Multilevel Pricing Schemes in a Deregulated Wireless Network Market - - PowerPoint PPT Presentation

Multilevel Pricing Schemes in a Deregulated Wireless Network Market

Yezekael Hayel CERI/LIA University of Avignon Workshop on New Avenues for Network Models IISc - 15/01/2014

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Plan

1

Introduction to Hierarchical Game Theory principles

2

Application: Spectrum market in Wireless Networks

3

Conclusions

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Plan

1

Introduction to Hierarchical Game Theory principles

2

Application: Spectrum market in Wireless Networks

3

Conclusions

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Introduction

Introduction

In a competition setting, there are several major concerns when some players may determine their actions after observing the actions of the other players. In game theory setting, we say that one of the players has the ability to enforce his strategy on the other players.

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Definition

Leader/Follower

In a game theoretic formulation of a hierarchical competition, we define two types of players: Leaders: players that take their decisions first, Followers: players that take their decisions after observing the leader’s decisions. IMPORTANT: The leaders know, ex ante, that the followers observe their action.

Stackelberg setting

The Stackelberg competition modela is a strategic game in economics in which the leader firm moves first and then the follower firms move

• sequentially. It considers only two players in competition.
• aH. von Stackelberg, Market Structure and Equilibrium: 1st Edition Translation into English,

Bazin, Urch & Hill, Springer 2011 (1934).

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Definition

Leader/Follower

In a game theoretic formulation of a hierarchical competition, we define two types of players: Leaders: players that take their decisions first, Followers: players that take their decisions after observing the leader’s decisions. IMPORTANT: The leaders know, ex ante, that the followers observe their action.

Stackelberg setting

The Stackelberg competition modela is a strategic game in economics in which the leader firm moves first and then the follower firms move

• sequentially. It considers only two players in competition.
• aH. von Stackelberg, Market Structure and Equilibrium: 1st Edition Translation into English,

Bazin, Urch & Hill, Springer 2011 (1934).

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Basic formulation

Utility functions

As the leaders know that the followers observe their action, the follower’s actions can be seen as implicit functions in the leader’s utility.

Stackelberg setting

We consider two players with the utility functions U1(a1, a2) and U2(a1, a2). Each player maximizes his utility function: max

a1

U1(a1, a2) and max

a2

U2(a1, a2).

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Basic formulation

Utility functions

As the leaders know that the followers observe their action, the follower’s actions can be seen as implicit functions in the leader’s utility.

Stackelberg setting

We consider two players with the utility functions U1(a1, a2) and U2(a1, a2). Each player maximizes his utility function: max

a1

U1(a1, a2) and max

a2

U2(a1, a2).

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Basic formulation

Simultaneous play

If both player takes his action without observing the action of the other, we look for a standard Nash equilibrium of this non-cooperative game.

Non-simultaneous play

We assume that player 1 decides first his action and player 2 takes his action after observing player 1’s action. player 1 is the leader, player 2 is the follower, player 1 knows that player 2 observes his action before taking his own decision.

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Basic formulation

Simultaneous play

If both player takes his action without observing the action of the other, we look for a standard Nash equilibrium of this non-cooperative game.

Non-simultaneous play

We assume that player 1 decides first his action and player 2 takes his action after observing player 1’s action. player 1 is the leader, player 2 is the follower, player 1 knows that player 2 observes his action before taking his own decision.

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Basic formulation

Maximization problem

For each action a1 of the leader, we denote by BR2(a1) the best-response function (or set) which determines the best action of the follower depending on the action of the leader, i.e. ∀a1, BR2(a1) = arg max

a2

U2(a1, a2). As the leader knows that the follower observes his action, the leader has the following utility function to maximize: max

a1

U1(a1, BR2(a1)). IMPORTANT: This function depends only on his own action a1, through the best-response function of the follower.

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Basic formulation

Maximization problem

For each action a1 of the leader, we denote by BR2(a1) the best-response function (or set) which determines the best action of the follower depending on the action of the leader, i.e. ∀a1, BR2(a1) = arg max

a2

U2(a1, a2). As the leader knows that the follower observes his action, the leader has the following utility function to maximize: max

a1

U1(a1, BR2(a1)). IMPORTANT: This function depends only on his own action a1, through the best-response function of the follower.

