# Subsidization Competition: Vitalizing the Neutral Internet

Richard T. B. Ma

National University of Singapore and  Advanced Digital Science Center
<EMAIL_ADDRESS>
###### Abstract

Unlike telephone operators, which pay termination fees to reach the users of
another network, Internet Content Providers (CPs) do not pay the Internet
Service Providers (ISPs) of users they reach. While the consequent cross
subsidization to CPs has nurtured content innovations at the edge of the
Internet, it reduces the investment incentives for the access ISPs to expand
capacity. As potential charges for terminating CPs’ traffic are criticized
under the net neutrality debate, we propose to allow CPs to voluntarily
subsidize the usage-based fees induced by their content traffic for end-users.
We model the regulated subsidization competition among CPs under a neutral
network and show how deregulation of subsidization could increase an access
ISP’s utilization and revenue, strengthening its investment incentives.
Although the competition might harm certain CPs, we find that the main cause
comes from high access prices rather than the existence of subsidization. Our
results suggest that subsidization competition will increase the
competitiveness and welfare of the Internet content market; however,
regulators might need to regulate access prices if the access ISP market is
not competitive enough. We envision that subsidization competition could
become a viable model for the future Internet.

##  1 Introduction

Driven by bandwidth-intensive video applications and pervasive Internet access
via mobile devices, Internet inter-domain traffic has been growing more than
$50\%$ per annum [20] and is expected to grow more than $100\%$ per annum for
mobile networks [34]. To sustain such rapid traffic growth, Internet Service
Providers (ISPs), especially the mobile access ISPs, need to upgrade their
network infrastructures and expand capacities. However, revenues from online
services are growing more than twice as fast as those from Internet access
[34], whose market capitalizations have stagnated as investors weigh high
capital requirements against continued margin pressure. This raises serious
challenges regarding the viability of the current Internet model in the
future.

Figure 1: Subsidization in a two-sided Internet market.

The fundamental problem lies in the two-sided market structure [5, 36] of the
Internet, where the ISPs provide a platform connecting end-users to the
Internet Content Providers (CPs). The CP-side transit ISPs are often the
backbone ISPs that provide IP transit services. Due to market competition,
recent years have witnessed a constant decrease in IP transit prices [32]. The
more serious problem lies in the last-mile connections toward the end-users,
which are often the bottlenecks of the Internet [10]. For these last-mile
access ISPs, including the mobile providers, the main source of revenue is
from their end-users. Traditionally, they also need to pay transit ISPs for
transit services, although settlement-free peering is common in practice.
Existing proposals to resolve this issue include: 1) allowing ISPs to impose
termination fees [6] on CPs’ traffic towards their end-users, and 2) allowing
ISPs to prioritize traffic, differentiate services and thus charge users non-
uniformly. However, both proposals have triggered a heated debate over network
neutrality [44, 11], whose advocates argue that zero-pricing [22] and the
neutrality of physical networks are needed to protect and encourage content
innovations of the Internet. After all, the limited capacities and rising
access costs will dampen user demand, if CPs cannot return the created value
to ISPs.

Traditionally, wireline access ISPs charge flat-rate prices and implicitly
subsidize CPs for content delivery; however, usage-based pricing is commonly
used by wireless providers. The FCC chairman has recently backed usage-based
pricing for broadband to penalize heavy Internet users [18, 37] and major U.S.
broadband providers, e.g., Verizon [38] and AT&T [42], have adopted tiered
schemes where charges are imposed to the metered usage above a predefined data
cap. We assume the adoption of usage-based pricing by access ISPs, under which
heavy users share the burden of subsidizing CPs. Our goal is to realign the
created value and stimulate the needed investment in the Internet value chain
and our high-level idea is illustrated in Figure 1, where we propose to allow
CPs to voluntarily subsidize the usage-based access costs caused by their
content traffic, partially or fully, for the end-users. Our solution
modularizes the issue along the tussle boundary such that the physical network
is kept neutral while innovative competition among CPs is enabled at higher
service layers so as to incentivize ISP investments. As a special case of full
subsidization, AT&T’s recent sponsored data [1] plan will likely be available
by the end of 2014. Nevertheless, it raised concerns over anti-competitive
practices, and the FCC says that they will be monitoring and prepared to
intervene if necessary [30].

To understand and demonstrate the potential impact of subsidization, we take
an analytical approach as no market data is available for an empirical study
yet. We develop a macroscopic Internet model and study the status quo one-
sided pricing of the access ISPs in Section 3. Building upon this framework,
we model the policy that regulates subsidization and analyze the competition
game among CPs in Section 4. We evaluate the policy implications on the ISP’s
revenue and system welfare in Section 5 and discuss the regulatory and
implementation issues, as well as the limitations of our model in Section 6.
Our analytical results include:

  * •

We characterize the impact of capacity, user population (Theorem 1) and one-
sided ISP pricing (Theorem 7) on the system utilization and the CPs’
throughput.

  * •

We analyze the CPs’ subsidization competition game and characterize the Nash
equilibrium (Theorem 3), its uniqueness and its dynamics (Theorem 4 and 6).

  * •

We characterize the impact of subsidization policy on the CPs’ throughput,
user population, and the system’s utilization (Theorem 17) and welfare
(Corollary 2).

Through our theoretical results and qualitative evaluations, we also identify
important policy implications as follows.

  * •

If allowed, profitable CPs tend to subsidize users more (Theorem 5) and
achieve higher throughput (Lemma 3), leading to higher system welfare
(Corollary 2).

  * •

For fixed ISP pricing, deregulation of subsidization will increase the
utilization and revenue of the access ISPs (Corollary 1), strengthening their
investment incentives.

  * •

Although the throughput of certain CPs might decrease, this mainly comes from
the ISP’s high price (Theorem 17) rather than the deregulation of
subsidization.

In short, subsidization provides a feedback channel for the highly profitable
CPs to repay their users and indirectly transfer value to the access ISPs. It
helps the access ISPs attract investment for capacity expansion while keeping
their physical networks neutral. Our results suggest that regulators should
promote this subsidization competition; however, they might need to regulate
the ISP’s pricing if the access market is not competitive enough. We envision
that subsidization competition could become a viable model for the future
Internet so as to accommodate the growing demand for data traffic.

##  2 Related Work

A report from A.T. Kearney [34] studied the performance, economic and policy
pressures of the Internet and identified four models for the future Internet:
1) modification of retail pricing schemes, 2) traffic-dependent charges, 3)
enhanced-quality services over the public Internet, and 4) bilateral service
agreements. We discuss the above existing models and their relationships with
our subsidization competition model.

###  2.1 Modification of Retail Pricing Schemes

Historically, the Internet adopted flat-rate prices [33, 39] for simplicity.
Economists [26] and computer scientists [9, 17, 19] advocated usage-based
pricing, which was shown to provide congestion control [17], quality of
service [19] and economic efficiency [9], and is adopted by mobile providers
[28]. Subsidization competition assumes the adoption of usage-based pricing by
access ISPs, under which the consumers’ usage-based charges could be
subsidized by the CPs.

###  2.2 Imposition of Termination Fees on CPs

One core question under the net neutrality debate [44] is whether ISPs should
be allowed to use two-sided pricing and charge CPs for terminating their
content traffic. Prior work [7, 29, 31] studied the investment incentives in
Internet-like two-sided markets and drew different conclusions. Njoroge et al.
[31] found that through CP-side pricing, ISPs could extract higher surplus and
maintain higher investment levels. However, Choi et al. [7] found that
expending capacity will decrease the CP-side sale price. Musacchio et al. [29]
found that although the returns on investment under one-sided or two-sided
pricings are comparable, the size of investment and profit depend on the
advertising rates and users’ price sensitivity. From an economic efficiency
point of view, [31] concluded that two-sided pricing results in higher social
welfare; however, Lee and Wu [22] argued that the zero pricing at the CP-side
could be optimal in theory [5, 36]. Also, [7] found that the short-run welfare
is higher under one-sided neutral regulation. Our subsidization mechanism does
not impose charges on the CPs by the ISPs, but provides an indirect mechanism
for them to voluntarily return the zero-pricing subsidy back to the ISPs via
their end-users.

###  2.3 Differentiated Services over the Internet

Another core debate under net neutrality is whether ISPs should be allowed to
differentiate services and prices. Wu [44] surveyed the discriminatory
practices of broadband operators and proposed solutions to manage bandwidth
and police ISPs. Sidak [41] focused on consumer utility and argued that
differential pricing is essential to maximize utility. Shetty et al. [40]
considered a two-class service model and studied capacity planning,
regulation, as well as differentiated pricing to consumers. Economides et al.
[13] compared various regulations for quality of service and price
discrimination, and drew conclusions about desirable regulation regimes. As
net neutrality is undergoing heated debate, regulation of desirable
differentiation is a rich area for further study. Under our solution, ISPs do
not make any physical or economic differentiation and CPs’ subsidies are made
voluntarily. Subsidization serves as a policy mechanism that encourages
competition and provides investment incentives and economic efficiency for the
Internet ecosystem.

###  2.4 Bilateral Settlements for Valued Services

A fundamental cause of investment shortage is the misalignment between the
value created and the revenue obtained in the Internet value chain. Clark et
al. [8] discussed various economic tussles among the ISPs, CPs and users. Ma
et al. [23] advocated the use of the Shapley profit-sharing mechanism for
multi-lateral ISP settlements and studied bilateral implementations [24].
Laskowski [21] identified that the network’s lack of accountability is an
obstacle to implement desirable contracting systems for the Internet. Because
accounting of traffic towards the end-users is feasible for access ISPs, our
subsidization mechanism can be regarded as a contracting system between the
CPs and access ISPs, even through they might not be directly connected to each
other. Through the feedback channel from CPs to end-users, subsidization
provide a new avenue for the Internet ecosystem to realign the profit
distribution among different stakeholders.

##  3 Macroscopic Internet Model

To understand the implications of policy on the Internet, we start with a
macroscopic model that captures the physical and business dynamics in the
status quo Internet among an access ISP, its users and the CPs used by these
consumers.

###  3.1 Basic Physical System Model

We denote $\mu$ as the capacity of an access ISP serving the users of a
certain geographical region and $\cal N$ as the set of CPs used by the ISP’s
consumers. For each CP $i\in\cal N$, we denote $m_{i}$ as its user population
and define $\theta_{i}\triangleq m_{i}\lambda_{i}$ as its total throughput,
where $\lambda_{i}$ denotes the CP’s average per user throughput. We define
$\theta\triangleq\sum_{k\in\cal N}\theta_{k}$ as the aggregate system
throughput. Under any fixed capacity $\mu$ and aggregate throughput $\theta$,
we define $\phi\triangleq\Phi(\theta,\mu)$ as the system utilization, a
function of $\theta$ and $\mu$. We assume the average user throughput is a
function of the utilization, i.e., $\lambda_{i}\triangleq\lambda_{i}(\phi)$.

######  Assumption 1

$\Phi(\theta,\mu)$ and any $\lambda_{i}(\phi)$ are differentiable.
$\Phi(\theta,\mu)$ is strictly decreasing in $\mu$, strictly increasing in
$\theta$ and satisfies $\lim_{\theta\rightarrow 0}\Phi(\theta,\mu)=0$ for all
$\mu>0$. $\lambda_{i}(\phi)$ is strictly decreasing in $\phi$ and satisfies
$\lim_{\phi\rightarrow\infty}\lambda_{i}(\phi)=0$.

Assumption 1 captures the physics of capacity utilization: utilization
increases when the system accommodates higher throughput or has less capacity,
and vice-versa. Because the system utilization and congestion are two sides of
the same coin (congestion happens when the system is over-utilized), it also
captures the physics of user throughput: users achieve higher throughput when
the system is less congested.

Figure 2: The basic system model $({\bf m},\mu)$.

We denote $\mathbf{m}$ as the vector of user populations of the CPs. Figure 2
illustrates our basic model $({\bf m},\mu)$, where the user populations ${\bf
m}$ and system capacity $\mu$ collectively determine the system utilization
$\phi$ and each CP $i$’s throughput $\theta_{i}$. Because utilization
increases with the accommodated throughput, which decreases with the
utilization (congestion), the resulting system utilization should be in an
equilibrium.

