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Wireless Network Pricing Chapter 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)

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Wireless Network Pricing Chapter 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 20, 2018 1 / 46

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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)

Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 7 November 20, 2018 2 / 46

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Chapter 7: Network Externalities

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Section 7.1: Theory: Network Externalities

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What is Externality?

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

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Examples: Negative Externality

Air Pollution (source: Internet)

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Examples: Negative Externality

Second-hand Smoke (source: Internet)

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Examples: Negative Externality

Traffic Congestion (source: Internet)

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Examples: Positive Externality

Lighthouse (source: Internet)

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Examples: Positive Externality

Bee Keeping (source: Internet)

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Examples: Positive Externality

Immunization (source: Internet)

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Impact of Externality

Can cause market failure without proper prices

◮ The market outcome will no longer be efficient. ◮ If market prices do not reflect the costs or benefits of externalities.

Example: negative externality of pollution

◮ The market price for steel reflects the cost labor, capital, and other

inputs, but may not include the cost due to air pollution.

◮ The steel manufacturer may produce more products than the socially

  • ptimal level.

Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 7 November 20, 2018 8 / 46

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Graphical Illustration of Market Failure

Quantity Price Q1 Q∗ MC(social) MC(private) MR MC(external) Quantity Price Q∗ Q1 MC MR(social) MR(private) MR(external)

Social optimal production level Q∗:

◮ Social Marginal Cost (MC) = Social Marginal Revenue (MR)

Left: negative production externality

◮ Private MC < Social MC ◮ Local optimal quality Q1 > Social optimal quality Q∗

Right: positive consumption externality

◮ Private MR < Social MR ◮ Local optimal quality Q1 < Social optimal quality Q∗ Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 7 November 20, 2018 9 / 46

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Negative Network Externality

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A Case Study: Water Pollution

The chemical company produces chemical products and discharges wastewater into the river. The water company produces bottle water by drawing water from the river. Water pollution increases the production cost of the water company.

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Graphical Illustration

Quantity Price Q1 Q∗ 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

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At Local Optimal Quality Q1

Quantity Price Q1 Q∗ A B C D E F $10 MC(social) MC(private) MC(external) MR

The chemical plant’s profit (i.e., revenue - cost): Q1 (MR − MCPrivate(Q)) dQ = A + B + E The water company’s profit due to externality (assuming 0 revenue): − Q1 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

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At Social Optimal Quality Q∗

Quantity Price Q1 Q∗ A B C D E F $10 MC(social) MC(private) MC(external) MR

The chemical plant’s profit (i.e., revenue - cost): Q∗ (MR − MCPrivate(Q)) dQ = A + B The water company’s profit due to externality (assuming 0 revenue): − Q∗ MCExternal(Q) dQ = −C Since C = B , the social surplus (sum of two profits): A + B − C = A

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Comparison

Social suplus at Q1 : A − D Social surplus at Q∗ : A With negative externally, individual profit maximization hurts the social surplus Solution: Pigovian tax

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Pigovian Tax

Quantity Price $10 A1 A2 B C Q1 Q∗ MC(social) MC(private)+Tax MC(private) MC(external) MR Tax

Charge chemical plant a tax

◮ Tax = external marginal cost at the optimal solution Q∗

Individual profit maximisation leads to production level of Q∗

◮ Chemical plant profit =

Q∗ (MR − MCPrivate(Q) − Tax) dQ = A1

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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 example

◮ If water company owns the water: it can charge the chemical plant a

price equal to the negative externally

◮ If chemical plant owns the water: it can demand a compensation from

water company for reducing the chemical production quantity

◮ Either way, it is possible to maximize social surplus Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 7 November 20, 2018 17 / 46

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Positive Network Externality

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A Case Study: Network Effect

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

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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 connecting with any one of the other N − 1 users. Total network value N(N − 1) ≈ N2.

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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 log N.

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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

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Section 7.2: Distributed Wireless Interference Compensation

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Wireless Power Control

Distributed power control in wireless ad hoc networks Elastic applications with no SINR targets Want to maximize the total network performance

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Network Model

T1 T2 T3 R1 R2 R3

h1

11

h2

11

h1

12

h2

12

Single-hop transmissions. A user = a transmitter/receiver pair. Transmit over one or multiple parallel channels. Interferences in the same channel (negative externality).

Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 7 November 20, 2018 25 / 46

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Single Channel Communications

2

σM

22

h σ2

12

h

11

h σ1

1

p Transmitters Receivers

21

h

M

p p

We focus on a single channel. For each user n ∈ N = {1, ..., N}:

◮ Power constraint: pn ∈ [Pmin

n

, Pmax

n

].

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

γn = pnhn,n σn +

m=n pmhn,m

.

◮ Utility function Un(γn): increasing, differentiable, strictly concave. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 7 November 20, 2018 26 / 46

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Network Utility Maximization (NUM) Problem

NUM Problem max

{Pmin

n

≤pn≤Pmax

n

,∀n}

  • n

Un(γn). Technical Challenges:

◮ Coupled across users due to interferences. ◮ Could be non-convex in power (check the Hessian matrix).

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

Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 7 November 20, 2018 27 / 46

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Benchmark - No Information Exchange

Each user picks power to maximize its own utility, given current interference and channel gain. Results in pn = Pmax

n

for all n.

◮ Maximum interference. ◮ Can be far from optimal. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 7 November 20, 2018 28 / 46

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Benchmark - No Information Exchange

Each user picks power to maximize its own utility, given current interference and channel gain. Results in pn = Pmax

n

for all n.

◮ Maximum interference. ◮ Can be far from optimal.

We propose algorithm with limited information exchange.

◮ Have nice interpretation as distributed Pigovian taxation. ◮ Analyze its behavior using supermodular game theory. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 7 November 20, 2018 28 / 46

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ADP Algorithm: Asynchronous Distributed Pricing

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

  • ∂Un(γn)

∂In

  • = ∂Un(γn)

∂γn γ2

n

pnhn,n . Power Updating: user n updates power pn to maximize surplus: Sn = Un(γn) − pn

  • m=n

πmhm,n. Repeat two phases asynchronously across users. Scalable and distributed: only need to announce single price, and know limited channel gains (hm,n).

Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 7 November 20, 2018 29 / 46

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ADP Algorithm

Interpretation of prices: Pigovian taxation

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ADP Algorithm

Interpretation of prices: Pigovian taxation ADP algorithm: distributed discovery of Pigovian taxes

◮ When does it converge? ◮ What does it converge to? ◮ Will it solve NUM Problem ? ◮ How fast does it converge? Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 7 November 20, 2018 30 / 46

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Convergence

Depends on the utility functions.

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Convergence

Depends on the utility functions. Coefficient of relative Risk Aversion (CRA) of U(γ): CRA(γ) = −γU′′(γ) U′(γ) .

◮ larger CRA ⇒ “more concave” U. Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 7 November 20, 2018 31 / 46

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Convergence

Depends on the utility functions. Coefficient of relative Risk Aversion (CRA) of U(γ): CRA(γ) = −γU′′(γ) U′(γ) .

◮ larger CRA ⇒ “more concave” U.

Theorem: If each user n ∈ N

◮ has a positive minimum transmission power Pmin

n

, and

◮ CRA(γn) ∈ [1, 2] for any values of γn,

then there is a unique optimal solution of NUM Problem, and the ADP algorithm globally converges to it. Proof: relating this algorithm to a fictitious supermodular game.

Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 7 November 20, 2018 31 / 46

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Supermodular Games

A class of games with strategic complementaries

◮ 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:

◮ A PNE exists. ◮ If the PNE is unique, then the asynchronous best response updates will

globally converge to it.

Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 7 November 20, 2018 32 / 46

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Convergence Speed

10 20 0.5 1 Power ADP Algorithm 200 400 600 0.5 1 Power Gradient−based Algorithm 5 10 15 20 20 40 60 80 Iterations Price 200 400 600 20 40 60 80 Iterations Price

10 users, log utilities. ADP algorithm (left figures) converges much faster than a gradient-based method (right figures).

Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 7 November 20, 2018 33 / 46

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Section 7.3: 4G Network Upgrade

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When To Upgrade From 3G to 4G?