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Basic formulation

Maximization problem

For each action a1 of the leader, we denote by BR2(a1) the best-response function (or set) which determines the best action of the follower depending on the action of the leader, i.e. ∀a1, BR2(a1) = arg max

a2

U2(a1, a2). As the leader knows that the follower observes his action, the leader has the following utility function to maximize: max

a1

U1(a1, BR2(a1)). IMPORTANT: This function depends only on his own action a1, through the best-response function of the follower.

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Stackelberg solution

Stackelberg equilibrium

A Stackelberg equilibrium is a vector of actions (a∗

1, a∗ 2) such that:

a∗

1 = arg max a1

U1(a1, BR2(a1)), and a∗

2 = BR2(a∗ 1).

Figure: H. Von Stackelberg

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Stackelberg model

Subgame perfect Nash equilibrium

The Stackelberg model can be solved to find the subgame perfect Nash equilibrium (SPNE), i.e. the strategy profile that serves best each player, given the strategies of the other player and that entails every player playing in a Nash equilibrium in every subgame.

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Stackelberg solution

In practice using Backward induction

In order to determine a Stackelberg equilibrium solution, there are three steps:

1

To compute the best-response function of the follower, for each action of the leader.

2

To solve the optimization problem for the leader, using the best-response function of the follower.

3

To determine the best action of the follower, using the best-response function, when the leader takes his decision determined in step 2.

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Stackelberg solution

In practice using Backward induction

In order to determine a Stackelberg equilibrium solution, there are three steps:

1

To compute the best-response function of the follower, for each action of the leader.

2

To solve the optimization problem for the leader, using the best-response function of the follower.

3

To determine the best action of the follower, using the best-response function, when the leader takes his decision determined in step 2.

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Stackelberg solution

In practice using Backward induction

In order to determine a Stackelberg equilibrium solution, there are three steps:

1

To compute the best-response function of the follower, for each action of the leader.

2

To solve the optimization problem for the leader, using the best-response function of the follower.

3

To determine the best action of the follower, using the best-response function, when the leader takes his decision determined in step 2.

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Matrix game example

First example (Normal Form Game)

We consider the following matrix game:

• K

U L (3, 1) (1, 3) R (2, 1) (0, 0)

• .

The (pure) NE is the couple (L, U). If line player is the leader and row player the follower, the SPNE when the leader takes action L (resp. action R) is the couple (L, U) (resp. (R, K)). The SE if the line player is the leader, is the couple (R, K). The payoff of the leader is increased.

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Matrix game example

First example (Normal Form Game)

We consider the following matrix game:

• K

U L (3, 1) (1, 3) R (2, 1) (0, 0)

• .

The (pure) NE is the couple (L, U). If line player is the leader and row player the follower, the SPNE when the leader takes action L (resp. action R) is the couple (L, U) (resp. (R, K)). The SE if the line player is the leader, is the couple (R, K). The payoff of the leader is increased.

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Matrix game example

First example (Normal Form Game)

We consider the following matrix game:

• K

U L (3, 1) (1, 3) R (2, 1) (0, 0)

• .

The (pure) NE is the couple (L, U). If line player is the leader and row player the follower, the SPNE when the leader takes action L (resp. action R) is the couple (L, U) (resp. (R, K)). The SE if the line player is the leader, is the couple (R, K). The payoff of the leader is increased.

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Matrix game example (cont.)

Second example (Normal Form Game)

We consider the other following matrix game:

• K

U L (2, 2) (1, 0) R (3, 1) (0, 1)

• .

The (pure) NE is the couple (R, K). If line player is the leader and row player the follower, the SPNE when the leader takes action L (resp. action R) is the couple (L, K) (resp. (R, K) OR (R, U)). The SE if the line player is the leader, is the couple (L, K). The payoff of the leader is decreased because the best-response sets of the follower is not a singleton.

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Matrix game example (cont.)

Second example (Normal Form Game)

We consider the other following matrix game:

• K

U L (2, 2) (1, 0) R (3, 1) (0, 1)

• .

The (pure) NE is the couple (R, K). If line player is the leader and row player the follower, the SPNE when the leader takes action L (resp. action R) is the couple (L, K) (resp. (R, K) OR (R, U)). The SE if the line player is the leader, is the couple (L, K). The payoff of the leader is decreased because the best-response sets of the follower is not a singleton.

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Matrix game example (cont.)

Second example (Normal Form Game)

We consider the other following matrix game:

• K

U L (2, 2) (1, 0) R (3, 1) (0, 1)

• .