######  Definition 1 (Utilization)

$\phi$ is the utilization of a system $(\mathbf{m},\mu)$ if it satisfies the
following condition:

$\phi=\Phi\left(\sum_{k\in\cal N}m_{k}\lambda_{k}(\phi),\mu\right).$ (1)

Definition 1 states that the utilization $\phi$ should induce the aggregate
system throughput $\theta=\sum_{k\in\cal N}m_{k}\lambda_{k}(\phi)$ such that
it leads to exactly the same level of utilization $\phi=\Phi(\theta,\mu)$.

We define $\Theta(\phi,\mu)\triangleq\Phi^{-1}(\phi,\mu)$ as the inverse
function of $\Phi(\theta,\mu)$ with respect to $\phi$. $\Theta(\phi,\mu)$ can
be interpreted as the implied amount of throughput that induces a utilization
level $\phi$ for a system with capacity $\mu$. By Assumption 1,
$\Theta(\phi,\mu)$ is strictly increasing in both $\phi$ and $\mu$. To
characterize the utilization of a system $(\mathbf{m},\mu)$, we define a gap
function $g(\phi)$ between the supply and demand of throughput under a fixed
level of utilization $\phi$ as
$g(\phi)\triangleq\Theta\left(\phi,\mu\right)-\sum_{k\in\cal
N}m_{k}\lambda_{k}(\phi)$.

######  Lemma 1

Given fixed capacity $\mu$ and user populations $\mathbf{m}$, $g(\phi)$ is a
strictly increasing function of $\phi$. The system operates at a unique level
of utilization $\phi$, which solves $g(\phi)=0$.

Lemma 1 characterizes the uniqueness of the system utilization $\phi$ under
which the throughput supply $\Theta(\phi,\mu)$ equals the aggregate throughput
demand $\sum_{k\in\cal N}m_{k}\lambda_{k}(\phi)$. Based on Lemma 1, for any
system $(\mathbf{m},\mu)$, we denote $\phi(\mathbf{m},\mu)$ as its unique
system utilization and $\theta_{i}(\mathbf{m},\mu)$ as CP $i$’s corresponding
throughput, i.e., $\theta_{i}(\mathbf{m},\mu)\triangleq
m_{i}\lambda_{i}\left(\phi(\mathbf{m},\mu)\right)$.

We define the marginal change of the throughput gap $g(\phi)$ due to the
marginal change in the system utilization $\phi$ as

$\frac{dg}{d\phi}\triangleq\frac{\partial\Theta(\phi,\mu)}{\partial\phi}-\sum_{k\in\cal
N}m_{k}\frac{d\lambda_{k}(\phi)}{d\phi}>0,$ (2)

where the first term (second summation) captures the change of throughput in
supply (total demand). We characterize the impact of user population $m_{i}$
and capacity $\mu$ on the utilization $\phi$ and throughput $\theta_{i}$, as a
function of $dg/d\phi$ as follows.

######  Theorem 1 (Capacity and User Effect)

The system utilization $\phi$ decreases if the capacity $\mu$ increases or any
user population $m_{i}$ decreases. In particular, we have

$\frac{\partial\phi}{\partial\mu}=-\left(\frac{dg}{d\phi}\right)^{-1}\frac{\partial\Theta(\phi,\mu)}{\partial\mu}<0,$
(3) $\text{and}\quad\frac{\partial\phi}{\partial
m_{i}}=\left(\frac{dg}{d\phi}\right)^{-1}\lambda_{i}>0,\quad\forall i\in\cal
N.$ (4)

Any CP $i$’s throughput $\theta_{i}$ increases with capacity $\mu$ and its
user population $m_{i}$; however, the increase in $m_{i}$ reduces the
throughput $\theta_{j}$ of all other CPs $j\neq i$. In particular,

$\frac{\partial\theta_{i}}{\partial\mu}=m_{i}\frac{d\lambda_{i}}{d\phi}\frac{\partial\phi}{\partial
m_{i}}>0,\quad\frac{\partial\theta_{i}}{\partial
m_{i}}=\lambda_{i}+m_{i}\frac{d\lambda_{i}}{d\phi}\frac{\partial\phi}{\partial
m_{i}}>0$ $\text{ and}\quad\frac{\partial\theta_{j}}{\partial
m_{i}}=m_{j}\frac{d\lambda_{j}}{d\phi}\frac{\partial\phi}{\partial
m_{i}}<0,\quad\forall j\neq i.$

Theorem 1 intuitively states that 1) when the system capacity becomes more
abundant, the throughput of any CP increases and the system utilization
decreases, and 2) when more users of a CP join the system, the system
utilization and the throughput of that CP increase, while that of any other CP
decreases. Equation (3) explicitly shows that the capacity effect on the
system utilization can be expressed as its impact on the feasible system
throughput $\partial\Theta(\phi,\mu)/\partial\mu$, normalized by $dg/d\phi$,
which re-balances the supply and demand of throughput so as to fill the
throughput gap $g(\cdot)$ under the new system utilization. Similarly in
Equation (4), $\lambda_{i}$ can be interpreted as the marginal increase in
throughput when a new user of CP $i$ joins the system. It also implies that
the user impact on the utilization is proportional to the per user throughput,
i.e., ${\partial\phi}/{\partial m_{i}}:{\partial\phi}/{\partial
m_{j}}=\lambda_{i}:\lambda_{j}$.

CPs’ throughput depends on whether their users are sensitive to congestion.
This can be characterized by utilization-elasticity of throughput or
$\phi$-elasticity of $\lambda_{i}$, defined as
$\epsilon_{\phi}^{\lambda_{i}}$.

######  Definition 2 (Elasticity)

The elasticity of $y$ with respect to $x$, or $x$-elasticity of $y$,
$\epsilon_{x}^{y}$ is defined as
$\displaystyle\epsilon_{x}^{y}\triangleq\frac{\partial y}{\partial
x}\frac{x}{y}$.

Elasticity is often expressed as $\epsilon_{x}^{y}=({\partial y}/y)/({\partial
x}/x)$ and interpreted as the percentage change in $y$ (the numerator) in
respond to the percentage change in $x$ (the denominator). In particular,
$\epsilon_{\phi}^{\lambda_{i}}$ captures the percentage change in
$\lambda_{i}$ in respond to the percentage change in the system utilization
$\phi$.

######  Lemma 2

If CP $i$ is replaced by a new CP $j$ which satisfies
$m_{j}\lambda_{j}(0)=m_{i}\lambda_{i}(0)$ and
$\epsilon_{\phi}^{{\lambda}_{j}}(\phi)=\epsilon_{\phi}^{{\lambda}_{i}}(\phi)$
for all $\phi>0$, then the system utilization does not change.

Lemma 2 characterizes CPs’ throughput by $\phi$-elasticity. Mathematically, by
keeping the same elasticity $\epsilon_{\phi}^{{\lambda}_{i}}$, we could scale
up the maximum throughput $\lambda_{i}(0)$ by
$\tilde{\lambda}_{i}(0)=\kappa\lambda_{i}(0)$ and scale down $m_{i}$ by
$\tilde{m}_{i}=\kappa^{-1}m_{i}$ for some constant $\kappa>0$ without
affecting the system utilization $\phi$ and the throughput of all other CPs.
Operationally, it implies that we could treat the traffic of a CP $i$ as if it
has a single big user, i.e., $\tilde{m}_{i}=1$, with the maximum throughput
$\tilde{\lambda}_{i}(0)=m_{i}\lambda_{i}(0)$. It also implies that for
multiple CPs, e.g., a set $\cal S\subset\cal N$, with the same
$\phi$-elasticity of throughput, we could conceptually aggregate them as a
single CP $k$ with parameters $m_{k}$ and $\lambda_{k}(0)$ that satisfy
$m_{k}\lambda_{k}(0)=\sum_{i\in\cal S}m_{i}\lambda_{i}(0)$. By this result, we
will use a single CP to model a group of CPs that have similar traffic
characteristics in our later numerical evaluations.

###  3.2 One-Sided ISP Pricing

In the current Internet, most access ISPs only charge their end-users. We
denote $t_{i}$ as the per-unit usage charge for the data traffic of CP $i$. We
assume that CP $i$’s user population $m_{i}$ is a function of its per-unit
charge $t_{i}$, i.e., $m_{i}=m_{i}(t_{i})$.

######  Assumption 2

$m_{i}(t_{i})$ is continuously differentiable and decreasing with
$\lim_{t_{i}\rightarrow\infty}m_{i}(t_{i})=0$.

Assumption 2 states that the user demand decreases with the per-unit charge on
the content traffic. This model is quite general and coincides with existing
models [5, 36] where the heterogeneity of users are modeled by the
distribution of their valuations on data traffic and only the users whose
valuations are higher than $t_{i}$ consume data traffic from CP $i$.

Because traffic from different CPs should not be differentiated under net
neutrality, ISPs often impose a uniform charge. We denote $p$ as this uniform
charge for all types of traffic. When CPs do not provide subsidies, we have
$t_{i}=p$ for any $i\in\cal N$. We define the ISP’s revenue as $R\triangleq
p\theta=p\sum_{i\in\cal N}m_{i}\lambda_{i}$ and each CP $i$’s utility as
$U_{i}=v_{i}\theta_{i}$, where $v_{i}$ denotes CP $i$’s average per-unit
traffic profit. Because the price $p$ affects the user demand $\mathbf{m}$, we
extend our basic system model to capture the one-sided ISP pricing as follows.

Figure 3: The system model with one-sided ISP pricing.

Figure 3 shows that the user populations $\mathbf{m}$ are influenced by the
price $p$, and that the utilization $\phi$ affects the ISP’s revenue
$R=p\sum_{i\in\cal N}m_{i}\lambda_{i}(\phi)$. ISPs can use revenue as feedback
to guide their pricing and capacity decisions; however, CPs’ utilities are
determined by throughput, which they have no means to influence in the one-
sided pricing model.

Because $t_{i}=p$ for any CP $i$, by Assumption 2, we express the user
populations as a function of price as $\mathbf{m}=\mathbf{m}(p)$. For a fixed
capacity $\mu$, we define $\phi(p)\triangleq\phi({\mathbf{m}}(p),\mu)$ as the
utilization of system $(\mathbf{m}(p),\mu)$ and $\theta_{i}(p)\triangleq
m_{i}(p)\lambda_{i}\left(\phi(p)\right)$ as the corresponding throughput of CP
$i$ under price $p$. By Assumption 1 and 2, both $\phi(p)$ and $\theta_{i}(p)$
are differentiable functions of $p$. We characterize the impact of price $p$
on the system utilization $\phi$ and each CP $i$’s throughput $\theta_{i}$ in
the following theorem. The price’s impact on $U_{i}$ is simply its impact on
$\theta_{i}$ scaled by $v_{i}$ and we will study its impact on the revenue $R$
under a more general setting in Section 5.

######  Theorem 2 (Price Effect)

The system utilization $\phi$ decreases with the price $p$. In particular, we
have

$\frac{\partial\phi}{\partial
p}=\left(\frac{dg}{d\phi}\right)^{-1}\sum_{k\in\cal
N}\frac{dm_{k}}{dp}\lambda_{k}\leq 0.$ (5)

The aggregate throughput $\theta$ decreases with the price $p$,

$\frac{\partial\theta}{\partial
p}=\frac{\partial\Theta}{\partial\phi}\left(\frac{\partial\Theta}{\partial\phi}-\sum_{i\in\cal
N}m_{i}\frac{d\lambda_{i}}{d\phi}\right)^{-1}\sum_{i\in\cal
N}\frac{dm_{i}}{dp}\lambda_{i}\leq 0,$ (6)

and any CP $i$’s throughput $\theta_{i}$ increases with $p$ if and only if

${\epsilon_{p}^{m_{i}}}/{\epsilon_{\phi}^{\lambda_{i}}}<-\epsilon_{p}^{\phi}.$
(7)

Theorem 7 states that when the price increases, due to its direct impact on
decreasing the user demand, the system utilization decreases, which implies
that the aggregate throughput of all the CPs also decreases. Similar to (3)
and (4), the price effect on the system utilization can be expressed as its
impact on the aggregate throughput $d\theta/dp$, normalized by $dg/d\phi$ in
Equation (5). However, a CP $i$’s throughput might increase or decrease,
depending on its utilization elasticity of (per user) throughput
$\epsilon_{\phi}^{\lambda_{i}}$ and price elasticity of user demand
$\epsilon_{p}^{m_{i}}$, i.e., the percentage change in the population $m_{i}$
in response to the percentage change in price $p$. In particular, $\theta_{i}$
increases with $p$ if its users are less price-sensitive and more congestion-
sensitive, i.e., $|{\epsilon_{p}^{m_{i}}}|$ is small and
$|{\epsilon_{\phi}^{\lambda_{i}}}|$ is large. Notice that the right hand side
of (7) measures the overall impact of price on the system utilization.

