Wireless Network Pricing Chapter 7: Network Externalitiesjianwei.ie.cuhk.edu.hk/.../WirelessNetworkPricing/... · Wireless Network Pricing Chapter 7: Network Externalities Jianwei

Wireless Network PricingChapter 7: Network Externalities

Jianwei Huang & Lin Gao

Network Communications and Economics Lab (NCEL)Information Engineering Department

The Chinese University of Hong Kong

Huang & Gao ( c©NCEL) Wireless Network Pricing: Chapter 7 November 3, 2014 1 / 46

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Chapter 7: Network Externalities


Section 7.1: Theory: Network Externalities


What is Externality?

Definition (Externality)

An externality is any side effect (benefit or cost) that is imposed by theactions of a player on a third-party not directly involved.


Examples: Negative Externality

Air Pollution (source: Internet)



Second-hand Smoke (source: Internet)



Traffic Congestion (source: Internet)


Examples: Positive Externality

Lighthouse (source: Internet)



Bee Keeping (source: Internet)



Immunization (source: Internet)


Impact of Externality

Can cause market failure without proper pricesI The market outcome will no longer be efficient.I If market prices do not reflect the costs or benefits of externalities.

Example: negative externality of pollutionI The market price for steel reflects the cost labor, capital, and other

inputs, but may not include the cost due to air pollution.I The steel manufacturer may produce more products than the socially

optimal level.


Graphical Illustration of Market Failure

Quantity

Price

0 Q1Q∗

MC(social)

MC(private)

MR

MC(external)

Quantity

Price

0 Q∗Q1

MC

MR(social)

MR(private)

MR(external)

Social optimal production level Q∗:I Social Marginal Cost (MC) = Social Marginal Revenue (MR)

Left: negative production externalityI Private MC < Social MCI Local optimal quality Q1 > Social optimal quality Q∗

Right: positive consumption externalityI Private MR < Social MRI Local optimal quality Q1 < Social optimal quality Q∗


Negative Network Externality


A Case Study: Water Pollution

The chemical company produces chemical products and dischargeswastewater into the river.

The water company produces bottle water by drawing water from theriver.

Water pollution increases the production cost of the water company.


Graphical Illustration

Quantity

Price

0 Q1Q∗

A

B

C

D

E

F

$10

MC(social)

MC(private)

MC(external)

MR

Constant MR per chemical product: $10.

Social MC = private MC (chemical plant) + external MC (pollution)

Social optimal quant Q∗ < local optimal quality Q1


At Local Optimal Quality Q1

Quantity

Price

0 Q1Q∗

AB

C

D

E

F

$10

MC(social)

MC(private)

MC(external)

MR

The chemical plant’s profit (i.e., revenue - cost):∫ Q1

0(MR −MCPrivate(Q)) dQ = A + B + E

The water company’s profit due to externality (assuming 0 revenue):

−∫ Q1

0(MCExternal(Q)) dQ = −(C + F )

Since C = B and F = D + E , the social surplus (sum of two profits):

A + B + E − (C + F ) = A− D


At Social Optimal Quality Q∗

Quantity

Price

0 Q1Q∗

AB

C

D

E

F

$10

MC(social)

MC(private)

MC(external)

MR

The chemical plant’s profit (i.e., revenue - cost):∫ Q∗

0(MR −MCPrivate(Q)) dQ = A + B

The water company’s profit due to externality (assuming 0 revenue):

−∫ Q∗

0(MCExternal(Q)) dQ = −C

Since C = B , the social surplus (sum of two profits):

A + B − C = A


Comparison

Social suplus at Q1 : A− D

Social surplus at Q∗ : A

With negative externally, individual profit maximization hurts thesocial surplus

Solution: Pigovian tax


Pigovian Tax

Quantity

Price

0

$10A1

A2 B

CQ1Q∗

MC(social)

MC(private)+Tax

MC(private)

MC(external)

MR

Tax

Charge chemical plant a taxI Tax = external marginal cost at the optimal solution Q∗

Individual profit maximisation leads to production level of Q∗

I Chemical plant profit =∫ Q∗

0(MR −MCPrivate(Q)− Tax) dQ = A1


The Coase Theorem

Nobel Laureate Ronald Coase proposes another view of externality

Assumptions: Transaction cost is negligible, property rights are clear

Result: Trade in externality will lead to efficient use of the resource

Back to the previous exampleI If water company owns the water: it can charge the chemical plant a

price equal to the negative externally

I If chemical plant owns the water: it can demand a compensation fromwater company for reducing the chemical production quantity

I Either way, it is possible to maximize social surplus


Positive Network Externality


A Case Study: Network Effect

More usage of the product by any user increases the product’s valuefor other users.


Metcalfe’s Law

Consider a network of N users.

Each user perceives a value increasing in N.

Each user attaches the same value to the possibility of connectingwith any one of the other N − 1 users.

Total network value N(N − 1) ≈ N2.


