# Profit and Strategic Analysis for MNO-MVNO Partnership

Nesrine Ben Khalifa, Amal Benhamiche, Alain Simonian, Marc Bouillon  Orange
Labs, France
Email<EMAIL_ADDRESS>
###### Abstract

We consider a mobile market driven by two Mobile Network Operators (MNOs) and
a new competitor Mobile Virtual Network Operator (MVNO). The MNOs can partner
with the entrant MVNO by leasing network resources; however, the MVNO can also
rely on other technologies such as free WiFi access points. Moreover, in
addition to its connectivity offer, the MVNO can also draw indirect revenues
from services due to its brand. In that framework including many access
technologies and several revenue sources, a possible partner MNO will then
have to decide which wholesale price to charge the MVNO for its resources.
This multi-actor context, added to the need to consider both wholesale and
retail markets, represents a new challenge for the underlying decision-making
process. In this paper, the optimal price setting is formulated as a multi-
level optimization problem which enables us to derive closed-form expressions
for the optimal MNOs wholesale prices and the optimal MVNO retail price. The
price attractivity of the MVNO is also evaluated in terms of its indirect
revenues and the proportion of resources leased from possible partner MNOs.
Finally, through a game-theoretical approach, we characterize the scenario
where both MNOs partner with the MVNO as the unique Nash equilibrium under
appropriate conditions.

###### Index Terms:

Network Economics, Strategic Partnership, Multi-Level Optimization, Non-
Cooperative Game Theory

††publicationid: pubid: ISBN 978-3-903176-08-9 © 2018 IFIP

##  I Introduction

Mobile telecommunication markets are usually covered by a few number of
operators because of high infrastructure and spectrum license costs. In
addition, mobile operators are facing the challenges of upgrading their
networks to 5G technology in order to cope with the increasing demand of
users’ traffic. This context, as well as the virtualization of wireless
networks, may lower the barrier for the entrance of Mobile Virtual Network
Operators (MVNO) to the market. While not possessing their own network
infrastructure, MVNOs can lease capacities from Mobile Network Operators
(MNOs) on the wholesale market to ensure wireless services to their customers.

A new generation of MVNOs has recently emerged, capable of leasing resources
from different MNOs while also taking advantage of the available WiFi hotspots
in order to propose a full connectivity offer to their customers. The latter
can therefore afford to blindly use multiple network/technology services to
establish the communications and use mobile Internet without any specific
setting. Typically, such MVNOs can launch their activity thanks to specific
inter-operator partnerships with MNOs for the cellular infrastructure
utilization while the WiFi offloading part remains a unilateral decision. A
recent example is offered by Google through its Project-Fi [1], launched in
the U.S. in partnership with three leading MNOs, namely Sprint, T-Mobile and
U.S. Cellular. This illustrates the case of a competitive MVNO able to absorb
a significant part of the retail market (on mobile and data services) from the
MNOs without being an expert in networking or even possessing the physical
infrastructures and the ability to manage them.

On the other hand, an entrant MVNO can be already positioned at a higher place
in the Value Chain and have income from its new customers through its current
“non telco” activity. In fact, this MVNO may have a good reputation as an OTT
(”Over the Top”) providing content or other high level services; it can thus
draw significant additional revenue, hereafter termed as indirect to
differentiate it from the telco offer, from its new customers.

This new multi-technological and multi-activity context therefore raises new
questions as to the interactions between MNOs and such a MVNO, namely the
optimal price setting adopted by all actors and the consequences on the
respective market shares and profits. The economic viability of a possible
partnership between MNOs and the MVNO should, in particular, be understood on
account of the existence of alternative technologies and other sources of
revenues.

###  I-A State-of-the-art

The ecosystem of MNO-MVNO relationships has been addressed in many studies. A
detailed description of the MVNOs’ classes in terms of their dependence to the
host operator is presented in [2], where it is shown that a MVNO can be
classified as either light or full. The authors describe, in particular, the
possible business models of MVNOs and examine the impact of different
parameters such as the MNO’s market share on the outcome of cooperation
between the host and virtual operators. In [3], the authors analyze the
incentives for MNOs to form a strategic cooperation with MVNOs. They
specifically examine the effects of the brand appeal of the MVNO and the
wholesale discount offered by the MNO on the fulfillment of mutually
beneficial partnerships.

Besides, the economic viability of MNO-MVNO relationship has been investigated
in [4, 5] and references therein. First in [4], it is argued that the business
of a MVNO may be profitable in a transitory phase when it partners with MNOs
with a small market share; however, it is shown in [5] that, in the long term,
the MVNO is better off when it preferably partners with a big MNO, say, the
incumbent. The latter point of view of a stabilized market and mature economic
actors will be adopted in the sequel.

A multi-stage game for modeling the MNO-MVNO interaction is presented in [6]
where the MNO investment, the MVNO’s decision on the leasing from the MNO and
the retail pricing are successively focused on at each stage. In [7], spectrum
leasing and pricing are studied with game-theoretic models. A comprehensive
study of the market share between MNO and MVNO based on the brand appeal of
the MNO is provided in [8] where the authors also use game theory tools.

A recent study of Google-Fi like MVNOs is provided in [9] where the price
setting between multiple service providers and the virtual operator is
examined. In that paper, the user defection rate to the MVNO is assumed to be
simply constant with no account of the impact of the MVNO price on its
customer base; further, the authors only optimize the MVNO price for given
wholesale MNO prices.

In contrast, we will here consider a more accurate economic model wherein (1)
using the so-called price-demand elasticity, the users reply to the MVNO offer
depends on the varying price difference between the MVNO offer and that of the
other MNOs; (2) beside the search of an optimal retail price for the MVNO, we
also determine the wholesale MNO prices by maximizing their respective profit.

###  I-B Addressed issues and contributions

In this paper, we make a thorough economic analysis of strategic MNO-MVNO
partnership. We consider the upcoming of a new MVNO, a potential competitor
proposing low-cost services and threatening the MNO market share.
Specifically, we address the following questions:

  * $\bullet$

When a MNO decides to conclude a partnership with the MVNO, what is the best
price setting for the partner MNO in order to maximize its profit?

  * $\bullet$

When the wholesale prices are fixed by partner MNOs, what is the optimal price
that should be charged by the MVNO to its customers ?

  * $\bullet$

What is the impact of the MVNO’s indirect revenues on its optimal retail price
?

  * $\bullet$

What is the best decision for the MNOs facing the entry of the MVNO?

In this aim, the optimal price setting is addressed via a Stackelberg Game
involving leaders and followers [5] (Section 2.3.6). In particular, we define
and study two different decision-making models for the optimal wholesale price
setting, namely a fully sequential model and a partially sequential model; we
then determine the optimal retail price of the MVNO. Furthermore, we study the
impact of the MVNO’s indirect revenue on its optimal retail price in both
decision-making models. Finally, we propose a game-theoretical approach to
determine the Nash equilibrium under sufficient conditions on this indirect
revenue and discuss its economic interpretation.

###  I-C Paper structure

In Section II, we formally introduce the economic model describing the
interactions between MNOs and the MVNO. In Section III, we formulate the price
setting problem and study the wholesale and retail price optimization for all
actors. A game-theoretical setting is discussed in Section IV. In Section V,
we comment our general results on economic grounds. Finally, some concluding
remarks are given in Section VI.

##  II Economic Model and Assumptions

We consider a mobile operator market composed of two MNOs that share the whole
customer base. The upcoming of a new MVNO proposing an attractive price
impacts the repartition of customers who can defect from their original
operators to the benefit of this MVNO. The MNOs must then identify the best
decisions they are able to make, that is, to decide whether to partner with
the MVNO. An operator that would decide to contract a partnership with the
MVNO would obviously not be immune to lose customers but might, nevertheless,
recover a part of the lost retail revenue via the wholesale income. Given this
ecosystem, the economic variables of the MNOs and MVNO activity can be
described as follows:

MNO’s profit: the profit of a MNO is the sum of its revenues obtained from the
retail market for end-users (and possibly from the Business-to-Business
wholesale market for partner MNOs) from which are subtracted the corresponding
costs. Denote by $p_{i}$ (resp. $w_{i}$) the unit price of MNO $i$ on the
retail (resp. the wholesale) market; the retail revenues then depend linearly
on the operator’s customer base, while the wholesale revenues depend linearly
on the amount of MVNO traffic accommodated by the MNO’s network. Besides, we
consider unit network costs $c_{i}$, non-network costs $\widetilde{c}_{i}$
(IT, commercial, etc.) and fixed costs $\overline{C}_{i}$; as a linear
approximation, the total network and non-network costs are assumed to depend
linearly on the customer base whereas fixed costs are independent of the
amount of the operator’s customer base;

MVNO’s profit: the total revenue of the MVNO is the sum of direct revenues
obtained from the retail market (with net unit price $p_{0}$) and of indirect
revenues (with unit price $r_{0}$) obtained from e.g. advertising. As to
network costs, we here consider that the MVNO enables its users’ devices
either to automatically connect to a free public WiFi Access Point (AP) or to
a partner MNO’s network, depending on the Quality of Service of each access
mode. The proportion of MVNO traffic handled by free WiFi APs is denoted by
$\gamma\in[0,1[$; the only network costs of the MVNO are therefore those
caused by the other proportion $1-\gamma$ of traffic dealt with MNOs through
partnerships. In addition to these network costs, the MVNO finally incurs non-
network and fixed costs denoted by $\widetilde{c}_{0}$ and $\overline{C}_{0}$,
respectively.

