be symmetric, with ρN
= R(ρN E ) = ρN E /N , and we must
i
have ρN E > 0, we conclude that if ρN E is a Nash equilibrium, then ρN E > 0 must be a solution to
the following equation:
g(ρ
NE
/N ; ρ
NE
NE ρ
ρN E ′ N E
1
′
NE
NE
ℓ (ρ ) − d
ρ
− ℓ(ρ ) +
= 0.
) = α + βf
1−
N
N
N
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e-companion to Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects
This is (6). Conversely, any solution ρN E > 0 to the preceding equation gives rise to a Nash equiE
= ρN E /N .
librium, by defining ρN
i
We now show that there exists exactly one solution to (6), i.e., to g(ρ/N ; ρ) = 0. We now note the
following; first, g(0; 0) = α + βf (0) − ℓ(0) − d′ (0), which is positive by Condition 1 in Assumption 4.
Second, g(ρ/N ; ρ) → −∞ as ρ → ∞; this follows by Condition 2 in Assumption 4. Finally, g(ρ/N ; ρ)
is a concave function of ρ, by Assumptions 1-3. Thus there must exist exactly one solution ρN E to
g(ρ/N ; ρ) = 0 and therefore to (6), and we must have ρN E > 0.
It remains to be shown that ui (ρN E ) > 0. To see this, note that by Assumption 3, we have
d′ (x) ≥ d(x)/x for x > 0. Since g(ρN E /N ; ρN E ) = 0 and ℓ′ (x) > 0 for all x > 0, we conclude that:
α + +βf
1−
1
N
d(ρN E /N )
> 0.
ρN E − ℓ(ρN E ) − N E
ρ /N
Multiplying by ρN E /N gives the individual utility at the Nash equilibrium, and thus we have
established that ui (ρN E) > 0 for all i, as required.
2
Proof of Lemma 1. Define the function G(ρ, N ) as:
ρ
1
ρℓ′ (ρ)
G(ρ, N ) = α + βf
1−
− d′
ρ − ℓ(ρ) −
.
N
N
N
(EC.3)
Under our assumptions, for every N ≥ 1, the function G is a concave function of ρ; and for every
ρ > 0, the function G is a strictly increasing function of N . (The strict monotonicity follows since
ℓ′ (ρ) > 0 for ρ > 0 by Assumption 2.) Further, G(0, N ) = α + βf (0) − ℓ(0) − d′ (0) > 0 by Assumption
4. But now observe from (6) that for any N ≥ 1 and N ′ > N , we have:
G(ρN E (N ′ ), N ′ ) = 0 = G(ρN E (N ), N ) < G(ρN E (N ), N ′ ).
Since G(0, N ′ ) > 0 and G(ρN E (N ), N ′ ) > 0, if ρN E (N ′ ) ≤ ρN E (N ) then we would have
G(ρN E (N ′ ), N ′ ) > 0, a contradiction to the stationarity condition (6). Thus ρN E (N ′ ) > ρN E (N ), as
required.
Next, we show that ρN E (N ) is bounded. It is clear that h(ρN E (N ), N ) ≥ 0; this follows because
every player can guarantee at least zero utility by choosing ρi = 0, so the Nash equilibrium social
e-companion to Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects
ec3
welfare must be nonnegative. So suppose that ρN E (N ) → ∞ as N → ∞; by Assumption 4, we
would have h(ρN E (N ), N ) → −∞, a contradiction. Therefore supN ≥1 ρN E (N ) < ∞.
2
Proof of Proposition 2. Note that as N changes, both N and ρN E (N ) change, and this creates
two changes in I(N ): first, the change in ui with respect to player i’s own effort; and second, the
change in ui with respect to the total effort of all players other than i. The first change is locally
zero, since we are considering a Nash equilibrium. We will show that the second change has the
same sign as J(N ), where:
J(N ) = βf
′
1
1−
N
ρ
NE
(N ) − ℓ′ (ρN E (N )).
(EC.4)
For notational simplicity, define ρ(N ) = ρN E (N ). We start by showing that ρ(N ) is differentiable.
Lemma EC.1. Suppose Assumptions 2-4 hold. Then ρN E (N ) is a differentiable function of N ,
with dρN E (N )/dN ≥ 0.
The lemma is proven below.
Next, define v(r, R) as follows:
v(r, R) = αr + βrf (R) − rℓ(r + R) − d(r).
Thus v(ρi , ρ−i ) is the individual utility of player i if he exerts effort ρi , and all other players exert
total effort ρ−i . In particular, we conclude that I(N ) = v(ρ(N )/N, (1 − 1/N )ρ(N )). Differentiating,
we have:
∂[ρ(N )/N ] ∂v
I (N ) =
∂N
∂r
′
ρ(N )
1
1
∂[(1 − 1/N )ρ(N )] ∂v ρ(N )
, 1−
, 1−
ρ(N ) +
ρ(N ) .
