# Profit Sharing Contracts between Content and Service Providers for Enhanced
Network Quality

Fehmina Malik, Manjesh K. Hanawal, and Yezekael Hayel Fehmina Malik and
Manjesh K. Hanawal are with IEOR, IIT Bombay, India. E-mail: {fehminam,
mhanawal<EMAIL_ADDRESS>Yezekael Hayel is with LIA/CERI, University of
Avignon, France. E-mail<EMAIL_ADDRESS>
###### Abstract

It has been a long demand of Internet Service Providers (ISPs) that the
Content Providers (CPs) share their profits for investments in network
infrastructure. In this paper, we study profit sharing contracts between a CP
with multiple ISPs. Each ISP commits to improving the Quality of Service (QoS)
for the end-users through higher investments efforts. The CP agrees to share
the profits due to the resulting higher demand for its content. We first model
non-cooperative interaction between the CP and the ISPs as a two-stage
Stackelberg game. CP is the leader that decides what fraction of its profits
will be shared with the ISPs. Each ISP then simultaneously decides the amount
of effort (investment) to enhance network quality. Here, CP cannot observe
individual effort by the ISPs, which poses a challenge for the CP to decide
how to share the profits with each ISP. Therefore, we also investigate a
cooperative scenario, where the CP only decides the total share it gives to
the ISPs, and each ISP then cooperatively shares the profit among themselves.
We study the effect of such cooperation between the ISPs by building a Nash
Bargaining based model. We show that the collaboration improves total effort
by the ISPs and the payoff of the CP.

###### Index Terms:

revenue sharing, net neutrality, coalitions, Nash Bargaining

##  I Introduction

In recent years, the growth of Internet traffic has been rapid, mainly due to
an increase in the usage of data-intensive services like online video
streaming, which accounts for over $65\%$ of the total downstream Internet
traffic, of which Youtube contributes for over $15\%$ and Netflix over $11\%$
[1]. Further, the COVID-19 pandemic led to the adoption of remote working,
e-teaching, online collaboration, gaming, video streaming, etc., and almost
$40\%$ growth in global traffic is seen within three months in early 2020 [2].
All these sudden changes have put unprecedented stress on the network, causing
congestion. Internet service providers (ISPs) have to upgrade the network
infrastructures to address the network performance issue in this fast-growing
Internet traffic environment. Besides increasing network bandwidth/capacity,
deployment of caching technologies is viewed as a critical solution for the
upgradation of networks to improve the Quality of Service (QoS) for end-users
[3].

The QoS perceived by users is the primary concern for content providers (CPs).
The high service quality directly increases users’ desire for content and
generates more demand for CPs, leading to higher revenue (mainly through
advertisements and subscriptions). As network congestion causes poor content
quality, resulting in decreased users’ willingness to browse the content thus,
CPs have incentives to pay for additional network investment to elevate the
QoS for end-users. With ISPs’ investment in the network, end users can enjoy a
better QoS, which increases the demand and attracts more traffic, resulting in
higher revenue of the CP. Thus, ISP’s investment in the network is beneficial
to the CP and may share its profit with ISP to cover the investment cost. This
increases network performance and, hence, covers the cost of the ISP’s network
investment. Such arrangements are becoming common between CPs and ISP, where
ISP helps CPs to increase its demand by deploying caching resources or through
premium peering ([4, 5]). For example, Netflix deploys a local cache of its
most popular content within the data centers of partner ISP’s network to
enhance video quality for end-users ([6, 7, 8]).

Profit-sharing contracts between CPs and ISPs may be mutually beneficial to
all and also benefit users. Through contracts, CPs may share their profits
with ISPs to invest in network infrastructure. With the improved networks,
users enjoy better quality over the Internet and increase demand for the
content. However, such private agreements may create disputes, raising policy
concerns related to net-neutrality regulations. In this paper, we investigate
the problem of profit-sharing between CP and multiple ISPs. In this case, the
CP can only observe the overall impact on traffic, which depends on
infrastructure enhancement (effort/action) by all ISPs, and cannot identify
and distinguish individual efforts levels by each ISP. This poses a challenge
for the CP to decide on profit sharing contract with each ISP. We also
consider the case of public ISP (owned by the government), which is non-
profitable. Further, we consider cooperative behavior where all players aim to
maximize social utility. Such cooperative setups are known to benefit all
players in [9, 10]). We consider a broad perspective on the behavior of the CP
and ISPs to provide insights into new ways of dealing with profit-sharing
between CP and ISPs.

In this paper, we consider different scenarios between ISPs. We first consider
the case involving non-profitable public ISP and private ISPs, which are part
of the marketplace. We then consider the case involving the private ISPs, but
they can act cooperatively or non-cooperatively. Our motivation to study these
different scenarios is to investigate which market framework is more
beneficial for end-users and with a better QoS. Our contributions and
observations are as follows:

  * •

We introduce Profit-sharing contracts between a CP with multiple ISPs. The CP
first offers a contract to each ISP, and ISPs accordingly determine their
optimal effort to improve demand for the CP’s content. This results in a
Stackelberg game with a single leader (CP) and multiple followers (ISPs).

  * •

We consider several market frameworks: (i) One public (non-profitable) and one
private ISP that maximizes its utility, and (ii) Private ISPs in competition
or cooperation.

  * •

We show that if there is no obligation on CP to sign a contract with Public
ISP, participation of public ISP has no effect. While if CP is obliged to sign
the profit contract with Public ISP, it will end up sharing more profit with
ISP while getting the same demand increment as in the case when CP is not
obliged to make a deal with the public ISP.

  * •

The equilibrium contracts is analyzed with private ISPs in non-cooperative and
cooperative scenarios. In the non-cooperative scenario, CP can offer a
customized profit share proportion for each ISP, and ISPs decide on their
efforts to maximize their profits. In the cooperation scenario, a mechanism is
proposed where CP makes a joint payment to ISPs where the ISPs bargain with
each other to make decisions on their individual efforts such that profit
share is in the proportion of their effort.

  * •

When the ISPs are symmetric with respect to per unit cost for investment, we
show both cooperative and non-cooperation scenarios results in the same level
of efforts and demand.

  * •

When the ISPs are asymmetric with respect to per unit cost for investment, the
cooperative mechanism imposed over ISPs alleviates the utility of all ISPs and
CP while also increasing the QoS for the content from CP.

This paper is organized as follows. In Section II we discuss the problem setup
and define the contracts. We study the equilibrium contracts for one public
and one private ISP case in Section III. We further studied competitive and
cooperative scenario symmetric ISP case in Section IV. The asymmetric ISP case
is studied in Section V and the comparison between public and private ISPs
(competition and cooperation) is discussed in Sections VI-VII. Conclusions are
discussed in Section VIII. Proofs of all stated results can be found in the
appendix.

###  I-A Related work:

There are a number of papers which explored the possibility of revenue sharing
or surplus transfer between service providers in the internet ([11, 12, 13,
14, 15, 16, 17, 18, 19, 20, 21, 9, 22, 23, 24, 25, 26, 27, 28, 29]). Content
charges by ISP to recover investment costs has been studied in [11, 15, 16,
17, 18, 19]. In [15] and [16], the authors investigated the feasibility of
ISPs charging the CPs, and evaluated its effect by modeling the Stackelberg
game between CPs and ISPs. In [17] and [18], a revenue-sharing scheme is
proposed when the ISP provides a content piracy monitoring service to CPs for
increasing the demand for their content. This work is extended to two ISPs
competing with each other in [19] where only one of them provides the content
piracy monitoring service. While some studies proposed cooperative settlement
between service providers through Shapley value [13, 14, 20]) or through
Bargaining concept ([12, 21, 9]). Some works ([22, 23, 24]) considered the
problem of peering agreements to extract surplus from CP to ISP, where CP pays
ISP to invest in peering link to improve QoS. [28] discussed the problem of
single CP with single non-strategic ISP, with both premium peering and
caching, while [29] extended it for multiple CPs. They considered ISP to be
non-strategic, and prices of resources are determined by market equilibrium.
Some works consider the payment for the deployment of CDNs for in-network
caching to enhance QoS ([25, 26, 27]).

Our previous work [10] considered the contract design problem between
monopolistic ISP and CPs, both in the presence and absence of neutrality
regulations. In the present paper, we focus on a scenario with multiple ISPs
and a single CP. The main problems of interest are:

  1. 1.

How is the profit share split between ISPs, as CP only observes the overall
impact on the demand due to sum investment by all ISPs?

  2. 2.

What cooperative mechanism between ISPs can help elevate the total investment
and hence the QoS for end-users without degrading ISPs’ utilities?

We consider a class of linear contracts between a CP and multiple ISPs that
leads to a tractable analysis. We study different types of interactions and
compare the impact of the contract on utilities of the CP and ISPs and QoS of
end-users under competitive and cooperative scenarios. We distinguish between
public and private ISPs and study how their participation benefits the end
users.

##  II Problem Description

Figure 1: Revenue flow between CPs, ISP and End-Users

We consider a single Content Provider (CP) whose content can be accessed by
multiple Internet Service Providers (ISPs). Let $n$ denote the number of ISPs
and $\mathcal{N}=\\{1,2,...,n\\}$ denote the set of ISPs. For each
$i\in\mathcal{N}$ we denote the i-th ISP as $ISP_{i}$. CP enters into an
profit sharing contract with the ISPs in which $ISP_{i},i\in\mathcal{N}$
chooses to make some networks investment quantified as
$a_{i}\in\mathcal{R}_{+};i\in\mathcal{N}$ to improve QoS for end-users. This
investment, for example, could be related to caching efforts or improving the
bandwidth of the network The resulting increase/gain in the revenue of CP over
a pre-specified horizon (say a billing cycle) is denoted by
$X\in\mathcal{R}_{+}$ and depends on the total effort of ISPs. In return, CP
shares part of its profits with ISPs to incentivize its investment; this share
for each $ISP_{i}$ is determined by a sharing function
$s_{i}:\mathcal{R}_{+}\rightarrow\mathcal{R}_{+}$, which is also referred to
as the contract/agreement. Specifically, the CP makes a payment of $s_{i}(X)$
to $ISP_{i}$ as part of this contract. Thus, the effective net revenue
increase of the CP is given by the difference
$\left(X-\sum_{i\in\mathcal{N}}s_{i}(X)\right)$. Figure 1 represents the
interactions between the agents in our model.