For both MNOs, we assume that long-distance (backhaul and core) network costs
are negligible compared to the access costs; the possible wholesale offer of
any MNO will therefore mainly account for the transportation through its
cellular access network of MVNO traffic.

TABLE I: Key Terms and Symbols Symbol | Definition ($i$ = 1, 2)
---|---
$p_{i}$, $c_{i}$, $\widetilde{c}_{i}$ | Unit retail price, network and non-network costs
| of MNO $i$
$\overline{C}_{i}$ | Fixed costs of MNO $i$
$Q_{i}$ | Customer base of MNO $i$ before the MVNO’s entry
$Q$ | Total customer base
$\pi_{i}=Q_{i}/Q$ | Market share of MNO $i$ before the MVNO’s entry
$Q_{i,0}$ | Customer base of MNO $i$ which defects to the MVNO
$Q_{0}$ | Total customer base of the MVNO
$r_{0}$ | Net unit indirect revenues of the MVNO
$p_{0}$, $\widetilde{c}_{0}$ | Unit retail price, non-network costs of the MVNO
$\overline{C}_{0}$ | Fixed costs of the MVNO
$w_{i}$ | Unit wholesale price charged by MNO $i$ to the MVNO
$\gamma$ | Proportion of MVNO traffic handled by
| free WiFi APs

Users Behavior: the users’ reply to the MVNO’s offer is assumed to depend only
on the relative price of that offer. Specifically, the behavior of users is
modeled according to the price-demand elasticity [10] so that the part of MNO
$i$ customers which defects to the MVNO is expressed by

$\displaystyle
Q_{i,0}=\varepsilon\,Q_{i}\left(\frac{p_{i}-p_{0}}{p_{i}}\right)$ (1)

where $Q_{i}$ denotes the customer base of MNO $i$ before the MVNO joins the
market and $\varepsilon>0$ is the price-demand elasticity coefficient
(although generally depending on prices, $\varepsilon$ is here assumed to be a
constant on the basis of a small price variability range). Without loss of
generality, we assume throughout the paper that $p_{2}\leqslant p_{1}$ and
$p_{0}\leqslant p_{2}$; this ensures that both $Q_{1,0}$ and $Q_{2,0}$ are
non-negative. The MVNO’s total customer base is then

$\displaystyle Q_{0}=Q_{1,0}+Q_{2,0}.$ (2)

MVNO traffic split: in the case when both MNOs partner with the MVNO, the MVNO
traffic which is not supported by free WiFi APs is split between MNOs networks
proportionally to their market share before the MVNO’s upcoming. Consequently,
the traffic transported for the MVNO on MNO $i$ network equals
$(1-\gamma)\pi_{i}Q_{0}$, where $\pi_{i}=Q_{i}/Q$ is the market share of MNO
$i$ before the MVNO’s entry (with $\pi_{1}+\pi_{2}=1$). This is motivated by
the fact that the MVNO will partner with a MNO all the more that the latter
has a large market share.

Note finally that a sample value of parameter $\gamma$ is given by the ratio
of the duration spent on WiFi access to the overall duration of a given
communication session; this ratio can then be averaged over all successive
sessions to provide the mean proportion $\gamma$; the latter is clearly
related to the given geographic density of the WiFi APs.

TABLE II: MVNO Traffic Split Traffic Amount | Accommodation
---|---
$\gamma Q_{0}$ | WiFi APs
$(1-\gamma)\pi_{1}Q_{0}$ | MNO 1’s Cellular BSs
$(1-\gamma)\pi_{2}Q_{0}$ | MNO 2’s Cellular BSs

Tables I and II sum up the notation used in the paper.

##  III Wholesale and Retail Price Setting

In this section, we address the optimization of MNOs and MVNO profits in order
to determine the optimal wholesale and retail prices. Specifically, we
introduce several optimization problems where $w_{i}$, $i\in\\{1,2\\}$, and
$p_{0}$ are the decision variables. In the sequel, we denote by ”Part” the
strategy consisting in partnering with the MVNO and by ”NonPart” the strategy
consisting in not partnering with the MVNO. In a competitive environment where
the MNOs do not collude with each other, two scenarios can be envisaged:

  * $\mathbf{A.}$

Only one operator, either MNO $1$ or MNO $2$, decides to contract with the
MVNO. We denote this scenario by (Part-NonPart) or (NonPart-Part),
respectively;

  * $\mathbf{B.}$

Both operators choose to contract with the MVNO. We denote this scenario by
(Part-Part).

These two scenarios are successively analyzed below.

###  III-A Scenario (Part-NonPart)

Consider first the case when only MNO $i$ proposes a wholesale offer to the
MVNO, while MNO $-i$ decides not to partner with the new entrant (by
convention, $i$ = 1 or 2 implies $-i$ = 2 or 1). In this case, the traffic
$\gamma Q_{0}$ generated by the MVNO’s users will be delivered through free
WiFi APs and the remaining $(1-\gamma)Q_{0}$ will be handled by the partner
MNO’s network. The MVNO’s profit is therefore given by

$\displaystyle{\cal{R}}_{0}(p_{0},w_{i})=(p_{0}+r_{0})Q_{0}-w_{i}(1-\gamma)Q_{0}-\widetilde{c}_{0}Q_{0}-\overline{C}_{0}.$
(3)

The optimal MVNO’s retail price $p_{0}^{*}$ can then be determined by solving
the following optimization problem:

$\displaystyle\underset{{0\leqslant p_{0}\leqslant p_{2}}}{\mbox{ max
}}{\cal{R}}_{0}(p_{0},w_{i}).$ (4)

######  Lemma 1.

In the (Part-NonPart) scenario, given the wholesale price $w_{i}$, the optimal
MVNO retail price equals

$\displaystyle p_{0}^{*}(w_{i})=\min(\widetilde{p}_{0}(w_{i}),p_{2})$ (5)

where

$\displaystyle\widetilde{p}_{0}(w_{i})=\frac{1-\gamma}{2}w_{i}+\frac{Q}{2\mathcal{S}}+\frac{\widetilde{c}_{0}-r_{0}}{2},$
(6)

with $\mathcal{S}=Q_{1}/p_{1}+Q_{2}/p_{2}$.

###### Proof.

First observe that, in view of definitions (1) and (2), $Q_{0}$ involved in
expression (3) is a linear function of variable $p_{0}$, thus making the
profit $\mathcal{R}_{0}$ a quadratic function of $p_{0}$. Given the price
$w_{i}$, the first order optimality condition
${\partial{\cal{R}}_{0}}(p_{0},w_{i})/{\partial p_{0}}=0$ for problem (4) then
provides the critical point $\widetilde{p}_{0}(w_{i})$ as given in (6).
Besides, the second derivative
$\partial^{2}{\cal{R}}_{0}(p_{0},w_{i})/\partial p_{0}^{2}$ is equal to the
negative constant $-2\varepsilon\mathcal{S}$; ${\cal{R}}_{0}(p_{0},w_{i})$ is
therefore a strictly concave function of $p_{0}$ with a unique maximum at
price $p_{0}^{*}(w_{i})$ given by (5). ∎