The (pure) NE is the couple (R, K). If line player is the leader and row player the follower, the SPNE when the leader takes action L (resp. action R) is the couple (L, K) (resp. (R, K) OR (R, U)). The SE if the line player is the leader, is the couple (L, K). The payoff of the leader is decreased because the best-response sets of the follower is not a singleton.

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Theorem

Theorem

For a given two-person finite game, if BR2(a1) is a singleton for each a1, then the payoff of the leader at the Stackelberg solution is higher or equal to its payoff at the Nash equilibrium solution.

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Example

Second example: Cournot game

We consider two companies (duopoly) that compete for quantity of product, i.e. q1 for the first firm and q2 for the second firm. The profit functions are: Π1(q1, q2) = P(q1 + q2)q1 − C1(q1), and Π2(q1, q2) = P(q1 + q2)q2 − C2(q2).

Linear structure

We assume a linear demand structure P(x) = a − bx and linear costs Ci(y) = αy.

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Example

Second example: Cournot game

We consider two companies (duopoly) that compete for quantity of product, i.e. q1 for the first firm and q2 for the second firm. The profit functions are: Π1(q1, q2) = P(q1 + q2)q1 − C1(q1), and Π2(q1, q2) = P(q1 + q2)q2 − C2(q2).

Linear structure

We assume a linear demand structure P(x) = a − bx and linear costs Ci(y) = αy.

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Example

Backward induction

Considering the quantity q1 fixed, the best-response of the follower is: BR2(q1) = a − bq1 − α 2b . Plugging this value in the profit function of the leader, we get: Π1(q1, BR2(q1)) = (a − b(q1 + a − bq1 − α 2b ))q1 − αq1. Maximizing this function in q1 yields: q∗

1 = a − α

2b . The SE is given by (q∗

1, q∗ 2) = ( a−α 2b , a−α 4b ).

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Example

Backward induction

Considering the quantity q1 fixed, the best-response of the follower is: BR2(q1) = a − bq1 − α 2b . Plugging this value in the profit function of the leader, we get: Π1(q1, BR2(q1)) = (a − b(q1 + a − bq1 − α 2b ))q1 − αq1. Maximizing this function in q1 yields: q∗

1 = a − α

2b . The SE is given by (q∗

1, q∗ 2) = ( a−α 2b , a−α 4b ).

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Example

Backward induction

Considering the quantity q1 fixed, the best-response of the follower is: BR2(q1) = a − bq1 − α 2b . Plugging this value in the profit function of the leader, we get: Π1(q1, BR2(q1)) = (a − b(q1 + a − bq1 − α 2b ))q1 − αq1. Maximizing this function in q1 yields: q∗

1 = a − α

2b . The SE is given by (q∗

1, q∗ 2) = ( a−α 2b , a−α 4b ).

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Example (contd)

Comparison to NE

The (symmetric) NE solution is: (qNE

1 , qNE 2 ) = (a − α

3b , a − α 3b ). The profits of the firms at the NE are: Π1(qNE

1 , qNE 2 ) = Π2(qNE 1 , qNE 2 ) = (a − α)2

9b . The profits of the firms at the SE are: Π1(q∗

1, q∗ 2) = (a − α)2

8b and Π2(q∗

1, q∗ 2) = (a − α)2

16b .

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Example (contd)

Comparison to NE

The (symmetric) NE solution is: (qNE

1 , qNE 2 ) = (a − α

3b , a − α 3b ). The profits of the firms at the NE are: Π1(qNE

1 , qNE 2 ) = Π2(qNE 1 , qNE 2 ) = (a − α)2

9b . The profits of the firms at the SE are: Π1(q∗

1, q∗ 2) = (a − α)2

8b and Π2(q∗

1, q∗ 2) = (a − α)2

16b .

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Example (contd)

Comparison to NE

The (symmetric) NE solution is: (qNE

1 , qNE 2 ) = (a − α

3b , a − α 3b ). The profits of the firms at the NE are: Π1(qNE

1 , qNE 2 ) = Π2(qNE 1 , qNE 2 ) = (a − α)2

9b . The profits of the firms at the SE are: Π1(q∗

1, q∗ 2) = (a − α)2

8b and Π2(q∗

1, q∗ 2) = (a − α)2

16b .

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Remarks

Remarks

There are several remarks about the Stackelberg solution concept. The model has been introduced before the Nash concept. Initially it was used in Economy to study competition between companies, but when one company has already the market. It is not a repeated game, each player takes only one action. To be the leader has some sort of advantage but not always. If the foliower’s response is not unique, there is ambiguity in the possible responses of the follower and thereby in the possible attainable utility levels of the leader. Hierarchical games: generalization of the Stackelberg equilibrium concept to several levels of leaders and followers.

