Multilevel Pricing Schemes in a DeregulatedWireless Network Market

Yezekael HayelCERI/LIA University of Avignon

Workshop on New Avenues for Network ModelsIISc - 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

IntroductionIn a competition setting, there are several major concerns when some playersmay 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 toenforce his strategy on the other players.

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Definition

Leader/FollowerIn a game theoretic formulation of a hierarchical competition, we define twotypes of players:

Leaders: players that take their decisions first,Followers: players that take their decisions after observing the leader’sdecisions.

IMPORTANT: The leaders know, ex ante, that the followers observe theiraction.

Stackelberg settingThe Stackelberg competition modela is a strategic game in economics inwhich the leader firm moves first and then the follower firms movesequentially. 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/FollowerIn a game theoretic formulation of a hierarchical competition, we define twotypes of players:

Leaders: players that take their decisions first,Followers: players that take their decisions after observing the leader’sdecisions.

IMPORTANT: The leaders know, ex ante, that the followers observe theiraction.

Stackelberg settingThe Stackelberg competition modela is a strategic game in economics inwhich the leader firm moves first and then the follower firms movesequentially. 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 functionsAs the leaders know that the followers observe their action, the follower’sactions can be seen as implicit functions in the leader’s utility.

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

maxa1

U1(a1,a2) and maxa2

U2(a1,a2).

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Basic formulation

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

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

maxa1

U1(a1,a2) and maxa2

U2(a1,a2).

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Basic formulation

Simultaneous playIf both player takes his action without observing the action of the other, welook for a standard Nash equilibrium of this non-cooperative game.

Non-simultaneous playWe assume that player 1 decides first his action and player 2 takes his actionafter 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 owndecision.

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Basic formulation

Simultaneous playIf both player takes his action without observing the action of the other, welook for a standard Nash equilibrium of this non-cooperative game.

Non-simultaneous playWe assume that player 1 decides first his action and player 2 takes his actionafter 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 owndecision.

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Basic formulation

Maximization problemFor each action a1 of the leader, we denote by BR2(a1) thebest-response function (or set) which determines the best action of thefollower depending on the action of the leader, i.e.

∀a1, BR2(a1) = arg maxa2

U2(a1,a2).

As the leader knows that the follower observes his action, the leader hasthe following utility function to maximize:

maxa1

U1(a1,BR2(a1)).

IMPORTANT: This function depends only on his own action a1, throughthe best-response function of the follower.

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Basic formulation

Maximization problemFor each action a1 of the leader, we denote by BR2(a1) thebest-response function (or set) which determines the best action of thefollower depending on the action of the leader, i.e.

∀a1, BR2(a1) = arg maxa2

U2(a1,a2).

As the leader knows that the follower observes his action, the leader hasthe following utility function to maximize:

maxa1

U1(a1,BR2(a1)).

IMPORTANT: This function depends only on his own action a1, throughthe best-response function of the follower.

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Basic formulation

Maximization problemFor each action a1 of the leader, we denote by BR2(a1) thebest-response function (or set) which determines the best action of thefollower depending on the action of the leader, i.e.

∀a1, BR2(a1) = arg maxa2

U2(a1,a2).

As the leader knows that the follower observes his action, the leader hasthe following utility function to maximize:

maxa1

U1(a1,BR2(a1)).

IMPORTANT: This function depends only on his own action a1, throughthe best-response function of the follower.

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Stackelberg solution

Stackelberg equilibriumA Stackelberg equilibrium is a vector of actions (a∗1,a

∗2) such that:

a∗1 = arg maxa1

U1(a1,BR2(a1)),

anda∗2 = BR2(a∗1).

Figure: H. Von Stackelberg

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Stackelberg model

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

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Stackelberg solution

In practice using Backward inductionIn order to determine a Stackelberg equilibrium solution, there are threesteps:

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

2 To solve the optimization problem for the leader, using the best-responsefunction of the follower.

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

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Stackelberg solution

In practice using Backward inductionIn order to determine a Stackelberg equilibrium solution, there are threesteps:

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

2 To solve the optimization problem for the leader, using the best-responsefunction of the follower.

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

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Stackelberg solution

In practice using Backward inductionIn order to determine a Stackelberg equilibrium solution, there are threesteps:

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

2 To solve the optimization problem for the leader, using the best-responsefunction of the follower.

3 To determine the best action of the follower, using the best-responsefunction, 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 UL (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 theleader 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 UL (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 theleader 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 UL (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 theleader 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 UL (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 theleader 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 ofthe 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 UL (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 theleader 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 ofthe 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 UL (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 theleader 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 ofthe follower is not a singleton.

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Theorem

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

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Example

Second example: Cournot gameWe 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),

andΠ2(q1,q2) = P(q1 + q2)q2 − C2(q2).

Linear structureWe assume a linear demand structure P(x) = a− bx and linear costsCi (y) = αy .

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Example

Second example: Cournot gameWe 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),

andΠ2(q1,q2) = P(q1 + q2)q2 − C2(q2).

Linear structureWe assume a linear demand structure P(x) = a− bx and linear costsCi (y) = αy .

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Example

Backward inductionConsidering 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−α4b ).

