The Economics of Competition and Cooperation Between MNOs and MVNOs

0.45 MB·

by  Mohammad Hassan Lotfi

The Economics of Competition and Cooperation
Between MNOs and MVNOs
Mohammad Hassan Lotfi Saswati Sarkar
Department of Electrical and System Engineering Department of Electrical and System Engineering
University of Pennsylvania University of Pennsylvania
Philadelphia, PA, 19104 Philadelphia, PA, 19104
Email: [email protected] Email: [email protected]
7 Abstract—In this work, we consider the economics of the is willing to lease. More importantly, it is not apriori clear
1 interactionbetweenMobileVirtualNetworkOperators(MVNOs) that under what conditions the MNO prefers to generate her
0 and Mobile Network Operators (MNOs). We investigate the revenue through the MVNO by leasing the resources to her,
2 incentives of an MNO for offering some of her resources to an
and therefore letting the MVNO to attract EUs.
MVNOinsteadofusingtheresourcesforherown.Weformulate
n the problem as a sequential game. We consider a market with Many works have considered the economics of re-
a one MNO and one MVNO, and a continuum of undecided end- source/spectrum sharing and the subsequent profit sharing
J
users. We assume that EUs have different preferences for the between SPs. Examples are [2]–[9]. In these works, authors
1 MNO and the MVNO. These preferences can be because of the
modeltheenvironmentusinggametheoryandseektoprovide
2 differences in the service they are offering or the reluctance of
intuitions for the pricing schemes, the split of EUs/benefits
an EU to buy her plan from one of them. We assume that the

] preferencesalsodependontheinvestmentleveltheMNOandthe between SPs, and the investment level of different SPs. In
I
N MVNO. We show that there exists a unique interior SPNE, i.e. this work, however, as mentioned before, we focus on the
theSPNEbywhichbothSPsreceiveapositivemassofEUs,and competition between the MNOs and MVNOs.
.
s characterizeit.Wealsoconsiderabenchmarkcaseinwhichthe We consider a market with one MNO (e.g. At&t) and one
c MNO and the MVNO do not cooperate, characterize the unique
MVNO(e.g.Google’sProjectFi).WeassumethattheMVNO
[ SPNE of this case, and compare the results of our model to the
benchmark case to assess the incentive of the MNO to invest in isactive,i.e.hasalreadysomeresources,andiswillingtolease
1
her infrastructure and to offer it to the MVNO. additional resources from the MNO in exchange of a fee. The
v
MNOdecideson theinvestmentlevel. Subsequently,knowing
1
5 I. INTRODUCTION this decision, the MVNO decides on the number of resources
0 Traditionally,wirelessserviceshavebeenofferedbyService shewantstoleasefromtheMNO.Then,simultaneously,both
6 Providers (SPs) that own the infrastructure they are operating MNO and MNVO decide on their pricing strategies for EUs.
0
on. Nowadays SPs are divided into (i) Mobile Network Op- Finally, EUs choose one of the MNO or MVNO to buy the
.
1 erators (MNOs) that own the infrastructure, and (ii) Mobile wireless plan from. We assume the per resource fee that the
0 Virtual Network Operators (MVNOs) that do not own the MVNO pays to the MNO to be fixed and discuss about the
7
infrastructure they are operating on, and use the resources of framework for determining this fee.
1
oneormoreMNOsbasedonabusinesscontract.MVNOscan To model EUs, we consider a continuum of undecided EUs
:
v distinguish their plans from MNOs by bundling their service that decide which of the SPs they want to buy their wireless
i
X with other products, offering different pricing plans for End- plan from. We assume that EUs have different preferences
Users (EUs), or building a good reputation through a better for each SP. These preferences can be because of different
r
a customer service. In recent years, the number of MVNOs is services that SPs offer. For example, the MVNO can bundle
rapidly growing. According to [1], between June 2010 and the wireless service with a free or cheap international call
June 2015, the number of MVNOs increased by 70 percent plan (through VoIP) to make her service more favorable for
worldwide,reaching1,017asofJune2015.EvensomeMNOs some EUs. Moreover, the preferences can be because of the
developedtheirownMVNOs.AnexampleofwhichisCricket reluctanceofanEUtobuyherwirelessplanfromaparticular
wirelesswhichisownedbyAt&tandoffersaprepaidwireless SP(e.g.anSPwithaninfamouscustomerservice).Weassume
service to EUs. thatthepreferencesalsodependontheinvestmentlevelofthe
In this work, we consider the economics of the interaction SPs. In other words, the higher the investment level of an SP,
between MVNOs and MNOs. We investigate the incentives the lower would be the reluctance of an EU for that SP.
