Stackelberg Game Modeling of Pricing for Mobile Virtual Network Operators

With rapid development of new network technologies, demands of network users grow dramatically. Primary mobile network operators (POs) cannot meet the demands of users. Therefore, secondary mobile virtual network operators (SOs) emerge to alleviate pressure on the network market. Obviously, the competition becomes more serious, for users have more choices for operators based on their requirements. Because all operators want to get their maximum profits, the main research of this paper is to obtain the maximum profits for one PO and one SO through pricing which uses Stackelberg leader-follower game. We have considered the changes of users’ demands, got the real-time equilibrium pricing point, and finally proved the correctness of our method.


INTRODUCTION
Wireless spectrum can be obtained in almost all countries all over the world, the utilization of wireless spectrum is indispensable for all businesses of wireless communications.The wireless spectrum refers to the continuous frequencies ranging from 3 kHz to 300 GHz typically.Concerning the management of wireless spectrum [1,2], spectrum management organizations for all countries have common characteristics.They divide wireless spectrum into continuous frequency bands.Each frequency band will be allocated to a particular primary mobile network operator (PO), and these operators will be granted absolute ownership for the given frequency bands [3].Other operators who don't get the license rights can't directly employ the given frequency bands.However, once an operator gets the spectrum license right, it possesses the right to lease portion of its spectrum to other operators, who also wish to entering into network market.This phenomenon can improve the social benefit [4].
Many articles have investigated the pricing competition in wireless network and game theory [5,6,7,8].Luis Grijarro establishes a model about the competition between one PO and one SO [9].He adopts a method of backward induction to analysis the interaction between PO and SO.Our paper is based on his model.We use a Stackelberg leader-follower game to resolve the competition game, and give the specific calculations for readers.
In this paper, the problem of how to maximize the profits of one primary mobile network operator (PO) and one secondary mobile virtual network operator (SO) through pric-ing is analyzed.We establish a model about the relationship of PO, SO and users, and use Stackelberg game to resolve the problem above [10].In order to develop technology innovation, PO leases portion of its spectrum license rights to SO [11,12].Both PO and SO provide service for users.Users choose PO or SO based on price, QoS (quality of service) and their basic willingness to pay [13].In this paper, It is assumed that there are only two operators in the network market, PO and SO respectively, and users have the right to choose one operator or neither.
The main contributions of this paper are as follows: (1) Analyze the change of numbers which users choose PO and SO through considering the distribution of users' basic willingness to pay for each bandwidth.
(2) Maximize the profits of PO and SO through pricing using Stackelberg game.
(3) Obtain the relationship between the equilibrium price point and users' demands and network supplies.
The remainder of this paper is organized as follows.Service model, users' utilities and definition of operators' profits are described in Section 2. Specific calculations process of game is derived in Section 3. Results of Stackelberg game are analyzed in Section 4. Section 5 concludes this paper and introduces the future work.The final part is the references.

SERVICE MODEL
The service model is depicted in Fig. 1.It is composed of PO, SO and users.We assume that there are only two network operators in the network market.PO possesses the spectrum license rights of W MHz frequency bandwidth in total, and leases b MHz to SO, which means that it leaves itself W − b MHz.SO pays for spectrum licenses at a price of p per Hz.Both PO and SO provide service to users.Each user should pay pp m.u.if he chooses PO, or pay ps m.u.if he chooses SO.Here, m.u. is monetary unit.These three prices p, pp and ps are referred at the same time period.We assume that there are n users in total, and the quantity of users is enough for competition.There are np users who choose PO and ns users who choose SO at state which users do not change their demands.It is assumed that the number of users who choose PO at next state which users' demands changed is n * Ωp, and for SO is n * Ωs.Ωp is the percent number of users who choose PO, so as Ωs.We study the moment which users' demands change, so the unit of time is 1s.
Users' utilities and definition of operators' profits will be represented in 2.1 and 2.2.