To understand the price effect more intuitively, we illustrate a numerical
example as follows. We consider a utilization function
$\Phi(\theta,\mu)=\theta/\mu$, which uses per capacity throughput as the
metric for system utilization. We consider the forms of throughput
$\lambda_{i}(\phi)=e^{-\beta_{i}\phi}$ and user demand
$m_{i}(t_{i})=e^{-\alpha_{i}t_{i}}$ that satisfy Assumption 1 and 2. Under
these exponential forms, the $\phi$-elasticity of throughput
$\epsilon_{\phi}^{\lambda_{i}}$ equals $-\beta_{i}\phi$ and the $p$-elasticity
of user population $\epsilon_{p}^{m_{i}}$ equals $-\alpha_{i}p$. Based on the
above setting, we can express a CP $i$’s throughput as
$\theta_{i}=e^{-(\alpha_{i}p+\beta_{i}\phi)}$, where $\alpha_{i}$ and
$\beta_{i}$ neatly characterize $\theta_{i}$’s sensitivity to price and
congestion, respectively. We obtain ${dg}/{d\phi}=\mu+\sum_{i\in\cal
N}\beta_{i}\theta_{i}$ and

$\frac{\partial\phi}{\partial
p}=\left(\frac{dg}{d\phi}\right)^{-1}\left(\sum_{i\in\cal
N}\frac{dm_{i}}{dp}\lambda_{i}\right)=-\frac{\sum_{i\in\cal
N}\alpha_{i}\theta_{i}}{\mu+\sum_{i\in\cal N}\beta_{i}\theta_{i}}.$

By Theorem 7, throughput $\theta_{i}$ increases at price $p$ if

$\frac{\alpha_{i}p}{\beta_{i}\phi}<\frac{\sum_{j\in\cal
N}\alpha_{j}\theta_{j}}{\mu+\sum_{k\in\cal N}\beta_{k}\theta_{k}}.$ (8)

We set the system capacity to be $\mu=1$ and consider a set of $9$ types of
CPs with values of $(\alpha_{i},\beta_{i})$ chosen from $\\{1,3,5\\}$.

Figure 4: Aggregate throughput $\theta$ and ISP’s revenue $R$.

Figure 4 plots the aggregate throughput $\theta$ (left) and the ISP’s revenue
$R$ (right) as a function of price $p$ that varies along the x-axis. We
observe that the aggregate throughput decreases with the price as indicated by
Theorem 7; however, the revenue $R=p\theta$ depends on both the price and
aggregate throughput and shows a single-peak pattern.

Figure 5: Throughput $\theta_{i}$ of different CPs.

Figure 5 shows the throughput $\theta_{i}$ of the $9$ individual CPs as a
function of the price $p$ in each sub-figure, respectively. In general, the
throughput is low with large values of $\alpha_{i}$ and $\beta_{i}$ (the lower
and right sub-figures), because the user population is more sensitive to price
and the per user throughput is more sensitive to congestion. By Theorem 7,
when $p$ increases, $\phi$ decreases, and therefore
$(\alpha_{i}/\beta_{i})(p/\phi)$ increases. As indicated by condition (8),
each $\theta_{i}$ decreases with $p$ eventually; however, when $p$ is small,
we observe that the CPs with a small ratio of $\alpha_{i}/\beta_{i}$ (the
upper and right sub-figures) demonstrate an increasing trend in throughput.
Intuitively, for these CPs, the increase in the per user throughput
$\lambda_{i}$ is much higher than the decrease in the user population $m_{i}$
so that the aggregate throughput $\theta_{i}$ could still increase.

Regulatory Implications: Under the existing one-sided pricing, an increase in
the access ISP’s price reduces the user demand (by Assumption 2), and
consequently reduces the system utilization and the total system throughput
(by Theorem 7). Regulators might want to regulate the price of an access ISP
if its high price induces low utilization and drives the total system
throughput too low (shown in Figure 4). However, as the existing wireless
capacities of many carriers are often under-provisioned and highly-loaded, and
regulations will further limit the ISPs’ profit margin, we do not see the need
for price regulation under the status quo.

##  4 Subsidization Competition

As the access ISPs can neither differentiate services nor charge CPs in the
one-sided pricing model in Figure 3, the limited profit margin does not
provide enough investment incentive for them to expand capacities. Neither do
the CPs have any means to express preferences and improve their utilities. In
this section, we propose to create a feedback channel for the CPs to influence
the system by allowing them to voluntarily subsidize the usage-based fees for
their users.

###  4.1 The Subsidization Competition Model

We denote $q\geq 0$ as a subsidization policy that limits the maximum subsidy
allowed. We denote $s_{i}\in[0,q]$ as the per-unit usage subsidy provided by
CP $i$ for its content traffic for users. We denote $\mathbf{s}$ as the vector
of subsidies of the CPs. We extend the definition of CP $i$’s utility as
$U_{i}\triangleq(v_{i}-s_{i})\theta_{i}$.

Figure 6: One-sided ISP pricing with CP subsidization.

Figure 6 illustrates the extended model where subsidization from CPs is
allowed. In this model, each CP $i$ could strategically choose its subsidy
$s_{i}$ to influence its throughput $\theta_{i}$ via its user population
$m_{i}$ so as to optimize its utility $U_{i}$. Regulators can also use a
certain welfare metric $W$ to determine the desirable policy regime $q$ for
the Internet industry.

Under price $p$ and subsidies $\mathbf{s}$, user population $m_{i}$ satisfies
$m_{i}(t_{i})=m_{i}(p-s_{i})$. Given any fixed ISP decision $(p,\mu)$, we
denote $\phi(\mathbf{s})\triangleq\phi({\mathbf{m}}(p,\mathbf{s}),\mu)$ as the
system utilization and define
$\theta_{i}(\mathbf{s})=m_{i}(p-s_{i})\lambda_{i}\left(\phi(\mathbf{s})\right)$
as CP $i$’s throughput.

######  Lemma 3

For any $s^{\prime}_{i}>s_{i}$, let
$\mathbf{s^{\prime}}=(s^{\prime}_{i},\mathbf{s}_{-i})$ and
$\mathbf{s}=(s_{i},\mathbf{s}_{-i})$ for some fixed strategy profile
$\mathbf{s}_{-i}$. For any price $p$,
$\phi(\mathbf{s^{\prime}})\geq\phi(\mathbf{s}),\
\theta_{i}(\mathbf{s^{\prime}})\geq\theta_{i}(\mathbf{s})$ and
$\theta_{j}(\mathbf{s^{\prime}})\leq\theta_{j}(\mathbf{s}),\ \forall j\neq i$.

Lemma 3 states that if a CP $i$ unilaterally increases its subsidy $s_{i}$,
its throughput $\theta_{i}$ and the system utilization $\phi$ increase;
however, the throughput of any other CP decreases. It shows that the
subsidization mechanism creates a competitive game among the CPs, where each
CP $i$ could use subsidy $s_{i}$ to maximize its utility $U_{i}$. Notice that
besides the policy constraint $q$, each CP’s optimal subsidy strategy depends
on all other CPs’ strategies as well as the ISP’s price $p$. In this section,
we study the policy implications of subsidization and therefore, we often
assume a fixed price $p$ for the access ISP. This setting corresponds to a
competitive access market where the ISPs cannot easily manipulate prices, or a
case where the ISP’s price is regulated. We will study an ISP’s pricing
strategy, its revenue and the policy implication on the system welfare in the
next section.

######  Definition 3 (Nash equilibrium)

Given any fixed ISP price $p$ and policy $q$, a strategy profile $\mathbf{s}$
is a Nash equilibrium  if each $s_{i}$ solves CP $i$’s utility maximization
problem:

$\displaystyle\max~{}~{}$ $\displaystyle
U_{i}(s_{i};\mathbf{s}_{-i})=(v_{i}-s_{i})\theta_{i}(\mathbf{s})$
$\displaystyle\mathrm{subject~{}to~{}}~{}$ $\displaystyle 0\leq s_{i}\leq q.$

We characterize the Nash equilibrium of the subsidization competition game in
the following theorem.

######  Theorem 3 (Characterization)

For any fixed $p$ and $q$, a strategy profile $\mathbf{s}$ is a Nash
equilibrium only if

$\quad s_{i}=\min\\{\tau_{i}(\mathbf{s}),q\\},\quad\forall\ i\in\cal N,$

where each $\tau_{i}$ denotes a threshold for CP $i$, defined by

$\displaystyle\tau_{i}(\mathbf{s})\triangleq\left(v_{i}-s_{i}\right)\epsilon_{s_{i}}^{m_{i}}\left(1+\epsilon_{\phi}^{\lambda_{i}}\epsilon_{m_{i}}^{\phi}\right).$
(9)

In particular, $v_{i}\leq\left({\partial\theta_{i}}/{\partial
s_{i}}\right)^{-1}\theta_{i}$ holds if $s_{i}=0$. Moreover, if
$U_{i}(\mathbf{s})=U_{i}(s_{i},{\mathbf{s}}_{-i})$ is concave in $s_{i}$ for
all $i\in\cal N$, then the above conditions are also sufficient.

Theorem 3 characterizes the Nash equilibrium in two ways. First, it indicates
that any CP $i$ does not subsidize if its profitability $v_{i}$ is lower than
$(\partial\theta_{i}/\partial s_{i})^{-1}\theta_{i}$. Second, if the policy
constraint is not tight, i.e., $s_{i}<q$, the equilibrium subsidy $s_{i}$ can
be characterized by its profit margin $v_{i}-s_{i}$ and the elasticity metrics
$\epsilon_{s_{i}}^{m_{i}}$, $\epsilon_{\phi}^{\lambda_{i}}$ and
$\epsilon_{m_{i}}^{\phi}$. Equation (9) can also be written as
$\tau_{i}(\mathbf{s})=(v_{i}-s_{i})\epsilon_{s_{i}}^{\theta_{i}}=(v_{i}-s_{i})\epsilon_{s_{i}}^{m_{i}}\left(1+\epsilon_{m_{i}}^{\lambda_{i}}\right)$.
This characterization also implies that a CP would subsidize its users more if
1) its traffic profitability $v_{i}$ (and therefore its profit margin
$v_{i}-s_{i}$) increases, or 2) its throughput elasticity
$\epsilon_{s_{i}}^{\theta_{i}}$ or user demand elasticity of subsidy
$\epsilon_{s_{i}}^{m_{i}}$ increases.

By Assumption 1 and 2, each $U_{i}(\mathbf{s})$ is differentiable in
$\mathbf{s}$. For any strategy profile $\mathbf{s}$, we define
$\mathbf{u}(\mathbf{s})=\\{u_{i}(\mathbf{s}):i\in\cal N\\}$ as the vector of
marginal utilities, where each $u_{i}(\mathbf{s})$ denotes the marginal
utility of CP $i$, defined as $u_{i}(\mathbf{s})\triangleq\partial
U_{i}(\mathbf{s})/\partial s_{i}$. Next, we characterize the uniqueness of
Nash equilibrium based on a condition on the marginal utilities
$\mathbf{u}(\mathbf{s})$.

######  Theorem 4 (Uniqueness)

For any fixed price $p$ and policy $q$, if for any distinct pair of feasible
strategy profiles $\mathbf{s^{\prime}\neq s}$, there always exist a CP
$i\in\cal N$ such that

$(s^{\prime}_{i}-s_{i})\left(u_{i}(\mathbf{s^{\prime}})-u_{i}(\mathbf{s})\right)<0,$
(10)

then there always exists a unique Nash equilibrium.