N
N
∂N
∂R
N
N
Now, since (ρ(N )/N, . . . , ρ(N )/N ) is a Nash equilibrium with N players, and ρ(N ) > 0, we must
have:
∂v
∂r
1
ρ(N )
, 1−
ρ(N ) = 0.
N
N
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Further, since ρ(N ) > 0 and ρ′ (N ) ≥ 0, we must have: ∂[(1 − 1/N )ρ(N )]/∂N > 0. Thus I ′ (N ) has
the same sign as:
∂v
∂R
ρ(N )
1
ρ(N )J(N )
ρ(N )
1
′
′
ρ(N ) =
βf
1−
ρ(N ) − ℓ (ρ(N )) =
, 1−
,
N
N
N
N
N
where J(N ) is defined as in (EC.4).
By Lemma 1, we know that ρ(N ) is strictly increasing in N ; and since f is concave and ℓ is
convex, we conclude that J(N ) is monotonically nonincreasing in N . Further, J(N ) is a continuous
function of N , since f ′ and ℓ′ are continuous, and ρ(N ) is continuous in N . Define m∗ and M ∗ as
follows:
m∗ = max{1, sup{m > 1 : J(m) > 0}}
(EC.5)
M ∗ = inf {m > 1 : J(m) < 0}.
(EC.6)
Since J is monotonically nonincreasing, we know m∗ ≤ M ∗ ; and clearly m∗ ≥ 1. Observe that if
either f is strictly concave, or ℓ is strictly convex, then J is strictly monotonic, so m∗ = M ∗ . Since
J is continuous, we have J(N ) > 0 for 1 ≤ N < m∗ ; J(N ) = 0 for m∗ ≤ N ≤ M ∗ ; and J(N ) < 0
for N > M ∗ . Using this observation, we conclude that I(N ) is strictly increasing for 1 ≤ N < m∗ ;
I(N ) is constant (and maximal) for m∗ ≤ N ≤ M ∗ ; and I(N ) is strictly decreasing for N > M ∗ .
The conclusion of the proposition now follows immediately.
2
Proof of Lemma EC.1. For notational simplicity, define ρ(N ) = ρN E (N ). Define G(ρ, N ) as in
(EC.3); then under our assumptions G is a concave function of ρ ≥ 0 for any fixed N ≥ 1. Suppose
that for some N ≥ 1, we have:
∂G
(ρ(N ), N ) = 0.
∂ρ
Then we conclude that ρ(N ) maximizes G(ρ, N ) over ρ ≥ 0. Further, since ρ(N ) satisfies
G(ρ(N ), N ) = 0, the maximal value of G(ρ, N ) in this region is zero. But this is impossible, since
G(0, N ) = α + βf (0) − ℓ(0) − d′ (0), which is positive by Assumption 4. We conclude that for all N ,
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e-companion to Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects
we have ∂G(ρ(N ), N )/∂ρ 6= 0, so by the implicit function theorem ρ(N ) is differentiable. Further,
since ρ(N ) is strictly increasing for N ≥ 1 (Lemma 1), we conclude that ρ′ (N ) ≥ 0.
2
Proof of Proposition 3. First consider, for fixed ρ ≥ 0, the following optimization problem:
maximize
W (ρ)
subject to
X
ρi = ρ;
i
ρ ≥ 0.
This is identical to (9)-(10), but with the total effort level predetermined. It is then clear from (2)
that ρ is an optimal solution to the preceding problem if and only if ρ is also an optimal solution
to the following problem:
minimize
X
i
subject to
X
ρi f (ρ − ρi ) − d(ρi )
ρi = ρ;
i
(EC.7)
ρ ≥ 0.
(EC.8)
But now observe that if f satisfies Assumption 1, then the function g(x) = xf (ρ − x) is strictly
concave over 0 ≤ x ≤ ρ. To see this we differentiate g:
g ′ (x) = f (ρ − x) − xf ′ (ρ − x).
Since f is strictly increasing, the first term is strictly decreasing; and since f is concave, the second
term is nonincreasing. Thus g ′ (x) is strictly decreasing over 0 ≤ x ≤ ρ. Thus the objective function
in (EC.7) is strictly concave and symmetric in ρ1 , . . . , ρN . Therefore the unique optimal solution
to (EC.7)-(EC.8) must satisfy ρi = ρ/N for all i.
Now note that if ρS is an optimal solution to (9)-(10), and we define ρS =
P
S
i ρi ,
then ρS must
also be an optimal solution to (EC.7)-(EC.8). From the preceding discussion, we conclude that in
fact ρSi = ρS /N for all i. In other words, any social optimum must be symmetric among the players.
In solving (9)-(10), therefore, it suffices to first optimize over the total effort ρ, then define
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e-companion to Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects
ρi = ρ/N . If we substitute the preceding relation in the expression (2), we find that (9)-(10) becomes
equivalent to finding a solution ρ to:
h
i
ρ
max αρ + βρf (N − 1)
− ρℓ(ρ) .
ρ≥0
N
Note that this is exactly (11).