of an MNO for offering some of her resources to an MVNO DifferentpreferencesofEUsforSPsenablesthepossibility
instead of using the resources for her own. Thus, we consider ofdifferentoutcomesforthemarket.Forexample,theMVNO
competitionandcooperationbetweenanMNOandanMVNO. whobundlesherservicewithfreeinternationalcallmayattract
NotethatitisnotaprioriknownhowmuchanMNOiswilling some of EUs even with a higher price than what MNO is
to invest on the infrastructure and how much an MVNO offering. Thus, instead of competing for EUs (by lowering
her price), the MNO can lease some of her resources to this tobeconvex,i.e.thecostofinvestmentperresourcesincreases
MVNO and receives her share of profit through the MVNO. with the amount of resources. For simplicity in analysis, we
We formulate the game as a sequential game, and seek the consider the cost of reserving resources to be quadratic with
Sub-gamePerfectNashEequilibrium(SPNE)ofthesequential respect to I .1
F
game using backward induction. We show that there exists a The utility of SP is:
L
uniqueinteriorSPNE,i.e.theSPNEbywhichbothSPsreceive
π =n (p −c)+sI2 −γI2 (2)
a positive mass of EUs, and characterize it. Moreover, we L L L F L
considerabenchmarkcaseinwhichtheMNOdoesnotinvest where nL, pL, and IL are the number of EUs with SPL,
inherinfrastructure,andsubsequentlytheMVNOcannotlease the access fee quoted by SPL for EUs, and the number of
any resources from the MNO. We characterize the unique resources that SPL add to her infrastructure. Note that γ is
SPNEofthiscase.Wecomparetheresultsofourmodeltothe the marginal cost of investment. Thus, the utility of SPL is
benchmark case to assess the incentive of the MNO to invest increasing with the revenue from EUs (nL(pL −c)) and the
in her infrastructure and to offer it to the MVNO. fee received from SPF (sIF2), and is decreasing with respect
Analytic and numerical results reveal that although the to the investment cost (γI2). Note that we consider the cost
L
number of EUs and the price that the MNO quotes for EUs of investment to be a quadratic function of IL.2
mightbelowerthanthoseoftheMVNO,theMNOwouldstill Trivially,weassumethatthefeeperresources(s)isgreater
generate a higher payoff than the MVNO if the per resource than or equal to the marginal cost of investment on the
fee that the MVNO pays to the MNO is sufficiently higher infrastructure(γ),i.e.s≥γ.Also,notethatIF ≤IL.Tohave
than the marginal cost of investment on the infrastructure. a non-trivial problem, we also assume IL >0 and IF ≥0.
In addition, results reveal that if the MNO knows that EUs The strategies of SPL is to choose the access fee for EUs
are reluctant joining her or have a high preference for the (pL) and the level of investment (IL). The strategies of SPF
MVNO, then a better strategy for her would be to invest in is to choose the access fee for EUs (pF) and the number of
her infrastructure, lease a portion (or the whole) of the new resources she leases from SPL (IF). We assume that the per
resources to the MVNO, and collect the benefit through the resourcefeesispredetermined,possiblythroughabargaining
fees she charges for the resources leased to the MVNO. framework between SPL and SPF.
The rest of the paper is organized as follows. In Section II,
EUs:
we present the model. In Section III, we find the SPNE
The strategy of an EU is to choose one of the SPs to buy
strategies. We present the SPNE outcome in Section IV. In
wirelessplanfrom.WemodeltheEUsusingahotellingmodel.
SectionV,weprovetheresultsofthebenchmarkcase.Finally,
We assume that the SP is located at 0, SP is located at 1,
in Section VI, we present the numerical results and discuss L F
andEUsaredistributeduniformlyalongtheunitinterval[0,1].
about the results.
The closer an EU to an SP, the more this EU prefers this SP
II. MODEL to the other. Note that the notion of closeness and distance
is used to model the preference of EUs, and may not be the
We consider one MNO and one MVNO that compete in
same as physical distance.
attracting a pool of undecided EUs. We assume that the
Specifically, the EU located at x∈[0,1] incurs a transport
MVNO is active, i.e. has already some resources. Thus, the
cost of t x (respectively, t (1 − x)) when joining SP
modelissetupsuchthatevenwhentheMVNOdoesnotlease L F L
(respectively,SP ),wheret (respectively,t )isthemarginal
any resources from the MNO, it can still attract some of EUs. F L F
transportcostforSP (respectively,SP ).Insum,weconsider
L F
SPs: t andt asthereluctanceofEUsforconnectingtoSP and
L F L
WedenotetheMNObySP (LrepresentsLeader,sincethis SPF, respectively.