Now, we consider the profit of MNO $i$ given by

$\displaystyle{\cal{R}}_{i}(p_{0},w_{i})=$
$\displaystyle\,p_{i}(Q_{i}-Q_{i,0})+w_{i}(1-\gamma)Q_{0}\;-$
$\displaystyle\,c_{i}(Q_{i}-Q_{i,0}+(1-\gamma)Q_{0})\;-$
$\displaystyle\,\widetilde{c}_{i}(Q_{i}-Q_{i,0})-\overline{C}_{i},$

that is,

$\displaystyle{\cal{R}}_{i}(p_{0},w_{i})=h_{i}(Q_{i}-Q_{i,0})+(w_{i}-c_{i})(1-\gamma)Q_{0}-\overline{C}_{i}$
(7)

where $h_{i}=p_{i}-c_{i}-\widetilde{c}_{i}\geqslant$ 0\. The partner MNO $i$
seeks to maximize its profit ${\cal{R}}_{i}$. To this end, we replace $p_{0}$
involved in expression (7) via $Q_{i,0}$ and $Q_{0}$ by its optimal value
$p_{0}^{*}(w_{i})$ derived in Lemma 1; in fact, the MNO anticipates the best
pricing strategy of the MVNO and thus sets its optimal wholesale price based
on this anticipation. Define then

$\displaystyle\mathcal{R}_{i}^{*}(w_{i})={\mathcal{R}}_{i}(p_{0}^{*}(w_{i}),w_{i})$
(8)

which is obtained through (7) with $Q_{i,0}=\varepsilon
Q_{i}(p_{i}-p_{0}^{*}(w_{i}))/p_{i}$ and $Q_{0}$ given by (2). The
optimization problem for MNO $i$ can then be expressed by

$\displaystyle\underset{{w_{i}\geqslant 0}}{\mbox{ max
}}\mathcal{R}_{i}^{*}(w_{i}).$ (9)

######  Proposition 1.

In the (Part-NonPart) scenario, the optimal MNO’s wholesale price equals

$\displaystyle\widehat{w}_{i}=\min(\overline{w}_{i},\widetilde{w}_{i})$ (10)

where we set

$\displaystyle\overline{w}_{i}=\frac{1}{1-\gamma}\left(2p_{2}-\frac{Q}{\mathcal{S}}+r_{0}-\widetilde{c}_{0}\right)$
(11)

and

$\displaystyle\widetilde{w}_{i}=\frac{c_{i}}{2}+\frac{1}{1-\gamma}\left(\frac{h_{i}Q_{i}}{2p_{i}\mathcal{S}}+\frac{Q}{2\mathcal{S}}+\frac{r_{0}-\widetilde{c}_{0}}{2}\right).$
(12)

The optimal MVNO’s retail price is then determined by

$\displaystyle\widehat{p}_{0}(\widehat{w}_{i})=\min(\widetilde{p}_{0}(\widehat{w}_{i}),p_{2}).$
(13)

Defining the constant

$\displaystyle\overline{r}_{i,0}=\frac{h_{i}Q_{i}}{p_{i}\mathcal{S}}+c_{i}(1-\gamma)+\frac{3Q}{\mathcal{S}}+\widetilde{c}_{0}-4p_{2},$
(14)

we then have $\overline{w}_{i}\leqslant\widetilde{w}_{i}\iff
p^{*}_{0}(\widehat{w}_{i})=p_{2}\iff r_{0}\leqslant\overline{r}_{i,0}$.

###### Proof.

We consider the two cases $(a)$ $\widetilde{p}_{0}(w_{i})>p_{2}$ and $(b)$
$\widetilde{p}_{0}(w_{i})\leqslant p_{2}$. First consider case $(a)$. This
corresponds to values of $w_{i}$ such that $w_{i}>\overline{w}_{i}$ where
$\overline{w}_{i}$ is given by (11). Relation (5) then yields
$p_{0}^{*}(w_{i})=p_{2}$ and by (8), we easily show that
$\mathcal{R}_{i}^{*}(w_{i})={\mathcal{R}}_{i}(p_{2},w_{i})$ is a linear and
increasing function of $w_{i}$. The optimal value of $\mathcal{R}_{i}^{*}$ is
thus obtained at $w_{i}=+\infty$; but this unbounded price giving the value
$-\infty$ for $\mathcal{R}_{0}$, case $(a)$ is thus eventually excluded.

Now consider case $(b)$. This corresponds to values of $w_{i}$ such that
$w_{i}\leqslant\overline{w}_{i}$. Relation (5) now yields
$p_{0}^{*}(w_{i})=\widetilde{p}_{0}(w_{i})$; using (8) again and writing now

$Q_{i,0}=\varepsilon\frac{Q_{i}}{p_{i}}(p_{i}-\widetilde{p}_{0}(w_{i})),\quad
Q_{0}=Q_{1,0}+Q_{2,0}$

as functions of $w_{i}$,
$\mathcal{R}_{i}^{*}(w_{i})={\mathcal{R}}_{i}(\widetilde{p}_{0}(w_{i}),w_{i})$
can then be easily expressed as a quadratic function of $w_{i}$. The first
order optimality condition $\partial\mathcal{R}_{i}^{*}(w_{i})/\partial
w_{i}=0$ for problem (9) yields the critical point $\widetilde{w}_{i}$ as
given in (12). Besides, the second derivative
$\partial^{2}\mathcal{R}_{i}^{*}(w_{i})/\partial w_{i}^{2}$ is equal to the
negative constant $-\varepsilon(1-\gamma)^{2}\mathcal{S}/2$. Therefore,
$\mathcal{R}_{i}^{*}(w_{i})$ is a strictly concave function of $w_{i}$ with a
unique maximum at $\widehat{w}_{i}$ given by (10).

The optimal MVNO’s retail price is obtained by replacing $w_{i}$ in (5) by
$\widehat{w}_{i}$ given in (10), hence (13). Finally, elementary algebra
reduces condition $\overline{w}_{i}\leqslant\widetilde{w}_{i}$ to
$r_{0}\leqslant\overline{r}_{i,0}$, with $\overline{r}_{i,0}$ given by (14). ∎

All previous results symmetrically hold for the (NonPart-Part) scenario.

###  III-B Scenario (Part-Part)

We now turn to the situation where both MNOs partner with the MVNO. Two models
can be proposed depending on the order in which decisions are taken, namely:

  * $\bullet$

A Fully Sequential (FS) model where a leader MNO (say, the one with the
highest market share) first chooses its wholesale price; then the second MNO,
the follower, determines its wholesale price accordingly; finally, the MVNO
chooses its retail price. This situation is illustrated in Fig. 1-Left.

  * $\bullet$

A Partially Sequential (PS) model where first the two MNOs (say, with
comparable weights) choose their respective wholesale price without
coordination; then, the MVNO chooses its retail price. This situation is
illustrated in Fig. 1-Right.

Leader MNO $i$ chooses $w_{i}$ MNO $-i$ chooses $w_{-i}$ MVNO chooses $p_{0}$

MNO $1$ chooses $w_{1}$ MNO $2$ chooses $w_{2}$ MVNO chooses $p_{0}$

Figure 1: Hierarchical decision models. Left,  Fully sequential model. Right,
Partially sequential model.

In both models, we consider that the MNOs move before the MVNO because they
own the resources to lease to the latter. They forecast the MVNO’s reply to
their pricing strategies and choose the wholesale prices that maximize their
profits, given the anticipated MVNO’s optimal pricing strategy. The MVNO
eventually chooses its retail price, given the wholesale prices fixed by the
MNOs.

The MVNO’s profit is now given by

$\displaystyle{\cal{R}}_{0}(p_{0},w_{1},w_{2})$ $\displaystyle=$
$\displaystyle(p_{0}+r_{0})Q_{0}-w_{1}(1-\gamma)\pi_{1}Q_{0}-$ (15)
$\displaystyle
w_{2}(1-\gamma)\pi_{2}Q_{0}-\widetilde{c}_{0}Q_{0}-\overline{C}_{0}$

and the optimization problem of the MVNO is formulated by

$\displaystyle\underset{{0\leqslant p_{0}\leqslant p_{2}}}{\mbox{ max
}}{\cal{R}}_{0}(p_{0},w_{1},w_{2}).$ (16)

######  Lemma 2.

In the (Part-Part) scenario, given the wholesale prices $w_{1}$ and $w_{2}$,
the optimal retail price of the MVNO equals

$\displaystyle
p_{0}^{*}(w_{1},w_{2})=\min(\widetilde{p}_{0}(w_{1},w_{2}),p_{2})$ (17)

where

$\displaystyle\widetilde{p}_{0}(w_{1},w_{2})=\frac{(1-\gamma)}{2}\left(\pi_{1}w_{1}+\pi_{2}w_{2}\right)+\frac{Q}{2\mathcal{S}}+\frac{\widetilde{c}_{0}-r_{0}}{2}.$
(18)

###### Proof.

Similar to that of Lemma 1. ∎

In this section, the notations for profit $\mathcal{R}_{i}$ should not be
confused with that of Section III-A. Now, consider the profit of MNO
$i\in\\{1,2\\}$ given by