Technically, the above sufficient condition for uniqueness requires
$-\mathbf{u}$ to be a $P$-function [27]. Notice that this uniqueness property
holds for any subset ${\cal Q}\subset[0,q]^{|\cal N|}$ of the strategy space
if condition (10) holds in the domain ${\cal Q}$.

###  4.2 Dynamics of Subsidies in Equilibrium

Since a Nash equilibrium is defined under a fixed price $p$ and policy $q$ by
Definition (3), we use $\mathbf{s}=\mathbf{s}(p,q)$ to indicate a Nash
equilibrium under $p$ and $q$. Equation (9) in Theorem 3 hinted that a CP with
higher profitability subsidizes users more in an equilibrium. We formally show
this as follows.

######  Theorem 5 (Profitability Effect on Subsidy)

If a CP $i$’s profitability increases from $v_{i}$ to $\hat{v}_{i}$
unilaterally, and under the condition (10) of Theorem 4, $\mathbf{s}$ and
$\mathbf{\hat{s}}$ are the corresponding Nash equilibria, then we must have
$\hat{s}_{i}\geq s_{i}$.

If a CP’s profitability increases, Theorem 5 tells that it will increase its
subsidy in equilibrium, and hints that its throughput will increase by Lemma
3. Through subsidization, CPs can respond to their profitability and influence
throughput.

After analyzing the impact of CPs’ profitability, we study how an equilibrium
$\mathbf{s}(p,q)$ is influenced by the price $p$ and policy $q$. We define
$\cal N_{-}$ and $\cal N_{+}$ as the set of CPs whose subsidies are $0$ and
$q$, respectively, and $\widetilde{\cal N}={\cal N}\backslash(\cal
N_{-}\cup\cal N_{+})$ as the remaining set of CPs whose subsidies are strictly
positive and less than $q$. We define $\mathbf{\tilde{s}}$ and
$\mathbf{\tilde{u}}$ as the subsidies and marginal utilities of the set
$\widetilde{\cal N}$ of CPs.

######  Theorem 6 (Equilibrium Dynamics)

If a Nash equilibrium $\mathbf{s}$ satisfies 1) $i\in\widetilde{\cal N}$ if
$u_{i}(\mathbf{s})=0$, and 2) the condition (10) for a neighborhood of
$\mathbb{R}^{|\cal N|}$ centered at $\mathbf{s}$ in the strategy space, then
in a neighborhood of $(p,q)$, a unique Nash equilibrium is a differentiable
function $\mathbf{s}(p,q)$, satisfying

$\frac{\partial s_{i}}{\partial q}=\left\\{\begin{array}[]{cl}\displaystyle
0&\text{if }\ i\in{\cal N}_{-};\\\ \displaystyle 1&\text{if }\ i\in{\cal
N}_{+};\\\ \displaystyle-\sum_{k\in\widetilde{\cal N}}\psi_{ik}\sum_{j\in{\cal
N}_{+}}\frac{\partial u_{k}}{\partial s_{j}},&\text{if }\ i\in\widetilde{\cal
N},\end{array}\right.$ (11) $\quad\text{and}\quad\frac{\partial
s_{i}}{\partial p}=\left\\{\begin{array}[]{cl}0&\text{if }\
i\notin\widetilde{\cal N};\\\ \displaystyle-\sum_{k\in\widetilde{\cal
N}}\psi_{ik}\frac{\partial u_{k}}{\partial p},&\text{if }\ i\in\widetilde{\cal
N},\end{array}\right.$ (12)

where
$\Psi=\\{\psi_{ij}\\}\triangleq(\nabla_{\mathbf{\tilde{s}}}\mathbf{\tilde{u}})^{-1}$,
i.e., $\psi_{ij}$ is at the $i$th row and $j$th column of the inverse of the
Jacobian matrix $\nabla_{\mathbf{\tilde{s}}}\mathbf{\tilde{u}}$.

Theorem 6 states that a marginal change in the ISP’s price or the regulatory
policy will not affect the behavior of the set ${\cal N}_{-}$ and ${\cal
N}_{+}$ of CPs: the CPs who do not subsidize remain the same and the CPs who
subsidize the amount $q$ keep subsidizing at the maximum level as $q$
increases. When $p$ changes, the set $\widetilde{\cal N}$ of CPs readjust
subsidies among themselves in a new equilibrium such that their marginal
utilities remain zero in (12); when $q$ changes, since the set ${\cal N}_{+}$
of CPs increase subsidies accordingly, their impacts have to be counted in the
new equilibrium in (11). By applying the KKT condition to Definition 3, we
know $u_{i}(\mathbf{s})=0$ for all $i\in\widetilde{\cal N}$, and the first
condition of Theorem 6 is a bit stronger which guarantees that the equilibrium
$\mathbf{s}$ is regular, i.e., locally differentiable. The second condition
only assumes (10) locally, which guarantees the local uniqueness of
equilibrium in a neighborhood centered at $\mathbf{s}$ by Theorem 4.

######  Corollary 1 (Deregulation)

Under a fixed ISP price $p$ and the conditions of Theorem 6, in a neighborhood
of $q$, we can express $\phi(q)$ and $R(q)$ as the system utilization and the
ISP’s revenue under the Nash equilibrium $\mathbf{s}(q)$. If $\mathbf{u}$ is
off-diagonally monotone, i.e., $\partial u_{i}(\mathbf{s})/\partial s_{j}\geq
0$ for all $i\neq j$,

$\frac{\partial\phi}{\partial q}\geq 0,\quad\frac{\partial R}{\partial q}\geq
0\quad\text{and}\quad\frac{\partial s_{i}}{\partial q}\geq 0,\quad\forall
i\in\cal N.$

The off-diagonally monotone property makes $-\mathbf{u}$ to be a Leontief type
[15] of $P$-matrix, which guarantees the stability of a macroeconomic system
due to Wassily Leontief [16]. Intuitively, it assumes although the utility of
a CP $i$ decreases when other CPs increase their subsidies, its marginal
benefit of subsidizing its own users, i.e., $u_{i}$, increases. Corollary 1
states that under this stability condition, when allowed to subsidize more,
CPs will increase subsidies, which will lead to an increase in the system
utilization and the ISP’s revenue.

Regulatory Implications: Under a competitive or price regulated access market,
deregulation of subsidization encourages CPs to subsidize users (by Theorem 5
and 6), and consequently increases the system utilization and the ISPs’
revenue (by Corollary 1). CPs’ user populations $\mathbf{m}$ increase because
the usage charges become cheaper after subsidization (by Assumption 2), but
the induced higher utilization (by Theorem 1) might cause congestion. The
total system throughput also increases (implied by Corollary 1 as $\phi$
increases), while the throughput of certain CPs might decrease. However, this
is caused by the negative network externality, i.e., the physics of congestion
of shared capacity. Although it might be unfavorable for the CPs whose traffic
is more congestion sensitive in the short term, as the ISPs improve profit
margins from higher utilization, they will have more incentives to expand
capacities so as to accommodate more traffic and relieve congestion in the
long term.

##  5 ISP Revenue and System Welfare

Although the policy $q$, the price $p$ and the subsidies $\mathbf{s}$ all
impact the system, these decisions are not made simultaneously and
independently. The system is fundamentally driven by a regulatory policy $q$,
under which the ISP determines its price $p(q)$ and then the CPs respond with
strategies $\mathbf{s}(p,q)$. Theorem 6 characterizes the dynamics of
$\mathbf{s}(p,q)$ and Corollary 1 characterizes the impact of policy $q$ when
$p$ is fixed. In practice, ISPs often use price to optimize revenue, and by
Theorem 6 the subsidies $\mathbf{s}$ are also influenced by $p$. In this
section, we study the impact of the ISP’s price on its revenue and the impact
of policy on the system welfare when the response of ISP’s pricing is taken
into account.

###  5.1 Impact of ISP’s Pricing on Its Revenue

Under a fixed policy $q$, by Theorem 6, we can write $\mathbf{s}(p)$ as the
induced subsidies under price $p$. Thus, the induced system utilization is
$\phi(\mathbf{s}(p))$ and the ISP’s induced revenue is $R(p)=p\sum_{i\in\cal
N}\theta_{i}(p)=p\sum_{i\in\cal
N}m_{i}(p-s_{i}(p))\lambda_{i}(\phi(\mathbf{s}(p)))$.

######  Theorem 7 (Marginal Revenue)

If the ISP’s revenue is differentiable at $p$, then its marginal revenue is

$\frac{dR(p)}{dp}=\sum_{i\in\cal N}\theta_{i}+\Upsilon\sum_{i\in\cal
N}\epsilon_{p}^{m_{i}}\theta_{i},$ (13)

where $\Upsilon$ and the $m_{i}$-elasticity of price $p$ satisfy

$\Upsilon=1+\sum_{j\in\cal
N}\epsilon_{m_{j}}^{\lambda_{j}}\quad\text{and}\quad\epsilon_{p}^{m_{i}}=\frac{p}{m_{i}}\frac{dm_{i}}{dt_{i}}\left(1-\frac{\partial
s_{i}}{\partial p}\right).$

Theorem 7 characterizes the change in revenue as the price varies and isolates
the effect of subsidization into the elasticities $\epsilon_{p}^{m_{i}}$,
where ${\partial s_{i}}/{\partial p}$ plays a role. It also generalizes the
case of one-sided pricing, i.e., ${\partial s_{i}}/{\partial p}=0$ for all
$i\in\cal N$. The second term in (13) measures the price’s impact on the
aggregate throughput, i.e., $p({\partial\theta}/{\partial p})$, which can be
factorized by $\Upsilon$, a parameter determined by the physical model in
Figure 2, because $\epsilon_{m_{j}}^{\lambda_{j}}$ can be decomposed as
$\epsilon_{m_{j}}^{\phi}\epsilon_{\phi}^{\lambda_{j}}$ and

$\epsilon_{m_{j}}^{\phi}\epsilon_{\phi}^{\lambda_{j}}=\frac{m_{j}}{\lambda_{j}}\frac{d\lambda_{j}}{d\phi}\frac{\partial\phi}{\partial
m_{j}}=m_{j}\frac{d\lambda_{j}}{d\phi}\left(\frac{dg}{d\phi}\right)^{-1}.$
(14)

###  5.2 Impact of Regulatory Policy on Welfare

By Theorem 7, we understand how the ISP’s pricing affects its revenue. We
assume that the ISP would adapt its price $p$ as a function of a given policy
$q$. The next theorem captures the impact of the policy $q$ on the system
states, i.e., $\phi$, $m_{i}$ and $\theta_{i}$, where the responses of both
the ISP’s price $p(q)$ and the CPs’ subsidies $\mathbf{s}(p,q)$ are taken into
account.

######  Theorem 8 (Policy Effect)

If the ISP’s price is a differentiable function $p(q)$ of policy $q$ and
$\mathbf{s}$ is a Nash equilibrium of $p(q)$ and $q$, under the conditions of
Theorem 6, in a neighborhood of $q$, we can express the Nash equilibrium as
$\mathbf{s}(q)\triangleq\mathbf{s}\left(p(q),q\right)$. The corresponding
system utilization $\phi(q)$, user population $m_{i}(q)$ and throughput
$\lambda_{i}(q)$ satisfy

$\frac{dm_{i}}{dq}=\frac{dm_{i}}{dt_{i}}\frac{dt_{i}}{dq}=\frac{dm_{i}}{dt_{i}}\left(\left(1-\frac{\partial
s_{i}}{\partial p}\right)\frac{dp}{dq}-\frac{\partial s_{i}}{\partial
q}\right),$ (15)
$\frac{d\phi}{dq}=\left(\frac{dg}{d\phi}\right)^{-1}\sum_{i\in\cal
N}\frac{dm_{i}}{dq}\lambda_{i}\quad\text{and}\quad\frac{d\lambda_{i}}{dq}=\frac{d\lambda_{i}}{d\phi}\frac{d\phi}{dq}.$
(16)

Any CP $i$’s throughput $\theta_{i}$ increases with $q$ if and only if

${\epsilon_{t_{i}}^{m_{i}}\epsilon_{q}^{t_{i}}}/{\epsilon_{\phi}^{\lambda_{i}}}<-\epsilon_{q}^{\phi}.$
(17)

Theorem 17 shows that the policy effect on utilization $d\phi/dq$ in (16) and
the condition (17) for throughput $\theta_{i}$ have similar forms as the price
effect $d\phi/dp$ in (5) and the condition (7). This is because the policy
effect is carried out by its impact on the user populations, i.e.,
$\epsilon_{q}^{m_{i}}=\epsilon_{t_{i}}^{m_{i}}\epsilon_{q}^{t_{i}}$, via its
impact on the price and subsidies. By Assumption 1 and 2, both
${\epsilon_{\phi}^{\lambda_{i}}}$ and $\epsilon_{t_{i}}^{m_{i}}$ are negative.
Inequality (17) tells that a CP’s throughput decreases if any only if
$(-\epsilon_{t_{i}}^{m_{i}})(-\epsilon_{q}^{t_{i}})<-{\epsilon_{\phi}^{\lambda_{i}}}\epsilon_{q}^{\phi}$,
which implies that the subsidy could help increase throughput via
$(-\epsilon_{q}^{t_{i}})$; however, $(-\epsilon_{t_{i}}^{m_{i}})$ could be
decreased due to the ISP’s increasing price, which will reduce the CP’s
throughput.