Define h(ρ, N ) according to:
h(ρ, N ) = α + βf
1
d(ρ/N )
1−
.
ρ − ℓ(ρ) −
N
ρ/N
When there is no ambiguity about N , for brevity we will drop the argument N and simply write
h(ρ). Observe that the objective function in (11) can be written as ρh(ρ).
If the optimal objective function value of (9)-(10) is positive, then any optimal solution ρS for
(11) must be positive; and at any such solution we must have h(ρS ) > 0. Thus to complete the
proof, we must characterize stationary points of ρh(ρ) in the region where h(ρ) > 0. The final
property required to establish uniqueness of ρS is provided in the following lemma; the proof of
which can be found immediately below this proof.
Lemma EC.2. Suppose that Assumptions 1-3 hold, and fix N ≥ 1. Let P = {ρ > 0 : h(ρ) >
0}. Then P is an open interval in (0, ∞) (possibly empty or unbounded), and ρh(ρ) is strictly
quasiconcave over P .
Since a strictly quasiconcave function has at most one maximizer over its domain, we conclude
ρh(ρ) has at most one maximizer in P . Thus, if W S > 0 and P is a bounded interval, then we can
conclude a unique solution ρS exists to (11). If P is unbounded, then h(ρ) must be nondecreasing
in P (since h is concave under our assumptions), and thus ρh(ρ) ↑ ∞ as ρ → ∞; so we conclude
the unique solution to (11) is ρS = ∞.
2
Proof of Lemma EC.2. We first observe that under our assumptions, h(ρ) is concave and continuous, so the set P = {ρ > 0 : h(ρ) > 0} must be an open interval in (0, ∞). Assume without loss
e-companion to Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects
ec7
of generality that P is nonempty. Since h is concave, h′ (ρ) is monotonically nonincreasing over I.
Define ρ0 as:
ρ0 = sup{ρ ∈ P : h′ (ρ) ≥ 0}.
(Note that if the set on the right hand side above is empty, then ρ0 = −∞.) Define P1 = {ρ ∈ P : ρ ≤
ρ0 }, and P2 = {ρ ∈ P : ρ > ρ0 }. Then by monotonicity of h′ , it follows that P1 and P2 are contiguous
intervals, and of course P1
S
P2 = P . Further, for all ρ ∈ P1 , we have h′ (ρ) ≥ 0, so:
∂
[ρh′ (ρ)] = h(ρ) + ρh′ (ρ) > 0,
∂ρ
so in this region ρh(ρ) is strictly increasing (since h(ρ) > 0 for all ρ ∈ P ).
For all ρ ∈ P2 , we have h′ (ρ) < 0, and so h is strictly decreasing over P2 . Consider ρ1 , ρ2 ∈ P2 such
that ρ1 < ρ2 ; then h(ρ1 ) > h(ρ2 ). Since h is concave, we have h′ (ρ2 ) ≤ h′ (ρ1 ) < 0. Thus ρ1 h′ (ρ1 ) ≥
ρ2 h′ (ρ2 ), and we conclude that:
h(ρ1 ) + ρ1 h′ (ρ1 ) > h(ρ2 ) + ρ2 h′ (ρ2 ).
Thus the derivative of ρh(ρ) is strictly decreasing in I2 , and thus ρh(ρ) is strictly concave in P2 .
When combined with the fact that ρh(ρ) is strictly increasing in P1 , we conclude that ρh(ρ) is
strictly quasiconcave over P .
2
Proof of Corollary 1. Define h(ρ) as in the proof of Proposition 3. The second condition in
Assumption 4 guarantees that ρh(ρ) → −∞ as ρ → ∞. The first condition ensures that ρ = 0
cannot be an optimal solution to (11), and further, that ǫh(ǫ) > 0 for sufficiently small ǫ > 0; this
establishes that W S > 0. By Proposition 3, the result follows.
2
Proof of Corollary 2. The proof follows by observing that as N increases, the feasible
region of the service manager only expands, so the optimal welfare cannot decrease. Now if
lim supN →∞ ρS (N ) = ∞, then by Assumption 4, it follows that h(ρS (N ), N ) → −∞ along a subsequence where N → ∞. Since W (ρS (N ), N ) = N h(ρS (N ), N ), this would imply W (ρS (N ), N ) → −∞
ec8
e-companion to Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects
along the same subsequence, a contradiction, since the service manager can always guarantee
nonnegative welfare for any N by choosing zero usage. Thus we conclude that ρS (N ) is bounded
above; this suffices to establish that W (ρS (N ), N ) must be bounded above as well.
2
Proof of Proposition 4. Existence of N ∗ is established via the following straightforward argument.
From the Nash equilibrium condition (6) and the definition of h in (4), we have:
h(ρ
NE
ρN E (N )ℓ′ (ρN E (N ))
(N ), N ) =
+ d′
N
ρN E (N )
N
−
d(ρN E /N )
.