$\displaystyle{\cal{R}}_{i}(p_{0},w_{i})$ $\displaystyle=$ $\displaystyle
p_{i}(Q_{i}-Q_{i,0})+w_{i}(1-\gamma)\pi_{i}Q_{0}-$ $\displaystyle
c_{i}(Q_{i}-Q_{i,0}+(1-\gamma)\pi_{i}Q_{0})-$
$\displaystyle\widetilde{c}_{i}(Q_{i}-Q_{i,0})-\overline{C}_{i},$

that is,

$\displaystyle{\cal{R}}_{i}(p_{0},w_{i})=$
$\displaystyle\;h_{i}(Q_{i}-Q_{i,0})\;+$ (19)
$\displaystyle\;(w_{i}-c_{i})(1-\gamma)\pi_{i}Q_{0}-\overline{C}_{i}$

where we set $h_{i}=p_{i}-c_{i}-\widetilde{c}_{i}$. Both MNO partners 1 and 2
seek to maximize their own profit ${\cal{R}}_{1}$ and ${\cal{R}}_{2}$. In
order to solve this optimization problem for either model (FS) or (PS), we
replace $p_{0}$ by its optimal value $p_{0}^{*}(w_{1},w_{2})$ derived above in
Lemma 2. Define then

$\displaystyle\mathcal{R}_{i}^{*}(w_{1},w_{2})={\mathcal{R}}_{i}(p_{0}^{*}(w_{1},w_{2}),w_{i})$
(20)

as obtained from (19) with $Q_{i,0}=\varepsilon
Q_{i}(p_{i}-p_{0}^{*}(w_{1},w_{2}))/p_{i}$ and $Q_{0}$ given by (2).

We now successively address the maximization of MNOs profits for the (FS) and
(PS) models.

####  III-B1 Fully Sequential Model

Assume that MNO $i\in\\{1,2\\}$ is the leader and MNO $-i$ is the follower (by
convention, $-i=2$ if $i=1$, and $-i=1$ if $i=2$). Given $w_{i}$, the follower
thus decides on the wholesale price $w_{-i}$ to charge the MVNO. The
optimization problem for both MNOs can then be expressed by the following
bilevel formulation

$\left\\{\begin{array}[]{l}\underset{w_{i}\geqslant 0}{\mbox{ max
}}\mathcal{R}_{i}^{*}(w_{1},w_{2})_{|w_{-i}=w_{-i}^{*}},\\\ \\\ \mbox{subject
to}\quad\quad w_{-i}^{*}=\underset{w_{-i}\geqslant 0}{\mbox{argmax
}}\mathcal{R}_{-i}^{*}(w_{1},w_{2})\end{array}\right.$ (21)

where the notation $|w_{-i}=w_{-i}^{*}$ means that function
$\mathcal{R}_{i}^{*}$ is evaluated for the variable $w_{-i}$ equal to
$w_{-i}^{*}$; note that $w_{-i}^{*}$ is a function of $w_{i}$. The symmetrical
case when MNO $-i$ is the leader is similarly defined. In order to solve
problem (21), we introduce the following definitions.

######  Definition 1.

We denote by $\Delta$ the closed triangular region defined by $\Delta$ =
$\\{(w_{1},w_{2})\in\mathbb{R}^{+}\times\mathbb{R}^{+}:\widetilde{p}_{0}(w_{1},w_{2})\leqslant
p_{2}\\}$ where $\widetilde{p}_{0}(w_{1},w_{2})$ is given by (18) (see Fig.
2).

Further denote by $\delta$ the boundary segment of $\Delta$ defined by
$\delta$ =
$\\{(w_{1},w_{2})\in\Delta:\widetilde{p}_{0}(w_{1},w_{2})=p_{2}\\}$.

$\Delta$$\mathbf{w}^{C}$$\mathbf{w}^{A}$$\mathbf{w}^{B}$$\delta$$D^{A}$$\delta$$D^{B}$$\widetilde{\mathbf{w}}^{A}$$\widetilde{\mathbf{w}}^{B}$$w_{1}$$w_{2}$
Figure 2: The region $\Delta$, its boundary $\delta$, and the points
$\mathbf{w}^{A}$, $\mathbf{w}^{B}$, $\widetilde{\mathbf{w}}^{A}$,
$\widetilde{\mathbf{w}}^{B}$ and $\mathbf{w}^{C}$ (lines $D^{A}$ and $D^{B}$
are invoked in Appendix VII-A).

######  Definition 2.

Let $\mathbf{w}^{A}=(w_{1}^{A},w_{2}^{A})$ and
$\mathbf{w}^{B}=(w_{1}^{B},w_{2}^{B})$ denote the pair of prices given by

$\left\\{\begin{array}[]{ll}w_{1}^{A}=\displaystyle\frac{\displaystyle\frac{h_{1}Q_{1}}{p_{1}}+c_{1}(1-\gamma)\pi_{1}\mathcal{S}+2Q-2\mathcal{S}p_{2}}{(1-\gamma)\pi_{1}\mathcal{S}},\\\
\\\
w_{2}^{A}=\displaystyle\frac{\displaystyle-\frac{h_{1}Q_{1}}{p_{1}}-c_{1}(1-\gamma)\pi_{1}\mathcal{S}-4Q+4p_{2}\mathcal{S}+\mathcal{T}}{(1-\gamma)\pi_{2}\mathcal{S}}\end{array}\right.$

and

$\left\\{\begin{array}[]{ll}w_{1}^{B}=\displaystyle\frac{\displaystyle-\frac{h_{2}Q_{2}}{p_{2}}-c_{2}(1-\gamma)\pi_{2}{\cal{S}}-4Q+4p_{2}{\cal{S}}+\mathcal{T}}{(1-\gamma)\pi_{1}{\cal{S}}},\\\
\\\
w_{2}^{B}=\displaystyle\frac{\displaystyle\frac{h_{2}Q_{2}}{p_{2}}+c_{2}(1-\gamma)\pi_{2}{\cal{S}}+2Q-2{\cal{S}}p_{2}}{(1-\gamma)\pi_{2}{\cal{S}}},\end{array}\right.$

respectively, with $\mathcal{T}=Q+(r_{0}-\widetilde{c}_{0})\mathcal{S}$ for
short.

Define also the function $\Omega_{-i}$, $i\in\\{1,2\\}$, by

$\displaystyle\Omega_{-i}(w_{i})$ $\displaystyle=$
$\displaystyle\frac{c_{-i}}{2}+\frac{h_{-i}Q_{-i}}{2p_{-i}(1-\gamma)\pi_{-i}\mathcal{S}}+\frac{Q}{2(1-\gamma)\pi_{-i}\mathcal{S}}-$
(22)
$\displaystyle\frac{\widetilde{c}_{0}-r_{0}}{2(1-\gamma)\pi_{-i}}-\frac{\pi_{i}}{2\pi_{-i}}w_{i}$

and the points
$\widetilde{\mathbf{w}}^{B}=(\widetilde{w}_{1},\Omega_{2}(\widetilde{w}_{1}))$,
$\widetilde{\mathbf{w}}^{A}=(\Omega_{1}(\widetilde{w}_{2}),\widetilde{w}_{2})$
where

$\displaystyle\widetilde{w}_{i}=\frac{\displaystyle\frac{h_{i}Q_{i}}{p_{i}}-\frac{h_{-i}Q_{-i}}{p_{-i}}+\mathcal{S}(1-\gamma)(c_{i}\pi_{i}-c_{-i}\pi_{-i})+\mathcal{T}}{2(1-\gamma)\pi_{i}\mathcal{S}}.$
(23)

Note that $\mathbf{w}^{A}\in\delta$ and $\mathbf{w}^{B}\in\delta$; a geometric
interpretation of the pairs $\mathbf{w}^{A}$, $\mathbf{w}^{B}$ and
$\widetilde{\textbf{w}}^{A}$, $\widetilde{\textbf{w}}^{B}$ is given in
Appendix VII-A. We can now state the following.

