Issues in Pricing Internet Services

Linhai He & Jean Walrand

{linhai, wlr}@eecs.berkeley.edu

Dept of EECS, U.C. Berkeley

March 8, 2004

2

Challenges

Stagnant telecommunication industry

“We know how to route packets; what we don’t

know how to do is route dollars.”

- David Clark, MIT

) Need efficient economic mechanisms to increase the profit of Internet service providers

3

Approach

• Combine economics with network protocol design

– Economics help identify utilities and strategies of users

– Protocols are designed to shape and enable the strategies

Goal: Networks mutually beneficial to both users and providers

• Two essential ingredients– More revenues from service differentiation/market

segmentation

Question: How to price differentiated services?

– Fair revenue distribution among the providers

Question: How should a provider price its share of service?

4

Outline

• Pricing Differentiated Services– Motivating examples

– Dynamic pricing schemes

• Pricing with Multiple Providers– Motivations

– Non-cooperative pricing

– Revenue sharing policy

– Implementation

• Pricing Wireless Access (with John Musacchio)

• Summary and Future Work

5

Pricing Differentiated Services: Base Model

p1

p2

strategic users

If users do not randomize their

choices, what kind of equilibrium

would happen?

If users do not randomize their

choices, what kind of equilibrium

would happen?

Users choose the service class which maximizes their net benefit

• Delay Ti: no preset targets; determined by users’ own

choices

− If equilibrium exists, higher price p ) smaller delay T• Congestion externality exists within and between the

classes

6

Outcome A. Prisoner’s Dilemma

H. P. L. P.

H. P.

L. P.

BA

p1

p2

A

B

f(T1) = 14

f(T2) = 5

f(T0) = 9

p1= 4

p2= 1

9-4 = 59-4 = 5

5-1 = 414-4 =10

14-4 =10 5-1 = 4

9-1 = 89-1 = 8

NE

H. P.

L. P.

7

Outcome B. No Pure-Strategy Equilibrium

H. P. L. P.

H. P.

L. P.

BA

p1

p2

A

B

p1= 4

p2= 1

9-4 = 59-4 = 5

7-1 = 611-4 = 7

13-4 = 9 5-1 = 4

9-1 = 89-1 = 8

f1

f2

T1 T0 T2

13 9 7

11 9 5

8

General Conditions for Two-Users Case

• If , both users will choose to

use high-price class ) Prisoners’ Dilemma

• If fa is convex and fb is concave, or vice versa,

then no pure-strategy equilibrium exists.

H. P. L. P.

H. P.

L. P.

BA

9

Extension to Many-User Case

• Model– Infinite number of atomic users making independent

choices

– User’s payoff function

willingness to pay;

with load density ()

• Equilibrium

load in class i

delay in class i

leave low-price class high-price class

2 10

10

Properties of Equilibrium: an example

• Utility function f is concave; strict-priority scheduling

1

p1-p2

stable butinefficientequilibrium

unstableequilibrium

1 ! x1 ! search 2 which satisfies

11

Properties of Equilibrium

− Perturbation around equilibrium cause change in users’ payoff

• Stability of the equilibrium

If M>0, then users with 2 B(1, ) has incentive to switch ) unstableThis might happen if congestion externality is significant

between classes.

• Multiple equilibria

if is not monotonic in

Example: small group of users move from L.P. into H.P. Consider

12

Challenge

• How to design the system so that it

is stable and efficient?

• Knobs one could turn:

– Scheduling policy

– Pricing scheme

13

To Stabilize…

• Scheduling policy: Paris-Metro model [Odlyzko]

– Inflexible in adapting to changes in user demand– Possible loss in revenue for being non-work-conserving

• Pricing Scheme: load-based pricing

– Let p1= p1(x1) while keep p2 constant, so that M<0 under

perturbation– Resulting equilibrium is stable, if

p1

p2user

s

No congestion externality

between classes ) always

stable

where k is a bound on

between class

within class

14

userD1

D2

agent(VCG)

bid:

charge: pi

To be more efficient…

Effect on last user in L.P.

Effect on last user in H.P. and L.P.