The increment in the demand for the content of CP denoted $D$ depends on the
total efforts $\sum_{i\in\mathcal{N}}a_{i}$’s made by all $ISP_{i}$,
$i\in\mathcal{N}$ jointly. Consistent with the law of diminishing returns, we
model the increment $D$ as an increasing concave function of $a_{i}$’s [30].
For analytical tractability, we assume that $D$ grows logarithmically in
$\sum_{i}a_{i}$’s and is given by $D:=\log(1+\sum_{i}a_{i})$. The resulting
profit (increment) for CP is assumed to be proportional to the demand
(increment) and is given by $X:=rD$, where $r$ is a constant that captures how
each additional unit of demand translates to earnings. Thus, the revenue
generated by CP is then $X=r\log(1+\sum_{i}a_{i})$. We restrict our analysis
to linear revenue sharing contracts between ISPs and the CP. Specifically,
these contracts are of the form $s_{i}(X)=\beta_{i}X$, where
$\beta_{i}\in[0,1]$ for all $i\in\mathcal{N}$. Thus the contracts are
paramterized by $\beta_{i},i\in\mathcal{N}$. The utility of the CP is denoted
as $U_{CP}$ and is given by

$U_{CP}=\left(1-\sum_{i\in\mathcal{N}}\beta_{i}\right)r\log\left(\sum_{i\in\mathcal{N}}a_{i}+1\right).$

Each ISP incurs a cost to improve the network quality. We assume that
$ISP_{i}$ incurs a cost of $c_{i}$ per unit efforts and hence the total cost
incurred by an effort level of $a_{i}$ is given by $c_{i}a_{i}$. Then the
utility of $ISP_{i}$ is given by

$U_{ISP_{i}}=\beta_{i}r\log\left(\sum_{i\in\mathcal{N}}a_{i}+1\right)-c_{i}a_{i}.$

We consider a leader-follower interaction as a Stackelberg game, with the CP
acting as leader and ISPs as followers. The CP leads the system by announcing
the revenue sharing contract, i.e., value $\beta_{i}$ to each
$ISP_{i},,i\in\mathcal{N}$. $ISP_{i}$ then responds to these contracts by
determining its effort $a_{i}$. As the leader in the game, CP is assumed to be
able to anticipate ISPs’ reactions to its action. Therefore, we analyze the
game using the backward induction method and the equilibrium obtained is
referred to as Stackelberg equilibrium. Finally, note that followers are
interacting through the demand increment $D$ and actions of each impacts the
utility function of all ISPs. Henceforth we refer to $D$ as simply demand.

##  III One public and one private ISP

In this section, we consider a special case with two ISPs, i.e., $n=2$ where
one of it is a public ISP while the other is a private ISP. Public ISP is
assumed to be nonprofit in nature and just aims to cover its marginal
investment, for example ISP runs as a public (social) service managed by a
government. On the other hand, private ISP is market oriented and chooses its
investment decisions (effort) to maximize its utility. Without loss of
generality, we suppose that $ISP_{1}$ is public and $ISP_{2}$ is private. The
hierarchical optimization simplifies to the following optimization problem:

CP (Leader):

$\max_{\beta_{i}\in[0,1],\sum_{i}\beta_{i}\leq
1}\quad(1-\beta_{1}-\beta_{2})r\log(\overline{a}_{1}+a_{2}+1)$

$ISP_{1}$ (follower: Public ISP)

$\overline{a}_{1}\quad\text{such
that}\quad\beta_{1}r\log(\overline{a}_{1}+a_{2}+1)-c_{1}\overline{a}_{1}=0$

$ISP_{2}$ (follower: Private ISP)

$\max_{a_{2}\geq 0}\quad\beta_{2}r\log(\overline{a}_{1}+a_{2}+1)-c_{2}a_{2}$

The best response by $ISP$’s in response to the contract offered by CP is
given by

$a_{1}+a_{2}=\frac{\beta_{1}r}{c_{1}}-1$ (1)

and

$a_{1}+a_{2}=\frac{\beta_{2}r}{c_{2}}-1$ (2)

This implies

$\frac{\beta_{1}}{\beta_{2}}=\frac{c_{1}}{c_{2}}$ (3)

The total effort $\overline{a}_{1}$ by $ISP_{1}$ is such that the cost of the
efforts is recovered from the profit shared by the CP, i.e., utility of
$ISP_{1}$ is zero, while $ISP_{2}$ aims to put efforts that maximizes its own
utility. Thus, the $ISP_{1}$ participates in the contract but only agrees to
put efforts just enough to cover its expenses. The optimal efforts by the ISPs
in response to the contract offered by CP, is given by the following
proposition.

###### Proposition 1

For a given contract $(\beta_{1},\beta_{2})$ offered by CP, optimal effort by
each ISP is given by:

$\overline{a}_{1}(\beta_{1},\beta_{2})=\frac{\beta_{1}r}{c_{1}}\log\left(\frac{\beta_{2}r}{c_{2}}\right)$

and

$a_{2}(\beta_{1},\beta_{2})=\frac{\beta_{2}r}{c_{2}}-\frac{\beta_{1}r}{c_{1}}\log\left(\frac{\beta_{2}r}{c_{2}}\right)-1,$

Aware of the response by the ISPs, the optimal strategy of CP is to offer a
contract $(\beta_{1}^{*},\beta_{2}^{*})$ that maximizes its utility, i.e.,
solution of the following maximization problem:

$\max_{\beta_{i}\in[0,1],\sum_{i}\beta_{i}\leq
1}\quad(1-\beta_{1}-\beta_{2})r\log\left(\frac{\beta_{2}r}{c_{2}}\right).$

Note that the increment depends only on $ISP_{2}$ parameters (in particular
$\frac{\beta_{2}r}{c_{2}}$) because the other ISP is public and aims to have
zero utility. Note that it is only interesting to consider the case $r>c_{2}$.
Because, if $r<c_{2}$, total effort in this case $\beta r/c_{2}<1$ as
$\beta\in(0,1)$, and it is not worthwhile for CP to make investments in order
to increase the demand. Also, by definition, total effort becomes zero when
$r<c_{2}$, thus $\beta_{2}$ does not impact the results discussed ahead.
Henceforth, we assume for the rest of the section that $r>c_{2}$.

The following result characterizes the equilibrium contracts between CP and
each ISP.

###### Theorem 1

The equilibrium contract between CP and each ISP is given by

$\beta_{1}^{*}=0\mbox{ and
}\beta_{2}^{*}=\frac{1}{W\left(\frac{r}{c_{2}}e\right)}.$ (4)

where $W(\cdot)$ is the LambertW function (see [31]). Since $W(\cdot)$ in
strictly increasing and $W(e)=1$, it follows that $\beta_{2}^{*}\in(0,1)$
whenever $r/c_{2}>1$. Moreover, note that equilibrium fraction $\beta_{2}^{*}$
is a strictly decreasing function of the ratio $r/c_{2}$ as might be expected:
the ratio $r/c_{2}$ captures the relative monetization power of the CP per
unit effort cost of the ISP. If CP’s ability to monetize the efforts improves,
then it needs lower efforts. Using Theorem 1, one can characterize the
equilibrium effort for all the players of the market.

###### Corollary 1

At equilibrium,

  * •

Efforts by the ISPs are given by

$\overline{a}_{1}^{*}=0\mbox{ and }\quad
a_{2}^{*}=\frac{r}{c_{2}W\left(\frac{r}{c_{2}}e\right)}-1.$ (5)

  * •

Utility of the CP is given by

$U_{CP}=\frac{r\left[W\left(\frac{r}{c_{2}}e\right)-1\right]^{2}}{W\left(\frac{r}{c_{2}}e\right)}.$
(6)

  * •

Utility of ISPs is given by

$\displaystyle U_{ISP_{1}}=0\mbox{ and }U_{ISP_{2}}$
$\displaystyle=r\left(1-\frac{2}{W\left(\frac{r}{c_{2}}e\right)}\right)+c_{2}$
(7)

Thus introducing a public ISP into the market appears to be a good idea to end
users perspective, as it can result in improvement of the investments done by
the private ISP. But in fact, when a CP makes a contract with a public and a
private ISP, the public ISP does not get any share from the CP at equilibrium
and hence the public ISP ends up not contributing anything to improving the
quality of service for end users. Effectively, public ISP participation in the
contract has no effect and problem reduces to a interaction between the CP and
private ISP. In following section we consider several private ISPs that
distribute the content traffic to end users.

If the regulator put restrictions that the CP has to make contract with Public
ISP (i.e. $\beta_{1}>0$), CP will end up sharing more revenue, while getting
same total demand. See Appendix IX-D for detail.

##  IV Symmetric Private ISPs

In this section we consider the symmetric case where cost incurred by all the
ISPs is the same, i.e., $c_{1}=c_{2}=\ldots=c_{n}:=c$. In other words, the
ISPs are symmetric with regards to the per unit cost incurred for investment
in QoS for CP. In this setting, we analyze the equilibrium contracts arising
in the competitive as well as cooperative setting, and the resulting surplus
of the CP and the ISPs.

###  IV-A Competitive scenario

In this competitive scenario between $n$ private ISPs, each ISP maximizes its
utility in order to determine its optimal effort. The CP shares optimally its
revenue with each ISP individually. Here, the two level optimization problem
is given as:
CP (Leader):

$\max_{\beta_{i}\in[0,1],\sum_{i}\beta_{i}\leq
1}\quad(1-\sum_{i=1}^{n}\beta_{i})r\log(\sum_{i=1}^{n}a_{i}+1)$

$ISP_{i};i\in\mathcal{N}$ (follower: Private ISP)

$\max_{a_{i}\geq 0}\quad\beta_{i}r\log(\sum_{i=1}^{n}a_{i}+1)-ca_{i}$

By symmetry, it must be such that $a_{1}=a_{2}=\cdots=a_{n}:=a$. Following
proportion gives best response of each ISP to the contract offered by the CP.

###### Proposition 2

The optimal effort $a$ of each ISP in response to the contract offered by the
CP, $\beta$ satisfies the following relationships:

$na+1=\frac{n\beta_{1}r}{c}=\frac{n\beta_{2}r}{c}=\cdots=\frac{n\beta_{n}r}{c}.$

Note that, because of symmetry between ISPs, ISP’s best response forces equal
share proportion by the CP for all ISPs. Now, given ISPs’ best response one
can characterize the equilibrium contracts between the CP and each ISP as
follows.

###### Theorem 2

Let $r>c$. The equilibrium contract between CP and $ISP_{i},i\in\mathbf{N}$
are given by

$\beta_{1}=\beta_{n}=\cdots=\beta_{n}:=\beta=\frac{1}{nW\left(\frac{r}{c}e\right)}.$

For $r\leq c$, $\beta=0$. This is because $W(\cdot)$ is strictly increasing
and $W(e)=1$. Moreover, total effort becomes $\beta r/c<1$, and it is not
worthwhile for CP to make investment for growth of the demand, and by
definition total effort becomes zero when $r<c$. Henceforth, we assume that
$r>c$. Using Theorem 2, one can characterize the equilibrium effort of ISPs
and utility of CP and ISPs as follows.