######  Proposition 2.

Define the constant $\overline{r}_{0}$ by

$\overline{r}_{0}=\frac{h_{1}Q_{1}}{p_{1}\mathcal{S}}+\frac{h_{2}Q_{2}}{p_{2}\mathcal{S}}+(1-\gamma)(\pi_{1}c_{1}+\pi_{2}c_{2})+\frac{7\,Q}{\mathcal{S}}+\widetilde{c}_{0}-8p_{2}.$

In the (Part-Part) scenario with FS model,

  * $\bullet$

if $r_{0}\leqslant\overline{r}_{0}$ and MNO 1 (resp. MNO 2) is the leader,
then the optimal wholesale price vector $(w_{1}^{*},w_{2}^{*})$ is given by
$\textbf{w}^{B}\in\delta$ (resp. $\textbf{w}^{A}\in\delta$) introduced above.
In either case, the optimal MVNO’s retail price is then

$p_{0}^{*}(w_{1}^{*},w_{2}^{*})=p_{2};$

  * $\bullet$

if $r_{0}>\overline{r}_{0}$ and MNO 1 (resp. MNO 2) is the leader, then the
optimal wholesale price vector $(w_{1}^{*},w_{2}^{*})$ is given by
$\widetilde{\mathbf{w}}^{B}\in\Delta\setminus\delta$ (resp.
$\widetilde{\mathbf{w}}^{A}\in\Delta\setminus\delta$) introduced above. In
either case, the optimal MVNO’s retail price is then

$p_{0}^{*}(w_{1}^{*},w_{2}^{*})=\widetilde{p}_{0}(w_{1}^{*},w_{2}^{*})<p_{2}.$

We defer the detailed proof to Appendix VII-A.

####  III-B2 Partially Sequential Model

Now assume that MNOs simultaneously choose their optimal wholesale prices.
Given (20), the joint optimization problem for both MNOs is thus expressed by

$\displaystyle\underset{w_{1}\geqslant 0}{\mbox{ max
}}{{\cal{R}}_{1}^{*}}(w_{1},w_{2})$ (24)

and

$\displaystyle\underset{w_{2}\geqslant 0}{\mbox{ max
}}{{\cal{R}}_{2}^{*}}(w_{1},w_{2}).$ (25)

######  Definition 3.

Let ${\bf{w}}^{C}=(w_{1}^{C},w_{2}^{C})$ denote the pair of prices given by

$\left\\{\begin{array}[]{l}w_{1}^{C}=\displaystyle\frac{\displaystyle\frac{2h_{1}\,Q_{1}}{p_{1}}-\frac{h_{2}\,Q_{2}}{p_{2}}+(1-\gamma)(2c_{1}\pi_{1}-c_{2}\pi_{2})\mathcal{S}+\mathcal{T}}{3(1-\gamma)\pi_{1}\mathcal{S}},\\\
\\\
w_{2}^{C}=\displaystyle\frac{\displaystyle\frac{2h_{2}\,Q_{2}}{p_{2}}-\frac{h_{1}\,Q_{1}}{p_{1}}+(1-\gamma)(2c_{2}\pi_{2}-c_{1}\pi_{1})\mathcal{S}+\mathcal{T}}{3(1-\gamma)\pi_{2}\mathcal{S}}\end{array}\right.$

with $\mathcal{T}=Q+(r_{0}-\widetilde{c}_{0})\mathcal{S}$.

The pair ${\bf{w}}^{C}$ is given a geometric interpretation in Appendix VII-B.
This enables us to state the following.

######  Proposition 3.

Define the constant $r_{0}^{\flat}$ by

$r_{0}^{\flat}=\frac{h_{1}Q_{1}}{p_{1}\mathcal{S}}+\frac{h_{2}Q_{2}}{p_{2}\mathcal{S}}+(1-\gamma)(\pi_{1}c_{1}+\pi_{2}c_{2})+\frac{5\,Q}{\mathcal{S}}+\widetilde{c}_{0}-6p_{2}.$

In the (Part-Part) scenario with PS model,

  * $\bullet$

if $r_{0}>r_{0}^{\flat}$, the optimal wholesale price vector
$(w_{1}^{*},w_{2}^{*})$ is given by ${\bf{w}}^{C}\in\Delta\setminus\delta$.
The optimal MVNO’s retail price is then

$p_{0}^{*}(w_{1}^{C},w_{2}^{C})=\widetilde{p}_{0}(w_{1}^{C},w_{2}^{C})<p_{2}$

as defined by (18);

  * $\bullet$

if $r_{0}=r_{0}^{\flat}$, the optimal wholesale price vector
$(w_{1}^{*},w_{2}^{*})$ is given by ${\bf{w}}^{C}\in\delta$. The optimal
MVNO’s retail price is then $p_{0}^{*}=p_{2}$;

  * $\bullet$

if $r_{0}<r_{0}^{\flat}$, the problem (24)-(25) admits no solution.

The proof of Proposition 3 is detailed in Appendix VII-B.

##  IV Game-Theoretic Model

In this section, we propose a non-cooperative game-theoretical model to
formalize the competition between MNO 1 and MNO 2.

######  Definition 4.

Introduce the two-player game where

  * $\\--$

the players are MNO $1$ and MNO $2$,

  * $\\--$

the strategies are either to contract with the new entrant, ”Part” strategy,
or not to contract, ”NonPart” strategy,

  * $\\--$

the payoffs are given by the matrix

$\bordermatrix{~{}&Part&NonPart\cr
Part&({\cal{R}}_{1}^{*}(w_{1}^{*},w_{2}^{*}),{\cal{R}}_{2}^{*}(w_{1}^{*},w_{2}^{*}))&({\cal{R}}_{1}^{*}(\widehat{w}_{1}),{\cal{R}}_{2}(\widehat{w}_{1}))\cr
NonPart&({\cal{R}}_{1}(\widehat{w}_{2}),{\cal{R}}_{2}^{*}(\widehat{w}_{2}))&({\cal{R}}_{1}^{0},{\cal{R}}_{2}^{0})\cr}.$

In this matrix, the pair $({\cal{R}}_{1}^{0},{\cal{R}}_{2}^{0})$ corresponds
to the scenario when no MNO partners with the MVNO, thus forbidding its
entrance into the market.

We recall that a Nash Equilibrium (NE) is a strategy profile such that no
player has an incentive to unilaterally deviate from this profile [11].
Besides, recall from definition (1) that $Q_{i,0}$ denotes the customer part
which defects from MNO $i$ to the new entrant in the scenario (Part, Part);
furthermore, let $\widehat{Q}_{i,0}$ denote the customer part which defects
from MNO $i$ to the MVNO in the scenario (NonPart, Part), that is, when only
MNO $-i$ contracts a partnership with the new entrant.

######  Lemma 3.

In the FS model with $r_{0}\leqslant\overline{r}_{0}$, a MNO loses more
customers when it is non-partner than when it is a partner of the MVNO, that
is, $\widehat{Q}_{i,0}\geqslant Q_{i,0}$.

###### Proof.

From Proposition 2 with $r_{0}\leqslant\overline{r}_{0}$, we have
$p_{0}^{*}(w_{1}^{*},w_{2}^{*})=p_{2}$ while Proposition 1 entails
$\widehat{p}_{0}(\widehat{w}_{-i})\leqslant p_{2}$. Using definition (1), we
then deduce that the difference $Q_{i,0}-\widehat{Q}_{i,0}=\frac{\varepsilon
Q_{i}}{p_{i}}\left(\widehat{p}_{0}(\widehat{w}_{-i})-p_{0}^{*}(w_{1}^{*},w_{2}^{*})\right)$
is non-positive, as claimed. ∎

As a consequence, we can formulate the subsequent result on the existence of a
Nash equilibrium.

######  Proposition 4.

In the FS model with $r_{0}\leqslant\overline{r}_{0}$ and for wholesale prices
higher than network costs,

(a) the scenario (Part, Part) is a NE;

(b) if $r_{0}\leqslant\min(\overline{r}_{0},\overline{r}_{2,0})$, (Part, Part)
is the unique NE.

###### Proof.

(a) Scenario (Part, Part) is a NE. Suppose that both operators partner with
the MVNO, that is, the scenario is (Part, Part); consider MNO $i$ and let
${\cal{R}}_{i}^{*}(w_{1}^{*},w_{2}^{*})$ denote its profit for this scenario.
Assume now that MNO $i$ unilaterally switches from strategy ”Part” to strategy
”NonPart”; we denote by ${\cal{R}}_{i}(\widehat{w}_{-i})$ the profit of MNO
$i$ when it applies strategy ”NonPart” while MNO $-i$ keeps strategy ”Part”.
We then have

$\displaystyle{\cal{R}}_{i}^{*}(w_{1}^{*},w_{2}^{*})=h_{i}(Q_{i}-Q_{i,0})+(w_{i}^{*}-c_{i})(1-\gamma)\pi_{i}Q_{0}-\overline{C}_{i}$

and
${\cal{R}}_{i}(\widehat{w}_{-i})=h_{i}(Q_{i}-\widehat{Q}_{i,0})-\overline{C}_{i}$,
so that

$\displaystyle{\cal{R}}^{*}_{i}(w_{1}^{*},w_{2}^{*})-{\cal{R}}_{i}(\widehat{w}_{-i})$
$\displaystyle\,=h_{i}({\widehat{Q}}_{i,0}-Q_{i,0})+(w_{i}^{*}-c_{i})\times$
$\displaystyle\,(1-\gamma)\pi_{i}Q_{0}.$ (26)

As $\widehat{Q}_{i,0}\geqslant Q_{i,0}$ by Lemma 3, the profit difference
${\cal{R}}^{*}_{i}(w_{1}^{*},w_{2}^{*})-{\cal{R}}_{i}(\widehat{w}_{-i})$ in
(26) is then non-negative after assumptions $h_{i}\geqslant 0$ and
$w_{i}^{*}-c_{i}\geqslant 0$; MNO $i$ then has no incentive to unilaterally
deviate from strategy ”Part” since this yields a profit decrease. (Part, Part)
is thus a NE.