• Goal– assignment rule which maximizes the sum of users’

utilities

• Mechanism-Design approach

– Socially efficient

▪ Assign users from H.P. to L.P. according to their bid

– Incentive compatible: charge a user by her externality effect

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Our Solution

• Congestion pricing

• Equilibrium

p1

p2

user

Di

pi

Users choose to join H.P. to L.P. in decreasing order of

two marginal users equilibrium prices

externality cost of the marginal users

16

Pricing with Multiple Providers: Outline

• Challenges

• Model and formulation

• Non-Cooperative Pricing

• Revenue Sharing

• Implementation

17

Challenges

• Internet is an interconnection of service providers

– An Internet service has to be jointly provided by a

group of service providers

– Providers are neither cooperative nor adversary; they

act strategically in their own interests

• Design requirements on pricing schemes

– Fair distribution of revenue

– Scalable implementation

– Robust against gaming or cheating

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A Possible Implementation

Provider 2

Provider 1

How should each provider price its share of service?

How should each provider price its share of service?

request

request $1

$2request $1

ACK $3

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Objectives

• Formulate an abstract model that summarizes common issues in various implementations

• Understand how providers would charge for their services when acting strategically

• Design a pricing mechanism which meets the aforementioned design requirements

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Model - Users

• Service Model

– QoS requirement ) limits on link load

• Users’ aggregate demand

– May be regulated by price p

– Demand d(p) is decreasing and differentiable

– Revenue pd(p) has a unique maximizer

– For use later, define

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Model - Providers

• Local capacity limit is private information

• QoS requirements and routes are fixed and are independent from prices

chargep1+ p2

provider 2

• Revenue = Price £ Demand

• Choose price to maximize its own revenue, while regulate the load to meet QoS requirement

demand p1+ p2

+ p1

provider 1

+ p2

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Formulation: an example

1 2p1 p2

D

• • •

demand= d(p1+p2) C1 C2

• A pricing game between two providers• Different solution concepts may apply,

depend on actual implementation• Nash game mostly suited for large

networks

Provider 1 Provider 2

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Outcome of the Nash Game

• Essentially a Cournot game with coupled local constraints

• Bottleneck providers get more share of revenue than others

• Bottleneck providers may not have incentive to upgrade

• Efficiency decreases quickly as network size gets larger

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Outcome of the Nash Game (cont)

• Bottleneck providers may lack incentive to upgrade

Again assume C1 > C2. It can be shown that when

provider 2’s constraint is active,

so that

may have a solution, i.e. a maximizer may exist, so

that J2 may not always increase with C2.

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Outcome of the Nash Game (cont)

Example: demand d(p) = Aexp(−Bp), >1

J2*

J1*

capacity unconstrained

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Improve Outcome of the Game

• Approach A: centralized allocation– Prices are chosen to maximize the total revenue

– Main challenge:▪ Individual provider’s benefit vs. social welfare

• Approach B: cooperative games– Pareto-efficient allocation among providers

– Fairness defined through set of axioms

▪ Generalized Nash’s bargaining solution

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Nash’s Bargaining Solution

• The equilibrium should satisfy

payoff J1

payoff J2

feasible payoff set

A

B

C

J2

J1

Pareto-efficient set

• Generalize to n-player case

Proportional Fairness Criteria

Proportional Fairness Criteria

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An Example

N

access

backbone

C

Solution:

where

Unfair allocation biased against backbone provider

Unfair allocation biased against backbone provider

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Modified Bargaining Solution

• A two-level bargaining approach

– Proportionally-fair split of revenue collected on each

route r

– Bargaining on per-provider basis for the total price per route

FACT: Equal sharing on each route.

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Modified Bargaining Solution: Example

C1

d(p)

p31

p1

C2

p32

100¢d(p)

10¢d(p)

p2

p3

d1

d3

d2

d3

In general, it is difficult to compute the solution in a decentralized way (not

scalable).

In general, it is difficult to compute the solution in a decentralized way (not

scalable).