###### Corollary 2

At equilibrium

  * •

The effort by each ISP is given by:

$a=\frac{\beta
r}{c}-\frac{1}{n}=\frac{r}{nW\left(\frac{r}{c}e\right)}-\frac{1}{n}$

  * •

The utility of CP is given by

$U_{CP}=(1-n\beta)r\log\left(\frac{\beta
r}{c}\right)=r\frac{\left[W\left(\frac{r}{c}e\right)-1\right]^{2}}{W\left(\frac{r}{c}e\right)}$

  * •

The utility of each ISP is given by

$\displaystyle U_{ISP}$ $\displaystyle=\beta r\log\left(\frac{\beta
r}{c}\right)-c\left(\frac{\beta r}{nc}-\frac{1}{n}\right)$
$\displaystyle=r\left(1-\frac{n+1}{nW\left(\frac{r}{c}e\right)}\right)+\frac{c}{n}$

From the above corollary, it is evident that the utility of CP is independent
of the number of ISPs, implying that in the case of symmetry between ISPs, CP
doesn’t care about how many ISPs enter the contract for infrastructure
investment. However, the individual effort and utility of each ISP depend on
the number of ISPs entering the contract with CP.

###  IV-B Cooperative mechanism

We now study the cooperative scenario, where ISPs cooperatively decide their
effort, and CP jointly makes a single revenue-sharing contract with all ISPs.
We propose that the division of revenue share between ISPs depends on the
ratio of their efforts. Having specified the CP’s behavior, a transparent
platform controlled by a regulatory authority facilitates the bargaining
between ISPs to ensure that the cooperative mechanism is adequately enforced.
Over here, ISPs can interact and bargain to arrive at individual effort to
offer to the CP. We apply the classical Nash Bargaining Solution (NBS) to
capture the outcome of this interaction [32]. Formally, the two levels
optimization problem with constraints is described as follows:
CP (Leader):

$\max_{\beta\in[0,1]}\quad(1-\beta)r\log(\sum_{i=1}^{n}a_{i}+1)$

ISPs:

$\max_{a_{i}\geq 0}\quad\prod_{i=1}^{n}\left(U_{ISP_{i}}(a_{i})-d\right)$

subjected to

$\frac{\beta_{i}}{\beta_{j}}=\frac{a_{i}}{a_{j}}\quad\forall\;i\neq j,\mbox{
and }\sum_{i=1}^{n}\beta_{i}=\beta$

where $U_{ISP_{i}}(a_{i})=\beta_{i}r\log(\sum_{i=1}^{n}a_{i}+1)-ca_{i}$ is the
utility of $ISP_{i}$ given the global CP’s revenue share $\beta$. The
disagreement point $d$ corresponds to the ISP utilities when they act non-
cooperatively, i.e., the Nash Equilibrium between the ISPs. The last two
constraints can be simplified to $\beta_{i}=\frac{\beta
a_{i}}{\sum_{j=1}^{n}a_{j}}\quad\forall i=1,2,...,n$. Because of the symmetry,
efforts and sharing are such that $a_{1}=a_{2}=...=a_{n}:=a$, and
$\beta_{1}=\beta_{2}=...=\beta_{n}=\frac{\beta}{n}$. Thus, the two levels
optimization problem with constraints can be reduced to:
CP (Leader):

$\max_{\beta\in[0,1]}\quad(1-\beta)r\log(na+1)$

ISPs:

$\max_{a\geq 0}\quad\left(\frac{\beta}{n}r\log(na+1)-ca-d\right)^{n}$

###### Proposition 3

The optimal effort by ISPs in response to the contract offered by the CP is
given by

$a(\beta)=\frac{1}{n}\left(\frac{\beta r}{c}-1\right).$ (8)

Given ISPs’ best response, we now characterize the equilibrium contract
between CP and ISPs in the following Theorem.

###### Theorem 3

The equilibrium contract between the CP and ISPs is given by

$\beta=\frac{1}{W\left(\frac{r}{c}e\right)}$ (9)

Note that the total profit share given by CP in competitive scenario is same
as that in the cooperative scenario. Further, from eqn (5), it can be easily
seen that effort is also the same. Hence, the competitive and cooperative
optimal solution coincides when ISPs are symmetric with respect to their cost.
Therefore, when ISPs are symmetric, competitive scenario is as good as the
cooperative scenario. Hence there is no need to force cooperation among ISPs
with costly regulations as long of all of them have same sensitivity to cost
of efforts.

###  IV-C The effect number of ISPs

We now study the impact of number of ISPs (or scaling in $n$) on the efforts
of ISPs’ at equilibrium, and the surplus of each agent. Our main result is the
following.

###### Theorem 4

Suppose that $r>c$, the non-zero equilibrium satisfies the following
properties.

  1. 1.

The fraction $\beta$ of revenue share each ISP gets is a strictly decreasing
function of $n$, however, total share $n\beta$ remain unchanged with change in
$n$.

  2. 2.

The effort by each ISP for CP is a strictly decreasing function of $n$, even
though the total effort is unaffected by the change in $n$.

  3. 3.

The utility of the CP does not depends on $n$.

  4. 4.

The utility of each ISP is strictly decreasing in $n$.

Theorem 4 highlights that an increase in the number of ISPs does not affect
CP. However, as more ISPs enter such profit sharing contract with the CP,
contribution by the CP gets ’split’ further between ISPs, thus each ISPs
earning reduces.

##  V Asymmetric ISPs

In this section we consider the asymmetric case where each ISPs cost of
investments is is not the same. This is captured by allowing $c_{i}\neq c_{j}$
for $i\neq j$. Our interest is to understand how disparity in the cost
influences preference of the players in the competitive and cooperative
scenarios of the access market. For simplicity and clarity of the main
insights, we focus on the case with two ISPs ($n=2$) and without loss of
generality, we assume that cost of $ISP_{2}$ is more than that of $ISP_{1}$,
i.e., $c_{2}>c_{1}$.

###  V-A Competitive scenario

We first consider the competitive scenario between two private ISPs, where
each provider determine its effort to maximizes its utility. This effort
impacts the CP revenue who shares part of it with each ISP individually. Here,
the bi-level optimization problem is given as:
CP (Leader):

$\max_{\beta_{i}\in[0,1],\sum_{i}\beta_{i}\leq
1}\quad(1-\beta_{1}-\beta_{2})r\log(a_{1}+a_{2}+1)$

$ISP_{i};i=1,2$ (follower: Private ISP)

$\max_{a_{i}\geq 0}\quad\beta_{i}r\log(a_{1}+a_{2}+1)-c_{i}a_{i}$

The best response of each ISP to the contract (sharing value) offered by the
CP can be obtained explicitly by solving the ISP optimization problem, given
that the sharing coefficients $\beta_{1}$ and $\beta_{2}$ are fixed.

###### Proposition 4

The optimal effort by ISPs in response to the contract offered by the CP is
given by one of the three following cases:

  * •

If $\frac{\beta_{1}}{\beta_{2}}=\frac{c_{1}}{c_{2}}$, then

$a_{1},a_{2}>0\quad\&\quad
a_{1}+a_{2}+1=\frac{\beta_{1}r}{c_{1}}=\frac{\beta_{2}r}{c_{2}}.$

  * •

If there exists $\lambda\geq 0$ such that
$\frac{\beta_{1}}{\beta_{2}}=\frac{c_{1}}{c_{2}+\lambda}$, then

$a_{1}>0,a_{2}=0\quad\&\quad
a_{1}+1=\frac{\beta_{1}r}{c_{1}}=\frac{\beta_{2}r}{c_{2}+\lambda}.$

  * •

If there exists $\lambda\geq 0$ such that
$\frac{\beta_{1}}{\beta_{2}}=\frac{c_{1}+\lambda}{c_{2}}$, then

$a_{1}=0,a_{2}>0,\quad\&\quad
a_{2}+1=\frac{\beta_{1}r}{c_{1}+\lambda}=\frac{\beta_{2}r}{c_{2}}.$

ISPs’ best response can then be plugged into the CP optimization problem to
characterize the equilibrium contracts between the CP and each ISP. Note that
the costs should satisfy the constraint $r>c_{1}+c_{2}$. Otherwise, total
effort becomes $\beta_{1}r/c_{1}<1$, and it is not worthwhile for CP to make
investment for growth of the demand. By definition total effort becomes zero
when $r<c_{1}+c_{2}$.

###### Theorem 5

Considering that $r>c_{1}+c_{2}$, CP utility is maximized when
$\frac{\beta_{1}}{\beta_{2}}=\frac{c_{1}}{c_{2}}$ and the equilibrium contract
between CP and $ISP_{i}$ is given by

$\beta_{i}^{*}=\frac{c_{i}}{(c_{1}+c_{2})W\left(\frac{r}{c_{1}+c_{2}}e\right)}\mbox{
for all }i=1,2$

Based on previous result in Theorem 5, the effort of each ISP at equilibrium
satisfy the following relationship:

$a_{1}^{*}+a_{2}^{*}=\frac{r}{(c_{1}+c_{2})W\left(\frac{r}{c_{1}+c_{2}}e\right)}-1.$

The change in strategy of one ISP will impact the strategy of the other ISP;
and there is a continuum of best efforts of the ISPs but the overall effort is
constant. There are many examples in literature with infinite number of
equilibrium especially where the actions o f the players are coupled [33].
There could be a unique responses of ISP if there is asynchronous decisions.
For example, one ISP can be the historical operator and determines its effort
first, before the decision of a new ISP. Therefore, the interaction structure
between the ISPs corresponds to a Stackelberg framework, assuming that the
historical ISP knows that a concurrent will come into the market after him.

It is interesting to note that if one of the ISPs decreases effort, it will
still get the same revenue share from the CP as the total effort is same at
equilibrium. As the competition results in continuum of equilibrium, some
specific equilibrium points may be preferred by each of the ISPs which leads
to instability. Also, such competitive scenario may lead to free riding by one
of the ISPs, therefore, in the next section we propose cooperation between
ISPs, whereby CP makes a joint payment to them and ISPs cooperatively decide
the effort and the division of revenue share between them.

Therefore, we deduce the results for symmetric ISPs as by symmetry their
individual effort will be the same. We characterize equilibrium contract and
utilities for symmetric ISPs as following.