######  Corollary 2 (Policy Impact on Welfare)

Under the conditions of Theorem 17, let $W(q)\triangleq\sum_{i\in\cal
N}\theta_{i}(q)v_{i}$ define the system welfare. Suppose ${d\phi}/{dq}$ in
(16) is positive, then the marginal welfare ${dW}/{dq}$ is positive if and
only if

$\sum_{i\in\cal N}\frac{w_{i}}{\sum_{k\in\cal N}w_{k}}v_{i}>\sum_{i\in\cal
N}-\epsilon_{m_{i}}^{\lambda_{i}}v_{i},\quad\text{where}\quad
w_{i}\triangleq\lambda_{i}\frac{dm_{i}}{dq}.$

We measure the system welfare in terms of the gross profit $W=\sum_{i\in\cal
N}{\theta_{i}v_{i}}$ of all the CPs for two reasons. First, it internalizes
the subsidy transfer from CPs to ISP. Second, because CP profits are often
positively correlated to their values to users, it also serves an estimate for
user welfare. In Corollary 2, $w_{i}v_{i}$ can be interpreted as the increase
in welfare due to the policy’s impact on the user population $m_{i}$, and
therefore, the left side of the inequality represents the normalized increase
in welfare due to the changes in the user populations. The right side of the
inequality represents the normalized decrease in welfare due to the policy’s
impact on each average throughput $\lambda_{i}$ via $m_{i}$. Corollary 2
states that the welfare increases if and only if the increasing component is
larger than the decreasing component. Notice that the decreasing component
only depends on the physical characteristics (14) of the CPs, while the weight
$w_{i}/\sum_{j\in\cal N}w_{j}$ for each $v_{i}$ tends to be large if $v_{i}$
is large, because profitable CPs will have stronger tendencies to subsidize
their users so as to attract a larger population $m_{i}$.

To understand the policy effect more intuitively, we perform a numerical
evaluation as follows. We use the same setting
$m_{i}(t_{i})=e^{-\alpha_{i}t_{i}}$, $\lambda_{i}(\phi)=e^{-\beta_{i}\phi}$
and $\Phi(\theta,\mu)=\theta/\mu$ as in Section 3. We model $8$ types of CPs
with $\alpha_{i},\beta_{i}\in\\{2,5\\}$ and $v_{i}\in\\{0.5,1\\}$. By Lemma 2,
each CP represents the aggregation of a group of CPs with similar
characteristics of traffic and user demand. In each of the following figures,
we vary the policy $q$ at $5$ levels from $0$ to $2.0$ and vary the ISP’s
price $p$ from $0$ to $2$ along the x-axis. When $q=0$, the figures show the
baseline case where subsidization is not allowed.

Figure 7: ISP’s Revenue $R$ and System Welfare $W$.

Figure 7 plots the ISP’s revenue $R$ (left) and the system welfare $W$
(right). We observe that under any fixed price $p$, both the ISP’s revenue and
the system welfare are higher when the CPs’ subsidization is less regulated,
i.e., $q$ is large. However, if the deregulation of subsidization will trigger
a higher ISP price, it also shows that the system welfare will decrease with
the price $p$ under any fixed policy $q$.

Figure 8: The subsidies $s_{i}$ of the $8$ types of CPs under equilibrium.
Figure 9: The user population $m_{i}$ of the $8$ types of CPs under
equilibrium. Figure 10: The throughput $\theta_{i}$ of the $8$ types of CPs
under equilibrium. Figure 11: The utility $U_{i}$ of the $8$ types of CPs
under equilibrium.

Figure 8 plots the subsidy $s_{i}$ of the $8$ types of CPs under equilibrium.
We observe that the CPs that have a higher profitability, i.e., $v_{i}=1$ in
the lower $4$ sub-figures, or a higher demand elasticity, i.e., $\alpha_{i}=5$
in the right $4$ sub-figures, provide much higher subsidies compared to their
counterparts. We also observe that when the price $p$ is small, except for the
two CPs with $\alpha_{i}=2$ and $v_{i}=0.5$, most CPs want to subsidize at the
maximum level $q$ constrained by the policy; however, when $p$ increases,
subsidies may stay flat and then decrease due to the decrease in profit
margin. By comparing Figure 8 with Figure 7, we observe that when $q=2$, the
ISP maximizes its revenue by setting $p$ a bit less than $1$, where the CPs’
subsidies are kept at a high level.

Figure 9 plots the corresponding user population $m_{i}$ of the $8$ types of
CPs under equilibrium. We observe that when $p$ increases, the user
populations of the CPs with higher demand elasticity in the right $4$ sub-
figures decrease steeper than those with lower demand elasticity. By comparing
the low-value CPs in the upper sub-figures with the high-value CPs in the
lower sub-figures, we observe that the user populations of the high-value CPs
decrease much slower so that they can retain higher user populations via
higher subsidies. In all cases, we observe that CPs always obtain higher user
population under a more relaxed policy $q$.

Figure 10 plots the individual throughput $\theta_{i}$ of the $8$ types of CPs
under equilibrium. In Figure 10, we observe that the CPs with higher
profitability, i.e., $v_{i}=1$, or lower congestion elasticity, i.e.,
$\beta_{i}=2$, achieve higher throughput compared to their counterparts. When
comparing the throughput with that of the baseline, i.e., $q=0$, we observe
that the CPs with high profitability achieve higher throughput, with the only
exception for the case $(\alpha_{i},\beta_{i},v_{i})=(2,5,1)$ when $p$ is
small. As $p$ is small, the system attracts large user demand and is
relatively congested. Because this CP is more congestion sensitive, i.e.,
$\beta_{i}$ is large, and by Corollary 1, subsidization will further increase
the system utilization, its throughput will be reduced. By comparing Figure 10
with Figure 7, we observe that with higher throughput for the CPs of higher
profitability, the system welfare increases when the policy $q$ is relaxed.

Figure 11 plots the resulting utility $U_{i}$ of the $8$ types of CPs under
equilibrium. As the individual CP’s utility is defined as
$U_{i}=(v_{i}-s_{i})\theta_{i}$, each $U_{i}$ shows similar trends as
$\theta_{i}$ in Figure 10, which is scaled by its profit margin $v_{i}-s_{i}$.
When $q$ increases, we observe that 1) CPs with high demand elasticity and
value, i.e., $\alpha_{i}=5$ and $v_{i}=1$, achieve higher utility via higher
subsidies, user population and throughput, and 2) CPs with low demand
elasticity and high congestion elasticity, i.e., $\alpha_{i}=2$ and
$\beta_{i}=5$, achieve lower utility due to the lower achievable throughput.
The utilities of other CPs under different policies are comparable.

Regulatory Implications: In a deregulated market where CPs are allowed to
subsidize their users, highly profitable CPs will provide greater subsidies
(by Theorem 5) and obtain more users (by Assumption 2), achieving higher
throughput (by Theorem 1) and leading to a higher system welfare (by Corollary
2). Due to negative network externality, some CPs may obtain lower throughput;
however, it can be resolved when ISPs obtain investment incentives and expand
capacity. Nonetheless, deregulation of subsidization might trigger an increase
in the ISP’s price, leading to a decrease in the system welfare (shown in
Figure 7). High ISP prices also discourage user demand (by Assumption 2) and
consequently decrease their throughput (shown in Figure 10). Our results
suggest that the subsidization competition will increase the ISP’s revenue and
system welfare (by Corollary 1); however, price regulations might be needed if
the access market is not competitive and the ISP’s price is set too high.

##  6 Issues and Model Limitations

Subsidization works with the usage-based pricing schemes of the access ISPs.
For mobile providers, implementing data caps and accounting for data traffic
consumptions are common in practice. Broadband providers in countries like
India and Australia have been using usage-based pricing, and major U.S. ISPs
like Verizon [38] and AT&T [42] have also adopted usage-based pricing after
gaining support from the FCC [37, 18]. To differentiate content and enable
subsidization from different CPs, platforms such as FreeBand [3] have been
developed to provide data traffic statistics to mobile carriers so as to lower
the bills of consumers, based on the type of applications. AT&T’s recent
sponsored data [1] plan allows CPs to fully subsidize their users. Although
our model allows partial subsidization, its technical feasibility is the same
as the proposed sponsored data applications.

Deregulation of subsidization might reduce the throughput of certain CPs,
e.g., the upper-left ones in Figure 10. These CPs either do not have
incentives to subsidize, because their users are not price-sensitive, i.e.,
the elasticity $\epsilon_{s_{i}}^{m_{i}}$ is small, or they cannot afford to
subsidize, because their profitability $v_{i}$ is low. In the former case, the
decrease in throughput is mainly due to a higher utilization of the system,
which will be relieved in the long term when the ISPs expand capacities.
Startup companies with low profits are of the latter case. The decrease in
throughput is mainly due to the ISP’s high price that limits the user demand,
and therefore, regulators might want to regulate ISP’s monopolistic pricing or
introduce competition, e.g., municipal infrastructures [25]. However, if they
are promising, we also expect that venture capital would provide the funding
source for them to subsidize their users and achieve their potential high
profitability. After all, we believe in a transparent and competitive market
where users can choose CPs based on quality and prices, and the businesses
drive the evolution of the Internet ecosystem.

Although our analyses imply that a deregulation of subsidization will provide
CPs more freedom to subsidize and increase the competitiveness of the content
market, regulators might still want to impose policies that prevent
discriminatory subsidization of the access ISPs. For example, Comcast’s Xbox
XFinity [2] service enables its users to access on-demand content on their
Xbox devices, which does not count against their $250$GB data cap. By
implicitly subsidizing their users, this service favors the ISP’s vertically
integrated business over other content competitors, e.g., Netflix. Some CPs
work with access ISPs to provide better services and cheaper prices for their
users, e.g., Google Apps for Verizon [4]; however, this might also inversely
affect the competitors of those favored CPs. We believe that the subsidization
option should be identical, and should be given to all CPs equally. Through
this, the access ISPs would treat CPs more neutrally and the subsidization
would create a transparent and uniform platform for the CPs to compete.

Our analytical framework is based on a macroscopic equilibrium model that
captures the interactions among an access ISP, a set of CPs and end-users via
the CPs’ subsidization and the ISP’s capacity and pricing decisions. Our
theoretical results are mostly qualitative, which provides understanding and
predictions of the changes in the system states when some of the driving
factors change. The limitation is that it might not be able to capture short-
term off-equilibrium types of system dynamics, where players’ decisions are
not rational or optimal. Our numerical evaluations are limited to a styled
model of CPs to capture qualitative trends; however, more detailed experiments
and validations could be challenging, because market data are needed so as to
obtain the characteristics of the CPs, e.g., their profitability and
elasticities, and an ISP needs to execute such a plan in the market. With the
emerging sponsored data plan from AT&T, we expect this type of market data
could be available for regulatory authorities as well as research communities.
This study focuses on a single access ISP; however, we believe that
competition between ISPs will also incentivize them to adopt subsidization
schemes, through which users can obtain subsidized services. Finally, our
result shows that under the subsidization mechanism, the access ISP could
increase utilization and revenue, which will further provide investment
incentives for them to expand capacity. We do not have sufficient space to
study the ISP’s capacity planning decision in detail, which is a direction of
our future work.