(b) (Part, Part) is the unique NE. Assume both MNOs use strategy ”Non Part”.
Now suppose MNO 2 unilaterally switches to ”Part”; we first have
${\cal{R}}_{2}^{0}=h_{2}Q_{2}-\overline{C}_{2}$ while
${\cal{R}}^{*}_{2}(\widehat{w}_{2})=h_{2}(Q_{2}-\widehat{Q}_{2,0})+(\widehat{w}_{2}-c_{2})(1-\gamma)Q_{0}-\overline{C}_{2}$
after (7), hence

${\cal{R}}^{*}_{2}(\widehat{w}_{2})-{\cal{R}}_{2}^{0}=-h_{2}\widehat{Q}_{2,0}+(\widehat{w}_{2}-c_{2})(1-\gamma)Q_{0}.$
(27)

From Proposition 1, we have $\widehat{Q}_{2,0}$ = 0 if
$r_{0}\leqslant\overline{r}_{2,0}$; the difference
${\cal{R}}^{*}_{2}(\widehat{w}_{2})-{\cal{R}}_{2}^{0}$ in (27) is then non-
negative after assumption $\widehat{w}_{2}-c_{2}\geqslant 0$ and (Non Part,
Non Part) is not a NE. We conclude that if $r_{0}\leqslant\overline{r}_{0}$
and $r_{0}\leqslant\overline{r}_{2,0}$, (Part,Part) is the only NE. ∎

Note that a preliminary study has given us hints for (Part,Part) to be still a
NE for $r_{0}>\overline{r}_{0}$. For the PS model, a similar analysis should
also be addressed for $r_{0}\leqslant r_{0}^{\flat}$.

##  V Economic Discussion

The results obtained in the previous sections allow us to provide the
following comments.

Scenario (Part-Part):  When both MNOs partner with the MVNO, Propositions 2
and 3 entail that, when $\gamma\rightarrow$ 1, all optimal wholesale prices
are of order $1/(1-\gamma)$; this increasing rate again confirms the effect of
economy of scale. Now, regarding the MVNO’s optimal retail price, we
distinguish two cases depending on the order in which decisions are taken by
MNOs. First, we have shown (Proposition 2) for the FS model that the entrant
MVNO charges its users the same price $p_{2}$ as the lowest MNO if
$r_{0}\leqslant\overline{r}_{0}$, thus attracting users only from the MNO with
the highest retail price; otherwise, it attracts users from both MNOs. Second,
we have shown (Proposition 3) for the PS model that the entrant MVNO cannot
set an optimal retail price strictly lower than both MNOs’ prices unless it
has sufficiently high indirect revenues, that is, $r_{0}>r_{0}^{\flat}$.
Otherwise, if $r_{0}<r_{0}^{\flat}$, there is no wholesale prices that jointly
optimize both MNOs profits.

For each class of actors, we can conclude the following:

For the MVNO: for all scenarios, the MVNO retail price $p_{0}^{*}$ decreases
with $r_{0}$; the MVNO can thus set a retail price strictly lower than that of
the MNOs if it has high enough indirect revenues. Besides, the threshold value
$\overline{r}_{0}$ (resp. $r_{0}^{\flat}$) in model FS (resp. model PS) is a
decreasing function of the proportion $\gamma$, so that the MVNO has an
optimal retail price strictly lower than that of both MNOs for large enough
$\gamma$. The technological independence of the MVNO from its partner MNOs due
to free WiFi access thus translates into an economic advantage on the retail
market (but this does not obviously account for better QoS and security levels
offered to its customers if a larger part of MVNO traffic were transferred
through optimized cellular networks);

For the MNOs: for the FS model and under sufficient conditions on the indirect
revenue of the MVNO, the scenario (Part, Part) defines the unique NE. This
means that the MNOs have then an incentive to partner with the new entrant: in
fact, the MNOs would in the first place prefer that the MVNO does not enter
the market in order to keep their customer base, but each MNO fears that its
competitor hosts the MVNO, in which case it would incur losses both on the
retail and wholesale markets. As a consequence, both MNOs will eventually
decide to partner with the MVNO. In addition, a non-partner MNO would incur
higher retail losses if it were non-partner than when it is a partner of the
MVNO. Cooperating with the MVNO therefore enables each MNO to compensate for a
part of its retail revenue losses.

##  VI Conclusion

In this paper, our contribution is twofold. First, we address the price
setting optimization problems for both MNOs and the entrant MVNO in the
framework of two distinct scenarios. In this aim, we propose several
mathematical programming formulations for the underlying problems, each one
corresponding to a specific decision-making scheme; we also discuss the
economic interpretation of the optimal solution in each case. Secondly, based
on the optimal price setting step, we provide a game-theoretical analysis of
the MNOs competition and show that (Part, Part) is the unique Nash
Equilibrium, and thus the most profitable scenario for both MNOs, provided
that appropriate conditions on the MVNO’s indirect revenue are fulfilled.

The particular case of only two competing MNOs has provided us with
interesting results, and a natural extension to this work would be to
generalize the results obtained for two MNOs to an arbitrary number
($n\geqslant$ 3) of MNOs. Indeed, the two-dimensional optimization problems
that we have here addressed may exhibit other features in the $n$-dimensional
case (e.g. several possible optimal points). Furthermore, the associated
$n$-player games could be amenable to cooperation schemes among players which
could be interestingly studied. On the other hand, other demand models
differing from that considered in this paper (customer defection due to the
price/demand elasticity) could also be envisaged; alternative models based on
interactions between users or on the MVNO brand appeal could capture other
preference sources of users towards each actor.

## Acknowledgment

The authors thank V. Chandrakumar, L. Le Beller and M. Touati at Orange Labs
for fruitful discussions, together with the anonymous referees for their
valuable comments.

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##  VII Appendix

###  VII-A Proof of Proposition 2.

Recall that MNO $i\in\\{1,2\\}$ (resp. MNO $-i$) is assumed to be the leader
(resp. the follower). We successively consider the two cases $(a)$
$(w_{1},w_{2})\notin\Delta$ and $(b)$ $(w_{1},w_{2})\in\Delta$.

First consider case $(a)$. In view of (17), this corresponds to values of
$(w_{1},w_{2})$ such that $p_{0}^{*}(w_{1},w_{2})=p_{2}$ and by (20), we
easily show that
${\cal{R}}_{-i}^{*}(w_{1},w_{2})={\cal{R}}_{-i}(p_{2},w_{-i})$ is a linear and
increasing function of $w_{-i}$. The optimal value of $\mathcal{R}_{-i}^{*}$
is thus obtained at $w_{-i}=+\infty$; as this unbounded price gives the value
$-\infty$ for $\mathcal{R}_{0}$, case $(a)$ can thus be excluded.