Briscore’s Refinement

Each user ranks other users in terms of decreasing importance.

Attach a value of 1/k to the kth important neighbour.

Total network value N(∑N−1

k=1 1/k)≈ N logN.


Different Types of Network Effect

Direct network effect: telephone, online social network

Indirect network effect: Office for Windows, DVDs for DVD players

Local network effect: instant messaging


Section 7.2: Distributed Wireless InterferenceCompensation


Wireless Power Control

Distributed power control in wireless ad hoc networks

Elastic applications with no SINR targets

Want to maximize the total network performance


Network Model

T1

T2

T3

R1

R2

R3

h111 h2

11

h112

h212

Single-hop transmissions.

A user = a transmitter/receiver pair.

Transmit over multiple parallel channels.

Interferences in the same channel (negative externality).

We focus on single channel here.


Single Channel Communications

2

σM

22h

σ212h

11h

σ1

1pTransmitters Receivers

21h

Mp

p

A set of N = {1, ..., n} users.

For each user n ∈ N :I Power constraint: pn ∈ [Pmin

n ,Pmaxn ].

I Received SINR (signal-to-interference plus noise ratio):

γn =pnhn,n

σn +∑

m 6=n pmhn,m.

I Utility function Un(γn): increasing, differentiable, strictly concave.


Network Utility Maximization (NUM) Problem

NUM Problem

max{Pmin

n ≤pn≤Pmaxn ,∀n}

∑

n

Un(γn).

Technical Challenges:I Coupled across users due to interferences.I Could be non-convex in power.

We want: efficient and distributed algorithm, with limited informationexchange and fast convergence.


Benchmark - No Information Exchange

Each user picks power to maximize its own utility, given currentinterference and channel gain.

Results in pn = Pmaxn for all n.

I Can be far from optimal.

We propose algorithm with limited information exchange.I Have nice interpretation as distributed Pigovian taxation.I Analyze its behavior using supermodular game theory.


Benchmark - No Information Exchange

Each user picks power to maximize its own utility, given currentinterference and channel gain.

Results in pn = Pmaxn for all n.

I Can be far from optimal.

We propose algorithm with limited information exchange.I Have nice interpretation as distributed Pigovian taxation.I Analyze its behavior using supermodular game theory.


ADP Algorithm: Asynchronous Distributed Pricing

Price Announcing: user n announces “price” (per unit interference):

πn =

∣∣∣∣∂Un(γn)

∂In

∣∣∣∣ =∂Un(γn)

∂γn

γ2n

pnhn,n.

Power Updating: user n updates power pn to maximize surplus:

Sn = Un(γn)− pn∑

m 6=n

πmhm,n.

Repeat two phases asynchronously across users.

Scalable and distributed: only need to announce single price, andknow limited channel gains (hm,n).


ADP Algorithm

Interpretation of prices: Pigovian taxation

ADP algorithm: distributed discovery of Pigovian taxesI When does it converge?I What does it converge to?I Will it solve NUM Problem ?I How fast does it converge?


ADP Algorithm

Interpretation of prices: Pigovian taxation

ADP algorithm: distributed discovery of Pigovian taxesI When does it converge?I What does it converge to?I Will it solve NUM Problem ?I How fast does it converge?


Convergence

Depends on the utility functions.

Coefficient of relative Risk Aversion (CRA) of U(γ):

CRA(γ) = −γU′′(γ)

U ′(γ).

I larger CRA ⇒ “more concave” U.

Theorem: If each user n has a positive minimum transmission powerand CRA(γn) ∈ [1, 2], then there is a unique optimal solution ofNUM Problem, and the ADP algorithm globally converges to it.

Proof: relating this algorithm to a fictitious supermodular game.


Convergence




U ′(γ).





Convergence




U ′(γ).





Supermodular Games

A class of games with strategic complementariesI Strategy sets are compact subsets of R; and each player’s pay-off Sn

has increasing differences:

∂2Sn∂xn∂xm

> 0,∀n,m.

Key properties:I A PNE exists.I If the PNE is unique, then the asynchronous best response updates will

globally converge to it.


Convergence Speed

0 10 200

0.5

1

Pow

er

ADP Algorithm

200 400 6000

0.5

1

Pow

er

Gradient−based Algorithm

5 10 15 200

20

40

60

80

Iterations

Pric

e

200 400 6000

20

40

60

80

IterationsP

rice

10 users, log utilities.

ADP algorithm (left figures) converges much faster than agradient-based method (right figures).


Section 7.3: 4G Network Upgrade


When To Upgrade From 3G to 4G?

Early upgrade:I More expensive, as cost decreases over timeI Starts with few users, hence a small initial revenue

Late upgrade:I Leads to a smaller market shareI Delays 4G revenues

Need to

I Capture the above tradeoffsI Consider the dynamics of users adopting 4G and switching providersI Understand the upgrade timing between competing cellular providers


Duopoly Model

Two competing operators

I Initially both using 3G technology

I Operator i decides to upgrade to 4G at time Ti

I Each operator wants to maximize its long-term profit

What will be the equilibrium of (T ∗1 ,T∗2 )?