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Remarks

Remarks

There are several remarks about the Stackelberg solution concept. The model has been introduced before the Nash concept. Initially it was used in Economy to study competition between companies, but when one company has already the market. It is not a repeated game, each player takes only one action. To be the leader has some sort of advantage but not always. If the foliower’s response is not unique, there is ambiguity in the possible responses of the follower and thereby in the possible attainable utility levels of the leader. Hierarchical games: generalization of the Stackelberg equilibrium concept to several levels of leaders and followers.

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Remarks

Remarks

There are several remarks about the Stackelberg solution concept. The model has been introduced before the Nash concept. Initially it was used in Economy to study competition between companies, but when one company has already the market. It is not a repeated game, each player takes only one action. To be the leader has some sort of advantage but not always. If the foliower’s response is not unique, there is ambiguity in the possible responses of the follower and thereby in the possible attainable utility levels of the leader. Hierarchical games: generalization of the Stackelberg equilibrium concept to several levels of leaders and followers.

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Remarks

Remarks

There are several remarks about the Stackelberg solution concept. The model has been introduced before the Nash concept. Initially it was used in Economy to study competition between companies, but when one company has already the market. It is not a repeated game, each player takes only one action. To be the leader has some sort of advantage but not always. If the foliower’s response is not unique, there is ambiguity in the possible responses of the follower and thereby in the possible attainable utility levels of the leader. Hierarchical games: generalization of the Stackelberg equilibrium concept to several levels of leaders and followers.

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Remarks

Remarks

There are several remarks about the Stackelberg solution concept. The model has been introduced before the Nash concept. Initially it was used in Economy to study competition between companies, but when one company has already the market. It is not a repeated game, each player takes only one action. To be the leader has some sort of advantage but not always. If the foliower’s response is not unique, there is ambiguity in the possible responses of the follower and thereby in the possible attainable utility levels of the leader. Hierarchical games: generalization of the Stackelberg equilibrium concept to several levels of leaders and followers.

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Plan

1

Introduction to Hierarchical Game Theory principles

2

Application: Spectrum market in Wireless Networks

3

Conclusions

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introduction

Description of MVNO

With the emergence of new wireless operators and standards, the spectrum becomes a market on which network operators compete. Mobile Virtual Network Operatora (MVNO): a wireless communications services provider that does not own the radio spectrum or wireless network infrastructure over which the MVNO provides services to its customers.

aAn MVNO enters into a business agreement with a mobile network operator to obtain bulk

access to network services at wholesale rates, then sets retail prices independently.

Objective

The goal is to study a wireless market on bandwidth, particularly the impact of pricing policies of the MVNO on this market.

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Multilevel Pricing Model

We consider a multilevel pricing model with 3 decision makers: end users, service providers (MVNO) and spectrum owner.

Top level

The spectrum owner maximizes his revenue vA(W, CW) = CWW depending

• n the tariff CW per unit of frequency bandwidth assigned to the service

provider, i.e. max

CW

CWW. (1)

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Multilevel Pricing Model

Middle level

The service provider determines the quantity of bandwidth W to license from the spectrum owner and the tariff CP the end users have to pay for: max

W,CP

{CP

n

• i=1

µ(Ti) − CWW}, (2) where the function µ(.) is the pricing scheme used by the service provider, which depends on user’s i transmission power Ti and n is the number of end users.

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Multilevel Pricing Model

Pricing policies

For service providers, a flat rate pricing scheme µ(x) = 1 determines an access to the Internet for all customers of the telco operator at a fixed tariff. In the power-based pricing schemea µ(x) = x, our model tends to limit the power of the end users as they have to pay the provider, not based on the throughput they have, but the power they consume.

aPower control game between the end users.

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Multilevel Pricing Model

Pricing policies

For service providers, a flat rate pricing scheme µ(x) = 1 determines an access to the Internet for all customers of the telco operator at a fixed tariff. In the power-based pricing schemea µ(x) = x, our model tends to limit the power of the end users as they have to pay the provider, not based on the throughput they have, but the power they consume.

aPower control game between the end users.

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Multilevel Pricing Model

Low level

Finally, each end user i determines his transmission power Ti in order to maximize his net utility ui(T1, . . . , Tn, CP) which is the difference between the throughput and the price imposed by the service provider, i.e. max

Ti {W ln (1 + γi) − CPµ(Ti)},

(3) where γi(T1, . . . , Tn) is the signal to noise ratio (SNR) on user’s i signal.