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Example

Backward inductionConsidering 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−α4b ).

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Example

Backward inductionConsidering 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−α4b ).

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Example (contd)

Comparison to NEThe (symmetric) NE solution is:

(qNE1 ,qNE

2 ) = (a− α

3b,

a− α3b

).

The profits of the firms at the NE are:

Π1(qNE1 ,qNE

2 ) = Π2(qNE1 ,qNE

2 ) =(a− α)2

9b.

The profits of the firms at the SE are:

Π1(q∗1 ,q∗2 ) =

(a− α)2

8band Π2(q∗1 ,q

∗2 ) =

(a− α)2

16b.

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Example (contd)

Comparison to NEThe (symmetric) NE solution is:

(qNE1 ,qNE

2 ) = (a− α

3b,

a− α3b

).

The profits of the firms at the NE are:

Π1(qNE1 ,qNE

2 ) = Π2(qNE1 ,qNE

2 ) =(a− α)2

9b.

The profits of the firms at the SE are:

Π1(q∗1 ,q∗2 ) =

(a− α)2

8band Π2(q∗1 ,q

∗2 ) =

(a− α)2

16b.

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Example (contd)

Comparison to NEThe (symmetric) NE solution is:

(qNE1 ,qNE

2 ) = (a− α

3b,

a− α3b

).

The profits of the firms at the NE are:

Π1(qNE1 ,qNE

2 ) = Π2(qNE1 ,qNE

2 ) =(a− α)2

9b.

The profits of the firms at the SE are:

Π1(q∗1 ,q∗2 ) =

(a− α)2

8band Π2(q∗1 ,q

∗2 ) =

(a− α)2

16b.

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Remarks

RemarksThere are several remarks about the Stackelberg solution concept.

The model has been introduced before the Nash concept. Initially it wasused in Economy to study competition between companies, but when onecompany 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 possibleresponses of the follower and thereby in the possible attainable utilitylevels of the leader.Hierarchical games: generalization of the Stackelberg equilibriumconcept to several levels of leaders and followers.

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Remarks

RemarksThere are several remarks about the Stackelberg solution concept.

The model has been introduced before the Nash concept. Initially it wasused in Economy to study competition between companies, but when onecompany 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 possibleresponses of the follower and thereby in the possible attainable utilitylevels of the leader.Hierarchical games: generalization of the Stackelberg equilibriumconcept to several levels of leaders and followers.

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Remarks

RemarksThere are several remarks about the Stackelberg solution concept.

The model has been introduced before the Nash concept. Initially it wasused in Economy to study competition between companies, but when onecompany 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 possibleresponses of the follower and thereby in the possible attainable utilitylevels of the leader.Hierarchical games: generalization of the Stackelberg equilibriumconcept to several levels of leaders and followers.

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Remarks

RemarksThere are several remarks about the Stackelberg solution concept.

The model has been introduced before the Nash concept. Initially it wasused in Economy to study competition between companies, but when onecompany 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 possibleresponses of the follower and thereby in the possible attainable utilitylevels of the leader.Hierarchical games: generalization of the Stackelberg equilibriumconcept to several levels of leaders and followers.

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Remarks

RemarksThere are several remarks about the Stackelberg solution concept.

The model has been introduced before the Nash concept. Initially it wasused in Economy to study competition between companies, but when onecompany 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 possibleresponses of the follower and thereby in the possible attainable utilitylevels of the leader.Hierarchical games: generalization of the Stackelberg equilibriumconcept 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 MVNOWith the emergence of new wireless operators and standards, the spectrumbecomes a market on which network operators compete.

Mobile Virtual Network Operatora (MVNO): a wireless communicationsservices provider that does not own the radio spectrum or wirelessnetwork infrastructure over which the MVNO provides services to itscustomers.

aAn MVNO enters into a business agreement with a mobile network operator to obtain bulkaccess to network services at wholesale rates, then sets retail prices independently.

ObjectiveThe goal is to study a wireless market on bandwidth, particularly the impact ofpricing 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 levelThe spectrum owner maximizes his revenue vA(W ,CW ) = CW W dependingon the tariff CW per unit of frequency bandwidth assigned to the serviceprovider, i.e.

maxCW

CW W . (1)

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Multilevel Pricing Model

Middle levelThe service provider determines the quantity of bandwidth W to license fromthe spectrum owner and the tariff CP the end users have to pay for:

maxW ,CP

{CP

n∑i=1

µ(Ti )− CW W}, (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 endusers.

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Multilevel Pricing Model

Pricing policiesFor service providers, a flat rate pricing scheme µ(x) = 1 determines anaccess to the Internet for all customers of the telco operator at a fixedtariff.

In the power-based pricing schemea µ(x) = x , our model tends to limitthe power of the end users as they have to pay the provider, not based onthe throughput they have, but the power they consume.

aPower control game between the end users.

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Multilevel Pricing Model

Pricing policiesFor service providers, a flat rate pricing scheme µ(x) = 1 determines anaccess to the Internet for all customers of the telco operator at a fixedtariff.