L
We assume the transport costs of EUs to depend on the
SP is the leader of the game), and denote the MVNO by SP
F
investment level of SPs. Thus, we assume that the higher the
(F represents Follower, since this SP is the follower of this
investment level of an SP in comparison to the other SP, the
leader/follower game). SP owns the infrastructure, invests in
L
higher would be the reluctance of EUs for joining the other
her infrastructure to attract EUs, and can lease parts of the
SP. This model is used to captures the factor of quality in the
new resources to SP in exchange of money.
F
decision of EUs. Thus, the transport cost of SP and SP
WedenotethefeeperresourcesthatSP paystoSP bys. L F
The utility of SPF is increasing with theFrevenue fromL EUs, are considered to be tL = IIFL and tF = 1−tL = ILI−LIF,
respectively.
whichdependsonthenumberofEUs(n )andtheaccessfee
F
Therefore, the utility of an EU located at distance x of
(p ).Thisutilityisdecreasingwithrespecttothefeeshepays j
F
to SP to reserve resources, i.e. sI2, where I is the number SPj, j ∈{L,F} is:
L F F
of resources that SPF reserves from SPL: uxj =v∗−tjxj −pj (3)
π =n (p −c)−sI2 (1)
F F F F 1Theoverallintuitionsofthemodelareexpectedtobethesameinthecase
, where c is the marginal cost associated with each user. Note ofconsideringanyconvexfunctionofIF
2Intuitionsofthemodelareexpectedtobethesamewhenconsideringany
that naturally, we expect the cost of investing in infrastructure convexfunctionofIL.
, where v∗ is a common valuation that captures the value of The EU located at x ∈ [0,1] is indifferent between
n
buying a wireless plan for EUs regardless of the SP chosen. connecting to SP ans SP if:
L F
Formulation: v∗−tF(1−xn)−pF =v∗−tLxn−pL
t +p −p (4)
WemodeltheinteractionbetweenSPsandEUsusingafour- ⇒x = F F L
n t +t
stage sequential game. Naturally, we assume that SP makes L F
L
the first move and is the leader of the game. The timing and Note that tF +tL =1. Substituting the value of tF yields:
the stages of the game are as follows: I −I
x = L F +p −p (5)
1) SP decidesontheinvestmentontheinfrastructure(I ). n I F L
L L L
2) SPF decidesontheinvestment,i.e.numberofresources Thus, the fraction of EUs with each SP (n and n ) is:
L F
to lease from SP (I such that I ≤I ).
L F F L 
0 if x <0
3) SP and SP determine the access fees for EUs (re-  n
L F n = IL−IF +p −p if 0≤x ≤1
spectively, pL and pF). L  1 IL F L if x >n1 (6)
4) EUs decide to subscribe to one of the SPs. n
n =1−n
We assumed the selection of investments by SPs (I and F L
L
I ) to happen before the selection of prices for EUs (p and Stage 3:
F L
pF) since investment decisions are long-term decision. These In this stage, SPL and SPF determine their prices for
decisions are expected to be kept constant over longer time EUs, p and p , respectively, to maximize their payoff.
L F
horizons in comparison to pricing decisions. We seek for Nash equilibrium (NE) strategies. Note that in
In the sequential game framework, we seek a Subgame general there might exist several NE strategies: some of them
Perfect Nash Equilibrium using backward induction: corner equilibria (an extreme case in which one of the SPs
receives zero EUs) and some interior equilibria (in which
Definition 1. Subgame Perfect Nash Equilibrium (SPNE):
both SPs receive a positive mass of EUs). In practice, the
A strategy is an SPNE if and only if it constitutes a Nash
latter equilibria are expected to occur more frequently. Thus,
Equilibrium (NE) of every subgame of the game.
henceforth, we seek to characterize the interior equilibria, i.e.
Definition 2. Backward Induction: Characterizing the equi- when 0<n <1 and 0<n <1.
L F
librium strategies starting from the last stage of the game and Thus, we look for all NE by which 0 < x < 1 (x
n n
proceeding backward. characterized in the previous stage of the game). In this case,
using (1), (2), and (6), the payoffs of SPs would be:
We also assume the full market coverage of EUs by SPs.
This means that each EU chooses exactly one SP to subscribe I
π =( F +p −p )(p −c)−sI2 (7)
to. This assumption is common in hotelling models and is F I L F F F
L
necessary to ensure competition between SPs. An equivalent
I −I
assumption is to consider the common valuation v∗ to be πL =( LI F +pF −pL)(pL−c)+sIF2 −γIL2 (8)
L
sufficientlylargesothattheutilityofEUsforbuyingawireless
2 In the following theorem, we prove that the NE uniquely
plan is positive regardless of the choice of SP.
exists and characterize it:
One might think that SP can eliminate competition by not
L
offering her resources to SP (or simply not investing in new Theorem 1. TheNEstrategiesofStage3bywhich0<n <
F L
infrastructure), and use the full market coverage to act as a 1 and 0<nF <1 are unique, and are pF =c+ IL3+ILIF and
mthoatnowpeolay.ssHumowedevethr,ethSiPs istnoothathvee craesseouirnceosurinmdeopdeenl.dRenetcaolfl pL =c+ 2IL3I−LIF.