Now consider case $(b)$. By (17), this corresponds to pairs $(w_{1},w_{2})$
such that $p_{0}^{*}(w_{1},w_{2})=\widetilde{p}_{0}(w_{1},w_{2})$. To solve
the optimization problem for follower MNO $-i$ in (21) with given $w_{i}$, we
introduce the associated Lagrange function given by

$\displaystyle{\cal{L}}_{-i}(w_{1},w_{2},\lambda_{-i})={\cal{R}}_{-i}^{*}(w_{1},w_{2})-\lambda_{-i}\left(\widetilde{p}_{0}(w_{1},w_{2})-p_{2}\right)$

where $\lambda_{-i}$ is the Lagrange multiplier associated to the constraint
$\widetilde{p}_{0}(w_{1},w_{2})\leqslant p_{2}$; from definition (20) and the
expression (18) of $\widetilde{p}_{0}(w_{1},w_{2})$,
${\cal{R}}_{-i}^{*}(w_{1},w_{2})$ is easily expressed as a quadratic function
of variable $w_{-i}$. The system of Karush-Kuhn-Tucker (KKT) ([12], Chap.4,
Sec. 4.2.13) conditions for the Lagrangian ${\cal{L}}_{-i}$ above can be
written as

$\left\\{\begin{array}[]{l}\displaystyle\frac{\partial{\cal{L}}_{-i}}{\partial
w_{-i}}(w_{1},w_{2},\lambda_{-i})=0,\\\ \\\ \lambda_{-i}\geqslant
0,\qquad\lambda_{-i}\big{(}\widetilde{p}_{0}(w_{1},w_{2})-p_{2}\big{)}=0,\\\
\\\ \widetilde{p}_{0}(w_{1},w_{2})\leqslant p_{2}.\end{array}\right.$ (28)

Two cases can intervene for the multiplier $\lambda_{-i}$:

(I) if $\lambda_{-i}=0$, the first KKT condition in system (28) reads

$\frac{\partial\mathcal{R}_{-i}^{*}}{\partial w_{-i}}(w_{1},w_{2})=0;$ (29)

function $\mathcal{R}^{*}_{-i}$ being quadratic, equation (29) is linear in
both variables $w_{1}$, $w_{2}$ and thus defines geometrically a line $D$
(displayed in Fig. 2 as line $D^{B}$ if $-i=2$ or line $D^{A}$ if $-i=1$).
Solving (29) for $w_{-i}$ then yields the unique maximum at point
$w_{-i}^{*}=\Omega_{-i}(w_{i})$ with function $\Omega_{-i}$ defined as in
(22).

Now, consider the profit maximization in problem (21) for the leader MNO $i$.
In order to solve it, we replace $w_{-i}$ in the profit
$\mathcal{R}^{*}_{i}(w_{1},w_{2})$ by the maximum
$w_{-i}^{*}=\Omega_{-i}(w_{i})$ derived above. Given (20), we thus define

$\displaystyle{\mathcal{R}}_{i}^{**}(w_{i})=\mathcal{R}_{i}^{*}(w_{1},w_{2})_{|w_{-i}=\Omega_{-i}(w_{i})}.$

Recall that we consider case $(b)$ for which $(w_{1},w_{2})\in\Delta$ and
$p_{0}^{*}(w_{1},w_{2})=\widetilde{p}_{0}(w_{1},w_{2})$, so that the
optimization problem for leader MNO $i$ eventually reads

$\left\\{\begin{array}[]{ll}\underset{w_{i}\geqslant
0}{\max~{}}{\cal{R}}_{i}^{**}(w_{i}),\\\ \\\ \mbox{subject to
}\quad\widetilde{p}_{0}(w_{1},w_{2})_{|w_{-i}=\Omega_{-i}(w_{i})}\leqslant
p_{2}.\end{array}\right.$ (30)

First, the constraint
$\widetilde{p}_{0}(w_{1},w_{2})_{|w_{-i}=\Omega_{-i}(w_{i})}\leqslant p_{2}$
in (30) is easily translated into $w_{i}\leqslant\overline{w}_{i}$ where we
set

$\displaystyle\overline{w}_{i}=\frac{\displaystyle-\frac{h_{-i}Q_{-i}}{p_{-i}}-c_{-i}(1-\gamma)\pi_{-i}\mathcal{S}-4Q+4p_{2}\mathcal{S}+\mathcal{T}}{(1-\gamma)\pi_{i}\mathcal{S}}$

with $\mathcal{T}=Q+(r_{0}-\widetilde{c}_{0})\mathcal{S}$. Second, the 1st
order condition $\partial\mathcal{R}_{i}^{**}(w_{i})/\partial w_{i}=0$ for
problem (30) yields the critical point $\widetilde{w}_{i}$, given as in (23).
The second derivative $\partial^{2}{\cal{R}}_{i}^{**}(w_{i})/\partial
w_{i}^{2}$ being equal to the negative constant
$-\varepsilon(1-\gamma)^{2}\pi_{i}^{2}\mathcal{S}/2$,
$\mathcal{R}_{i}^{**}(w_{i})$ is therefore a strictly concave function of
$w_{i}$ and has a unique maximum on $\mathbb{R}^{+}$ at $\widetilde{w}_{i}$.
It thus follows from the latter discussion that

$w_{i}^{*}=\min(\widetilde{w}_{i},\overline{w}_{i})$

is the unique solution to problem (30). Now, we easily verify that
$\widetilde{w}_{i}\geqslant\overline{w}_{i}\iff$
$r_{0}\leqslant\overline{r}_{0}$ where $\overline{r}_{0}$ is expressed in
Proposition 2. We thus conclude that if $r_{0}\leqslant\overline{r}_{0}$, the
optimal solution is the intersection point $\textbf{w}^{*}$ =
$(\overline{w}_{1},\Omega_{2}(\overline{w}_{1}))$ = $\textbf{w}^{B}\in
D^{B}\cap\delta$ when $i=1$, or
$\textbf{w}^{*}=(\Omega_{1}(\overline{w}_{2}),\overline{w}_{2})$ =
$\textbf{w}^{A}\in D^{A}\cap\delta$ when $i=2$; otherwise, if
$r_{0}>\overline{r}_{0}$, the optimal solution is the intersection point
$\textbf{w}^{*}$ =
$(\widetilde{w}_{1},\Omega_{2}(\widetilde{w}_{1}))=\widetilde{\textbf{w}}^{B}\in
D^{B}\cap\Delta$ when $i=1$, or $\textbf{w}^{*}$ =
$(\Omega_{1}(\widetilde{w}_{2}),\widetilde{w}_{2})=\widetilde{\textbf{w}}^{A}\in
D^{A}\cap\Delta$ when $i=2$;.

(II) otherwise, if $\lambda_{-i}>0$, the first KKT condition
$\partial{\cal{L}}_{-i}(w_{1},w_{2},\lambda_{-i})/\partial w_{-i}=0$ in system
(28) yields

$\displaystyle w_{-i}=\bar{\omega}_{-i}(\lambda_{-i},w_{i}),$ (31)

with
$\overline{\omega}_{-i}(\lambda_{-i},w_{i})=A_{i}(w_{i})/\varepsilon\mathcal{S}(1-\gamma)\pi_{-i}$
where we set

$\displaystyle A_{i}(w_{i})=$
$\displaystyle\;\varepsilon\frac{h_{-i}Q_{-i}}{2p_{-i}}+\varepsilon\frac{Q}{2}-\varepsilon\mathcal{S}w_{i}\frac{1-\gamma}{2}\pi_{i}-\varepsilon\mathcal{S}\frac{\tilde{c}_{0}-r_{0}}{2}\;+$
$\displaystyle\;\varepsilon
c_{-i}(1-\gamma)\pi_{-i}\frac{\mathcal{S}}{2}-\frac{\lambda_{-i}}{2}.$

The complementary slackness condition $\widetilde{p}_{0}(w_{1},w_{2})=p_{2}$
then eventually reduces to $\lambda_{-i}=\Lambda_{-i}(w_{i})$ where

$\displaystyle\Lambda_{-i}(w_{i})=$
$\displaystyle\;\varepsilon\mathcal{S}(1-\gamma)\pi_{i}w_{i}+\varepsilon\frac{h_{-i}Q_{-i}}{p_{-i}}+3\varepsilon
Q\;+$
$\displaystyle\;\varepsilon\mathcal{S}(\tilde{c}_{0}-r_{0})+\varepsilon\mathcal{S}c_{-i}(1-\gamma)\pi_{-i}-4\varepsilon\mathcal{S}p_{2}.$

By inserting this expression of $\lambda_{-i}=\Lambda_{-i}(w_{i})$ into the
right-hand side of (31), we finally get $w_{-i}=\overline{\Omega}_{-i}(w_{i})$
where

$\overline{\Omega}_{-i}(w_{i})=\frac{-(1-\gamma)\pi_{i}\mathcal{S}w_{i}-Q-\mathcal{S}(\tilde{c}_{0}-r_{0})+2\mathcal{S}p_{2}}{\mathcal{S}(1-\gamma)\pi_{-i}}.$

Now, consider the profit of leader MNO $i$ given by

$\displaystyle\mathcal{R}_{i}^{**}(w_{i})=\mathcal{R}_{i}^{*}(w_{1},w_{2})_{|w_{-i}=\overline{\Omega}_{-i}(w_{i})}.$

The first derivative of ${\cal{R}}_{i}^{**}(w_{i})$ with respect to $w_{i}$
equals $d{\cal{R}}_{i}^{**}(w_{i})/dw_{i}=\pi_{i}(1-\gamma)Q_{0}$ and is
strictly positive; but in the present case, $w_{i}\in[0,w_{i}^{0}]$ where
$w_{i}^{0}$ is either the abcissa or the ordinate of the intersection point of
line $\delta$ with the axis $w_{-i}$ = 0. The corresponding optimal unit price
$w_{-i}^{*}$ for the follower MNO $-i$ is therefore 0; as having no economic
relevance for this partner MNO, this case (II) is eventually excluded.