##  7 Conclusions

The Internet is facing a dilemma of increasing traffic demand and decreasing
incentives for the access ISPs to expand capacity, because service
prioritization and two-sided pricing are not allowed under the net neutrality
principles. We propose to allow CPs to voluntarily subsidize the data usage
costs incurred at the access ISPs for their end-users. Our solution
modularizes the issue along the tussle boundary such that the physical network
is still kept neutral while the innovative competition among CPs is enabled at
higher service layers. Through our macroscopic modeling and analyses, we show
that a deregulated subsidization policy will incentivize highly profitable CPs
to subsidize users and increase the system welfare and the revenue of the
access ISPs. With the improved profit margins, the access ISPs would obtain
more investment incentives under subsidization. However, the deregulation of
subsidization might trigger an ISP to increase price, which would decrease the
system welfare as well as the throughput of startup CPs. We suggest that price
regulation might be needed if the access market is not competitive enough and
the price is too high. Because subsidization creates a feedback channel for
CPs to return value to the access ISPs and realign the created value and
investment in the value chain, we believe that subsidization competition could
vitalize neutral Internet for the future.

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##  Appendix A Proofs

Proof of Lemma 1: By Assumption 1, we know $\Theta(\phi,\mu)$ is strictly
increasing in $\phi$ and $\lambda_{i}(\phi)$ is strictly decreasing in $\phi$;
therefore, $g(\phi)=\Theta(\phi,\mu)-\sum_{i\in\cal N}m_{i}\lambda_{i}(\phi)$
is strictly increasing in $\phi$. Because $\lim_{\phi\rightarrow
0}\Theta(\phi,\mu)=0$ and $\lambda_{i}(0)>0$ for all $i\in\cal N$,
$lim_{\phi\rightarrow 0}g(\phi)<0$; because
$\lim_{\phi\rightarrow\infty}\lambda_{i}(\phi)=0$ for all $i\in\cal N$,
$lim_{\phi\rightarrow\infty}g(\phi)=\lim_{\phi\rightarrow\infty}\Theta(\phi,\mu)>0$.
As a result, $g(\phi)=0$ must have a unique solution.

Finally, when $g(\phi)=0$, $\Theta(\phi,\mu)=\sum_{i\in\cal
N}m_{i}\lambda_{i}(\phi)$, which is equivalent to Equation (1) of Definition
1.

Proof of Lemma 2: Let $\kappa=m_{j}/m_{i}$. Given CP $i$’s characterization
$\lambda_{i}(\cdot)$, we verify that
$\lambda_{j}(\phi)=\kappa^{-1}\lambda_{i}(\phi)$ for all $\phi>0$. This is
because when $\lambda_{j}(0)=\kappa^{-1}\lambda_{i}(0)$, the above implies
$\epsilon_{\phi}^{\lambda_{j}}(\phi)=\left(\frac{d\lambda_{j}}{d\phi}\right)\left(\frac{\phi}{\lambda_{j}}\right)=\left(\frac{1}{\kappa}\frac{d\lambda_{i}}{d\phi}\right)\left(\frac{\phi}{\kappa^{-1}\lambda_{i}}\right)=\epsilon_{\phi}^{\lambda_{i}}(\phi)$
for all $\phi>0$. Thus, $\theta_{j}(\phi)=m_{j}\lambda_{j}(\phi)=\kappa
m_{i}\kappa^{-1}\lambda_{i}(\phi)=m_{i}\lambda_{i}(\phi)=\theta_{i}(\phi)$ for
all $\phi>0$. Finally, because given $\phi$, the throughput of CP $i$ and $j$
are the same, the system will induce the same utilization $\phi$, and
therefore, $\phi$ is also the unique system utilization when CP $i$ is
replaced by CP $j$.

Proof of Theorem 1: Because $g(\phi)$ is strictly increasing in $\phi$ by
Lemma 1 and by Equation (2), we have

$\frac{dg}{d\phi}=\frac{\partial\Theta}{\partial\phi}-\sum_{k\in\cal
N}m_{k}\frac{d\lambda_{k}}{d\phi}>0.$

When considering the capacity effect, we write the system utilization
$\phi(\mu)$ as a function of the system capacity. By Lemma 1,
$g(\phi(\mu))=\Theta(\phi(\mu),\mu)-\sum_{k\in\cal
N}m_{k}\lambda_{k}(\phi(\mu))=0$. By taking derivative of $\mu$ on both sides,
we obtain

$\frac{\partial\Theta}{\partial\phi}\frac{\partial\phi}{\partial\mu}+\frac{\partial\Theta}{\partial\mu}-\sum_{k\in\cal
N}m_{k}\frac{d\lambda_{k}}{d\phi}\frac{\partial\phi}{\partial\mu}=0,$

which is equivalent to

$\left(\frac{\partial\Theta}{\partial\phi}-\sum_{k\in\cal
N}m_{k}\frac{d\lambda_{k}}{d\phi}\right)\frac{\partial\phi}{\partial\mu}=-\frac{\partial\Theta}{\partial\mu}\quad\text{or}\quad\frac{dg}{d\phi}\frac{\partial\phi}{\partial\mu}=-\frac{\partial\Theta}{\partial\mu}.$

Since $\Theta(\phi,\mu)$ is increasing in $\mu$,
$\partial\Theta/\partial\mu>0$, which implies

$\frac{\partial\phi}{\partial\mu}=-\left(\frac{dg}{d\phi}\right)^{-1}\frac{\partial\Theta}{\partial\mu}<0.$

When considering the user effect, we write the system utilization $\phi({\bf
m})$ as a function of the user populations. By Lemma 1, $g(\phi({\bf
m}))=\Theta(\phi({\bf m}),\mu)-\sum_{k\in\cal N}m_{k}\lambda_{k}(\phi({\bf
m}))=0$. By taking derivative of $m_{i}$ on both sides, we obtain

$\frac{\partial\Theta}{\partial\phi}\frac{\partial\phi}{\partial
m_{i}}-\sum_{k\in\cal
N}m_{k}\frac{d\lambda_{k}}{d\phi}\frac{\partial\phi}{\partial
m_{i}}-\lambda_{i}=0,$

which is equivalent to

$\left(\frac{\partial\Theta}{\partial\phi}-\sum_{k\in\cal
N}m_{k}\frac{d\lambda_{k}}{d\phi}\right)\frac{\partial\phi}{\partial
m_{i}}=\lambda_{i}\quad\text{or}\quad\frac{dg}{d\phi}\frac{\partial\phi}{\partial
m_{i}}=\lambda_{i}.$

Since $\frac{dg}{d\phi}>0$, the above implies $\frac{\partial\phi}{\partial
m_{i}}=\left(\frac{dg}{d\phi}\right)^{-1}\lambda_{i}>0$. Finally, the results
for $\partial\theta_{i}/\partial\mu$, $\partial\theta_{i}/\partial m_{i}$ and
$\partial\theta_{j}/\partial m_{i}$ can be derived by using
$\theta_{i}(\phi)=m_{i}\lambda_{i}(\phi)$ and chain rules, and applying the
above results for $\partial\phi/\partial\mu$ and $\partial\phi/\partial
m_{i}$.

Proof of Theorem 7: When considering the price effect, we write the system
utilization $\phi(p)=\phi({\bf m}(p))$ as a function of the price. By Lemma 1,
$g(\phi(p))=\Theta(\phi(p),\mu)-\sum_{k\in\cal
N}m_{k}(p)\lambda_{k}(\phi(p))=0$. By taking derivative of $p$ on both sides,
we obtain

$\frac{\partial\Theta}{\partial\phi}\frac{\partial\phi}{\partial
p}-\sum_{k\in\cal
N}\left(\frac{dm_{k}}{dp}\lambda_{k}+m_{k}\frac{d\lambda_{k}}{d\phi}\frac{\partial\phi}{\partial
p}\right)=0,$

which is equivalent to

$\left(\frac{\partial\Theta}{\partial\phi}-\sum_{k\in\cal
N}m_{k}\frac{d\lambda_{k}}{d\phi}\right)\frac{\partial\phi}{\partial
p}=\sum_{k\in\cal N}\frac{dm_{k}}{dp}\lambda_{k}.$

By Assumption 2, $m_{k}(p)$ is decreasing in $p$, which implies

$\frac{\partial\phi}{\partial
p}=\left(\frac{dg}{d\phi}\right)^{-1}\sum_{k\in\cal
N}\frac{dm_{k}}{dp}\lambda_{k}\leq 0.$

By taking derivatives of $p$ on $\theta_{i}(p)=m_{i}(p)\lambda_{i}(\phi(p))$,

$\frac{d\theta_{i}}{dp}>0\Leftrightarrow{\frac{dm_{i}}{dp}}{{\lambda_{i}}}+m_{i}\frac{d\lambda_{i}}{d\phi}\frac{\partial\phi}{\partial
p}>0,$

which is equivalent to

$\frac{1}{m_{i}}{\frac{dm_{i}}{dp}}>-\frac{1}{{\lambda_{i}}}\frac{d\lambda_{i}}{d\phi}\frac{\partial\phi}{\partial
p}\quad\text{or}\quad{\epsilon_{p}^{m_{i}}}/{\epsilon_{\phi}^{\lambda_{i}}}<-\epsilon_{p}^{\phi}.$

Finally,

$\frac{d\theta}{dp}=\sum_{i\in\cal N}\frac{d\theta_{i}}{dp}=\sum_{i\in\cal
N}{\frac{dm_{i}}{dp}}{{\lambda_{i}}}+m_{i}\frac{d\lambda_{i}}{d\phi}\frac{\partial\phi}{\partial
p}.$

By substituting $\partial\phi/\partial p$ of (5) into the above, we obtain

$\frac{d\theta}{dp}=\sum_{i\in\cal
N}{\frac{dm_{i}}{dp}}{{\lambda_{i}}}\left(1+\left(\frac{dg}{d\phi}\right)^{-1}\sum_{k\in\cal
N}{\frac{d\lambda_{k}}{d\phi}}{{m_{k}}}\right).$

By substituting $dg/d\phi$ of (2) into the above, we obtain (6).

Proof of Lemma 3: Because $s^{\prime}_{i}>s_{i}$,
$t^{\prime}_{i}=p-s^{\prime}_{i}<p-s_{i}=t_{i}$ and by Assumption 2,
$m^{\prime}_{i}=m_{i}(t^{\prime}_{i})\geq m_{i}(t_{i})=m_{i}$.

By Theorem 1, we know that $\partial\phi/\partial m_{i}>0$, and therefore,
$\phi(\mathbf{s}^{\prime})\geq\phi(\mathbf{s})$. Also by Theorem 1, we know
that $\partial\theta_{i}/\partial m_{i}>0$ and $\partial\theta_{j}/\partial
m_{i}<0$ for all $j\neq i$. As a result,
$\theta_{i}(\mathbf{s}^{\prime})\geq\theta_{i}(\mathbf{s})$ and
$\theta_{j}(\mathbf{s}^{\prime})\leq\theta_{j}(\mathbf{s})$ for all $j\neq i$,
which implies
$U_{j}(\mathbf{s}^{\prime})=v_{j}\theta_{j}(\mathbf{s}^{\prime})\leq
v_{j}\theta_{j}(\mathbf{s})=U_{j}(\mathbf{s})$ for all $j\neq i$.

