wireless network pricing chapter 6: oligopoly...

Wireless Network PricingChapter 6: Oligopoly Pricing

Jianwei Huang & Lin Gao

Network Communications and Economics Lab (NCEL)Information Engineering DepartmentThe Chinese University of Hong Kong

Huang & Gao ( c�NCEL) Wireless Network Pricing: Chapter 6 November 15, 2016 1 / 69

The Book

E-Book freely downloadable from NCEL website: http://ncel.ie.cuhk.edu.hk/content/wireless-network-pricing

Physical book available for purchase from Morgan & Claypool(http://goo.gl/JFGlai) and Amazon (http://goo.gl/JQKaEq)


http://ncel.ie.cuhk.edu.hk/content/wireless-network-pricing


http://goo.gl/JFGlai

http://goo.gl/JQKaEq

Section 6.3:Wireless Service Provider Competition Revisited


Network Model

Provider 1

Provider 2

Provider 3

A set J = {1, . . . , J} of service providersI Provider j has a supply Qj of resource (e.g., channel, time, power)I Providers operate on orthogonal spectrum bands

A set I = {1, . . . , I} of usersI User i can obtain resources from multiple providers: q i = (qij , 8j 2 J )

I User i ’s utility function is ui⇣PJ

j=1

qijcij⌘: increasing and strictly

concave


An Example: TDMA

Each provider j has a total spectrum band of Wj .

qij : the fraction of time that user i transmits on provider j ’s bandI Constraints:

Pi qij 1, for all j 2 J .

cij : the data rate achieved by user i on provider j ’s band

cij = Wj log(1 +Pi |hij |2�2

ijWj)

I Pi : user i ’s peak transmission power.I hij : the channel gain between user i and network j .I �2

ij : the Gaussian noise variance for the channel.

ui⇣PJ

j=1

qijcij⌘: user i ’ utility of the total achieved data rate


Two-Stage Game

Stage I: each provider j 2 J announces a unit price pjI Each provider i wants to maximize his revenueI Denote p = (pj , 8j 2 J ) as the price vectors of all providers.

Stage II: each user i 2 I chooses a demand vector q i = (qij , 8j 2 J )I Each user i wants to maximize his payo↵ (utility minus payment)I Denote q = (q i , 8i 2 I) as the demand vector of all users.

Analysis based on backward induction


Goal: Derive the SPNE

A price demand tuple (p⇤,q⇤(p⇤)) is a SPNE if no player has anincentive to deviate unilaterally at any stage of the game.

I Each user i maximizes its payo↵ by choosing the optimal demandq⇤i (p⇤), given prices p⇤.

I Each provider j maximizes its revenue by choosing price p⇤j , given otherproviders’ prices p⇤�j = (p⇤k , 8k 6= j) and the user demands q⇤(p⇤).


Stage II: User’s Demand Optimization

Each user i 2 I solves a user payo↵ maximization (UPM) problem:

UPM : maxq i�0

0

@ui

0

@JX

j=1

qijcij

1

A�JX

j=1

pjqij

1

A .

Problem UPM may have more than one optimization solution q⇤i

I Since it is not strictly concave maximization problem in q⇤i

Problem UPM has a unique solution of the e↵ective resource xi

Lemma (6.16)

I For each user i 2 I, there exists a unique nonnegative value x⇤i , such thatPj2J cijq⇤ij = x⇤i for every maximizer q⇤

i of the UPM problem.

I For any provider j such that q⇤ij > 0, pj/cij = mink2J pk/cik .


Decided vs. Undecided Users

Definition (Preference set)

For any price vector p, user i ’s preference set is

Ji (p) =⇢j 2 J :

pjcij

= mink2J

pkcik

�.

A decided user has a singleton preference set.

An undecided user has a preference set that includes more than oneprovider.

One can use a bipartite graph representation (BGR) to uniquelydetermine the demands of undecided users.

This will lead to all users’ optimal demand q⇤(p) = (q⇤i (p), 8i 2 I) in

Stage II.


Stage I: Provider’s Revenue Optimization

Each provider j 2 J solves a provider revenue maximization (PRM)problem

PRM : maxpj�0

pj ·min

Qj ,X

i2Iq⇤ij(pj , p�j)

!

Solving the PRM problem requires the consideration of otherproviders’ prices p�j .


Benchmark: Social Welfare Optimization (Ch. 4)

SWO: Social Welfare Optimization Problem

maximizeX

i2Iui (xi )

subject toX

j2Jqijcij = xi , 8i 2 I,

X

i2Iqij = Qj , 8j 2 J ,

variables qij , xi � 0, 8i 2 I, j 2 J .


Stage I: Provider’s Revenue Optimization

Theorem

Under proper technical assumptions, the unique socially optimal demandvector q⇤ and the associated Lagrangian multiplier vector p⇤ of the SWOproblem constitute the unique SPNE of the provider competition game.