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Interference free model

General wireless model

We consider the situation where the end users have access to the network without interfering with the other end users.

SNR description

Then, the signal-to-noise ratio (SNR) of each end user i is given by γi = LhTi (W/n)σ2 where Ti is user’s i transmit power, L is the crosstalk coefficient and h the same channel gain.

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Interference free model

General wireless model

We consider the situation where the end users have access to the network without interfering with the other end users.

SNR description

Then, the signal-to-noise ratio (SNR) of each end user i is given by γi = LhTi (W/n)σ2 where Ti is user’s i transmit power, L is the crosstalk coefficient and h the same channel gain.

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Interference free model

Hierarchical solution

User i’s utility is given as follows: ui(Ti) = (W/n) ln

• 1 + LhTi/(Wσ2/n)
• − CPµ(Ti).

Provider tariff

User i accepts the serve if his (net) utility is positive. Then, we get the value of the maximum service provider tariff Cp.

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Interference free model

Hierarchical solution

User i’s utility is given as follows: ui(Ti) = (W/n) ln

• 1 + LhTi/(Wσ2/n)
• − CPµ(Ti).

Provider tariff

User i accepts the serve if his (net) utility is positive. Then, we get the value of the maximum service provider tariff Cp.

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Results

Table: The optimal solution for interference-free model

Power based pricing Flat rate pricing CW

1 4

≈ 0.468 W

nLhT σ2

≈ 0.462nLhT

σ2

CP

Lh 2σ2

≈ 0.532LhT

σ2

T (T, . . . , T) (T, . . . , T) vP

nLhT 4σ2

≈ 0.316nLhT

σ2

vA

nLhT 4σ2

≈ 0.216nLhT

σ2

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Several remarks

General remarks

The bandwidth tariffs CW do not depend on the number of end users n. Considering the power-based pricing scheme, the provider and the spectrum owner share profit equally, i.e. vA = vP. With the flat rate pricing scheme, the service provider’s payoff is essentially greater than the spectrum owner’s one. The total payoff is higher with the flat rate pricing scheme compared with the power-based pricing mechanism. End user chooses also to transmit using maximum power, even if a power-based pricing is used by the provider instead of a flat rate pricing scheme.

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Several remarks

General remarks

The bandwidth tariffs CW do not depend on the number of end users n. Considering the power-based pricing scheme, the provider and the spectrum owner share profit equally, i.e. vA = vP. With the flat rate pricing scheme, the service provider’s payoff is essentially greater than the spectrum owner’s one. The total payoff is higher with the flat rate pricing scheme compared with the power-based pricing mechanism. End user chooses also to transmit using maximum power, even if a power-based pricing is used by the provider instead of a flat rate pricing scheme.

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Several remarks

General remarks

The bandwidth tariffs CW do not depend on the number of end users n. Considering the power-based pricing scheme, the provider and the spectrum owner share profit equally, i.e. vA = vP. With the flat rate pricing scheme, the service provider’s payoff is essentially greater than the spectrum owner’s one. The total payoff is higher with the flat rate pricing scheme compared with the power-based pricing mechanism. End user chooses also to transmit using maximum power, even if a power-based pricing is used by the provider instead of a flat rate pricing scheme.

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Several remarks

General remarks

The bandwidth tariffs CW do not depend on the number of end users n. Considering the power-based pricing scheme, the provider and the spectrum owner share profit equally, i.e. vA = vP. With the flat rate pricing scheme, the service provider’s payoff is essentially greater than the spectrum owner’s one. The total payoff is higher with the flat rate pricing scheme compared with the power-based pricing mechanism. End user chooses also to transmit using maximum power, even if a power-based pricing is used by the provider instead of a flat rate pricing scheme.

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Several remarks

General remarks

The bandwidth tariffs CW do not depend on the number of end users n. Considering the power-based pricing scheme, the provider and the spectrum owner share profit equally, i.e. vA = vP. With the flat rate pricing scheme, the service provider’s payoff is essentially greater than the spectrum owner’s one. The total payoff is higher with the flat rate pricing scheme compared with the power-based pricing mechanism. End user chooses also to transmit using maximum power, even if a power-based pricing is used by the provider instead of a flat rate pricing scheme.