Users Switching

W.L.O.G., assume T1 < T2

Three time periods: [0,T1], (T1,T2], and (T2,∞)

When t ∈ [0,T1]: No user switching.


Users Switching

W.L.O.G., assume T1 < T2

Three time periods: [0,T1], (T1,T2], and (T2,∞)

When t ∈ [0,T1]: No user switching.


Users Switching

When t ∈ (T1,T2]: both inter- and intra- operator user switching

Duan, Huang, WalrandCTW 5/2012 17

3G TO 4GWhen should an operator upgrade from 3G to 4G?

Customers switch providers to get 4G, at rate �⇥, � < 1.Customers of one provider upgrade to 4G at rate �.

1, 3G

1, 4G

2, 3G2, 4G

T1 T2

�

↵�

�

Model: Customer migrationsProvider 1 Provider 2

3G 3G

4G

�

↵�

3G 3G

4G

� �

4G

Provider 1 Provider 2

When t ∈ (T2,∞): only intra-operator user switching




1, 3G

1, 4G

2, 3G2, 4G

T1 T2

�

↵�

�


3G 3G

4G

�

↵�

3G 3G

4G

� �

4G



Users Switching

When t ∈ (T1,T2]: both inter- and intra- operator user switching




1, 3G

1, 4G

2, 3G2, 4G

T1 T2

�

↵�

�


3G 3G

4G

�

↵�

3G 3G

4G

� �

4G


When t ∈ (T2,∞): only intra-operator user switching




1, 3G

1, 4G

2, 3G2, 4G

T1 T2

�

↵�

�


3G 3G

4G

�

↵�

3G 3G

4G

� �

4G



Network Value (Revenue)

Network value depends on the number of subscribers

I Assume that operator i has Ni 4G users, i = 1, 2

I Total 4G network value is (N1 + N2) log(N1 + N2) (network effect)

I Operator i ’s network value (revenue) is Ni log(N1 + N2)

Later upgrade ⇒ take advantage of existing 4G population

The revenue for 3G network is similar, with an coefficient γ ∈ (0, 1)


















Revenue and Market Share



Model: Revenue

Ri(t) = �N itN + (1 � �)N i⇤

t N⇤t

N it := number of users of provider i

N i⇤t := number of 4G users of provider i

N⇤t := N1⇤

t + N2⇤t

4G calls cost 1

Other calls cost � < 1

1, 3G

1, 4G

2, 3G2, 4G

R1(t)

R2(t)

T1 T2

MS "4G "

4G "MS #

4G "

Profit

�i =

Z 1

0

e��tRi(t)dt � Ke�UTi

U > ⇥ + �⇤U = decrease rate of technology cost

� = discounting rateHuang & Gao ( c©NCEL) Wireless Network Pricing: Chapter 7 November 3, 2014 40 / 46

Upgrade Cost and Time Discount

One-time upgrade cost:I K at time t = 0I Discounted over time: K exp(−Ut)

Revenue is also discounted over time by exp(−St)

Earlier upgrade ⇒ larger revenue and larger cost


Equilibrium Timings

0 0.05 0.1 0.15 0.2 0.25 0.30

0.05

0.1

0.15

0.2

0.25

0.3

Operator 1’s equilibrium time T1*

Ope

rato

r 2’

s eq

uilib

rium

tim

e T

2*

NE 1: T1* ≤T

2*

NE 2: T1* ≥T

2*

Low cost regime:0=T

1*=T

2* as K↑

Medium cost regime:0=T

1*<T

2*↑ as K↑

High cost regime:0<T

1*↑<T

2*↑: as K↑


Equilibrium Profits

0.4 0.45 0.5 0.55

0.42

0.44

0.46

0.48

0.5

0.52

0.54

0.56

Operator 1’s equilibrium profit π1*

Ope

rato

r 2’

s eq

uilib

rium

pro

fit π

2*

NE 1: T1*≤T

2*

NE 2: T1*≥T

2*

Medium cost regime:π

1*↑<π

2*↓ as K↑

High cost regime:π

1*↑<π

2*↑ as K↑

Low cost regime:π

1*↓=π

2*↓ as K↑


Section 7.4: Chapter Summary


Key Concepts

TheoryI Positive and negative ExternalityI Market failureI Pigovian taxI Network effect

ApplicationI Distributed wireless power control based on Pigovian taxI Cellular network upgrade considering network effect


References and Extended Reading

J. Huang, R. Berry and M. Honig, “Distributed Interference Compensationfor Wireless Networks,” IEEE Journal on Selected Areas in Communications,vol. 24, no. 5, pp. 1074-1084, 2006

L. Duan, J. Huang, and J. Walrand, “Economic Analysis of 4G Network

Upgrade,” IEEE Transactions on Mobile Computing, accepted 2014




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