F
SPL. Thus, regardless of the strategy of SPL, SPF would be Remark 1. Note that ddpILL ≥ 0 and ddpILF ≤ 0. Thus, as we
expect intuitively, p (respectively, p ) is increasing (respec-
still active in the market, and the full market coverage would L F
tively, decreasing) with respect to I . Also, p (respectively,
not lead to the monopoly of SPL. L L
p )isdecreasing(respectively,increasing)withrespecttoI .
F F
III. THESUB-GAMEPERFECTNASHEQUILIBRIUM
Proof. In this case, every NE should satisfy the first order
In the backward induction, we start with Stage 4:
condition. Thus, p∗ and p∗ should be determined such that
F L
Stage 4: dπF =0 and dπL =0. The first order conditions yield:
dpF dpL
In this subsection, we characterize the division of EUs
I I −I
between SPs in the equilibrium, i.e. nL and nF, using the 2p∗F −p∗L = IF +c & 2p∗L−p∗F = LI F +c
L L
knowledge of the strategies chosen by the SPs in Stages 1,
Thus,
2, and 3. To do so, we characterize the location of the EU
thatisindifferentbetweenjoiningeitheroftheSPs,xn.Thus, p∗ =c+ IL+IF & p∗ =c+ 2IL−IF (9)
EUs located at [0,xn) join SPL, and those located at (xn,1] F 3IL L 3IL
joins the non-neutral SP (using the full market coverage Therefore,p∗ andp∗ aretheuniqueNEstrategiesiftheyyield
F F L
assumption). 0 < x < 1 and no unilateral deviation is profitable for SPs.
n
In Case A, we check the former, and in Case B, we check the s> 1 :
9I2
latter. L
In this case, the payoff of SP is concave. Thus, the first
F
Case A: We first check the condition that with p∗L and p∗F, order condition yields the possible optimum investment level,
0<xn <1. Using (5) and (9): IˆF:
dπ 2(Iˆ +I )
F| =0⇒ F L −2sIˆ =0
xn = ILI−LIF +p∗F −p∗L = 2IL3I−LIF dIF IˆF ⇒Iˆ =9IL2 IL F (12)
F 9sI2 −1
, which is greater than zero since I ≥ I and I > 0. x L
L F L n
is also clearly less than one (note that I ≥ 0). Thus, the Thus, the optimum investment level, I∗ is:
F F
condition 0<xn <1 holds. I∗ =min{Iˆ ,I } (13)
Case B: Note that d2πF < 0 and d2πL < 0. Thus, any F F L
solutions to the first orddpe2Fr conditions dwpo2Luld maximize the Note that from the assumption of this case 9sIL2 > 1. Thus,
payoff of the SPs when 0 < xn < 1, and no unilateral IˆF > 0 (using (12)). In addition, from the expression of IˆF
deviation by which 0<x <1 would be profitable for SPs. (12):
n Iˆ <I ⇐⇒ 9sI2 >2
Now,wediscussthatanydeviationbySPssuchthatn =0 F L L
L
and nL =1 (which subsequently yields nF =1 and nF =0, Thus, the optimum investment level for SPF would be:
respectively) is not profitable. Note that the payoff of SPs,
 (cid:113)
(s1u)bsaenqdue(n2t)l,yayrieelcdosnntiFnu↑ou1s aansdnnLF↓↓00,,arnedspencLtiv↑el1y).(wThhiucsh, IF∗ = I9sIIL2L−1 iiff IIL >≤(cid:113)922s (14)
thepayoffsofbothSPswhenselectingp∗ andp∗ (solutionsof L L 9s
L F
first order conditions) are greater than or equal to the payoffs s< 1 :
9I2
when n = 0 and n = 1. Thus, any deviation by SPs such L
L L In this case, π would be convex. Thus, the optimum level
F
that n =0 or n =1 is not profitable for SPs.
L L of investment would be on the boundaries of the feasible set.
This complete the proof that p∗F and p∗L are the unique NE Thus, IF∗ would be either 0 or IL, whichever yields a higher
strategies by which both SPs are active, i.e. 0<xn <1. payoff. Note that:
1 4
π | = & π | = −sI2
Stage 2: F IF=0 9 F IF=IL 9 L
Thus, I∗ = I if and only if 4 −sI2 ≥ 1 (Note that we
In this stage of the game, SP decides on the investment F L 9 L 9
F assumed that if I = 0 and I = I yield the same payoff,
F F L
level,i.e.thenumberofresourcestobeleasedfromSPL (IF), then SP chooses I = I ). Thus, s ≤ 1 yields I∗ = I .
with the condition that IF ≤IL to maximize πF: Note thaFt from the FassumpLtion of the ca3sIeL2, s < 1 F< 1L.