Users' Utilities
There are three factors which may influence one user's choice: user's basic willingness to pay for each bandwidth, QoS, and price respectively.
Firstly, this paper defines τi as user's basic willingness to pay for each bandwidth.i=[1, 2, . . ., n]. τi is distributed uniformly.It's probability density function f (x) is positive and continuous on [0, φ] and φ>0.It's cumulative density function is defined as The unit of τi is m.u./bit.We assume that users' demands change at the first second on the next state.Secondly, this paper defines Qp and Qs as the representative of QoS which is offered by PO and SO, respectively.Qp and Qs can be described by spectral efficiencies k (p) and k (s) [14].k (p) and k (s) are assumed fixed and k (p) <k (s) .SO has a more efficient operation than PO, thus it can compete with PO.The spectrum supplied by each operator is evenly divided by users, so QoS can be influenced by np and ns.The more users choose PO, the worse QoS becomes.While QoS becomes worse, less users will choose PO.The functions of Qp and Qs are described as ns ).This indicates that the QoS is affected by supplies and demands.We define that Qp > Qs for PO has more resources than SO.
The third factor is price.The higher the price is, the lower the utility is.Users intend to get high quality of service with low price.Considering the discussions above, user's utility functions are obtained.We use ui,p and ui,s to represent the utility functions for PO and SO individually.
The unit of Qp is bit, so the unit of ui,p is m.u.. Users expect to obtain corresponding or more services for their net charge.Therefore, user i chooses PO when ui,p>ui,s and ui,p>0, chooses SO when ui,p<ui,s and ui,s>0, and chooses neither when ui,p<0 and ui,s<0.

Definition of Operators' Profits
In this paper, the expressions of the profits of PO and SO are described as PO can get profits from users and SO.Ωp is the percent number of users who choose PO after our game, pp is the net charge that each user should pay.SO pays the rent to PO. SO buys b MHz spectrum license rights from PO. p is the unit price.Cp is the basic cost of PO.The composition of the profit of SO is similar.We assume that p, b, Cp and Cs are fixed in this paper.But at the future work, p and b will be considered to affect the profit of operators.
The notions used throughout this paper are summarized as shown in Table 1.

SPECIFIC PROCESS OF GAME
The general process of our game can be summarized in two steps.First, we should calculate the number of users who choose two operators through considering the distribution of their basic willingness to pay for each bandwidth.The second step is to maximize the profits of two operators through Stackelberg game.For Qp>Qs, the slope of users' utilities who choose PO and SO can be got.If τi,p<τi,s, we can get that τ 0 i <τi,p<τi,s for Qs Qp−Qs > 0. If τi,s<τi,p, τi,s<τi,p<τ 0 i can be got.These can be proved furthermore by Fig. 2. When ui,p = ui,s > 0, we can get the game equilibrium.So the condition that τi,s<τi,p<τ 0 i is reasonable.We have known that τi ∈ [0, φ], so 0<τi,s<τi,p<τ 0 i <φ.It can be obtained the distribution of users through analyzing Fig. 2(b).

Users' Choices
When τ 0 i < τi < φ, it can be got that ui,p > ui,s.When τi,s < τi < τ 0 i , ui,p < ui,s.Ωp and Ωs are the percent number which users choose PO or SO and they are the functions of pp and ps.Therefore, the profit of each operator is the function of pp and ps.If we adopt the fixed price, the number of users who choose PO is n*Ωp, so as SO which is n*Ωs, and they are also fixed.Because we want to obtain the maximum profits of two operators, we use game theory to change the price to get an equilibrium point.That is the reason why we consider Ωp, Ωs, Πp and Πs as the functions of pp and ps.