###### Corollary 3

For symmetric ISPs with $c_{1}=c_{2}:=c$, at equilibrium we have

  * •

revenue share by CP for each ISP is

$\beta^{*}=\frac{1}{2W\left(\frac{r}{2c}e\right)}$ (10)

  * •

effort by each ISP is

$a=\frac{1}{2}\left(\frac{r}{2W\left(\frac{r}{2c}e\right)}-1\right)$ (11)

  * •

utility of each ISP is given by

$U_{ISP}=\frac{r}{2}\left(1-\frac{3}{2W\left(\frac{r}{2c}e\right)}\right)+\frac{c}{2}$
(12)

  * •

the utility of CP utility is

$U_{CP}=r\frac{\left[W\left(\frac{r}{2c}e\right)-1\right]^{2}}{W\left(\frac{r}{2c}e\right)}$
(13)

###  V-B Cooperative mechanism

Figure 2: Graphical plots of objective function of NBP

We now study the cooperative scenario in which ISPs cooperatively decide their
individual effort and CP makes a revenue sharing contract jointly with all
ISPs. Formally, we define the bi-levels optimization problem as follows:
CP (Leader):

$\max_{\beta\in[0,1]}\quad(1-\beta)r\log(a_{1}+a_{2}+1)$

ISPs:

$\max_{a_{1},a_{2}\geq
0}\quad\left(U_{ISP_{1}}(a_{1},a_{2})-d_{1}\right)\left(U_{ISP_{2}}(a_{1},a_{2})-d_{2}\right)$

subjected to

$\frac{\beta_{1}}{\beta_{2}}=\frac{a_{1}}{a_{2}},\mbox{ and
}\beta_{1}+\beta_{2}=\beta$

where
$U_{ISP_{i}}(a_{1},a_{2})=\beta_{i}r\log(a_{1}+a_{2}+1)-c_{i}a_{i};i=1,2$ is
the utility of $ISP_{i}$ given the CP’s revenue share $\beta_{i}$. The
disagreement point $d_{i}$ correspond to the $ISP_{i}$ utility in competitive
scenario, i.e., the Nash equilibrium between the ISPs. Simplifying the last
two constraints as $\beta_{i}=\frac{\beta a_{i}}{a_{1}+a_{2}}\quad\forall
i=1,2$, the Bargaining problem for ISPs reduces to

$\displaystyle\max_{a_{1},a_{2}\geq 0}\quad$ $\displaystyle\left(\frac{\beta
a_{1}}{a_{1}+a_{2}}r\log(a_{1}+a_{2}+1)-c_{1}a_{1}-d_{1}\right)\times$
$\displaystyle\left(\frac{\beta
a_{2}}{a_{1}+a_{2}}r\log(a_{1}+a_{2}+1)-c_{2}a_{2}-d_{2}\right)$

###### Theorem 6

There exist a non-zero NBS, i.e. $a_{1}^{B},a_{2}^{B}>0$. Furthermore, the
non-zero solution $(a_{1}^{B},a_{2}^{B})$ is unique.

We also illustrate the concavity of objective function graphically in Figure
2. The graphs are three dimensional plots of the objective function with
varying $a_{1}$ and $a_{2}$. The plots show unique peak, implying concave
nature of objective function. The computation of NBS for this case is not
analytically tractable because of the complex objective function arising due
to asymmetrical costs of ISPs, but existence of non-zero solution and and
uniqueness are established in the appendix. We the next section we empirically
study how the cooperative mechanism affects all the players. The gradient
methods can be used to numerically determine the NBS.

  * •

Stage 1: CP first announces a total share $\beta$ for both ISPs and ISPs
maximizes their sum utility to find the optimal effort for the CP.

  * •

Stage 2: ISPs cooperatively decide the division of revenue share $\beta$, that
depends on both effort, between them. We consider the outcome of division of
the share proportion $\beta$ by using the well-known cooperative solution
concept of the Nash Bargaining solution (NBS). Nash Bargaining problem helps
two players decide how to share a surplus generated by their joint actions.
This solution concept satisfies important axioms which are: invariant to
affine transformations, Pareto optimality, independence of irrelevant
strategies and symmetry [8].

####  V-B1 Stage 1

Stage 1 problem constitutes joint payment of revenue share by CP to ISPs, and
joint utility maximization by ISPs to make effort decision. The problem for
Stage 1 is given as follows:
CP:

$\max_{\beta\in[0,1]}\quad(1-\beta)r\log(a_{1}+a_{2}+1)$

ISPs:

$\max_{a_{1},a_{2}}\quad\beta r\log(a_{1}+a_{2}+1)-c_{1}a_{1}-c_{2}a_{2}$

###### Proposition 5

The optimal effort by ISPs in response to the contract offered by the CP is
given by

$a_{1}=0,\quad a_{2}(\beta)=\frac{\beta r}{c_{2}}-1$ (14)

From the above proposition, it can be seen that when ISPs jointly maximize
their utility, it is always best for them that only ISP with higher cost
coefficient put the effort. Now, we characterize the equilibrium share
proportion for ISPs, which depends upon the action by ISPs.

###### Theorem 7

For $r>c_{2}$, the equilibrium contract between CP and ISPs given by

$\beta=\frac{1}{W\left(\frac{re}{c_{2}}\right)}$ (15)

####  V-B2 Stage 2

Under this stage, ISPs can interact and bargain with one another to arrive at
a vector of revenue sharing contracts offered by CP,
$(\beta_{1}^{B},\beta_{2}^{B})$, known as Nash Bargaining solution (NBS).
Formally, the NBS $(\beta_{1}^{B},\beta_{2}^{B})$ is defined as the solution
of the following optimization problem:

$\max_{\beta_{1},\beta_{2}\in[0,1]}\quad\left(U_{ISP_{1}}(\beta_{1})-d_{ISP_{1}}\right)\left(U_{ISP_{2}}(\beta_{2})-d_{ISP_{1}}\right)$

subjected to

$U_{ISP_{1}}(\beta_{1})\geq d_{ISP_{1}},$ $U_{ISP_{2}}(\beta_{2})\geq
d_{ISP_{2}}.$

The disagreement point $(d_{ISP_{1}},d_{ISP_{2}})$ is taken to the utilities
of ISPs corresponding to the case of competitive scenario in previous section.

The NBS gives the division of $\beta$ given in eqn. (15) between ISPs.
Specifically, the NBS ($\beta_{1}^{B},\beta_{2}^{B}$) can be written as
$\beta_{1}^{B},\beta-\beta_{2}^{B}$ since $\beta=\beta_{1}^{B}+\beta_{2}^{B}$.
Thus, the Nash Bargaining problem can be written as follows:

$\displaystyle\max_{\beta_{1}\geq 0}\hskip
1.42262pt\\{\beta_{1}r\log\left(\frac{\beta
r}{c_{2}}\right)-d_{1}\\}\\{(\beta$
$\displaystyle-\beta_{1})r\log\left(\frac{\beta r}{c_{2}}\right)-\beta
r-d_{2}\\}$ subjected to $\displaystyle\beta_{1}r\log\left(\frac{\beta
r}{c_{2}}\right)$ $\displaystyle\geq d_{1}$
$\displaystyle(\beta-\beta_{1})r\log\left(\frac{\beta r}{c_{2}}\right)-\beta
r$ $\displaystyle\geq d_{2}$

where $d_{i}=d_{ISP_{i}},i=1,2$ The NBS can be computed only if we have
utility values of each ISP in competitive scenario. Therefore, we consider the
special case where
$\frac{a_{1}^{*}}{a_{2}^{*}}=\frac{\beta_{1}^{*}}{\beta_{2}^{*}}$, i.e., ratio
of individual effort by each ISP is in proportion to the revenue share
proportion shared by CP with them at equilibrium.

###### Theorem 8

The NBS for the division of revenue share proportion between ISPs is given by

$\displaystyle\beta_{1}^{B}=$
$\displaystyle\frac{\beta}{2}-\frac{c_{2}a_{2}+d_{2}-d_{1}}{2r\log\left(\frac{\beta
r}{c_{2}}\right)},$ $\displaystyle\beta_{2}^{B}=$
$\displaystyle\frac{\beta}{2}+\frac{c_{2}a_{2}+d_{2}-d_{1}}{2r\log\left(\frac{\beta
r}{c_{2}}\right)}$

or

$\beta_{1}^{B}=\frac{d_{1}}{r\log\left(\frac{\beta
r}{c_{2}}\right)},\beta_{2}^{B}=\frac{c_{2}a_{2}+d_{2}}{r\log\left(\frac{\beta
r}{c_{2}}\right)}$

##  VI Comparison between public/private ISP

We first compare the cases when CP makes contract with public and private ISPs
verses both private ISPs. Our main result is the following.

###### Theorem 9

The following statements hold with two competing ISPs:

  1. 1.

Total profit shared by the CP is higher when both ISPs are private as compared
to the case where one ISP is public.

  2. 2.

Total effort is higher in the presence of public ISP as compared to the case
when both ISPs are private.

  3. 3.

CP Utility is higher in the presence of public ISP as compared to the case
when both ISPs are private.

The above Theorem implies that the introduction of public ISP makes private
ISP invest more for the quality enhancement for end-users compared to the case
of two private ISPs in competition. The higher investment results in higher
demand for the CP, hence the higher utility for the CP. This means that
competition between profit-maximizing ISPs skews the best effort by each ISP.
Therefore, we proposed cooperation between ISPs as an intervention and studied
its impact in the next section.

##  VII Competition or Cooperation

(a)

(b)

(c)

(d)

(e)

Figure 3: Comparison between three cases: One public ISP, two private ISPs in
competition and cooperation. Here, we fix $c_{2}=2$, $c_{1}=1$ and vary $r$
from 6 to 12.

(a)

(b)

(c)

(d)

(e)

Figure 4: Comparison between three cases: One public ISP, two private ISPs in
competition and cooperation. Here, we fix $c_{2}=0.5=1$, $r=10$ and vary
$c_{2}$ from 1 to 5.

In this section, we compare the competitive and cooperative scenarios. We
focus on the case of asymmetric ISPs, as in the section IV we already show
that for symmetric ISPs, both the scenarios give the same solution. Since
analytical solution in cooperative solution is not tractable, we illustrate
the impact of the proposed cooperative mechanism via numerical illustrations
in Figures 3-4. Fig. 3(a)-(d) compares the three scenarios: public ISP,
competition, and cooperation between private ISPs. We plot share proportion,
total effort, and utilities of all agents by varying $r$, other parameters
being fixed. Because of the infinite number of equilibrium between ISPs in
competitive case, we choose the equilibrium for numerical illustrations such
that $\frac{a_{1}^{*}}{a_{2}^{*}}=\frac{\beta_{1}^{*}}{\beta_{2}^{*}}$. As
seen in figs. 3(b),(c) that total effort and CP’s utility is highest when
cooperative mechanism is imposed over ISPs. Note that the cooperative
mechanism elevates the utility of both ISPs from the case when they act non-
cooperatively. We also observe that share proportion is less in the
cooperative case than in the competitive case and is least in the presence of
public ISP.

From Fig. 4(a)-(d), it is also clear that the share proportion is increasing
in $c_{2}$, which is evident as the investment cost for ISP increases; the CP
would have to contribute more revenue share to cover the cost of the ISPs.
Further, the total effort by the ISPs decreases with $c_{2}$, implying higher
the cost, the lesser the effort will be. The utility of all players decreases
with an increase in $c_{2}$. The main reason for the decline is the decrease
in total effort resulting in less demand, and hence CP earns less revenue
resulting in less share from ISPs.

On the other hand, share proportion is decreasing in $r$, which implies that
when the monetization power of CP is high, CP will have less incentive in
revenue share contract for the investment cost of the ISP, as might be
expected. The total effort by ISPs increases with $r$, as with the high
monetization power of CP, ISPs have more incentive to invest in the network.
Moreover, the utility of all players increases with an increase in $r$, which
is because with higher monetization of CP, total effort increases resulting in
higher demand and more revenue for the CP, hence the higher utilities for
ISPs.

As observed in the plots in Figures 3-4, it is evident that all the players
are benefited in the cooperative mechanism, including the CP. Moreover, in all
three scenarios, the total effort by ISPs is highest when ISPs cooperate,
implying higher QoS for end-users. In fact, the cooperative mechanism elevates
the utilities of ISPs and benefits CP and end-users. Therefore, everyone is
better off by cooperation between ISPs.