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Several conclusions

Conclusions

On one hand, a power-based pricing scheme will not help the spectrum

• wner to control the total power consumption of the network but will

permit for him to get more market share. On the other hand, the flat rate pricing scheme is preferable for the service provider since it brings him higher profit compared to using a power based pricing scheme.

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Several conclusions

Conclusions

On one hand, a power-based pricing scheme will not help the spectrum

• wner to control the total power consumption of the network but will

permit for him to get more market share. On the other hand, the flat rate pricing scheme is preferable for the service provider since it brings him higher profit compared to using a power based pricing scheme.

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Interference channels

SNR description

We consider that the access to the network is performed considering user’s

• interference. The signal to interference and noise ratio (SINR) γi of end user i

is given by γi(T1, . . . , Tn) = LhTi

• 

Wσ2 + h

n

• j=1,j=i

Tj   .

Non-cooperative game between end-users

The end user are competing through a power control game with the following utility functions: ui(T) = W ln (1 + γi(T)) − CPµ(Ti).

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Interference channels

SNR description

We consider that the access to the network is performed considering user’s

• interference. The signal to interference and noise ratio (SINR) γi of end user i

is given by γi(T1, . . . , Tn) = LhTi

• 

Wσ2 + h

n

• j=1,j=i

Tj   .

Non-cooperative game between end-users

The end user are competing through a power control game with the following utility functions: ui(T) = W ln (1 + γi(T)) − CPµ(Ti).

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Interference channels

Table: The optimal solution for interference channels model

Power based pricing Flat rate pricing W

(n+L−1)hT σ2

not explicit CW

nL 4(n+L−1)

CW = arg maxCW <n ln(1+L/(n−1)) CWW(CW) CP

Lh 2σ2

W(CW) ln

• 1 +

LhT W(CW )σ2+(n−1)hT

• T

(T, . . . , T) (T, . . . , T) vP

nLhT 4σ2

nW(CW) ln

• 1 +

LhT W(CW )σ2+(n−1)hT

• − CW W(CW)

vA

nLhT 4σ2

CW W(CW)

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Multilevel Pricing Model

Results

The major contributions of this application are described in the following items. The power-based pricing mechanism implies the same profit for the service provider and the spectrum owner. The flat rate pricing mechanism induces a higher profit for the service provider compared to the profit of the spectrum owner. Finally, we obtain that the power-based pricing mechanism, assuming high SINR, leads to zero profit for the service provider. Whereas the flat rate pricing mechanism induces non-zero profit for the service provider.

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Multilevel Pricing Model

Results

The major contributions of this application are described in the following items. The power-based pricing mechanism implies the same profit for the service provider and the spectrum owner. The flat rate pricing mechanism induces a higher profit for the service provider compared to the profit of the spectrum owner. Finally, we obtain that the power-based pricing mechanism, assuming high SINR, leads to zero profit for the service provider. Whereas the flat rate pricing mechanism induces non-zero profit for the service provider.

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Multilevel Pricing Model

Results

The major contributions of this application are described in the following items. The power-based pricing mechanism implies the same profit for the service provider and the spectrum owner. The flat rate pricing mechanism induces a higher profit for the service provider compared to the profit of the spectrum owner. Finally, we obtain that the power-based pricing mechanism, assuming high SINR, leads to zero profit for the service provider. Whereas the flat rate pricing mechanism induces non-zero profit for the service provider.

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Plan

1

Introduction to Hierarchical Game Theory principles

2

Application: Spectrum market in Wireless Networks

3

Conclusions

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Conclusions

Hierarchical games (mainly Stackelberg game) are very useful for the study of competition between decision makers with non-symmetric informations on the actions of the other players. The assumptions about perfect information observed by the follower, and knowledge of the leader about follower’s strategy, is necessary. This framework can be applied in several application domains where such hierarchy between several decision makers is natural (telecommunication, economics, etc).

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Conclusions

Hierarchical games (mainly Stackelberg game) are very useful for the study of competition between decision makers with non-symmetric informations on the actions of the other players. The assumptions about perfect information observed by the follower, and knowledge of the leader about follower’s strategy, is necessary. This framework can be applied in several application domains where such hierarchy between several decision makers is natural (telecommunication, economics, etc).

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Conclusions

Hierarchical games (mainly Stackelberg game) are very useful for the study of competition between decision makers with non-symmetric informations on the actions of the other players. The assumptions about perfect information observed by the follower, and knowledge of the leader about follower’s strategy, is necessary. This framework can be applied in several application domains where such hierarchy between several decision makers is natural (telecommunication, economics, etc).

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THANK YOU !

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