ρN E /N
(EC.9)
Since ρN E (N ) remains bounded as N → ∞, it follows that limN →∞ h(ρN E (N ), N ) = 0. Since the
aggregate welfare in Nash equilibrium is ρN E (N )h(ρN E (N ), N ) and rhoN E (N ) remains bounded,
we conclude the aggregate welfare at the Nash equilibrium approaches zero as well. Thus there
must be a finite N ∗ at which Nash equilibrium welfare is maximized.
We now show that N ∗ ≥ M ∗ . Write W (ρN E (N ), N ) = N I(N ). Since I(N ) is quasiconcave and
positive, at any optimal solution to (12), we must have I ′ (N ∗ ) = −I(N ∗ )/N ∗ < 0. Thus I is strictly
decreasing at N ∗ . Proposition 2 (quasiconcavity of I) then implies that N ∗ > M ∗ .
2
Proof of Proposition 5. As noted above, the social welfare at the social optimum must be nondecreasing in N ; thus ρS (N )h(ρS (N ), N ) is nondecreasing in N . Let A = supN ≥1 ρS (N )h(ρS (N ), N ).
We claim that 0 < A < ∞. Condition 2 of Assumption 4 guarantees that ρS (N ) takes values in
a compact set as N → ∞, and as a result h(ρS (N ), N ) must remain bounded as well (since h is
continuous). Further, A > 0 by Corollary 1.
We thus have:
ρN E (N )h(ρN E (N ), N )
ρN E (N )h(ρN E (N ), N )
≤
Γ(N
)
≤
.
ρS (1)h(ρS (1), 1)
A
Thus to prove the result of the proposition, it suffices to show that for all N ≥ 1 and for some
K1 , K2 > 0:
K2
K1
≤ ρN E (N )h(ρN E (N ), N ) ≤
.
N
N
(EC.10)
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e-companion to Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects
We now return to (EC.9):
h(ρ
NE
ρN E (N )ℓ′ (ρN E (N ))
(N ), N ) =
+ d′
N
ρN E (N )
N
−
d(ρN E /N )
.
ρN E /N
Since ρN E (N ) is strictly increasing and bounded, and ℓ′ is continuous, we can choose K1′ , K2′ > 0
such that:
ρN E (N )2 ℓ′ (ρN E (N )) K2′
K1′
<
<
.
N
N
N
Assume for the moment that d is twice continuously differentiable. Then from Assumption 3,
the convexity assumptions on d and d′ imply that d′ (x) ≥ d(x)/x ≥ d′ (0) for x > 0; and that
d′ (x) − d′ (0) ≤ xd′′ (x) for x > 0. Thus:
0≤d
Again,
since
ρN E (N )
′
ρN E (N )
N
is
NE
d(ρN E /N )
ρ (N )
′
− NE
− d′ (0)
≤d
ρ /N
N
ρN E (N )d′′ (ρN E (N )/N )
≤
.
N
bounded,
we
can
choose
a
constant
K2′′ > 0
such
that
ρN E (N )2 d′′ (ρN E (N )/N ) ≤ K2 for all N . Taking K1 = K1′ and K2 = K2′ + K2′′ then yields the result
of the proposition. In the case that d′ is not continuously differentiable, a similar argument can be
carried out by considering the subgradients of d′ ; we omit the details.
2
Proof of Theorem 1. Our approach will be to compare the stationarity conditions for the social
optimum and Nash equilibrium. Define H(ρ, N ) and G(ρ, N ) as follows:
ρ
1
1
+ρ β 1−
ρ − ℓ(ρ) − d′
f′
1−
ρ − ℓ′ (ρ) ;
N
N
N
(EC.11)
′
ρ
ρℓ (ρ)
1
ρ − ℓ(ρ) − d′
−
.
(EC.12)
G(ρ, N ) = α + βf
1−
N
N
N
H(ρ, N ) = α + βf
1−
1
N
(Note (EC.12) is equivalent to (EC.3).) As in the proof of Proposition 5, the condition (6) implies
ρN E (N ) > 0 is the unique solution to G(ρN E (N ), N ) = 0. Since N > 1, we conclude from (11)
that ρS (N ) > 0 is the unique solution to H(ρS (N ), N ) = 0. Since h, f ′ , ℓ′ , and d′ are continuous,
and by Assumption 4 and Lemma 1 both ρS (N ) and ρN E (N ) take values in a compact set, we
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immediately conclude that ρN E (N ) and ρS (N ) are continuous if N ≥ 1. Since ρS (N )h(ρS (N ), N ) >
0, we conclude that Γ(N ) is continuous in N if N ≥ 1.
We immediately see from the definitions of H and G that:
H(ρ
NE
(N ), N ) = H(ρ
NE
(N ), N ) − G(ρ
NE
(N ), N ) = ρ
NE
1
(N ) 1 −
J(N ),
N
(EC.13)
where J is given by (EC.4). recalling the properties of J cited subsequent to ((EC.13)), we have
the following observations:
1. 1 < N < m∗ if and only if H(ρN E (N ), N ) > 0.