##  VIII Conclusion

We studied the problem of revenue sharing between single CP and multiple ISP
on the Internet using the leader-follower interactions, with CP acting as a
leader and the ISPs as followers. We consider the case where CP enters the
profit sharing contract with both public (non-profitable) and private ISPs and
show that having public ISP makes private ISP invest more in the network
compared to the case of both being private ISPs. Further, putting compulsion
on CP to enter the contract with public ISP reduces CP’s surplus while having
the same amount of QoS for end-users. We also studied the case where all ISPs
are private, under competitive and cooperative setting. Cooperative mechanism
is imposed over ISPs, where ISPs work together (like a coalition) and
’bargain’ on a transparent platform to develop a mutually binding contract
with the CP. We compared the total effort and utility of CP in these cases. We
show that as expected, it is beneficial for everyone to have private ISPs in
cooperation as it gives higher QoS for end-users and utility of each player as
well. On the other hand, competition between private ISPs leads to less effort
by ISPs which worsens the payoff for everyone.

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##  IX Appendix

###  IX-A Proof of Proposition 1

The optimization can be re-written as

$\displaystyle\max_{\beta_{i}\in[0,1],\sum_{i}\beta_{i}\leq 1}\quad$
$\displaystyle(1-\beta_{1}-\beta_{2})r\log(\overline{a}_{1}+a_{2}+1)$
subjected to $\displaystyle\overline{a}_{1}$
$\displaystyle=\frac{\beta_{1}r\log(\overline{a}_{1}+a_{2}+1)}{c_{1}}$
$\displaystyle\max_{a_{2}\geq 0}\quad\beta_{2}r$
$\displaystyle\log(\overline{a}_{1}+a_{2}+1)-c_{2}a_{2}$

Now, first order optimality condition for $ISP_{2}$ maximization problem is
given by:

$\frac{\beta_{2}r}{\overline{a}_{1}+a_{2}+1}-c_{2}=0\implies\overline{a}_{1}+a_{2}+1=\frac{\beta_{2}r}{c_{2}}$
(16)

Using this, we get
$\overline{a}_{1}=\frac{\beta_{1}r\log\left(\frac{\beta_{2}r}{c_{2}}\right)}{c_{1}}$.
Substituting this $\overline{a}_{1}$ in eqn. (16), we get
$a_{2}=\frac{\beta_{2}r}{c_{2}}-\frac{\beta_{1}r}{c_{1}}\log\left(\frac{\beta_{2}r}{c_{2}}\right)-1,$

###  IX-B Proof of Theorem 1

Using proposition 1, the CP’s optimization problem reduces to

$\max_{\beta_{i}\in[0,1],\sum_{i}\beta_{i}\leq
1}\quad\left(1-\beta_{1}-\beta_{2}\right)r\log\left(\frac{\beta_{2}r}{c_{2}}\right)$

It is optimized when $\beta_{1}=0$ and then CP’s problem reduces to

$\max_{\beta_{2}\in[0,1]}\quad\left(1-\beta_{2}\right)r\log\left(\frac{\beta_{2}r}{c_{2}}\right)$

From CP optimization problem, it can be observed that for $r/c_{2}<1$, CP has
no incentive to share a fraction of their revenue with the $ISP_{2}$ and
$\beta_{2}=0$ is the equilibrium. Now assume $r/c_{2}\geq 1$. For this case
the optimal value of $\beta_{2}$ will be such that $r\beta_{2}/c_{2}\geq 1$.
The first order optimality condition $\partial U_{CP}/\partial\beta_{2}=0$
then gives:

$\log\left(\frac{\beta_{2}r}{c}\right)=\frac{1-\beta_{2}}{\beta_{2}}$

Solving and rearranging the terms, we get:

$\displaystyle\frac{1}{\beta_{2}}e^{{\frac{1}{\beta_{2}}}}$
$\displaystyle=\frac{re}{c_{2}}$

Using the definition of the LamebertW function we get

$\beta_{2}=\frac{1}{W\left(\frac{r}{c_{2}}e\right)}$

###  IX-C Proof of Corollary 1

Substituting optimal $\beta_{1},\beta_{2}$ obtained in Theorem 1 in optimal
effort, we get

$\displaystyle a_{1}=0,a_{2}$
$\displaystyle=\frac{r}{c_{2}W\left(\frac{r}{c_{2}}e\right)}-1$ $\displaystyle
U_{CP}$
$\displaystyle=\left(1-0-\frac{1}{W\left(\frac{r}{c_{2}}e\right)}\right)\log\left(\frac{r}{c_{2}W\left(\frac{r}{c_{2}}e\right)}\right)$
$\displaystyle=\left(1-\frac{1}{W\left(\frac{r}{c_{2}}e\right)}\right)\log\left(\frac{\frac{re}{c_{2}}}{W\left(\frac{re}{c_{2}}\right)e}\right)$
$\displaystyle=\frac{W\left(\frac{r}{c_{2}}e\right)-1}{W\left(\frac{r}{c_{2}}e\right)}\left(W\left(\frac{r}{c_{2}}e\right)-1\right)$
$\displaystyle\left(\because\log(\left(\frac{x}{W(x)}\right)=W(x)\right)$
$\displaystyle=\frac{\left[W\left(\frac{r}{c_{2}}e\right)-1\right]^{2}}{W\left(\frac{r}{c_{2}}e\right)}$

$U_{ISP_{1}}=0$

$\displaystyle U_{ISP_{2}}$
$\displaystyle=\frac{r}{W\left(\frac{r}{c_{2}}e\right)}\log\left(\frac{\frac{r}{c_{2}}}{W\left(\frac{r}{c_{2}}e\right)}\right)-c_{2}\left(\frac{r}{c_{2}W\left(\frac{r}{c_{2}}e\right)}-1\right)$
$\displaystyle=\frac{r}{W\left(\frac{r}{c_{2}}e\right)}\left(W\left(\frac{r}{c_{2}}e\right)-1\right)-\frac{r}{W\left(\frac{r}{c_{2}}e\right)}+c_{2}$
$\displaystyle=r\left(1-\frac{2}{W\left(\frac{r}{c_{2}}e\right)}\right)+c_{2}$

###  IX-D If regulator imposes that $\beta_{1}>0$

We have the following problem:

$\displaystyle\max_{\beta_{i}\in[0,1],\sum_{i}\beta_{i}\leq 1}\quad$
$\displaystyle(1-\beta_{1}-\beta_{2})r\log(\overline{a}_{1}+a_{2}+1)$
subjected to $\displaystyle\overline{a}_{1}$
$\displaystyle=\frac{\beta_{1}r\log(\overline{a}_{1}+a_{2}+1)}{c_{1}}$
$\displaystyle\max_{a_{2}\geq 0}\quad\beta_{2}r$
$\displaystyle\log(\overline{a}_{1}+a_{2}+1)-c_{2}a_{2}$

Now, first order optimality condition for $ISP_{2}$ maximization problem is
given by:

$\frac{\beta_{2}r}{\overline{a}_{1}+a_{2}+1}-c_{2}=0\implies\overline{a}_{1}+a_{2}+1=\frac{\beta_{2}r}{c_{2}}$
(17)

Using this, we get

$\overline{a}_{1}=\frac{\beta_{1}r\log\left(\frac{\beta_{2}r}{c_{2}}\right)}{c_{1}}$
(18)

Then, the CP’s optimization problem reduces to

$\max_{\beta_{i}\in[0,1],\sum_{i}\beta_{i}\leq
1}\quad\left(1-\beta_{1}-\beta_{2}\right)r\log\left(\frac{\beta_{2}r}{c_{2}}\right)$

Since, $\beta_{1}>0$, it will be set by the CP such that

$\displaystyle\overline{a}_{1}c_{1}={\beta_{1}r\log\left(\frac{\beta_{2}r}{c_{2}}\right)}$
$\displaystyle\implies\beta_{1}=\frac{\overline{a}_{1}c_{1}}{r\log\left(\frac{\beta_{2}r}{c_{2}}\right)}$
(19)

Then CP’s problem reduces to

$\max_{\beta_{2}\in[0,1]}\quad\left(1-\beta_{2}\right)r\log\left(\frac{\beta_{2}r}{c_{2}}\right)-c_{1}a_{1}$

From CP optimization problem, it can be observed that for $r/c_{2}<1$, CP has
no incentive to share a fraction of their revenue with the $ISP_{2}$ and
$\beta_{2}=0$ is the equilibrium. Now assume $r/c_{2}\geq 1$. For this case
the optimal value of $\beta_{2}$ will be such that $r\beta_{2}/c_{2}\geq 1$.
The first order optimality condition $\partial U_{CP}/\partial\beta_{2}=0$
then gives:

$\log\left(\frac{\beta_{2}r}{c_{2}}\right)=\frac{1-\beta_{2}}{\beta_{2}}$

Solving, we get:

$\beta_{2}=\frac{1}{W\left(\frac{r}{c_{2}}e\right)}$

Substituting this, we get

$\displaystyle a_{2}$
$\displaystyle=\frac{r}{c_{2}W\left(\frac{r}{c_{2}}e\right)}-\overline{a}_{1}-1$
$\displaystyle\overline{a}_{1}$
$\displaystyle=\frac{\beta_{1}r}{c_{1}}\left(W\left(\frac{r}{c_{2}}e\right)-1\right)$

## Cooperative scenario between public and private ISP

Here, we study the case of a cooperative scenario where given the constant
effort by $ISP_{1}$, $ISP_{2}$ maximizes the joint utility of both ISPs to
make the effort decision to improve the QoS for CP’s content. Further, CP pays
a revenue share to ISPs jointly, and ISPs cooperatively decide the split
between them. There are two stages under this case.

####  IX-D1 Stage 1

Stage 1 problem constitutes joint payment of revenue share by CP to ISPs, and
joint utility maximization by $ISP_{2}$ to make effort decision (given fixed
$\overline{a}_{1}$). The problem for Stage 1 is given as follows:
CP:

$\max_{\beta\in[0,1]}\quad(1-\beta)r\log(\overline{a}_{1}+a_{2}+1)$

Since, $ISP_{1}$ puts constant effort $\overline{a}_{1}$, the joint
maximization problem for $ISP_{2}$ is given by

$\max_{a_{2}}\quad\beta
r\log(\overline{a}_{1}+a_{2}+1)-c_{1}\overline{a}_{1}-c_{2}a_{2}$

The best response $a_{2}(\beta)$ by $ISP_{2}$ in response to the contract
offered by CP is given such that:

$a_{2}(\beta)=\frac{\beta r}{c_{2}}-1-\overline{a}_{1}.$ (20)

Following Theorem gives equilibrium revenue share proportion by CP.