Proof of Theorem 3: If $\mathbf{s}$ is a Nash equilibrium, each
$s_{i}\in[0,q]$ maximizes
$U_{i}(s_{i};\mathbf{s}_{-i})=(v_{i}-s_{i})m_{i}(p-s_{i})\lambda_{i}\left(\phi\left(\mathbf{s}\right)\right)$.
By Karush-Kuhn-Tucker condition, we can derive that

$\frac{\partial U_{i}}{\partial s_{i}}\left\\{\begin{array}[]{cl}\leq
0&\text{if }\ s_{i}=0;\\\ \geq 0&\text{if }\ s_{i}=q;\\\ =0&\text{if }\
0<s_{i}<q;\end{array}\right.\quad\forall\ i\in\cal N.$ (18)

For CP $i$ with $s_{i}=0$, the above condition tells that

$\frac{\partial U_{i}}{\partial
s_{i}}=(v_{i}-s_{i})\frac{\partial\theta_{i}}{\partial
s_{i}}-\theta_{i}=v_{i}\frac{\partial\theta_{i}}{\partial
s_{i}}-\theta_{i}\leq 0.$

By Lemma 3, we know ${\partial\theta_{i}}/{\partial s_{i}}>0$ and the above
implies $v_{i}\leq\left({\partial\theta_{i}}/{\partial
s_{i}}\right)^{-1}\theta_{i}$. By taking partial derivatives of $s_{i}$ on
$U_{i}=(v_{i}-s_{i})m_{i}\lambda_{i}$, we deduce that

$\frac{\partial U_{i}}{\partial
s_{i}}=(v_{i}-s_{i})\left(m_{i}\frac{\partial\lambda_{i}}{\partial
s_{i}}+\lambda_{i}\frac{\partial m_{i}}{\partial
s_{i}}\right)-m_{i}\lambda_{i}.$

Dividing $m_{i}\lambda_{i}$ on both sides of (18), we have

$(v_{i}-s_{i})\left(\frac{1}{\lambda_{i}}\frac{\partial\lambda_{i}}{\partial
s_{i}}+\frac{1}{m_{i}}\frac{\partial m_{i}}{\partial
s_{i}}\right)\left\\{\begin{array}[]{cl}\leq 1&\text{if }\ s_{i}=0;\\\ \geq
1&\text{if }\ s_{i}=q;\\\ =1&\text{if }\ 0<s_{i}<q;\end{array}\right.$

When we multiply $s_{i}$ on both sides of the above, the left hand side
becomes

$(v_{i}-s_{i})\left(\epsilon_{s_{i}}^{m_{i}}+\epsilon_{s_{i}}^{\lambda_{i}}\right)=(v_{i}-s_{i})\left(\epsilon_{s_{i}}^{m_{i}}+\epsilon_{\phi}^{\lambda_{i}}\epsilon_{m_{i}}^{\phi}\epsilon_{s_{i}}^{m_{i}}\right)=\tau_{i}(\bf
s).$

When $s_{i}=0$, $\tau_{i}({\bf s})=0=s_{i}$, and therefore, we have

$\tau_{i}({\bf s})\left\\{\begin{array}[]{cl}\geq s_{i}&\text{if }\
s_{i}=q;\\\ =s_{i}&\text{if }\ 0\leq
s_{i}<q,\end{array}\right.\quad{\text{o}r}\quad
s_{i}=\min\\{\tau_{i}(\mathbf{s}),q\\}.$

Finally, if $U_{i}(\mathbf{s})$ is concave in $s_{i}$ for all $i\in\cal N$,
local optimality also guarantees the global optimality and therefore, the
above first order necessary conditions become sufficient condition for
$\mathbf{s}$ being a Nash equilibrium.

Proof of Theorem 4: The condition (10) implies that
$U_{i}(s_{i};\mathbf{s}_{-i})$ is concave in $s_{i}$ for any
$\mathbf{s}_{-i}$. Because the strategy space $[0,q]$ is compact, it
guarantees the existence of Nash equilibrium. Suppose there exists two
distinct Nash equilibria $\mathbf{\hat{s}}$ and $\mathbf{\tilde{s}}$. By
concavity of $U_{i}(s_{i};\mathbf{s}_{-i})$ in $s_{i}$ and the maximum
principle, for any $i\in\cal N$ and any $x_{i}\in[0,q]$,

$(x_{i}-\hat{s}_{i})u_{i}(\mathbf{\hat{s}})\leq
0,\quad\text{and}\quad(x_{i}-\tilde{s}_{i})u_{i}(\mathbf{\tilde{s}})\leq 0.$

By substituting $\mathbf{x}=\mathbf{\tilde{s}}$ in the first inequality and
$\mathbf{x}=\mathbf{\hat{s}}$ in the second inequality, we deduce for any
$i\in\cal N$,

$(\tilde{s}_{i}-\hat{s}_{i})u_{i}(\mathbf{\hat{s}})\leq
0,\quad\text{and}\quad(\hat{s}_{i}-\tilde{s}_{i})u_{i}(\mathbf{\tilde{s}})\leq
0.$

By adding the above inequalities, we further deduce that

$(\tilde{s}_{i}-\hat{s}_{i})(u_{i}(\mathbf{\hat{s}})-u_{i}(\mathbf{\tilde{s}}))\leq
0,\quad\forall i\in\cal N.$

The above is equivalent to
$(\tilde{s}_{i}-\hat{s}_{i})(u_{i}(\mathbf{\tilde{s}})-u_{i}(\mathbf{\hat{s}}))\geq
0$ for all $i\in\cal N$, which contradicts the condition (10).

Proof of Theorem 5: For any $j\in{\cal N}\backslash\\{i\\}$, we show that

$\left(\hat{s}_{j}-s_{j}\right)\left(u_{j}(\mathbf{\hat{s}})-u_{j}(\mathbf{s})\right)\geq
0.$

Because the above is symmetric in $\mathbf{s}$ and $\mathbf{\hat{s}}$, without
loss of generality, we can consider the cases of $s_{j}\leq\hat{s}_{j}$. By
using the KKT condition of (18), we have the following four cases.

  1. 1.

$s_{j}=\hat{s}_{j}$:
$\left(\hat{s}_{j}-s_{j}\right)\left(u_{j}(\mathbf{\hat{s}})-u_{j}(\mathbf{s})\right)=0$.

  2. 2.

$s_{j}=0$ and $\hat{s}_{j}\in(0,q)$: $u_{j}(\mathbf{s})<0$ and
$u_{j}(\mathbf{\hat{s}})=0$; and therefore, the above becomes
$\hat{s}_{j}\left(-u_{j}(\mathbf{s})\right)\geq 0$.

  3. 3.

$s_{j}=0$ and $\hat{s}_{j}=q$: $u_{j}(\mathbf{s})<0$ and
$u_{j}(\mathbf{\hat{s}})>0$; and therefore, the above becomes
$q\left(u_{j}(\mathbf{\hat{s}})-u_{j}(\mathbf{s})\right)\geq 0$.

  4. 4.

$s_{j}\in(0,q)$ and $\hat{s}_{j}=q$: $u_{j}(\mathbf{s})=0$ and
$u_{j}(\mathbf{\hat{s}})>0$; and therefore, the above becomes
$\hat{(}q-s_{j})u_{j}(\mathbf{\hat{s}})\geq 0$.

By the condition (10), we deduce
$\left(\hat{s}_{i}-s_{i}\right)\left(u_{i}(\mathbf{\hat{s}})-u_{i}(\mathbf{s})\right)<0$.
Suppose $\hat{s}_{i}<s_{i}$, the above implies that
$u_{i}(\mathbf{\hat{s}})>u_{i}(\mathbf{s})$. However, if $\hat{s}_{i}<s_{i}$,
we must have $s_{i}>0$ and by the KKT condition of (18), we have
$u_{i}(\mathbf{s})\geq 0$, which further implies that
$u_{i}(\mathbf{\hat{s}})>u_{i}(\mathbf{s})\geq 0$. Again by the KKT condition
of (18), $u_{i}(\mathbf{\hat{s}})>0$ implies $\hat{s}=q$ which contradicts
with the assumption $\hat{s}_{i}<s_{i}$. Consequently, we must have
$\hat{s}_{i}\geq s_{i}$.

Proof of Theorem 6: The condition (10) of Theorem 4 implies the local
concavity of the utility functions. By Proposition 1.4.2 of [14], the Nash
equilibrium $\mathbf{s}$ can be equivalently characterized as the solution of
a variational inequality, denoted as $VI(F,K)$, where $F\triangleq-\mathbf{u}$
and $K\triangleq[0,q]^{|\cal N|}$. $\mathbf{s}\in\cal K$ is a solution of
$VI(F,K)$ if $(\mathbf{x}-\mathbf{s})^{T}F(\mathbf{s})\geq 0$ for all
$\mathbf{x}\in K$. We apply the sensitivity analysis [43, 12] of variational
inequalities to obtain the dynamics of the Nash equilibrium $\mathbf{s}(p,q)$
as the price $p$ or the policy $q$ changes.

For each CP $i$, the constraint $s_{i}\in[0,q]$ can be written as two linear
constraints $g^{-}_{i}(\mathbf{s})\triangleq s_{i}\geq 0$ and
$g^{+}_{i}(\mathbf{s})\triangleq q-s_{i}\geq 0$. For any $\mathbf{s}$, the set
of binding constraints are

${\cal G}=\\{g^{-}_{i}(\mathbf{s}):i\in{\cal
N_{-}}\\}\cup\\{g^{+}_{i}(\mathbf{s}):i\in{\cal N_{+}}\\}.$ (19)

Because ${\cal N_{-}}\cap{\cal N_{+}}=\emptyset$, the gradients of the binding
constraints are linearly independent. Because $u_{i}(\mathbf{s})=0$ implies
$s_{i}\in\widetilde{\cal N}$ for all $i\in\cal N$, by the KKT condition of
(18), we know that the strict complementary slackness condition of Theorem 3.1
of [43] holds. By Theorem 4, the Nash equilibrium is locally unique, and
therefore, by Theorem 3.1 of [43], in a neighborhood of $(p,q)$, the Nash
equilibrium can be written as a differentiable function $\mathbf{s}(p,q)$.

Let $m\triangleq|\cal N_{-}|+|\cal N_{+}|$ be the number of binding
constraints. We define $G$ as a $m\times n$ matrix, whose $i$th row is the
gradient of the $i$th binding constraint with respect to $\mathbf{s}$. We
define $E_{p}$ and $E_{q}$ as $m\times 1$ vectors, whose $i$th component are
the partial derivatives of the $i$th binding constraint with respect to $p$
and $q$, respectively. Following the framework of [12], we define
$Q=G^{T}(GG^{T})^{-1}G$ and $M$ to be an $n\times m$ matrix satisfying $MG=Q$
and $QM=M$. Given the set $\cal G$ of linear constraints in (19), we can
deduce that $Q=\\{q_{ij}\\}$ is a diagonal matrix satisfying $q_{ii}=1$ if
$i\notin\widetilde{\cal N}$ and $q_{ii}=0$ otherwise, and $M=G^{T}$.

When $q$ is the sensitivity parameter, $E_{q}=\\{e_{j}\\}$ satisfies that
$e_{j}=1$ if the $j$th constraint is a type of $g^{+}_{i}(\mathbf{s})=q-s_{i}$
constraint and $e_{j}=0$ otherwise. Because the constraints $\cal G$ are
linear, Theorem 3.1 of [12] implies that $Q\nabla_{q}\mathbf{s}(q)=-ME_{q}$
and
$(I-Q)(\nabla_{\mathbf{s}}F\nabla_{q}\mathbf{s}(q)+\nabla_{q}F)=\mathbf{0}$.
Because when multiplying $\nabla_{q}\mathbf{s}(q)$ by $Q$, the values
associated with the non-binding strategies vanish, the first equation implies

$\partial s_{i}/\partial q=0,\ \ \forall i\in{\cal N}_{-}\ \ \text{and}\ \
\partial s_{i}/\partial q=1,\ \ \forall i\in{\cal N}_{+}.$ (20)

Because $F\triangleq-\mathbf{u}$ and $\mathbf{u}$ does not depend on $q$, the
second equation can be written as
$(I-Q)\nabla_{\mathbf{s}}\mathbf{u}\nabla_{q}\mathbf{s}=\mathbf{0}$. Because
when multiplied by $I-Q$, the components associated with the binding
strategies vanish, by substituting (20) into the above, we obtain
$\nabla_{\mathbf{\tilde{s}}}\mathbf{\tilde{u}}\nabla_{q}\mathbf{\tilde{s}}+\nabla_{\mathbf{s}}\mathbf{\tilde{u}}E_{q}=\mathbf{0}$.
By the condition (10) of Theorem 4, we deduce that $F\triangleq-\mathbf{u}$ is
a $P$-function [27]. If we restrict to the CPs in $\widetilde{\cal N}$, we
know that the Jacobian $\nabla_{\mathbf{\tilde{s}}}(\mathbf{-\tilde{u}})$ is a
$P$-matrix [27], which is always non-singular. Therefore, we deduce
$\nabla_{q}\mathbf{\tilde{s}}=-(\nabla_{\mathbf{\tilde{s}}}\mathbf{\tilde{u}})^{-1}\nabla_{\mathbf{s}}\mathbf{\tilde{u}}E_{q}=-\Psi\nabla_{\mathbf{s}}\mathbf{\tilde{u}}E_{q}$,
the same as the third case of (11).