9I2 3I2
max π = max (cid:0)IL+IF(cid:1)2−sI2 (10) Thus, IF∗ =IL. L L
IF≤IL F IF≤IL 3IL F s= 9I12:
L
In this case, π would be an increasing linear function of
Note that for the last equality, we used (7) and Theorem 1. F
I . Thus, I∗ =I .
F F L
Theorem 2. The optimum investment level of SP is: Putting all the cases together, the result of the theorem
F
follows.
 (cid:113)
IF∗ = 9sIIL2L−1 if IL >(cid:113)92s (11) Stage 1:
 IL if IL ≤ 92s In this stage, SPL decides on the level of investment IL to
maximize her payoff, π :
L
2I −I∗
Remark 2. If the fee per resources (s) or the investment by maxπ =max( L F)2+sI∗2−γI2 (15)
SPL (IL)islow,thenSPF reservesalltheavailableresources. IL L IL 3IL F L
I(IfF∗no<t,ItLh)e.nNSoPteFthraesteirnvethsisacfarsaec,tioddIInFL∗o<f a0v.aTilhaubsl,ethreesohuigrhceers iwshcehraerfaocrtetrhiezeldastineq(u1a1l)i.tyI,nwtheeusneedxt(8th)eaonrdemTh,ewoeremcha1r,aacntedriIzF∗e
the number of available resources, the lower would be the the candidate optimum answers:
number of resources reserved by SP .
F Theorem 3. The optimum solution to (15) is I∗ =
(cid:113) L
Pπrooifs. cNoontveexthaitf isn (<10),1πF, aisndcoinscalivneeaorveifr IsF=if s1>.9WI1L2e, cmoinnd{ition92so,InˆL:}, where IˆL is the solution of the first order
F 9I2 9I2
characterize the optimum iLnvestment level, i.e. I∗ undeLr each 1 1 I
of these conditions: F πL,IF∗ = 9(2− 9sI2 −1)2+s9sI2L−1 −γIL2 (16)
L L
(cid:113)
Remark 3. Theorem yields that the minimum optimum level A. If I∗ > 2 :
(cid:113) L 9s
of investment by SPL is 92s. Corollary 1 (Outcome A). If I∗ >(cid:113) 2 , then:
L 9s
Remark 4. Note that the first term in (16) is increasing with
I (since the number of EUs is increasing with I ). The • Stage 1:TheoptimumlevelofinvestmentofSPL isIL∗ =
L L Iˆ (characterized in Theorem 3).
secondtermisthepaymentfromSP whichisdecreasingwith L
respect to IL (since when IL > (cid:113)F92s, IF∗ is decreasing with • SI∗tag=e 2:IL∗The.optimum level of investment by SPF is:
respect to IL). The third term is the cost of investment which F 9sIL∗2−1 1+ 1
is decreasing with IL. Thus, when either s or γ is sufficiently • Stage 3: Prices for EUs are: p∗F = c+ 9s3IL2−1 and
large, we expect the utility (16) to be decreasing with respect 2− 1
(cid:113) p∗ =c+ 9sIL2−1.
to I , and the optimum answer to be I∗ = 2 . On the L 3
otheLr hand, IˆL is expected to be the optiLmum an9sswer when • S1+ta9gseIL214−:1Tahnedfrna∗ct=ion2s−o9fsIEL21−U1s.with each SP are: n∗F =
both s and γ are sufficiently small. We discuss about this in 3 L 3
Numerical Results (Section VI). Numerical results reveal that Proof. The outcome of Stage 2 directly follows from the
(cid:113)
for large sets of parameters, we expect I∗ = 2 . results of Stage 2. The outcome of Stage 3 follows from (9)
F 9s
and the outcome of Stage 2. The outcome of Stage 4 follows
Proof. To find the optimum level of investment, I∗, consider
L from (6) and the outcomes of Stage 2 and 3.