2. m∗ ≤ N ≤ M ∗ if and only if H(ρN E (N ), N ) = 0.
3. N > M ∗ if and only if H(ρN E (N ), N ) < 0.
To conclude the proof of the theorem, we recall from Lemma EC.2 that for fixed N , ρh(ρ, N )
is strictly quasiconcave over the region I(N ) = {ρ : h(ρ, N ) > 0}. Further, ρS (N ) is the unique
local optimum (in fact, local maximum) of ρh(ρ, N ) over I(N ). From Proposition 1, we know that
ρN E (N ) > 0 and h(ρN E (N ) > 0, so ρN E (N ) ∈ I(N ) for all N .
We know that for ρ > 0:
H(ρ, N ) =
∂
[ρh(ρ, N )],
∂ρ
and ρS (N ) is the unique value of ρ ∈ I(N ) such that H(ρ, N ) = 0. Thus strict quasiconcavity of
ρh(ρ, N ) over I(N ) implies that ρN E (N ) < ρS (N ) if H(ρN E (N ), N ) > 0, and ρN E (N ) > ρS (N ) if
H(ρN E (N ), N ) < 0. Finally, if H(ρN E(N ), N ) = 0, then ρN E (N ) = ρS (N ). Since ρS (N ) is unique,
we conclude that Γ(N ) < 1 if and only if ρN E (N ) 6= ρS (N ).
2
Proof of Theorem 2. Let ρN E (N ; β) denote the total Nash equilibrium effort when N players are
in the system, and the positive externality coefficient is β, and let J(N ; β) denote the function
defined in (EC.4). Then in the case where f (x) = x, we have:
J(N ; β) = β − ℓ′ (ρN E (N ; β)).
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ec11
It follows from the proof of Theorem 1 that if J(0; β) < 0, then M ∗ (β) = 1; in this case, the positive
externality is not strong enough to improve individual utility at the Nash equilibrium as the number
of agents in the system increases. However, we note that from Condition 2 of Assumption 4, it must
be the case of that ρN E (N ; β) is uniformly bounded above, as both N and β approach infinity; and
thus as β → ∞, eventually we will have J(0; β) > 0, which is sufficient to imply M ∗ (β) > 1. Thus
in this proof we assume without loss of generality that M ∗ (β) > 1.
We first establish the claimed properties of ǫ and φ(·). We start by noting that since ℓ is convex
and nonnegative, there holds xℓ′ (x) ≥ ℓ(x) − ℓ(0). Since ℓ(x) → ∞ as x → ∞, we conclude ǫ ≥ 1. To
establish that φ(c) is monotonically nonincreasing, note that if c′ > c, then d′ (x/c) < d′ (x/c′ ) since
d is convex. Taking limits in the definition of φ(c) establishes the result.
We now turn our attention to the limiting behavior of M ∗ (β). M ∗ (β) is determined by two
conditions: first, the condition that J(M ∗ (β)) = 0; and second, the stationarity condition for the
Nash equilibrium. We now use these two conditions to construct a fixed point equation for M ∗ (β).
For notational simplicity, let R(β) = ρN E (M ∗ (β); β). Since M ∗ (β) must be the unique solution to
J(N ) = 0, we have β = ℓ′ (R(β)). Let L denote the inverse of ℓ′ on [0, ∞); then we conclude that
R(β) = L(β).
Next, from the Nash equilibrium stationarity condition (6) and the fact that f (x) = x, we have:
1
R(β) ′
R(β)
′
α+β 1− ∗
R(β) − ℓ(R(β)) + ∗
= 0.
ℓ (R(β)) − d
M (β)
M (β)
M ∗ (β)
Since R(β) = L(β), we can substitute and rearrange to obtain:
1+
ℓ(L(β))
2
d′ (L(β)/M ∗ (β))
α
−
− ∗
=
.
βL(β)
βL(β)
M (β)
βL(β)
(EC.14)
Since ℓ′ is strictly increasing, we know βL(β) → ∞ as β → ∞. Further, note that as β → ∞, we
have:
ℓ(L(β))
1
ℓ(x)
= lim ′
= .
x→∞ xℓ (x)
β→∞ βL(β)
ǫ
lim
Note the right hand side of (EC.14) is nonnegative; thus, taking the limit as β → ∞ of the left
hand side, we obtain limβ→∞ −2/M ∗ (β) = 0, which is only possible if M = ∞. This establishes the
first case considered in the theorem.
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We assume for the remainder of the proof, therefore, that ǫ > 1. Suppose that M is finite. Then
from (EC.14), taking the limit as β → ∞, we conclude:
1−
1
d′ (L(β)/M ∗ (β))
2
= lim
−
.