Pricing Game
In this paper, Stackelberg leader-follower game is used to obtain the maximum profits of PO and SO.PO is the leader, and SO is the follower.PO makes its strategy first, and then SO makes the corresponding strategy to maximize its own profit.PO knows what strategies SO would make.The specific analysis using mathematics is as follow: Firstly, assuming that pp is fixed, so that Πs is only the function of ps.Then we calculate the partial deriving for Πs to ps.Let the partial derivative function equal to zero, then the best response of SO is got.pp is expressed by ps.
Next, we substitute ps into the function of Πp.Πp will be the function of pp only.So the maximum profit of Πp can be got through deriving Πp to pp and let this equal to zero.The equilibrium pricing point is obtained.Finally, the maximum values of both Πp and Πs are also got.The specific process of the calculations are as follows: Calculating partial deriving for Πs to ps and let it equal to zero.They can be shown as Eq. ( 7) and Eq. ( 8): ps is expressed as the function of pp base on our mathematical calculation.
Next, by substituting ps into the function of Πp, we can express Πp as the function of pp only.Furthermore, we make the derivation for Πp to pp, and let the derived function equal to zero.They can be expressed as Eq. ( 10) and Eq.(11).
pp can be got as Eq. ( 12) Furthermore, ps can be got as Eq. ( 13) According to Eq. ( 12) and Eq. ( 13), the value of pp and ps can be got.Therefore the max values of Πp and Πs are obtained from Eq. (3) and Eq. ( 4) under the situation that variables p, b, Cp and Cs are known.Section 4 will analyzes our results.

ANALYSIS RESULTS
In this paper, we analyze the results from two aspects: 1) The relationship between the equilibrium price point and users' demands and network supplies.2) Demonstration of the correctness of our method.

Analysis of Equilibrium Point
It can be proved that the equilibrium price p * p and p * s are changed with users' demands and network supplies.p * p and p * s relate to Qp and Qs from Eq. ( 12) and Eq. ( 13).But Qp and Qs are the functions of W , b, np and ns which are the specific reflections of supplies and demands.
p * p and p * s are the linear functions for φ, and they are strictly monotone increasing with φ.That is, the higher users' maximum willingness to pay for each bandwidth, the higher the price.It can be proved through As Eq. ( 12) shows, the higher the Qp, the higher the p * p .Considering that p * p as the function of Qp, we can validate our conclusion as Eq. ( 14) In We have known that Qp>Qs, so that There is a point that Qs = (2 − √ 2)Qp when Eq. ( 17) reflects that the price should be decreased when the supply is overmuch.Those above analyze the relationship between the equilibrium price and users' demands and network supplies.It shows that our equilibrium pricing point is real-time with the supplies and demands.The correctness of our method will be proved next.4) prove the correctness of our method that using Stackelberg game can maximize operators' profits.According to Eq. (18), we can realize that SO gets its best reaction with PO's price to acquire its maximum profit.We can also realize that PO can obtain its best strategy from Eq. ( 19) as PO has known SO's response.If SO does't take its best response as Eq. ( 9), it can't get the maximum profit.In the same way, if SO takes the best strategy while PO has't chosen the price p * p , PO also can't obtain the maximum profit.

CONCLUSIONS AND FUTURE WORK
In this paper, a model that using Stackelberg game to maximize the profits of network operators including both PO and SO is proposed.It is considered first the changes of users' number which they choose operators based on their basic willingness to pay for each bandwidth.Then the specific process of the game are given.In addition, Analysis is conducted to verify the real-time of equilibrium pricing point and the correctness of the proposed method.At the Stackelberg equilibrium point, the following conclusions can be made: 1) The profits of both PO and SO are maximized.
2) Neither the user has incentive to change its operator, nor the operator has the incentive to change its price.3) Our equilibrium pricing point is real-time with users' demands and network supplies.
The important factors that p, b and the basic cost Cp, Cs will be considered to maximize the profits of operators at future work.The variables p and b reflect further the relationship between PO and SO as the leader and the follower.

Table 1 :
Basic notations used in our game model notation unitThe meaning of each notation