Optimization, Game, and Algorithm

Social Welfare Optimization

Section 4.3.2maximizing vector q⇤

Lagrange multipliers p⇤

(q⇤,p⇤)

Provider Competition Game

Section 6.3.1

equilibrium price p⇤equilibrium user demand q⇤

(q⇤,p⇤)

Primal-Dual Algorithm

Section 4.3.3limt!1(q(t),p(t)) = (q⇤,p⇤)

(q⇤,p⇤)

Figure: Relationship among di↵erent concepts


Section 6.4:Competition with Spectrum Leasing


Network Model

Secondary users (transmitter-receiver pairs)

Spectrum owner

Operator i Operator j

Investment (leasing bandwidth)

Pricing (selling bandwidth)

Spectrum owner


Three-Stage Multi-leader-follower Game

Stage I: Leasing Game

Leasing Bandwidth B1 and B2

(with unit costs C1 and C2)

Stage II: Pricing Game

Pricing π1 and π2

Stage III

User k Chooses One

Operator i and Demand wki

Operators(leaders)

Users(followers)

Backw

ard

Ind

uctio

n


Stage IIII: Users’ Bandwidth Demands

User k ’s payo↵ of choosing operator i = 1, 2

uk(⇡i ,wki ) = wki ln

✓Pmax

i hin0

wki

◆� ⇡iwki

I High SNR approximation of OFDMA systemI Optimal demand: w⇤

ki (⇡i ) = argmaxwki�0

uk(⇡i ,wki ) = gke�(1+⇡i )

I Optimal payo↵: uk(⇡i ,w⇤ki (⇡i ))

User k prefers the “better” operator: i⇤ = argmaxi=1,2 uk(⇡i ,w⇤ki (⇡i ))

Users demands may not be satisfied due to limited resourceI Di↵erence between preferred demand and realized demand


Stages II: Pricing Game

Players: two operators

Strategies: ⇡i � 0, i = 1, 2

Payo↵s: profit Ri for operator i = 1, 2:

Ri (Bi ,Bj ,⇡i ,⇡j) = ⇡iQi (Bi ,Bj ,⇡i ,⇡j)� BiCi


Stage II: Pricing Equilibrium

Symmetric equilibrium: ⇡⇤1

= ⇡⇤2

.

Threshold structure:I Unique positive equilibrium exists B

1

+ B2

Ge�2.

( 1)M

( 3)M

( )L

( 2)M

( )H

0

2Ge−

1Ge−

iB

jB

2Ge− 1Ge−

(L) : Unique nonzero equilibrium

(M1)‐(M3) : No equilibrium

(H) : Unique zero equilibrium


Stage I: Leasing Game

Players: two operators

Strategies: Bi 2 [0,1), i = 1, 2, and B1

+ B2

Ge�2.

Payo↵s: profit Ri for operator i = 1, 2:

Ri (Bi ,Bj) = Bi

✓ln

✓G

Bi + Bj

◆� 1� Ci

◆


Stage I: Leasing Equilibrium

Linear in wireless characteristics G =P

i gi ;

Threshold structure:I Low costs: infinitely many equilibriaI High comparable costs: unique equilibriumI High incomparable costs: unique monopoly equlibrium

1j iC C= +

( )L

1

10 iC

jC

( )HC

( )HI

( ')HI

1j iC C= −

(L) : Infinitely many equilibria

(HC) : Unique equilibrium

(HI)‐(HI’) : Unique equilibrium


Equilibrium Summary (Assuming Ci Cj)

CostsLOW HC HI

Ci + Cj 1 Ci + Cj > 1, Cj > 1 + Ci

Cj � Ci 1

equilibria Infinite Unique Unique

(B⇤i ,B

⇤j )

(⇢Ge�2, (1� ⇢)Ge�2),

✓(1+Cj�Ci )G

2eCi+Cj+3

2

, (1+Ci�Cj )G

2eCi+Cj+3

2

◆(Ge�(2+Ci ), 0)

⇢ 2 [Cj , (1� Ci )]

(⇡⇤i ,⇡

⇤j ) (1, 1)

⇣Ci+Cj+1

2

, Ci+Cj+1

2

⌘(1 + Ci ,N/A)

User SNR e2 eCi+Cj+3

2 e2+Ci

User Payo↵ gke�2 gke�⇣

Ci+Cj+3

2

⌘

gke�(2+Ci )

Users achieve the same SNR

User k ’s payo↵ is linear in gk


Robustness of Results

To obtain closed form solutions, we have assumedI All users achieve high SNR

Previous observations still hold in the general case

I Users operate in general SNR regime: rki (wki ) = wki ln⇣1 + Pmax

k hkn0

wki

⌘


Impact of Duopoly Competition on OperatorsBenchmark: Coordinated Case

I Operators cooperate in investment and pricing to maximize total profit

Define

E�ciency Ratio =Total Profit in Competition Case

Total Profit in Coordinated CasePrice of Anarchy = minCi ,Cj E�ciency Ratio= 0.75.


Section 6.5: Chapter Summary


Key Concepts

Theory: Game TheoryI Dominant StrategyI Pure and Mixed Strategy Nash EquilibriumI Subgame Perfect Nash Equilibrium

Theory: OligopolyI Cournot competitionI Bertrand competitionI Hotelling competition

Application: Wireless Network Competition Revisited

Application: Competition with Spectrum Leasing


References and Extended Reading

J. Huang, “How Do We Play Games?” online video tutorial, on YouKu(http://www.youku.com/playlist_show/id_19119535.html) andiTunesU (https://itunes.apple.com/hk/course/how-do-we-play-games/id642100914)

V. Gajic, J. Huang, and B. Rimoldi, “Competition of Wireless Providers for

Atomic Users,” IEEE Transactions on Networking, vol. 22, no. 2, pp. 512 -

525, April 2014

L. Duan, J. Huang, and B. Shou, “Duopoly Competition in DynamicSpectrum Leasing and Pricing,” IEEE Transactions on MobileComputing, vol. 11, no. 11, pp. 1706 1719, November 2012



http://www.youku.com/playlist_show/id_19119535.html

https://itunes.apple.com/hk/course/how-do-we-play-games/id642100914

https://itunes.apple.com/hk/course/how-do-we-play-games/id642100914


wireless network pricing chapter 6: oligopoly...

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