ǫ M β→∞
βL(β)
If M < M , then for all sufficiently large β, there holds d′ (L(β)/M ∗ (β)) < d′ (L(β)/M ). Thus for
M ≥ 1 such that M < M , we have:
d′ (L(β)/M )
d′ (L(β)/M ∗ (β))
≤ lim
= φ(M ),
β→∞
β→∞
βL(β)
βL(β)
lim
where the last equality follows by defining x = L(β). A similar result holds for M > M , with the
inequality reversed.
We conclude that for all M ≥ 1 such that M < M , we have:
1−
1
2
−
≤ φ(M ).
ǫ M
1−
2
1
−
≥ φ(M ).
ǫ M
Further, for all M > M , we have:
(Note that for both of these inequalities, we use the fact that 1/M is strictly decreasing in M .)
This result implies that if M is finite, then:
2
1
M = inf M ≥ 1 :
+ φ(M ) ≤ 1 −
,
M
ǫ
as required.
On the other hand, if M is infinite, then an argument analogous to the preceding two paragraphs
shows that for all M ≥ 1:
1
φ(M ) ≥ 1 − ,
ǫ
so that:
2
1
M ≥1:
+ φ(M ) ≤ 1 −
= ∅.
M
ǫ
This concludes the proof.
2
e-companion to Johari and Kumar: The Interaction of Positive Externalities and Congestion Effects
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Proof of Theorem 3. If ǫ = 1 then M ∗ (β) → ∞ by Theorem 2; since N ∗ (β) ≥ M ∗ (β) for all β (by
Proposition 4), we conclude N ∗ (β) → ∞ as well. Thus for the remainder of the proof we assume
that ǫ > 1.
Throughout the proof we let T (β) = ρ(N ∗ (β); β), where ρ(N ; β) is the Nash equilibrium total
effort with N players when the positive externality coefficient is β. Taking subsequences if necessary,
we assume without loss of generality that N ∗ (β) converges (possibly to infinity); we will show the
limit is uniquely defined. For notational simplicity, we also suppress the dependence of T and N ∗
on β.
The proof is divided into several steps.
Step 1: limβ→∞ T (β) = ∞. By definition of M ∗ , we know that ℓ′ (ρ(M ∗ (β); β)) = β. Thus
as β → ∞, we conclude ρ(M ∗ (β); β) → ∞ as well. Recall that N ∗ (β) ≥ M ∗ (β) for each β by
Proposition 4, and that T (β) ≥ ρ(M ∗ (β); β), since ρ(N, β) is strictly increasing in N . Thus we
must have T (β) → ∞ as well.
For the remainder of the proof, we define the following functions:
T (β)ℓ′ (T (β))
ℓ′ (T (β)) + T (β)ℓ′′ (T (β))
; E(β) =
;
ℓ(T (β))
ℓ′ (T (β))
d′ (T (β)/N ∗ (β))
d′′ (T (β)/N ∗ (β))
; Φ(β) = ′
;
Φ(β) =
′
T (β)ℓ (T (β))
ℓ (T (β)) + T (β)ℓ′′ (T (β))
(T (β)/N ∗ (β))d′ (T (β)/N ∗ (β))
∆(β) =
.
d(T (β)/N ∗ (β))
E(β) =
(EC.15)
(EC.16)
(EC.17)
As before, we suppress the dependence of these functions on β for notational simplicity. Since the
total Nash effort T (β) is always nonzero, the denominators are all strictly positive above.
Step 2: The derivative of ρ(N ; β) at N ∗ is given by:
∂ρ(N ∗ ; β)
T :=
=
∂N
T
N∗
1 + Φ · E − (1 − 1/∆)ΦN ∗
1 + E(1 + Φ)
.
(EC.18)
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To see this, observe that (6) can be rewritten as:
ρ(N ; β)ℓ′ (ρ(N ; β))
+ d′
h(ρ(N ; β), N ; β) =
N
ρ(N ; β)
N
−
N d(ρ(N ; β)/N )
,
ρ(N ; β)
where h(ρ, N ; β) = α + β(1 − 1/N )ρ − ℓ(ρ) − d(ρ/N )/(ρ/N ). Since the total Nash welfare is given by
ρ(N ; β)h(ρ(N ; β), N ; β), we conclude that N ∗ satisfies the following necessary first order optimality
condition:
ρ(N ; β)2 ℓ′ (ρ(N ; β))
N ∈ arg max
+ ρ(N ; β)d′
N ≥1
N
∗
ρ(N ; β)
N
ρ(N ; β)
− Nd
N
.
Letting T (β) = ∂ρ(N ∗ ; β)/∂N , we conclude N ∗ must satisfy the following necessary condition for
optimality:
T
2T ℓ′ (T ) + T 2 ℓ′′ (T )
N∗
T 2 ℓ′ (T )
−
+ T d′
(N ∗ )2
T
T
−d
+
N∗
N∗
T
T
T
T
′
T d′′
−
N
d
= 0.
−
N∗
N∗
N ∗ (N ∗ )2
Rearranging and dividing through by T /N ∗ yields:
T
T
T
T ℓ′ (T ) d(T /N ∗ )
T
′′
′
′
′
′′
−
−
−
d
+d
= 0.