two scenarios:
(cid:113)
IL ≤ 92s: speNcottteotIh∗atainndthsi.sIncaasded,iatsiodni,scguivsesnedt,hIaFt∗Iisdisecfirxeeads,inpg∗wainthdrpe∗-
In this case, I∗ =I by (11). Thus, the maximization (15) L L F L
F L are decreasing and increasing with respect to the per resource
would become:
fee, s, respectively. Also, given that s is fixed, p∗ and p∗
m√ax (s−γ)IL2 are decreasing and increasing with I∗, respectivelyF. RegardL-
IL≤ 92s less of p∗ (respectively, p∗) being dLecreasing (respectively,
Thus: F L
(cid:40) (cid:113) increasing), the number of EUs with SP (respectively, SP ),
2 if s≥γ F L
I∗ = 9s (17) i.e. n∗ (respectively, n∗) is still decreasing (respectively,
L F L
0 if s<γ increasing) with respect to I∗ (if s fixed). This is because
L
Note that we assumed that if IL > 0 and IL = 0 yield the of the increase (respectively, decrease) in tF (respectively,
same payoff for SPL, then this SP chooses IL > 0. This is tL), i.e. the transport cost of SPF (respectively, SPL). To
the reason that for s=γ, I >0 is chosen. Also, recall that understand these changes in the transport costs, recall that
L
to have a non-trivial problem, we assumed s ≥ γ. Thus, if increasing IL∗, decreases IF∗, and subsequently increases tF.
IL ≤(cid:113)92s, then IL∗ =(cid:113)92s. ThIuns,StehcetiorenlatVioIn,swhiep obbetswereveentshaatnIdL∗thies oduetpceonmdeenitsomnorse.
(cid:113)
I > 2 : complicated.
L 9s
In this case, IF∗ = 9sII2L−1. Thus, the maximization (15) B. If I∗ ≤(cid:113) 2 :
would become: L L 9s
(cid:113)
tIhLeNIL→pom>rtee√avx(cid:113)i9t2ohsu9aπ2sstL,cI,IaπF∗F∗sLe=,(.I1F∗T119h)(→u2iss−,πcth9oLesn|IItoLiL21nn=−uly√o1up9)s2os2s,a+stwibsIhl9LeiscoIh=ILp2Lti−is(cid:113)m1cu9o2m−sn.sγaiTdnILeh2srwuesed,(r1ai8inss) Co••••roSSSSllttttaaaaarggggyeeee24123::::(OnIIpLF∗F∗∗∗Fut==c==om(cid:113)Ic23L∗e+.9a2sBn23.)d.annIf∗LdI=pL∗∗L≤13=. c+92s,13t.hen:
the solution to the first order condition on πL,I∗, i.e. IˆL if Proof is similar to the proof of the previous corollary.
(cid:113) F
IˆL > 92s. Results of the theorem follows. In this case, SPF reserves all available resources, and the
investment level of SP (which is equal to the number of
L
resourcesreservedbySP )isadecreasingfunctionofthefee
F
IV. THEOUTCOMEOFTHEGAME per resources, i.e. s. SP quotes a higher price for EUs in
F
In this section, we characterize the equilibrium outcome comparison to SP . In spite of the higher price, SP would
L F
of the game using the results of the previous section and be able to attract more EUs given the better investment level
discuss about them. In Corollaries 1 and 2, we characterize in comparison to SP which translates into a lower transport
(cid:113) (cid:113) L
the equilibrium outcome when I∗ > 2 and I∗ ≤ 2 , cost.
L 9s L 9s
respectively. Note that in each case, there exists a unique We calculate πL and πF using Corollary 2 and (2):
outcome and the two cases are mutually exclusive. Thus, the
1 2γ 2
outcome of the game exists and is unique. π∗ = − & π∗ = (19)
L 3 9s F 9
Thus, the payoff of SP would be higher than the payoff of , which is greater than zero since t > 0 and t > 0. x is
L F L n
SP ,i.e.π∗ >π∗,ifandonlyifs>2γ.Inthiscase,although also clearly less than one. Thus, the condition 0 < x < 1
F L F n
in comparison to SP , SP offers her service to EUs with holds.
F L
lower price and attracts a lower number of EUs, she can still Case B: Note that d2πF < 0 and d2πL < 0. Thus, any
dp2 dp2
obtain a higher payoff through the per resource fee that she solutions to the first ordeFr conditions woLuld maximize the
collectsfromSPF.Thus,SPL leasestheresourcestoSPF and payoff of the SPs when 0 < xn < 1, and no unilateral
instead generates revenue through the fee she charges to SPF. deviation by which 0<xn <1 would be profitable for SPs.
Now,wediscussthatanydeviationbySPssuchthatn =0
L
V. BENCHMARKCASE and n =1 (which subsequently yields n =1 and n =0,
L F F
Now, we consider the case in which SP does not invest respectively) is not profitable. Note that the payoff of SPs,
L
in her infrastructure. Similar to the original model, both SP (22) and (23), are continuous as n ↓ 0, and n ↑ 1 (which
L L L
and SP compete to attract a pool of undecided users. The subsequently yields n ↑ 1 and n ↓ 0, respectively). Thus,
F F F
hotelling model for EUs is the same as before. The only thepayoffsofbothSPswhenselectingp∗ andp∗ (solutionsof
L F
differenceisthetransportcost.Here,weconsiderthetransport first order conditions) are greater than or equal to the payoffs
coststobeparameters,anddenotethembyt >0andt >0 when n = 0 and n = 1. Thus, any deviation by SPs such
L F L L
for SP and SP , respectively. We discuss about the effects that n =0 or n =1 is not profitable for SPs.