The equilibrium contract between CP and ISPs given by

$\beta=\frac{1}{W\left(\frac{re}{c_{2}}\right)}$ (21)

Given, equilibrium share proportion, one can now characterize the equilibrium
effort by $ISP_{2}$ given by The effort by $ISP_{2}$ at equilibrium is given
by

$a_{2}=\frac{r}{c_{2}W\left(\frac{re}{c_{2}}\right)}-\overline{a}_{1}-1$ (22)

####  IX-D2 Stage 2

Formally, the NBS $(\beta_{1}^{B},\beta_{2}^{B})$ is defined as the solution
of the following optimization problem:

$\max_{\beta_{1},\beta_{2}\in[0,1]}\quad\left(U_{ISP_{1}}(\beta_{1})-d_{ISP_{1}}\right)\left(U_{ISP_{2}}(\beta_{2})-d_{ISP_{1}}\right)$

subjected to

$U_{ISP_{1}}(\beta_{1})\geq d_{ISP_{1}},$ $U_{ISP_{2}}(\beta_{2})\geq
d_{ISP_{2}}.$

Here, the disagreement point $(d_{1},d_{2})$ is taken to the utilities of ISPs
corresponding to the case of competitive scenario. The NBS gives the split of
$\beta$ between ISPs. Specifically, the NBS ($\beta_{1}^{B},\beta_{2}^{B}$)
can be written as $\beta_{1}^{B},\beta-\beta_{2}^{B}$ since
$\beta=\beta_{1}^{B}+\beta_{2}^{B}$. Thus, the Nash Bargaining problem can be
written as follows:

$\displaystyle\max_{\beta_{1}^{B}}\quad\\{\beta_{1}^{B}r\log(\overline{a}_{1}+a_{2}+1)$
$\displaystyle-c_{1}\overline{a}_{1}\\}\times$
$\displaystyle\\{(\beta-\beta_{1}^{B})$ $\displaystyle
r\log(\overline{a}_{1}+a_{2}+1)-c_{2}a_{2}-d_{2}\\}$ subjected to
$\displaystyle\beta_{1}^{B}r\log(\overline{a}_{1}+a_{2}+1)$ $\displaystyle-
c_{1}\overline{a}_{1}\geq 0$
$\displaystyle(\beta-\beta_{1}^{B})r\log(\overline{a}_{1}+a_{2}+1)$
$\displaystyle-c_{2}a_{2}\geq d_{2}$

Note that $d_{1}=0$ and $\beta_{1}=0$ in competitive scenario. $\beta$ here is
same as $\beta_{2}$ in competitive scenario and $\beta_{1}=0$. NBP gives same
or better utility to each player than that of at disagreement point. Here,
there is no scope of improvement in utilities of both ISPs simultaneously,
therefore, NBS coincides with that of disagreement point (competitive
equilibrium in this case).

###  IX-E Proof of Proposition 2

Each $ISP_{i}$ optimizes over $a_{i}$ and by the symmetry we must have
$a_{1}=a_{2}=...=a_{n}:=a$. Thus, the problem for $ISP_{i}$ is given as

$\max_{a\geq 0}\quad\beta_{i}r\log(na+1)-ca$

The first order necessary condition gives

$na+1=\frac{n\beta_{i}r}{c}\quad\forall i=1,2,...,n$

Then we must have

$\frac{n\beta_{1}r}{c}=\frac{n\beta_{2}r}{c}=...=\frac{n\beta_{n}r}{c}\implies\beta_{1}=\beta_{2}=...=\beta_{n}$

###  IX-F Proof of Theorem 2

Given ISPs’ best response, the problem for CP reduces to

$\max_{\beta\in(0,1/n)}(1-n\beta)r\log\left(\frac{n\beta r}{c}\right)$

The first order optimality condition $\partial U_{CP}/\partial\beta=0$ then
gives:

$\log\left(\frac{\beta r}{c}\right)=\frac{1-n\beta}{n\beta}$

Solving and rearranging terms, we get:

$\implies\frac{1}{n\beta}e^{{\frac{1}{n\beta}}}=\frac{re}{c}$

Using the definition of the LamebertW function we get

$\frac{1}{n\beta}=W\left(\frac{re}{c}\right)\implies\beta=\frac{1}{nW\left(\frac{r}{c}e\right)}$

###  IX-G Proof of Corollary 2

The proof is on the same lines as that of Corollary 1.

###  IX-H Proof of Proposition 3

First order condition for NBP between ISPs gives

$\frac{\beta}{n}\frac{n}{na+1}-c=0\implies na+1=\frac{\beta r}{c}$

###  IX-I Proof of Theorem 3

Given ISPs’ best response, CP’s problem reduces to

$\max_{\beta\in[0,1]}(1-\beta)r\log\left(\frac{\beta r}{c}\right)$

The first order optimality condition $\partial U_{CP}/\partial\beta=0$ then
gives:

$\log\left(\frac{\beta r}{c}\right)=\frac{1-\beta}{\beta}$

Solving and rearranging terms, we get:

$\implies\frac{1}{\beta}e^{{\frac{1}{\beta}}}=\frac{re}{c}$

Using the definition of the LamebertW function we get

$\frac{1}{\beta}=W\left(\frac{re}{c}\right)\implies\beta=\frac{1}{W\left(\frac{r}{c}e\right)}$

###  IX-J Proof of Theorem 4

Part 1: We have

$\beta(n)=\frac{1}{nW\left(\frac{re}{c}\right)}$

It can be seen clearly that $\beta$ decreases with $n$, however,
$n\beta=\frac{1}{W\left(\frac{re}{c}\right)}$ remain unchanged with change in
$n$.
Part 2: The effort by each ISP is

$a(n)=\frac{1}{n}\left(\frac{r}{cW\left(\frac{re}{c}\right)}-1\right)$

which is clearly decreasing function in $n$. The total effort
$na=\frac{r}{cW\left(\frac{re}{c}\right)}-1$ remain unchanged with change in
$n$.
Part 3: Now, the utility of CP is given as

$U_{CP}=(1-n\beta)r\log\left(\frac{n\beta r}{c}\right)$

We know that $n\beta$ remain unchanged with change in $n$, therefore, from
above expression it is clear that utility of CP also remains unchanged.
Part 4: Sum utility of all ISPs is given by

$nU_{ISP}=n\beta r\log\left(\frac{n\beta r}{c}\right)-c\left(\frac{n\beta
r}{c}-1\right)$

Since, $n\beta$ remains same with change in $n$, from above expression it can
be clearly seen that $nU_{ISP}$ also remains same with change in $n$. Hence,
$U_{ISP}$ is strictly decreasing in $n$.

###  IX-K Proof of Proposition 4

The problem for $ISP_{i}$ is given as

$\max_{a_{\geq}0}\quad\beta_{i}r\log(a_{1}+a_{2}+1)-c_{1}a_{i}$

Lagrangian function for the problem is given as

$L_{i}=\beta_{i}r\log(a_{1}+a_{2}+1)-c_{i}a_{i}-\lambda_{i}a_{i}$

The corresponding KKT conditions are given as:

$\frac{\partial L_{i}}{\partial a_{i}}=0\implies a_{1}+a_{2}+1=\frac{\beta
r}{c_{i}+\lambda_{i}}$ $\lambda_{i}a_{i}=0,\lambda_{2}a_{2}=0,\mbox{ and
}\lambda_{1},\lambda_{2},a_{1},a_{2}\geq 0.$

Now, we have the following cases: Case 1:
$a_{1},a_{2}>0\implies\lambda_{1}=\lambda_{2}=0$, then we get

$a_{1}=\frac{\beta_{1}r}{c_{1}}-a_{2}-1,\quad
a_{2}=\frac{\beta_{2}r}{c_{2}}-a_{1}-1$

Solving, we get $\frac{\beta_{1}}{\beta_{2}}=\frac{c_{1}}{c_{2}}$
Case 2: $a_{1}=0,a_{2}>0\implies\lambda_{1}>0,\lambda_{2}=0$, then we have

$a_{2}+1=\frac{\beta_{1}r}{c_{1}+\lambda_{1}},\quad
a_{2}+1=\frac{\beta_{2}r}{c_{2}}$

Solving, we get
$\frac{\beta_{1}}{\beta_{2}}=\frac{c_{1}+\lambda_{1}}{c_{2}}\implies\frac{\beta_{1}}{\beta_{2}}>\frac{c_{1}}{c_{2}}$
Case 3: $a_{1}>0,a_{2}=0\implies\lambda_{1}=0,\lambda_{2}>0$, then we have

$a_{1}+1=\frac{\beta_{1}r}{c_{1}},\quad
a_{1}+1=\frac{\beta_{2}r}{c_{2}+\lambda_{2}}$

Solving, we get
$\frac{\beta_{1}}{\beta_{2}}=\frac{c_{1}}{c_{2}+\lambda_{2}}\implies\frac{\beta_{1}}{\beta_{2}}<\frac{c_{1}}{c_{2}}$
Case 4: $a_{1}=0,a_{2}=0\implies\lambda_{1}>0,\lambda_{2}>0$, This case is not
possible because (0,0) cannot be equilibrium as it will give zero utility.

###  IX-L Proof of Theorem 5

For Case 1: $\frac{\beta_{1}}{\beta_{2}}=\frac{c_{1}}{c_{2}}$, CP’s problem is
given as

$\max_{\beta_{i}\in[0,1],\sum_{i}\beta_{i}\leq
1}(1-\beta_{1}-\beta_{2})r\log\left(\frac{\beta_{1}r}{c_{1}}\right)$
$\implies\max_{\beta_{1}\in[0,1]}\quad\left(1-\left(\frac{c_{1}+c_{2}}{c_{1}}\right)\beta_{1}\right)r\log\left(\frac{\beta_{1}r}{c_{1}}\right)$

The first order optimality condition $\partial U_{CP}/\partial\beta_{1}=0$
then gives:

$\log\left(\frac{\beta_{1}r}{c_{1}}\right)=\frac{1-\left(\frac{c_{1}+c_{2}}{c_{1}}\right)\beta_{1}}{\left(\frac{c_{1}+c_{2}}{c_{1}}\right)\beta_{1}}$

Solving and using definition of LambertW function,we get:

$\beta_{1}=\frac{c_{1}}{(c_{1}+c_{2})W\left(\frac{re}{c_{1}+c_{2}}\right)}$

Using relation $\frac{\beta_{1}}{\beta_{2}}=\frac{c_{1}}{c_{2}}$, we get

$\beta_{2}=\frac{c_{2}}{(c_{1}+c_{2})W\left(\frac{re}{c_{1}+c_{2}}\right)}$
$U_{CP}^{I}=\left(1-\frac{1}{W\left(\frac{re}{c_{1}+c_{2}}\right)}\right)\log\left(\frac{\frac{r}{c_{1}+c_{2}}}{W\left(\frac{re}{c_{1}+c_{2}}\right)}\right)$

Similarly for Case 2:
$\frac{\beta_{1}}{\beta_{2}}=\frac{c_{1}+\lambda_{1}}{c_{2}}$, we get