T 2ℓ (T ) + T ℓ (T ) + d
N∗
N∗
T /N ∗
N∗
N∗
N∗
Noting that d′′ (T /N ∗ )/ℓ′ (T ) = Φ · E, we obtain exactly the expression (EC.18).
Step 3: There holds:
ℓ′ (T ) α/(βT ) + 1 − 1/N ∗
=
.
β
1/E + 1/N ∗ + Φ
(EC.19)
T
T ℓ′ (T )
1
′
.
+d
α + β 1 − ∗ T = ℓ(T ) +
N
N∗
N∗
(EC.20)
Observe from (6) that for each β:
Rewriting, we obtain:
1
α
+1− ∗ =
βT
N
ℓ′ (T )
β
d′ (T /N ∗ )
ℓ(T )
∗
+ 1/N +
.
T ℓ′ (T )
T ℓ′ (T )
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The result follows.
Step 4: There holds:
βT
1
T
1
ℓ′ (T )
d(T /N ∗ )
′
+
R
1
−
1
−
+
d
−
= 0,
N∗
N∗
β
N∗
T /N ∗
(EC.21)
1
(1/∆ − 1)ΦN ∗
+
.
1 + E/(1 + Φ · E)
1 + E(1 + Φ)
(EC.22)
where
R :=
To see this, we use an alternate optimality condition for N ∗ . We write W (ρ(N ; β), N ) =
ρ(N ; β)h(ρ(N ; β), N ; β). The first order necessary condition for optimality is:
T
1
βT 2
T
T
T
T
′
′
′
α + 2βT 1 − ∗ − ℓ(T ) − T ℓ (T ) − d
+
+
d
= 0,
−d
N
N∗
(N ∗ )2
N∗
N∗
N∗
where T = ∂ρ(N ∗ ; β)/∂N ∗ . If we substitute from (EC.20) and divide through by T /N ∗ , we obtain
exactly (EC.21).
Step 5: As β → ∞, there holds E(β) → ǫ; E(β) → ǫ; Φ(β) → 0; Φ(β) → 0; and lim inf β→∞ ∆(β) ≥
1. The claim regarding ∆(β) follows because Assumption 3 implies xd′ (x) ≥ d(x) for all x. By Step
1, since T → ∞ as β → ∞, the claim regarding E follows. The claim regarding E then follows by
L’Hôpital’s rule.
Next, observe that since d′ is convex, there holds:
d′′ (T /N ∗ )
d′′ (T )
≤
.
ℓ′ (T ) + T ℓ′′ (T ) ℓ′ (T ) + T ℓ′′ (T )
If d′′ (T ) remains bounded as T → ∞, then the right hand side above converges to zero. Otherwise,
d′′ (T ) is unbounded, and thus d′ (T ) is unbounded as well, as T → ∞. In this case, we conclude the
right hand side of the above inequality converges to φ, which is zero by assumption. Thus Φ → 0
as well.
Next, observe that for all β:
d′ (T )
d′ (T /N ∗ )
≤
,
T ℓ′ (T )
T ℓ′ (T )
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since d is convex. By L’Hôpital’s rule the right hand side converges to φ = 0 as β → ∞, so the
claim regarding Φ follows.
Step 6: If ǫ = ∞, then N ∗ (β) → ∞ as β → ∞. We use a proof by contradiction; suppose, instead,
that N ∗ (β) remains bounded. Taking subsequences if necessary, assume that N ∗ (β) converges to
N < ∞.
We first note that since d is convex and d(0) = 0 (by Assumption 3), we have d′ (T /N ∗) ≥
d(T /N ∗)/(T /N ∗). From Step 5, observe that R approaches zero as β → ∞; to see this, rewrite the
denominator of the first term of (EC.22) as 1 + 1/(1/E + Φ), and note that the latter quantity
approaches zero as β → ∞. Further, from Steps 3 and 5, ℓ′ (T )/β approaches a finite limit as
β → ∞. Thus the left hand side of (EC.21) approaches infinity as β → ∞, contradicting (EC.21)
for large β. This proves the claim.
Step 7: There holds:
(1/∆ − 1)ΦN ∗
= 0.
β→∞ 1 + E(1 + Φ)
lim
(EC.23)
We start by showing that limβ→∞ ΦN ∗ = 0. We consider two cases. First, if T /N ∗ → ∞ as β → ∞,
then letting y = T /N ∗ , we have:
N ∗Φ =
N ∗ d′ (T /N ∗ )
d′ (y)
d′ (y)
=
≤
,
T ℓ′ (T )
yℓ′ (N ∗ y) yℓ′ (y)
since N ∗ ≥ 1 and ℓ′ is increasing. As y → ∞, the right hand side above approaches φ(1), which is
zero by assumption. Thus in this case we have N ∗ Φ → 0.