Early upgrade:

◮ More expensive, as cost decreases over time ◮ Starts with few users, hence a small initial revenue

Late upgrade:

◮ Leads to a smaller market share ◮ Delays 4G revenues

Need to

◮ Capture the above tradeoffs ◮ Consider the dynamics of users adopting 4G and switching providers ◮ Understand the upgrade timing between competing cellular providers Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 7 November 20, 2018 35 / 46

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Duopoly Model

Two competing operators

◮ Initially both using 3G technology ◮ Operator i decides to upgrade to 4G at time Ti ◮ Each operator wants to maximize its long-term profit

What will be the equilibrium of (T ∗

1 , T ∗ 2 )?

Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 7 November 20, 2018 36 / 46

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Users Switching

W.L.O.G., assume T1 < T2 Three time periods: [0, T1], (T1, T2], and (T2, ∞)

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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.

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Users Switching

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

λ αλ λ Provider 1 Provider 2 3G 3G 4G

λ αλ

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Users Switching

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

λ αλ λ Provider 1 Provider 2 3G 3G 4G

λ αλ

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

1. λ.

T T

λ αλ λ 1 2

λ αλ

3G 3G 4G

λ λ

4G Provider 1 Provider 2

Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 7 November 20, 2018 38 / 46

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Network Value (Revenue)

Network value depends on the number of subscribers

◮ Assume that operator i has Ni 4G users, i = 1, 2 ◮ Total 4G network value is (N1 + N2) log(N1 + N2) (network effect) ◮ Operator i’s network value (revenue) is Ni log(N1 + N2) Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 7 November 20, 2018 39 / 46

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Network Value (Revenue)

Network value depends on the number of subscribers

◮ Assume that operator i has Ni 4G users, i = 1, 2 ◮ Total 4G network value is (N1 + N2) log(N1 + N2) (network effect) ◮ Operator i’s network value (revenue) is Ni log(N1 + N2)

Later upgrade ⇒ take advantage of existing 4G population

Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 7 November 20, 2018 39 / 46

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Network Value (Revenue)

Network value depends on the number of subscribers

◮ Assume that operator i has Ni 4G users, i = 1, 2 ◮ Total 4G network value is (N1 + N2) log(N1 + N2) (network effect) ◮ 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)

Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 7 November 20, 2018 39 / 46

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Revenue and Market Share

∗

2∗

1 1

1, 3G 1, 4G 2, 3G 2, 4G

R1(t) R2(t) T1

T2

MS ↑ 4G ↑

4G ↑

MS ↓ 4G ↑ αλ

  • st

Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 7 November 20, 2018 40 / 46

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Upgrade Cost and Time Discount

One-time upgrade cost:

◮ K at time t = 0 ◮ Discounted over time: K exp(−Ut)

Revenue is also discounted over time by exp(−St) Earlier upgrade ⇒ larger revenue and larger cost

Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 7 November 20, 2018 41 / 46

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Equilibrium Timings

0.05 0.1 0.15 0.2 0.25 0.3 0.05 0.1 0.15 0.2 0.25 0.3 Operator 1’s equilibrium time T1

*

Operator 2’s equilibrium time T2

*

NE 1: T1

*≤T2 *

NE 2: T1

*≥T2 *

Low cost regime: 0=T1*=T2* as K↑ Medium cost regime: 0=T1*<T2*↑ as K↑ High cost regime: 0<T1*↑<T2*↑: as K↑ Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 7 November 20, 2018 42 / 46

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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

*

Operator 2’s equilibrium profit π2

*

NE 1: T1*≤T2* NE 2: T1*≥T2*

Medium cost regime: π1*↑<π2*↓ as K↑ High cost regime: π1*↑<π2*↑ as K↑ Low cost regime: π1*↓=π2*↓ as K↑ Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 7 November 20, 2018 43 / 46

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Section 7.4: Chapter Summary

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Key Concepts

Theory

◮ Positive and negative Externality ◮ Market failure ◮ Pigovian tax ◮ Network effect

Application

◮ Distributed wireless power control based on Pigovian tax ◮ Cellular network upgrade considering network effect Huang & Gao ( c NCEL) Wireless Network Pricing: Chapter 7 November 20, 2018 45 / 46

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References and Extended Reading

  • J. Huang, R. Berry and M. Honig, “Distributed Interference Compensation

for 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, vol. 14, no. 5, pp. 975 - 989, 2015

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

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