When taking $p$ as the sensitivity parameter and applying Theorem 3.1 of [12],
we can obtain $Q\nabla_{p}\mathbf{s}(p)=-ME_{p}$ and
$(I-Q)(\nabla_{\mathbf{s}}F\nabla_{p}\mathbf{s}(p)+\nabla_{p}F)=\mathbf{0}$.
Because the constraints in $\cal G$ do not depend on $p$, $E_{p}$ is a zero
vector, therefore, the first equation implies that $\partial s_{i}/\partial
p=0$ for all $i\notin\widetilde{\cal N}$. By substituting
$F\triangleq-\mathbf{u}$ into the second equation, we have
$(I-Q)(\nabla_{\mathbf{s}}\mathbf{u}\nabla_{p}\mathbf{s}+\nabla_{p}\mathbf{u})=\mathbf{0}$.
Similarly, by substituting $\partial s_{i}/\partial p=0$ for all
$i\notin\widetilde{\cal N}$ into the above, we obtain
$\nabla_{\mathbf{\tilde{s}}}\mathbf{\tilde{u}}\nabla_{p}\mathbf{\tilde{s}}+\nabla_{p}\mathbf{\tilde{u}}=\mathbf{0}$.
Because $\nabla_{\mathbf{\tilde{s}}}(\mathbf{-\tilde{u}})$ is non-singular, it
is equivalent to the second case of (12) in a matrix form as

$\nabla_{p}\mathbf{\tilde{s}}=-(\nabla_{\mathbf{\tilde{s}}}\mathbf{\tilde{u}})^{-1}\nabla_{p}\mathbf{\tilde{u}}=-\Psi\nabla_{p}\mathbf{\tilde{u}}.$

Proof of Corollary 1: By (11) of Theorem 6, $\partial s_{i}/\partial q\geq 0$
is immediate for $i\notin\widetilde{\cal N}$. For the CPs in $\widetilde{\cal
N}$, by Theorem 6, we have
$\nabla_{q}\mathbf{\tilde{s}}=-(\nabla_{\mathbf{\tilde{s}}}\mathbf{\tilde{u}})^{-1}\nabla_{\mathbf{s}}\mathbf{\tilde{u}}E_{q}=-\Psi\nabla_{\mathbf{s}}\mathbf{\tilde{u}}E_{q}$.
Because $\mathbf{u}$ is off-diagonally monotone, we know that
$\nabla_{\mathbf{\tilde{s}}}(-\mathbf{\tilde{u}})$ is a $P$-matrix with non-
positive off-diagonal entries or an $M$-matrix [35], which implies that all
the entries of $-\Psi$ are nonnegative. By Theorem 6, we know that
$\nabla_{\mathbf{s}}\mathbf{\tilde{u}}E_{q}$ is a vector of $\\{\partial
u_{i}/\partial s_{j}:i\in{\widetilde{\cal N}},j\in{\cal N}_{+}\\}$. Because
$\mathbf{u}$ is off-diagonally monotone, the components of
$\nabla_{\mathbf{s}}\mathbf{\tilde{u}}E_{q}$ are all non-negative, and
therefore,
$\nabla_{q}\mathbf{\tilde{s}}=-\Psi\nabla_{\mathbf{s}}\mathbf{\tilde{u}}E_{q}\geq\mathbf{0}$.

Now we have shown that $\partial s_{i}/\partial q\geq 0$ for all $i\in{\cal
N}$. Since all the subsidies are monotonic, by Lemma 3, we deduce that
$\partial\phi/\partial q\geq 0$. Finally, because the ISP’s revenue can be
written as $R=p\Theta(\phi,\mu)$ and
$\partial\Theta(\phi,\mu)/\partial\phi\geq 0$, we have

$\frac{\partial R}{\partial q}=\frac{\partial
R}{\partial\phi}\frac{\partial\phi}{\partial
q}=p\frac{\partial\Theta(\phi,\mu)}{\partial\phi}\frac{\partial\phi}{\partial
q}\geq 0.$

Proof of Theorem 7: By Equation (4) and (5), we deduce

$\frac{\partial\phi}{\partial p}=\frac{\partial\phi}{\partial
m_{i}}\frac{1}{\lambda_{i}}\sum_{j\in\cal N}\frac{\partial m_{j}}{\partial
p}\lambda_{j},\quad\forall i\in\cal N.$

By substituting the above into
$\displaystyle\frac{\partial\theta_{i}}{\partial p}=\frac{\partial
m_{i}}{\partial
p}\lambda_{i}+m_{i}\frac{d\lambda_{i}}{dp}\frac{\partial\phi}{\partial p}$, we
obtain $\displaystyle\frac{\partial\theta_{i}}{\partial p}=\frac{\partial
m_{i}}{\partial
p}\lambda_{i}+m_{i}\frac{d\lambda_{i}}{dp}\frac{\partial\phi}{\partial
m_{i}}\frac{1}{\lambda_{i}}\sum_{j\in\cal N}\frac{\partial m_{j}}{\partial
p}\lambda_{j}$, or

$\displaystyle\epsilon_{p}^{\theta_{i}}\theta_{i}=$ $\displaystyle
p\frac{\partial\theta_{i}}{\partial p}=\frac{p}{m_{i}}\frac{\partial
m_{i}}{\partial
p}\theta_{i}+m_{i}\frac{d\lambda_{i}}{dp}\frac{\partial\phi}{\partial
m_{i}}\frac{1}{\lambda_{i}}\sum_{j\in\cal N}\frac{p}{m_{j}}\frac{\partial
m_{j}}{\partial p}\theta_{j}$ $\displaystyle=$
$\displaystyle\epsilon_{p}^{m_{i}}\theta_{i}+\epsilon_{m_{i}}^{\phi}\epsilon_{\phi}^{\lambda_{i}}\sum_{j\in\cal
N}\epsilon_{p}^{m_{j}}\theta_{j}$

Because $R(p)=p\sum_{i\in\cal N}\theta_{i}(p)$, the marginal revenue is

$\displaystyle\frac{dR}{dp}=$ $\displaystyle\sum_{i\in\cal
N}\theta_{i}+p\sum_{i\in\cal N}\frac{\partial\theta_{i}}{\partial
p}=\sum_{i\in\cal N}\theta_{i}+\sum_{i\in\cal
N}\epsilon_{p}^{\theta_{i}}\theta_{i}$ $\displaystyle=$
$\displaystyle\sum_{i\in\cal N}\theta_{i}+\sum_{i\in\cal
N}\epsilon_{p}^{m_{i}}\theta_{i}+\epsilon_{m_{i}}^{\phi}\epsilon_{\phi}^{\lambda_{i}}\sum_{j\in\cal
N}\epsilon_{p}^{m_{j}}\theta_{j}$ $\displaystyle=$
$\displaystyle\sum_{i\in\cal N}\theta_{i}+\sum_{i\in\cal
N}\epsilon_{p}^{m_{i}}\theta_{i}+\sum_{i\in\cal
N}\epsilon_{m_{i}}^{\phi}\epsilon_{\phi}^{\lambda_{i}}\sum_{j\in\cal
N}\epsilon_{p}^{m_{j}}\theta_{j}$ $\displaystyle=$
$\displaystyle\sum_{i\in\cal N}\theta_{i}+\Upsilon\sum_{i\in\cal
N}\epsilon_{p}^{m_{i}}\theta_{i}.$

Proof of Theorem 17: By Theorem 6, the Nash equilibrium is locally unique,
i.e., $s_{i}(q)\triangleq s_{i}\left(p(q),q\right)$, we deduce

$\frac{ds_{i}}{dq}=\frac{\partial s_{i}}{\partial q}+\frac{\partial
s_{i}}{\partial p}\frac{dp}{dq}.$ (21)

Thus, we can obtain (15) by substituting the above into

$\frac{dm_{i}}{dq}=\frac{dm_{i}}{dt_{i}}\frac{dt_{i}}{dq}=\frac{dm_{i}}{dt_{i}}\left(\frac{dp}{dq}-\frac{ds_{i}}{dq}\right).$

Similarly, we can obtain (16) by substituting (4) into

$\frac{\partial\phi}{\partial q}=\sum_{i\in\cal N}\frac{\partial\phi}{\partial
m_{i}}\frac{dm_{i}}{dq}.$

Finally, $\displaystyle\frac{\partial\theta_{i}}{\partial q}>0$ can be written
as
$\displaystyle{\frac{dm_{i}}{dq}}{{\lambda_{i}}}+m_{i}\frac{d\lambda_{i}}{d\phi}\frac{d\phi}{dq}>0$.
By Assumption 1, $\epsilon_{\phi}^{\lambda_{i}}<0$ and because
$\epsilon_{q}^{\theta_{i}}=\epsilon_{q}^{m_{i}}+\epsilon_{q}^{\lambda_{i}}=\epsilon_{q}^{t_{i}}\epsilon_{t_{i}}^{m_{i}}+\epsilon_{q}^{\phi}\epsilon_{\phi}^{\lambda_{i}}$
and ${\partial\theta_{i}}/{\partial q}>0$ is equivalent to
$\epsilon_{q}^{\theta_{i}}>0$,

$\epsilon_{q}^{\theta_{i}}>0\Leftrightarrow\epsilon_{q}^{t_{i}}\epsilon_{t_{i}}^{m_{i}}>-\epsilon_{q}^{\phi}\epsilon_{\phi}^{\lambda_{i}}\Leftrightarrow\epsilon_{q}^{t_{i}}\epsilon_{t_{i}}^{m_{i}}/\epsilon_{\phi}^{\lambda_{i}}<-\epsilon_{q}^{\phi}.$

Proof of Corollary 2: Because $W(q)\triangleq\sum_{i\in\cal
N}\theta_{i}(q)v_{i}$,

$\frac{dW}{dq}=\sum_{i\in\cal
N}m_{i}\frac{d\lambda_{i}}{dq}v_{i}+\sum_{i\in\cal
N}\lambda_{i}\frac{dm_{i}}{dq}v_{i}.$

By substituting $w_{i}\triangleq\lambda_{i}\frac{dm_{i}}{dq}$ and (16) into
the above,

$\displaystyle\frac{dW}{dq}=$ $\displaystyle\sum_{i\in\cal
N}m_{i}\frac{d\lambda_{i}}{d\phi}\frac{d\phi}{dq}v_{i}+\sum_{i\in\cal
N}w_{i}v_{i}$ $\displaystyle=$ $\displaystyle\sum_{i\in\cal
N}m_{i}\frac{d\lambda_{i}}{d\phi}\left(\frac{dg}{d\phi}\right)^{-1}\left(\sum_{j\in\cal
N}w_{j}\right)v_{i}+\sum_{i\in\cal N}w_{i}v_{i}.$

Because $d\phi/dq>0$, by (16) we know that $\sum_{j\in\cal N}w_{j}>0$.
Therefore, $dW/dq>0$ is equivalent to

$\sum_{i\in\cal
N}m_{i}\frac{d\lambda_{i}}{d\phi}\left(\frac{dg}{d\phi}\right)^{-1}v_{i}+\sum_{i\in\cal
N}\frac{w_{i}}{\sum_{j\in\cal N}w_{j}}v_{i}>0.$

By substituting
$\left({dg}/{d\phi}\right)^{-1}=\lambda_{i}^{-1}\frac{\partial\phi}{\partial
m_{i}}$ into the above,

$\sum_{i\in\cal
N}\frac{m_{i}}{\lambda_{i}}\frac{d\lambda_{i}}{d\phi}\frac{\partial\phi}{\partial
m_{i}}v_{i}+\sum_{i\in\cal N}\frac{w_{i}}{\sum_{j\in\cal N}w_{j}}v_{i}>0,$

which is equivalent to $\sum_{i\in\cal N}\frac{w_{i}}{\sum_{k\in\cal
N}w_{k}}v_{i}>\sum_{i\in\cal N}-\epsilon_{m_{i}}^{\lambda_{i}}v_{i}$.