###  VII-B Proof of Proposition 3.

We successively consider the two cases $(a)$ $(w_{1},w_{2})\notin\Delta$ and
$(b)$ $(w_{1},w_{2})\in\Delta$.

First consider case $(a)$. In view of (17), this corresponds to pairs
$(w_{1},w_{2})$ such that $p_{0}^{*}(w_{1},w_{2})=p_{2}$ and by (20), we
easily show that
$\mathcal{R}_{i}^{*}(w_{1},w_{2})={\mathcal{R}}_{i}(p_{2},w_{i})$ is a linear
and increasing function of $w_{i}$, $i\in\\{1,2\\}$. The optimal value of
$\mathcal{R}_{i}^{*}$ is thus obtained at $w_{i}=+\infty$; as this unbounded
price gives the value $-\infty$ for $\mathcal{R}_{0}$, case $(a)$ can
therefore be excluded.

Now consider case $(b)$. In view of (17), this corresponds to values of
$(w_{1},w_{2})$ such that
$p_{0}^{*}(w_{1},w_{2})=\widetilde{p}_{0}(w_{1},w_{2})$. From definition (20)
and the expression (18) of $\widetilde{p}_{0}(w_{1},w_{2})$, each function
${\cal{R}}_{1}^{*}(w_{1},w_{2})$ and ${\cal{R}}_{2}^{*}(w_{1},w_{2})$ is again
expressed as a quadratic function of variable $w_{1}$ and $w_{2}$,
respectively. The KKT conditions associated with optimization problem (24) for
MNO $1$, and optimization problem (25) for MNO $2$, read

$\left\\{\begin{array}[]{l}\displaystyle\frac{\partial{\cal{R}}_{i}^{*}}{\partial
w_{i}}(w_{1},w_{2})=\lambda_{i}\frac{(1-\gamma)\pi_{i}}{2},\\\ \\\
\lambda_{i}\geqslant
0,\qquad\lambda_{i}(\widetilde{p}_{0}(w_{1},w_{2})-p_{2})=0,\\\ \\\
\widetilde{p}_{0}(w_{1},w_{2})\leqslant p_{2};\end{array}\right.$ (32)

for $i=1$ and $i=2$, respectively. At this point, we need to distinguish three
cases according to the values of the Lagrange multipliers (I) $\lambda_{1}=0$
and $\lambda_{2}=0$, (II) $\lambda_{1}>0$ and $\lambda_{2}>0$, (III)
$\lambda_{1}=0$ and $\lambda_{2}>0$ (or reversely $\lambda_{1}>0$ and
$\lambda_{2}=0$).

(I) First assume $\lambda_{1}=0$ and $\lambda_{2}=0$. The simultaneous
conditions

$\frac{\partial{\cal{R}}_{1}^{*}}{\partial
w_{1}}(w_{1},w_{2})=0,\quad\frac{\partial{\cal{R}}_{2}^{*}}{\partial
w_{2}}(w_{1},w_{2})=0$

yield the critical pair ${\bf{w}}^{C}\in D^{A}\cap D^{B}$, intersection point
of lines $D^{A}$ and $D^{B}$ defined in (29). For each $i\in\\{1,2\\}$, the
second derivative $\partial^{2}\mathcal{R}^{*}_{i}(w_{1},w_{2})/\partial
w_{i}^{2}$ equals the negative constant
$-\varepsilon(1-\gamma)^{2}\pi_{i}^{2}\mathcal{S}$; $\mathcal{R}^{*}_{i}$ is
thus a strictly concave function of the variable $w_{i}$ and the point
$\mathbf{w}^{C}$ is therefore the unique joint maximum for
$\mathcal{R}^{*}_{1}$ and $\mathcal{R}^{*}_{2}$. Besides, it is readily shown
that $\widetilde{p}_{0}(w_{1}^{C},w_{2}^{C})<p_{2}$ if and only if
$r_{0}>r_{0}^{\flat}$; in such a case, we then have
${\bf{w}}^{C}\in\Delta\setminus\delta$ and this point ${\bf{w}}^{C}$ is the
optimal pair of wholesale prices.

(II) Now assume $\lambda_{1}>0$ and $\lambda_{2}>0$. Solving each KKT system
(32) for $i=1$ and $i=2$, we get

$\left\\{\begin{array}[]{l}w_{1}(\lambda_{1})=\displaystyle\frac{\displaystyle\frac{h_{1}Q_{1}}{p_{1}}+c_{1}(1-\gamma)\pi_{1}\mathcal{S}+2Q-2\mathcal{S}p_{2}-\frac{\lambda_{1}}{\varepsilon}}{(1-\gamma)\pi_{1}\mathcal{S}},\\\
w_{2}(\lambda_{1})=\displaystyle\frac{\displaystyle\frac{h_{2}Q_{2}}{p_{2}}+c_{2}(1-\gamma)\pi_{2}\mathcal{S}+2Q-2\mathcal{S}p_{2}-\frac{\lambda_{2}}{\varepsilon}}{(1-\gamma)\pi_{2}\mathcal{S}},\\\
\lambda_{1}+\lambda_{2}=\varepsilon\mathcal{S}(r_{0}^{\flat}-r_{0}).\end{array}\right.$

We have $\lambda_{1}+\lambda_{2}>0$ if and only if $r_{0}<r_{0}^{\flat}$.
Calculating the respective values of $\varphi_{1}(\lambda_{1})$ =
$\mathcal{R}_{1}^{*}(w_{1}(\lambda_{1}),w_{2}(\lambda_{1}))$ and
$\varphi_{2}(\lambda_{1})$ =
$\mathcal{R}_{2}^{*}(w_{1}(\lambda_{1}),w_{2}(\lambda_{1}))$ easily shows that
$\varphi_{1}$ and $\varphi_{2}$ are linear functions over interval
$[0,\varepsilon\mathcal{S}(r_{0}^{\flat}-r_{0})]$ with respective maximum at
$\lambda_{1}$ = 0 and $\lambda_{1}$ =
$\varepsilon\mathcal{S}(r_{0}^{\flat}-r_{0})$ (see Fig. 3). Consequently,
there cannot be a joint solution that maximizes both
$\mathcal{R}_{1}^{*}(w_{1},w_{2})$ and $\mathcal{R}_{2}^{*}(w_{1},w_{2})$
unless $r_{0}=r_{0}^{\flat}$, in which case the optimal point coincides with
$\mathbf{w}^{C}$.

$\varphi_{1}(\lambda_{1})$$\varphi_{2}(\lambda_{1})$$\varepsilon\mathcal{S}(r_{0}^{\flat}-r_{0})$0$\lambda_{1}$
Figure 3: Variations of linear functions $\varphi_{1}$ and $\varphi_{2}$ on
interval $[0,\varepsilon\mathcal{S}(r_{0}^{\flat}-r_{0})]$.

(III) Finally assume $\lambda_{1}=0$ and $\lambda_{2}>0$. Solving then each
KKT system (32) for $i=1$ and $i=2$ yields the critical pair ${\bf{w}}^{A}$
and the value $\lambda_{2}=\varepsilon\mathcal{S}(r_{0}^{\flat}-r_{0})$; in
particular, we have $\lambda_{2}>0$ if and only if $r_{0}<r_{0}^{\flat}$.
Therefore, ${\bf{w}}^{A}$ is the pair of optimal wholesale prices when
$r_{0}<r_{0}^{\flat}$. Symmetrically, the case $\lambda_{1}>$ 0 and
$\lambda_{2}=$ 0 gives the optimal pair $\textbf{w}^{B}$ if
$r_{0}<r_{0}^{\flat}$. In view of the above properties of functions
$\varphi_{1}$ and $\varphi_{2}$, however, we easily show that neither
$\textbf{w}^{A}$ nor $\textbf{w}^{B}$ can be a joint solution that
simultaneously maximizes $\mathcal{R}_{1}^{*}(w_{1},w_{2})$ and
$\mathcal{R}_{2}^{*}(w_{1},w_{2})$, as required in (24) and (25). This joint
problem has consequently no solution when $r_{0}<r_{0}^{\flat}$.