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Our Approach

• Trade Pareto-efficiency with scalability– Providers still share revenue on a per-route basis

– but compute equilibrium total price pr through Nash

game

• Advantages– No need of knowing individual capacity constraints

– Can be implemented by a distributed protocol (scalable)

– Can eliminate drawbacks of non-cooperative pricing

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Example Revisited

C1

d2(p)

p31

p1

C2

p32

d1(p)

d3(p)

p2

p3

Provider 1

Provider 2

Best-response:

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Optimality Condition

• For a route r on link i (general network topology)

marginal cost on

link i“locally optimal”

total price for the route

sum of prices charged by other

providers

hop count

A system of N such equations for each flowA system of N such equations for each flow

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Optimal Price: solution

feasible set of , there is a unique solution to the price that links should mark for flows on a route r

if link i has the largest i,

on all other links,

) Only the most “congested” link on a route marks price

• Each provider solves its i based on local constraints

– A Nash game with i as strategy

– Pure-strategy Nash equilibrium exists in this game

(proof by Brower’s fixed-point theorem)

36

Properties of the Equilibrium

• Compare with centralized approach

Centralized:

Sharing:

• Incentive to upgrade– Upgrade will always increase bottleneck providers’

revenue

• Efficient when capacities are adequate– It is the same as that in centralized allocation

– Revenue per provider strictly dominates that in Nash game

37

Distributed Implementation

flows on

route r

1

i

N

Nr = 0 rs = 0

…… Nr =Nr + 1r

s = max(rs, i)

Can be shown to converge to the Nash equilibrium, by using Lyapunov function

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A Numerical Example

C1=2 C2=5 C3=3

demand = 10 exp(-p2) on all routes

r1

r2

r3

r4

i

link 1

link 3

link 2

prices

p2

p3

p1

p4

s1 = s2= s3 =1

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What about cost?

• Net-benefit of a provider = revenue – unit cost £ load

– Weighted proportionally-fair allocation on each route) Equal return

on investment

New objective function

New optimal price How to solicit true cost info from theproviders?

How to solicit true cost info from theproviders?

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Summary and Ongoing Work

• Summary– Non-cooperative pricing between providers may be

unfair, inefficient and discourage the evolution of the

Internet

– Cooperative pricing help increase providers’ revenue

and lead to more efficient use of the network resources

• Ongoing work

– Bounds on the loss of efficiency due to Nash

implementation

– Adding competition (routing) to the models

– Efficient architecture for revenue distribution

June 28, 2002 Preliminary Results

Pricing Wireless Access

• How can they conduct their transaction?– Pre-pay? Access Point might take the money and run.

– Post-pay? Client might enjoy service and not pay.

– Pay as she goes?

• Will this payment model work?– Will the access point charge a fixed price over session

duration?– Will client and access point accept this payment model at all?

ClientAccess Point

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General Formulation

pt Access Point

Discrete time slot model: 1 2 t...

Access point proposes priceat the start of a slot:

Accept

Quit GameClient’s Choices:

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Web Browsing Model of Client Utility

• Client’s session utility :

• Note: Asymmetric information:

– Access Point knows the distribution of (U, )

– Client knows the sample value of (U, )

U: utility per slot

T: # slots client ends up buying

: # slots client interested in buying

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File Transfer Model of Client Utility

• Client’s utility a step function.

Utility

#slots connected

• Asymmetric information:– Access Point knows distribution of – Client knows the sample value of

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Summary of Results

• Web Browsing Model– Access point charges a constant price.

– Clients with high enough utilities connect.

• File Transfer Model:– Clients are “pessimistic” and refuse to pay

anything until the last time slot.

– Access Point price not constant.

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References

1. Linhai He and Jean Walrand. Internet Service Differentiation and Market Segmentation, in preparation.

2. Linhai He and Jean Walrand. Pricing Internet Service with Multiple Providers, the 41st Annual Allerton Conference on Communication, Control and Computing, Monticello, IL, Oct. 2003. Available at http://www.eecs.berkeley.edu/ ~linhai/publications/Allerton03.pdf

3. John Musacchio and Jean Walrand. Game Theoretic Modeling of Wi-Fi pricing, 41st Annual Allerton Conference on Communication, Control and Computing, Monticello, IL, Oct. 2003. Available at http://robotics.eecs.berkeley.edu/~wlr/ Papers/allerton2003_WiFi.pdf

4. Andrew Odlyzko, Paris Metro Pricing for the Internet, ACM Conference on Electronic Commerce, 1998.

5. Geoff Huston. Interconnection and Peering, the Internet Protocol Journal, March 1999.

6. C. Courcoubetis and R.R. Weber. Pricing Network Services, Springer Verlag, 2003.

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Thank you!