In the power-based pricing schemea µ(x) = x , our model tends to limitthe power of the end users as they have to pay the provider, not based onthe throughput they have, but the power they consume.

aPower control game between the end users.

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Multilevel Pricing Model

Low levelFinally, each end user i determines his transmission power Ti in order tomaximize his net utility ui (T1, . . . ,Tn,CP) which is the difference between thethroughput and the price imposed by the service provider, i.e.

maxTi

{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 modelWe consider the situation where the end users have access to the networkwithout interfering with the other end users.

SNR descriptionThen, 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 thesame channel gain.

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Interference free model

General wireless modelWe consider the situation where the end users have access to the networkwithout interfering with the other end users.

SNR descriptionThen, 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 thesame channel gain.

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Interference free model

Hierarchical solutionUser i ’s utility is given as follows:

ui (Ti ) = (W/n) ln(1 + LhTi/(Wσ2/n)

)− CPµ(Ti ).

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

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Interference free model

Hierarchical solutionUser i ’s utility is given as follows:

ui (Ti ) = (W/n) ln(1 + LhTi/(Wσ2/n)

)− CPµ(Ti ).

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

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Results

Table: The optimal solution for interference-free model

Power based pricing Flat rate pricing

CW14 ≈ 0.468

W nLhTσ2 ≈ 0.462nLhT

σ2

CPLh

2σ2 ≈ 0.532LhTσ2

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

vPnLhT4σ2 ≈ 0.316nLhT

σ2

vAnLhT4σ2 ≈ 0.216nLhT

σ2

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Several remarks

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

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Several remarks

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

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Several remarks

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

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Several remarks

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

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Several remarks

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

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Several conclusions

ConclusionsOn one hand, a power-based pricing scheme will not help the spectrumowner to control the total power consumption of the network but willpermit for him to get more market share.On the other hand, the flat rate pricing scheme is preferable for theservice provider since it brings him higher profit compared to using apower based pricing scheme.

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Several conclusions

ConclusionsOn one hand, a power-based pricing scheme will not help the spectrumowner to control the total power consumption of the network but willpermit for him to get more market share.On the other hand, the flat rate pricing scheme is preferable for theservice provider since it brings him higher profit compared to using apower based pricing scheme.

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Interference channels

SNR descriptionWe consider that the access to the network is performed considering user’sinterference. The signal to interference and noise ratio (SINR) γi of end user iis given by

γi (T1, . . . ,Tn) = LhTi/Wσ2 + h

n∑j=1,j 6=i

Tj

.

Non-cooperative game between end-usersThe end user are competing through a power control game with the followingutility functions:

ui (T ) = W ln (1 + γi (T ))− CPµ(Ti ).

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Interference channels

SNR descriptionWe consider that the access to the network is performed considering user’sinterference. The signal to interference and noise ratio (SINR) γi of end user iis given by

γi (T1, . . . ,Tn) = LhTi/Wσ2 + h

n∑j=1,j 6=i

Tj

.

Non-cooperative game between end-usersThe end user are competing through a power control game with the followingutility 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)hTσ2 not explicit

CWnL

4(n+L−1) CW = arg maxCW<n ln(1+L/(n−1)) CW W (CW )

CPLh

2σ2 W (CW ) ln(

1 + LhTW (CW )σ2+(n−1)hT

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

vPnLhT4σ2 nW (CW ) ln

(1 + LhT

W (CW )σ2+(n−1)hT

)− CW W (CW )

vAnLhT4σ2 CW W (CW )

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Multilevel Pricing Model

ResultsThe major contributions of this application are described in the followingitems.

The power-based pricing mechanism implies the same profit for theservice provider and the spectrum owner.The flat rate pricing mechanism induces a higher profit for the serviceprovider compared to the profit of the spectrum owner.Finally, we obtain that the power-based pricing mechanism, assuminghigh SINR, leads to zero profit for the service provider. Whereas the flatrate pricing mechanism induces non-zero profit for the service provider.

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Multilevel Pricing Model

ResultsThe major contributions of this application are described in the followingitems.

The power-based pricing mechanism implies the same profit for theservice provider and the spectrum owner.The flat rate pricing mechanism induces a higher profit for the serviceprovider compared to the profit of the spectrum owner.Finally, we obtain that the power-based pricing mechanism, assuminghigh SINR, leads to zero profit for the service provider. Whereas the flatrate pricing mechanism induces non-zero profit for the service provider.

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Multilevel Pricing Model

ResultsThe major contributions of this application are described in the followingitems.

The power-based pricing mechanism implies the same profit for theservice provider and the spectrum owner.The flat rate pricing mechanism induces a higher profit for the serviceprovider compared to the profit of the spectrum owner.Finally, we obtain that the power-based pricing mechanism, assuminghigh SINR, leads to zero profit for the service provider. Whereas the flatrate 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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ConclusionsHierarchical games (mainly Stackelberg game) are very useful for thestudy of competition between decision makers with non-symmetricinformations on the actions of the other players.The assumptions about perfect information observed by the follower, andknowledge of the leader about follower’s strategy, is necessary.This framework can be applied in several application domains wheresuch hierarchy between several decision makers is natural(telecommunication, economics, etc).

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

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

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THANK YOU !

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