L F L L
of these parameters in the Numerical Results (Section VI). This complete the proof that p∗ and p∗ are the unique NE
F L
The analysis of stage 4 of the game would be the same as strategies by which both SPs are active, i.e. 0<x <1.
n
before, and by (4):
Note that in the absence of investments, Stage 2 and 1
t +p −p would be of no importance. In the following corollary, we
x = F F L (20)
n t +t characterize the outcome of the game in the benchmark case
L F
(subscript B stands for Benchmark):
Thus,

0 if x <0
 n Corollary 3. Theequilibriumoutcomeofthebenchmarkcase
nL = 1tFt+Lp+Ft−FpL iiff 0x≤>xn1≤1 (21) is as follows:
n
nF =1−nL • cN+EptrLi+ci2ntgF.strategiesare:p∗F,B =c+2tL3+tF andp∗L,B =
3
decNisoitoen.thTahtuisn, tthheepbaeynocfhfsmoafrkSPcsaswe,outhlderbeeismondoifiinevdeastsm: ent • Fractions of EUs with each SP: n∗L,B = 32(ttFF++ttLL) and
n∗ = tF+2tL .
F,B 3(tF+tL)
π =n (p −c) (22) Using (22) and (23), the payoffs of SPs in these cases are:
F F F
t +2t 2t +t
πL =nL(pL−c) (23) π∗ =( F L )2 & π∗ =( F L )2
F,B 3(t +t ) L,B 3(t +t )
F L F L
Thus, the only decision of SPs is to determine their pricing (25)
for EUs. We use the same approach as Stage 3 of the game Note that if t >t , then p∗ >p∗ , n∗ >n∗ , and
L F F,B L,B F,B L,B
to characterize the NE pricing strategies for SPs by which subsequently π∗ >π∗ , and vice versa. Thus, the SP that
F,B L,B
0<x <1: EUs have lower reluctance for, receives the highest payoff.
n
Theorem 4. The NE pricing strategies of SPs are: VI. NUMERICALRESULTSANDDISCUSSIONS
p∗ =c+ 2tL+tF In this section, we use numerical simulations (i) to de-
F 3 termine whether and under what conditions the outcome in
t +2t
p∗ =c+ L F Corollary 1 (Outcome A) would emerge, and (ii) to provide
L 3
insights for results under different parameters of the model.
Proof. Proof is similar to the proof of Theorem 3. p∗F and p∗L For all results, we consider c=1.3
should be determined such that dπF =0 and dπL =0: In Figure 1, we plot the optimum level of investment of
dpF dpL
SP (I∗), the number of resources that SP reserves (I∗),
p∗F =c+ 2tL3+tF & p∗L =c+ 2tF3+tL (24) andLtheLminimumoptimumlevelofinvestmenFtbySPL ((cid:113)92Fs),
Therefore, p∗ and p∗ are the unique NE strategies if they when γ = 0.1 (left) and γ = 0 (right) (recall that, in (2), γ
F L
yield 0<x <1 and no unilateral deviation is profitable for is the marginal cost of investment.). The discontinuities are
n
SPs.InCaseA,wechecktheformer,andinCaseB,wecheck because of the transition of theoutcome of the game from the
the latter. Outcome A to Outcome B (outcome in Corollary 2). Results
Case A: We first check the condition that with p∗ and p∗, reveal that for each outcome, IL∗ and IF∗ are decreasing with
L F
0<x <1. Using (20) and (24): the per resource fee, s. Thus, the higher the fee per resources
n
x = 2tF +tL 3Notethatthechoiceofcbarelyaffectstheresults.Itmayonlyshiftsome
n 3(t +t ) oftheresults(e.g.thepricechargedtoEUs)byonlyaconstant.
F L
Fig. 1: Investment decisions of SPs vs. per resource fee, s Fig. 2: Pricing decisions of SPs for EUs versus s
that SP pays to SP , the lower would be the number of
F L
resourcesthatSP reserves,andsubsequentlythelowerwould
F
be the investment of SP . Also, when the marginal cost of
L
investment (γ) is zero, SP investment is significantly more
L
in comparison to γ = 0.1. Thus, intuitively, the higher the
investment cost, the lower would be the investment.