$\beta_{1}=\frac{c_{1}+\lambda_{1}}{(c_{1}+c_{2}+\lambda)W\left(\frac{re}{c_{1}+c_{2}+\lambda_{1}}\right)}$
$\beta_{2}=\frac{c_{2}}{(c_{1}+c_{2}+\lambda_{1})W\left(\frac{re}{c_{1}+c_{2}+\lambda_{1}}\right)}$
$U_{CP}^{II}=\left(1-\frac{1}{W\left(\frac{re}{c_{1}+c_{2}+\lambda_{1}}\right)}\right)\log\left(\frac{\frac{r}{c_{1}+c_{2}+\lambda_{1}}}{W\left(\frac{re}{c_{1}+c_{2}+\lambda_{1}}\right)}\right)$

and for Case 3: $\frac{\beta_{1}}{\beta_{2}}=\frac{c_{1}}{c_{2}+\lambda_{2}}$,
we get

$\beta_{1}=\frac{c_{1}}{(c_{1}+c_{2}+\lambda_{2})W\left(\frac{re}{c_{1}+c_{2}+\lambda_{2}}\right)}$
$\beta_{2}=\frac{c_{2}+\lambda_{2}}{(c_{1}+c_{2}+\lambda_{2})W\left(\frac{re}{c_{1}+c_{2}+\lambda_{2}}\right)}$
$U_{CP}^{III}=\left(1-\frac{1}{W\left(\frac{re}{c_{1}+c_{2}+\lambda_{2}}\right)}\right)\log\left(\frac{\frac{r}{c_{1}+c_{2}+\lambda_{2}}}{W\left(\frac{re}{c_{1}+c_{2}+\lambda_{2}}\right)}\right)$

As $W(x)$ and $g(x):=\frac{x}{W(x)}$ is increasing in $x$
($\because\frac{\partial g(x)}{\partial x}=\frac{1}{W(x)(1+W(x))}>0$), then we
have

$U_{CP}^{I}>U_{CP}^{II},U_{CP}^{I}>U_{CP}^{III}$

###  IX-M Proof of Theorem 6

Existence: For $\beta\in(0,1)$, we have

$\displaystyle\max_{a_{1},a_{2}\geq 0}\quad$ $\displaystyle\left(\frac{\beta
a_{1}}{a_{1}+a_{2}}r\log(a_{1}+a_{2}+1)-c_{1}a_{1}-d_{1}\right)\times$
$\displaystyle\left(\frac{\beta
a_{2}}{a_{1}+a_{2}}r\log(a_{1}+a_{2}+1)-c_{2}a_{2}-d_{2}\right)$
$\left(\frac{\beta
a_{1}}{a_{1}+a_{2}}r\log(a_{1}+a_{2}+1)-c_{1}a_{1}-d_{1}\right)=F_{1}\text{(say)}$

and

$\left(\frac{\beta
a_{2}}{a_{1}+a_{2}}r\log(a_{1}+a_{2}+1)-c_{2}a_{2}-d_{2}\right)=F_{2}\text{(say)}$

Only possible solution to the NBP is $a_{1},a_{2}>0$, because $a_{1}=0\implies
F_{1}<0$ and $a_{2}=0\implies F_{2}<0$. Then, the NBS exists if and only if
$F_{1}>0$ and $F_{2}>0$.
First consider $F_{1}>0$

$\displaystyle\iff$ $\displaystyle\frac{\beta
a_{1}}{a_{1}+a_{2}}r\log(a_{1}+a_{2}+1)>d_{1}+c_{1}a_{1}$ $\displaystyle\iff$
$\displaystyle\frac{\beta}{a_{1}+a_{2}}r\log(a_{1}+a_{2}+1)>\frac{d_{1}}{a_{1}}+c_{1}\quad(a_{1}>0)$
$\displaystyle\iff$
$\displaystyle\beta>\frac{a_{1}+a_{2}}{\log(a_{1}+a_{2}+1)}\left(\frac{d_{1}}{a_{1}r}+\frac{c_{1}}{r}\right)$

Since, $\beta\in(0,1)$, the above inequality holds iff

$\displaystyle\frac{a_{1}+a_{2}}{\log(a_{1}+a_{2}+1)}\left(\frac{d_{1}}{a_{1}r}+\frac{c_{1}}{r}\right)$
$\displaystyle<1$ $\displaystyle\iff\frac{d_{1}}{a_{1}r}+\frac{c_{1}}{r}<$
$\displaystyle\frac{\log(a_{1}+a_{2}+1)}{a_{1}+a_{2}}$
$\displaystyle\iff\frac{c_{1}}{r}<$
$\displaystyle\frac{\log(a_{1}+a_{2}+1)}{a_{1}+a_{2}}-\frac{d_{1}}{a_{1}r}$

Similarly, $F_{2}>0$

$\displaystyle\iff\frac{c_{2}}{r}$
$\displaystyle<\frac{\log(a_{1}+a_{2}+1)}{a_{1}+a_{2}}-\frac{d_{2}}{a_{2}r}$

Since, $c_{1}<c_{2}$, we get

$\frac{c_{1}}{r}<\frac{\log(a_{1}+a_{2}+1)}{a_{1}+a_{2}}-\frac{d_{2}}{a_{2}r}$

Now, $\frac{\log(a_{1}+a_{2}+1)}{a_{1}+a_{2}}<1$, also $c_{1}/r<1\forall
i=1,2$. Therefore, from above inequality, it can be easily concluded that
there will exist $a_{1},a_{2}>0$ which satisfies this condition.
Uniqueness: Let $(\overline{a}_{1},\overline{a}_{2})$ and
$(\hat{a}_{1},\hat{a}_{2})$ be two different Nash Bargaining solutions, i.e,
two different optimizers of NBP. Then, we should have

$F_{1}(\overline{a}_{1},\overline{a}_{2})F_{2}(\overline{a}_{1},\overline{a}_{2})=F_{1}(\hat{a}_{1},\hat{a}_{2})F_{2}(\hat{a}_{1},\hat{a}_{2})$
$\displaystyle\implies\left(\frac{\beta
r}{\overline{a}_{1}+\overline{a}_{2}}r\log(\overline{a}_{1}+\overline{a}_{2}+1)\right)^{2}\overline{a}_{1}\overline{a}_{2}$
$\displaystyle-
c_{2}\frac{\beta\overline{a}_{1}}{\overline{a}_{1}+\overline{a}_{2}}r\log(\overline{a}_{1}+\overline{a}_{2}+1)$
$\displaystyle-
d_{2}\frac{\beta\overline{a}_{1}}{\overline{a}_{1}+\overline{a}_{2}}r\log(\overline{a}_{1}+\overline{a}_{2}+1)$
$\displaystyle-
c_{1}\frac{\beta\overline{a}_{2}}{\overline{a}_{1}+\overline{a}_{2}}r\log(\overline{a}_{1}+\overline{a}_{2}+1)$
$\displaystyle-
d_{1}\frac{\beta\overline{a}_{2}}{\overline{a}_{1}+\overline{a}_{2}}r\log(\overline{a}_{1}+\overline{a}_{2}+1)$
$\displaystyle+c_{1}c_{2}\overline{a}_{1}\overline{a}_{2}+c_{1}d_{2}\overline{a}_{1}+c_{2}d_{1}\overline{a}_{2}+d_{1}d_{2}$
$\displaystyle=\left(\frac{\beta
r}{\hat{a}_{1}+\hat{a}_{2}}r\log(\hat{a}_{1}+\hat{a}_{2}+1)\right)^{2}\hat{a}_{1}\hat{a}_{2}$
$\displaystyle-
c_{2}\frac{\beta\hat{a}_{1}}{\hat{a}_{1}+\hat{a}_{2}}r\log(\hat{a}_{1}+\hat{a}_{2}+1)$
$\displaystyle-
d_{2}\frac{\beta\hat{a}_{1}}{\hat{a}_{1}+\hat{a}_{2}}r\log(\hat{a}_{1}+\hat{a}_{2}+1)$
$\displaystyle-
c_{1}\frac{\beta\hat{a}_{2}}{\hat{a}_{1}+\hat{a}_{2}}r\log(\hat{a}_{1}+\hat{a}_{2}+1)$
$\displaystyle-
d_{1}\frac{\beta\hat{a}_{2}}{\hat{a}_{1}+\hat{a}_{2}}r\log(\hat{a}_{1}+\hat{a}_{2}+1)$
$\displaystyle+c_{1}c_{2}\hat{a}_{1}\hat{a}_{2}+c_{1}d_{2}\hat{a}_{1}+c_{2}d_{1}\hat{a}_{2}+d_{1}d_{2}$

Substituting the values and comparing the coefficient of terms on both the
sides, we get

$c_{1}d_{2}\overline{a}_{1}=c_{1}d_{2}\hat{a_{1}}\implies\overline{a}_{1}=\hat{a}_{1}$
$c_{2}d_{1}\overline{a}_{2}=c_{2}d_{1}\hat{a_{2}}\implies\overline{a}_{2}=\hat{a}_{2}$

###  IX-N Proof of Theorem 9

We use superscript $Pr$ and $Pu$ for the case where both ISPs are private and
one ISP is public, respectively.
We have $\beta^{Pr}=\frac{1}{W\left(\frac{re}{c_{1}+c_{2}}\right)}$ and
$\beta^{Pu}=\frac{1}{W\left(\frac{re}{c_{2}}\right)}$
Since, $W(x)$ is slowly increasing function in $x$, we have

${W\left(\frac{re}{c_{1}+c_{2}}\right)}<{W\left(\frac{re}{c_{2}}\right)}\implies\beta^{Pr}>\beta^{Pu}$
(23)

The total effort
$TA^{Pr}=\frac{\frac{r}{c_{2}}}{W\left(\frac{re}{c_{2}}\right)}-1$ and
$TA^{Pu}=\frac{\frac{r}{c_{1}+c_{2}}}{W\left(\frac{re}{c_{1}+c_{2}}\right)}-1$
Now, as $g(x):=\frac{x}{W(x)}$ is increasing in $x$, then we have

$TA^{Pu}>TA^{Pr}$ (24)

We can write the CP utility as $U_{CP}=(1-\beta)\log\left(TA+1\right)$ Now,
using relations (23) and (24), we get

$U_{CP}^{Pu}>U_{CP}^{Pr}$

## Proof of Theorem LABEL:ISP1

$\displaystyle U_{ISP_{2}}\geq U_{ISP_{1}}\quad\text{iff}$
$\displaystyle\frac{c_{2}r}{(c_{1}+c_{2})}\left(1-\frac{2}{W\left(\frac{r}{c_{1}+c_{2}}e\right)}\right)+c_{2}\hat{a}_{1}^{*}+c_{2}\geq\frac{c_{1}r}{(c_{1}+c_{2})}\left(1-\frac{1}{W\left(\frac{r}{c_{1}+c_{2}}e\right)}\right)-c_{1}a_{1}^{*}$
$\displaystyle\iff(c_{1}-c_{2})a_{1}^{*}\geq\frac{r}{c_{1}+c_{2}}\left[c_{1}\left(1-\frac{1}{W\left(\frac{r}{c_{1}+c_{2}}e\right)}\right)-c_{2}\left(1-\frac{2}{W\left(\frac{r}{c_{1}+c_{2}}e\right)}\right)\right]-c_{2}$
$\displaystyle\iff
a_{1}^{*}\geq\frac{r}{c_{1}^{2}-c_{2}^{2}}\left[c_{1}\left(1-\frac{1}{W\left(\frac{r}{c_{1}+c_{2}}e\right)}\right)-c_{2}\left(1-\frac{2}{W\left(\frac{r}{c_{1}+c_{2}}e\right)}\right)\right]-\frac{c_{2}}{c_{1}-c_{2}}$