Suppose instead that T /N ∗ remains bounded as β → ∞. Then d′ (T /N ∗ ) remains bounded as
well, but since we have already shown T → ∞, we must have ℓ′ (T ) → ∞. Thus we again have that
N ∗ Φ → 0 as β → ∞. This fact, together with Step 5, establishes the desired result.
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Step 8: If ǫ < ∞, then: limβ→∞ N ∗ (β) = N , where N is the unique positive solution to (16). We
first analyze the last two terms of (EC.21). First observe that:
d′ (T /N ∗ ) − d(T /N ∗ )/(T /N ∗ ) d′ (T /N ∗ )(1 − 1/∆) d′ (T /N ∗ ) ℓ′ (T )
=
≤
·
.
βT
βT
T ℓ′ (T )
β
Now from Steps 3 and 5, if ǫ is finite it follows that ℓ′ (T )/β remains bounded as β → ∞. Further:
d′ (T /N ∗ )
d′ (T )
≤
,
T ℓ′ (T )
T ℓ′ (T )
where the latter inequality follows by convexity of d. The right hand side of the preceding inequality
converges to φ(1), which is zero by assumption. Thus we conclude that as β → ∞:
d′ (T /N ∗ ) − d(T /N ∗ )/(T /N ∗ )
→ 0.
βT
(EC.24)
We now prove the claim. As in the previous step, the proof is by contradiction; suppose to the
contrary (taking subsequences if necessary) that N ∗ (β) approaches infinity as β → ∞. Note that
the result of step 6 implies that as β → ∞, we have R → 1/(1 + ǫ), from (EC.22) and (EC.23). If
we divide through (EC.21) by βT and consider the limit as β → ∞, the left hand side converges to
(1 − ǫ)/(1 + ǫ), which is strictly negative. This contradicts (EC.21) for large β, so we cannot have
N ∗ → ∞ as β → ∞. Thus if ǫ < ∞, then N ∗ remains bounded as β → ∞.
We now need to show that N ∗ converges to a unique limit. Consider a convergent subsequence
if necessary. Dividing through (EC.21) and taking the limit as β → ∞ yields:
1
+
N
1
1+ǫ
1
1−
N
1 − 1/N
1−
1/ǫ + 1/N
= 0.
Simplifying this equation yields (16), and completes the proof.
2
Proof of Theorem 4. Our approach is to identify the scaling behavior of I(N ) from the stationarity
condition (6), which we reproduce here for the specific model under consideration:
1
α+β 1−
N
ρN E (N ) −
ρN E (N )
1
−
= 0.
KN γ − ρN E (N ) N (KN γ − ρN E (N ))2
(EC.25)
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We begin by showing that ρN E (N ) → ∞. Suppose not; then it follows that the left hand side of
(EC.25) remains bounded away from zero (since α > 0), contradicting the stationarity condition.
We conclude that ρN E (N ) → ∞ as N → ∞. Dividing through (EC.25) by ρN E (N ), we conclude
that:
1
lim
N →∞ ρN E (N )(KN γ
− ρN E (N ))
+
N (KN γ
1
= β.
− ρN E (N ))2
(EC.26)
In particular, the limit is a positive constant.
The limit is somewhat easier to manipulate if we define η(N ) = ρN E (N )/(KN γ ). In this case we
obtain:
1
1
1
1
+
=
+
.
ρN E (N )(KN γ − ρN E (N )) N (KN γ − ρN E (N ))2 N 2γ η(N )(1 − η(N )) N 1+2γ (1 − η(N ))2
If both η(N ) and 1 − η(N ) remain bounded away from zero as N → ∞, then the limit in (EC.26)
cannot hold. Thus either η(N ) → 0, or 1 − η(N ) → 0. In the former case, we conclude η(N ) ∼
N −2γ /β, i.e., βN 2γ η(N ) approaches 1 as N → ∞.3 But in this case, we must have that N γ ρN E (N )
approaches a constant as N → ∞, which in turn implies that ρN E (N ) → 0 as N → ∞, a contradiction.
Thus we must have η(N ) → 1 as N → ∞. It now follows that 1 − η(N ) must decay quickly enough
to offset either N 2γ or N 1/2+γ , whichever is smaller. Thus 1 − η(N ) ∼ N −γ−min{γ,1/2} /β as N → ∞.
We conclude that ρN E (N ) ∼ KN γ − KN − min{γ,1/2} /β. Recall that the individual welfare at the
Nash equilibrium is given by:
1
ρN E (N )
NE
NE
α+β 1−
ρ (N ) − ℓ(ρ (N )) .
I(N ) =
N
N
Substituting and simplifying, we have that as N → ∞:
I(N ) ∼ N γ−1 βK 2 N γ − βK 2 N min{γ,1/2} + o(N γ ) .
It now follows that if γ < 1/2, then I(N ) → 0 as N → ∞; on the other hand, if γ > 1/2, then
I(N ) → ∞ as N → ∞.
3
Here and throughout the proof, we adopt the notation that f (N ) ∼ g(N ) if and only if limN →∞ f (N )/g(N ) = 1.
2