Results also confirm the intuitions presented in Remark 4
thatforsufficientlysmallsandγ,theoptimumlevelofinvest-
mentbySP wouldbeequaltoIˆ (introducedinTheorem3)
L L
(cid:113)
which is higher than 2 (Outcome A). Numerical results
9s
alsorevealthatforγ >0.12,OutcomeAwouldnotoccureven Fig. 3: Payoffs of SPs for EUs versus s
for small s. Thus, the outcome in which Iˆ is the optimum
L
level of investment only occurs for a small set of parameters.
Therefore,forawiderangeofparameters,weexpectOutcome yield that when γ is extremely small (thus, I∗ is large), the
L
B to be the outcome of the game. rate of change in I∗ with s dominates the rate of change in
L
In Figure 2, we plot the pricing decisions of SPs for EUs, s. Thus, as s increases, p∗ increases and p∗ decreases.
F L
i.e.p∗ andp∗,forγ =0.1(left)andγ =0(right).Similarto NotethatbyCorollaries1and2,thedependencyofn∗ and
L F L
Figure 1, the discontinuities in the figures are because of the n∗ to parameters of the model is similar to the dependency
F
transition of the outcome of the game from Outcome A to B. of p∗ and p∗. The only difference is the exclusion of c from
L F
Note that in Outcome B (when s is sufficiently large), p∗ and the expressions.
F
p∗ are constant (given that c is a constant) independent of γ InFigure3,weplotthepayoffsofSPs,i.e.π∗ andπ∗,and
L F L
and s, and the price that SP charges is twice the price that the payoff of SP in the benchmark case for three scenarios
F L
SP charges. However, in Region A, if γ, i.e. the marginal (i) t = 0.5 and t = 0.5, (ii) t = 1 and t = 0, and
L F L F L
cost of investment, is extremely small (e.g. zero), then SP (iii) t =0 and t =1. Results reveal that although for some
L F L
would be able to charge a higher price than SP (Figure 2- parameters,thepricethatSP quotesforEUsandthenumber
F F
right). The reason is that for γ small, SP will invest more ofEUsthatshecanattractishigherthanthepriceofSP and
L L
(I∗ small). We also stated in Section IV after Corollary 1 that the number of EUs of this SP, SP can still fetch a higher
L L
p∗ andp∗ aredecreasingandincreasingwithI∗,respectively. payoff than SP . The reason is the fee that SP pays to SP .
F L L F F L
Thus,smallinvestmentcostyieldsahigherinvestmentbySPL In addition, we can only expect an investment by SPL if
and as a result a higher price for EUs of this SP. thisinvestmentyieldsahigherpayoffthanthebenchmarkcase
Also note that in Outcome A, depending on γ, prices can with no investment. Results in Figure 3 yield that the higher
be increasing or decreasing with s: If the marginal cost of the transport cost of SP in comparison to SP , the lower
L F
investment is extremely small, then p∗ is increasing with s would be the payoff of SP in the benchmark case, and the
F L
and p∗ is decreasing with s (Figure 2-right). The opposite is higher would be the incentive of this SP to invest and lease
L
truewhenγ isnotsmall(Figure2-left).Todescribethereason her resources to SP . Thus, the preferences of EUs for SPs,
F
behind this result, note that from Corollary 1, p∗ and p∗ are i.e. the transport costs, are one of the main factors in gauging
L F
increasing and decreasing, respectively, with respect to both the plausibility of a cooperation between an MNO and an
s and I∗. On the other hand, I∗ is itself decreasing with s MVNO. If an MNO knows that EUs are reluctant joining her,
L L
(Figure 1). Thus, as s increases two factors one increasing (s) i.e.EUshaveahightransportcostsfortheMNO,thenshemay
and one decreasing (I∗) affect the prices. Numerical results investinherinfrastructure,leaseaportionofthenewresources
L
to an independent MVNO, which EUs have lower reluctance
for, and collect the profit through the fees she charges to the
MVNO.
In Figure 3, note that if s is small, then SP fetches a
F
higher payoff than SP since she can get a higher revenue
L
from EUs (Figure 2-left and recalling that n∗ has the same
F
pattern as p∗). Overall, increasing s increases the payoff of
F
SP . Counter intuitively, when s is large enough such that
L
OutcomeBoccurs(Corollary2)SP receivesahigherpayoff
F
in comparison to her payoffs when s is small. This occurs
since in this case, SP reserves all the available resources
F
from SP . This enables SP to charge a high price for EUs
L F
while still attracting EUs. Thus, when s is large enough, both
SPs receive high revenue at the expense of EUs paying more.
Thus, the welfare of EUs would be low. This highlights the
importance of a framework by which the per resource fee is
determinedsuchthatallentitiesofthemarketcanbenefit.This
frameworkcouldbeabargaininggamebetweenSPswithsome
restrictions imposed by a regulator. This is a topic of future
work.
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