## Proof of Theorem LABEL:thm:beta-2

Here CP’s problem reduces to

$\max_{\beta\in[0,1]}\quad(1-\beta)r\log\left(\frac{\beta r}{c_{2}}\right)$

Solving first order optimality condition, we get

$\beta=\frac{1}{W\left(\frac{re}{c_{2}}\right)}$

## Proof of Theorem LABEL:thm:nbs-1

The first order optimality condition for the problem is given by

$\displaystyle\frac{\partial}{\partial\beta_{1}}\\{\beta_{1}r\log(a_{1}+a_{2}+1)-c_{1}a_{1}-d_{1}\\}$
$\displaystyle\\{(\beta-\beta_{1})r\log(a_{1}+a_{2}+1)-c_{2}a_{2}-d_{2}\\}=0$
$\implies
r\log(a_{1}+a_{2}+1)\\{(\beta-2\beta_{1})r^{2}\log^{2}(a_{1}+a_{2}+1)-c_{2}a_{2}-d_{2}+c_{1}a_{1}+d_{1}\\}=0$
$\implies\beta_{1}=\frac{\beta}{2}+\frac{c_{1}a_{1}-c_{2}a_{2}+d_{1}-d_{2}}{2r\log(a_{1}+a_{2}+1)}$

and

$\beta_{2}=\beta-\beta_{1}=\frac{\beta}{2}-\frac{c_{1}a_{1}-c_{2}a_{2}+d_{1}-d_{2}}{2r\log(a_{1}+a_{2}+1)}$

## Proof of Theorem LABEL:thm:SV-1

Coalition $\\{ISP_{1}\\}$
Problem is given by
CP:

$\max_{\beta\in[0,1]}\quad(1-\beta)r\log(a_{1}^{*}+1)$

$ISP_{1}$ chooses $a_{1}^{*}$ such that its utility is maximized, i.e.,

$\max_{a_{1}^{*}}\quad\beta r\log(a_{1}^{*}+1)-c_{1}a_{1}^{*}$

The first order optimality condition for $ISP_{1}$ gives

$a_{1}^{*}=\frac{\beta r}{c_{1}}-1$

The best response of CP to $ISP_{1}$’s strategy is then given by

$\max_{\beta}\quad\beta r\log\left(\frac{\beta
r}{c_{1}}\right)\implies\beta=\frac{1}{W\left(\frac{re}{c_{1}}\right)}\\\
Therefore,\begin{aligned}
v(ISP_{1})&=\frac{r}{W\left(\frac{re}{c_{1}}\right)}\log\left(\frac{r}{(c_{1})W\left(\frac{re}{c_{1}}\right)}\right)-c_{1}\left(\frac{r}{c_{1}W\left(\frac{re}{c_{1}}\right)}-1\right)\\\
&=\frac{r}{W\left(\frac{re}{c_{1}}\right)}\left[W\left(\frac{re}{c_{1}}\right)-1\right]-\frac{r}{W\left(\frac{re}{c_{1}}\right)}+c_{1}\\\
&=r\left(1-\frac{2}{W\left(\frac{re}{c_{1}}\right)}\right)+c_{1}\end{aligned}\textbf{Coalition
$\\{ISP_{2}\\}$}\\\ Problemisgivenby\\\ CP:\\\ $

max_β∈[0,1] (1-β)rlog(a_2+1)

max_a_2 βrlog(a_2+1)-c_2a_2

$ThebestresponsebyISP_{2}inresponsetothecontractofferedbyCPisgivenby$

a_2=βrc2-1

$Now,problemforCPreducesto$

max_β∈[0,1] (1-β)rlog(βrc2)

$thisgives$

β_2=1W(rec2)

$Therefore,\begin{aligned}
v(ISP_{2})&=\frac{r}{W\left(\frac{re}{c_{2}}\right)}\log\left(\frac{r}{(c_{2})W\left(\frac{re}{c_{2}}\right)}\right)-c_{2}\left(\frac{r}{c_{2}W\left(\frac{re}{c_{2}}\right)}-1\right)\\\
&=\frac{r}{W\left(\frac{re}{c_{2}}\right)}\left[W\left(\frac{re}{c_{2}}\right)-1\right]-\frac{r}{W\left(\frac{re}{c_{2}}\right)}+c_{2}\\\
&=r\left(1-\frac{2}{W\left(\frac{re}{c_{2}}\right)}\right)+c_{2}\end{aligned}\textbf{Coalition
$\\{ISP_{1},ISP_{2}\\}$
:}BothISPjointlymaximizetheirprofittodeterminetotalefforta_{1}^{*}+a_{2},wherea_{1}^{*}isfixed.ThenproblematISPlevelisgivenby$

max_a_2 βrlog(a_1^*+a_2+1)-c_1a_1^*-c_2a_2

$Thefirstorderoptimalityconditiongives$

a_2=βrc2-a_1^*-1

$thenCP^{\prime}sproblemreducesto$

max_β∈[0,1] (1-β)rlog(βrc2)

$Solvingfirstorderoptimalitycondition,weget$

β=1W(rec2)

$Thus,wehave$

a_2=rc2W(rec2)-a_1^*-1

$Then,\begin{aligned}
v({ISP_{1}},ISP_{2})&=\frac{r}{W\left(\frac{re}{cc_{2}}\right)}\log\left(\frac{2r}{c_{2}W\left(\frac{re}{c_{2}}\right)}\right)-c_{1}a_{1}^{*}-c_{2}\left(\frac{r}{c_{2}W\left(\frac{re}{c_{2}}\right)}-a_{1}^{*}-1\right)\\\
&=\frac{r}{W\left(\frac{re}{c_{2}}\right)}\left[\log\left(\frac{re}{c_{2}W\left(\frac{re}{c_{2}}\right)}\right)-\log(e)\right]-\frac{r}{W\left(\frac{re}{c_{2}}\right)}-c_{1}a_{1}^{*}+c_{2}a_{1}^{*}+c_{2}\\\
&=\frac{r}{W\left(\frac{e}{c_{2}}\right)}\left[W\left(\frac{re}{c_{2}}\right)-1\right]-\frac{r}{W\left(\frac{re}{c_{2}}\right)}+a_{1}^{*}(c_{2}-c_{1})+c_{2}\\\
&=r\left(1-\frac{2}{W\left(\frac{re}{c_{2}}\right)}\right)+a_{1}^{*}(c_{2}-c_{1})+c_{2}\end{aligned}\par
Now,\begin{aligned}
\Phi_{ISP_{1}}&=\frac{0!.(2-0-1)!}{2!}\\{v(ISP_{1})-v(\phi)\\}+\frac{1!.(2-1-1)!}{2!}\\{v(ISP_{1},ISP_{2})-v(ISP_{2})\\}\\\
&=\frac{1}{2}\\{v(ISP_{1})-v(\phi)\\}+\frac{1}{2}\\{v(ISP_{1},ISP_{2})-v(ISP_{2})\\}\\\
&=\frac{1}{2}\left[r\left(1-\frac{2}{W\left(\frac{re}{c_{1}}\right)}\right)+c_{1}\right]\\\
&+\frac{1}{2}\left[r\left(1-\frac{2}{W\left(\frac{re}{c_{2}}\right)}\right)+a_{1}^{*}(c_{2}-c_{1})+c_{2}-r\left(1-\frac{2}{W\left(\frac{re}{c_{2}}\right)}\right)-c_{2}\right]\\\
&=\frac{1}{2}\left[r\left(1-\frac{2}{W\left(\frac{re}{c_{1}}\right)}\right)+c_{1}+a_{1}^{*}(c_{1}-c_{1})\right]\end{aligned}\par\begin{aligned}
\Phi_{ISP_{2}}&=\frac{0!.(2-0-1)!}{2!}\\{v(ISP_{2})-v(\phi)\\}+\frac{1!.(2-1-1)!}{2!}\\{v(ISP_{1},ISP_{2})-v(ISP_{1})\\}\\\
&=\frac{1}{2}\\{v(ISP_{2})-v(\phi)\\}+\frac{1}{2}\\{v(ISP_{1},ISP_{2})-v(ISP_{1})\\}\\\
&=\frac{1}{2}\left[r\left(1-\frac{2}{W\left(\frac{re}{c_{2}}\right)}\right)+c_{2}\right]\\\
&+\frac{1}{2}\left[r\left(1-\frac{2}{W\left(\frac{re}{c_{2}}\right)}\right)+a_{1}^{*}(c_{2}-c_{1})+c_{2}-r\left(1-\frac{2}{W\left(\frac{re}{c_{1}}\right)}\right)-c_{1}\right]\\\
&=\frac{1}{2}\left[r\left(1-\frac{4}{W\left(\frac{re}{c_{2}}\right)}+\frac{2}{W\left(\frac{re}{c_{1}}\right)}\right)+(a_{1}^{*}+1)(c_{2}-c_{1})\right]\end{aligned}\par\par$

## Proof of Theorem LABEL:thm:cmparision-1

Part 1: Total Effort when ISPs don’t cooperate is given by

${TA}^{NC}=\frac{\frac{r}{(c_{1}+c_{2})}}{W\left(\frac{re}{(c_{1}+c_{2})}\right)}$

Total Effort when ISPs cooperate is given by

$TA^{C}=\frac{\frac{r}{c_{2}}}{W\left(\frac{re}{c_{2}}\right)}$

By Using LambertW property, it can be easily shown that $\frac{x}{W(x)}$ is
increasing in $x$. Therefore, using this we get

$\frac{\frac{r}{c_{2}}}{W\left(\frac{re}{c_{2}}\right)}\geq\frac{\frac{r}{(c_{1}+c_{2})}}{W\left(\frac{re}{(c_{1}+c_{2})}\right)}$

Part 2: Now, CP Utility when ISPs don’t cooperate is given by

$U_{CP}^{NC}=\frac{\left(W\left(\frac{re}{c_{1}+c_{2}}\right)-1\right)^{2}}{W\left(\frac{re}{c_{1}+c_{2}}\right)}$

And, CP Utility when ISPs cooperate is given by

$U_{CP}^{C}=\frac{\left(W\left(\frac{re}{c_{2}}\right)-1\right)^{2}}{W\left(\frac{re}{c_{2}}\right)}$

Using Labertw property, it can be shown that $\frac{(W(x)-1)^{2}}{W(x)}$ is
decreasing in $x$, so we get

$\frac{\left(W\left(\frac{re}{c_{2}}\right)-1\right)^{2}}{W\left(\frac{re}{c_{2}}\right)}\geq\frac{\left(W\left(\frac{re}{c_{1}+c_{2}}\right)-1\right)^{2}}{W\left(\frac{re}{c_{1}+c_{2}}\right)}$