Communication Networks

1
Communication Networks

• A Second Course

Jean Walrand Department of EECS University of
California at Berkeley
2
Non-Cooperative Games

• Comparing Nash Equilibria
• Three Problems in Networks
• Service Differentiation
• Multi-Provider
• Pricing Wi-Fi

3
Comparing Nash Equilibria

• Examples
• Pareto
• Minimax
• Risk-Dominant

4
Comparing Nash Equilibria

• Examples

L R
T 4, 4 1, 2
B 2, 1 3, 3
L R
T 3, 2 1, 1
B 1, 1 2, 3
L R
T 4, 4 2, 4
B 5, 2 2, 3
L R
T 2, 2 -100, 0
B 0, -100 1, 1
5
Comparing Nash Equilibria

• Pareto

L R
T 4, 4 1, 2
B 2, 1 3, 3
A

• Both players are strictly better off with the NE
  (T, L) than with the NE (B, R).
• They can safely assume that (T, L) will be
  played.
• (T, L) is the unique Pareto NE (the vector of
  rewards is not dominated component-wise by a
  different NE)

6
Comparing Nash Equilibria

• Minimax

L R
T 4, 3 1, 2
B 2, 1 3, 4

• Both (T, L) and (B, R) are Pareto There is no
  other equilibrium where both players improve
  their rewards.
• What will happen?
• Minimax (B, R) Both A and B maximize their
  minimum reward. Should be called maximin .

7
Comparing Nash Equilibria

• Minimax

L R
T 4, 3 1, 1
B 2, 2 3, 4

• The Minimax is (B, L)
• However, this is not very credible .
• If P1 thinks P2 chooses L, then P1 chooses T
• Also, if P2 thinks P1 chooses B, then P2 chooses
  R
• What will happen?

8
Comparing Nash Equilibria

• Minimax ?? Pareto

L R
T 2, 2 2, 1
B 1, 2 5, 5

• Minimax (T, L)
• Unique Pareto (B, R)
• Would the players choose (B, R)?

9
Comparing Nash Equilibria

• Risk-Dominant

L R
T 2, 2 -5, 0
B 0, -5 1, 1

• (T, L) is the unique Pareto NE
• However, by playing L, P2 faces a big risk if P1
  plays B
• P2 reduces his risk by playing R
• Knowing this, P1 may prefer to play B
• P1 faces a smaller risk by playing B than P2 does
  by playing L
• This suggests that (B, R) is a less risky NE.

10
Comparing Nash Equilibria

• Risk-Dominant

L R
T 2, 2 -5, 0
B 0, -5 1, 1
Concept Risk-Dominance Define p as follows If
P(P2L) p, then P1 T is preferable. ? If
P(P2R) 1 - p, then P1 B is preferable If p
gt 0.5, we say that (B, R) is risk-dominant
else (T, L) is risk-dominant In this case, p
0.75 ? (B, R) is risk-dominant
11
Comparing Nash Equilibria
L R
T 2, 2 -5, 0
B 0, -5 1, 1

• Risk-Dominant

Lets do the math If P(P2L) p, then P1 T
is preferable U1(T) 2p 5(1 p) U1(B) 0p
1(1 p) Minimum value of p is s.t. 1 p 2p
5(1 p) ? p 0.75
12
Comparing Nash Equilibria

• Risk-Dominant

L R
T 2, 2 -5, 0
B 0, -5 ? 1, 1
A B
A ? 2, 2 -0.5, 0
B 0, -0.5 1, 1
13
Comparing Nash Equilibria

• Risk-Dominant

A B
A 9, 9 0, 8
B 8, 0 7, 7
Related Definition ? Linear-Tracing
Define G(t) Game where P2 randomizes his
strategy with probability 1 t N(t) set
of NE for G(t) If there is a continuous graph t,
f(t) in N(t), then f(1) defines a
Linear-Tracing Risk-Dominant NE
14
Three Problems

1. Service Differentiation
2. Multi-Provider Network
3. Pricing Wi-Fi
4. Conclusions

15
Three Problems

• Service Differentiation
• Market segmentation
• Capture willingness to pay more for better
  services

16
Three Problems (cont.)

1. Multi-Provider Network

Incentives for better services through all
providers ? Improved Services Revenues
17
Three Problems (cont.)

1. Wi-Fi Access

Incentives to open private Wi-Fi access
points ? Ubiquitous Access
18
Three Problems (cont.)

• Note Related problems
• P2P Incentives
• Incentives for Security
• Fair sharing among content and network providers

19
Service Differentiation

• Model
• Examples
• Proposal

Joint work with Linhai He
20
Service Differentiation

• Model ?
• Examples
• Proposal

Joint work with Linhai He
21
Model

• Two possible outcomes
• Users occupy different queues (delays T1 T2)
• Users share the same queue (delay T0)
• If users do not randomize their choices, which
  outcome will happen?

22
Model (cont)
T1 lt T0 lt T2
fi(.) nonincreasing


f1(T1) p1
f2(T2) p2
f1(T0) p2
f1(T2) p2
f2(T1) p1
f2(T0) p2
TOC Service Differentiation ? Model
23
Service Differentiation

• Model
• Examples ?
• Proposal

Joint work with Linhai He
24
Example 1
Here, fi(.) f(.)
f(T1) 14 f(T0) 9 f(T2) 5
p1 4 p2 1
B
H
L
A


9 4 5 9 4 5
14 4 10 5 1 4
H
9 1 8 9 1 8
5 1 4 14 4 10
L
25
Example 1
Assume A picks H. Should B choose H or L?
Assume A picks H. B should choose H.
Assume A picks L. Should B choose H or L?
Assume A picks L. B should choose H.
B ? H.
Since B chooses H, A should also choose H.
B
H
L
A


NE
5 5
10 4
H
NE Nash Equilibrium
8 8
4 10
L
26
Example 1
A and B choose H, get rewards equal to 5. If they
had both chosen L, their rewards would have been
8!
Prisoners Dilemma!
B
H
L
A


NE
5 5
10 4
H
8 8
4 10
L
27
Example 2
f0 f1
T1 13, 11 T0 9, 9 T2 7, 5
p1 4 p2 1
B
H
L
No Pure Equilibrium
A


9 4 9 - 4
13 4 5 - 1
H
9 1 9 - 1
7 1 11 - 4
L
28
Example 3

• Extension to many users

Also, the other users prefer L.
Note T1 and T2 depend on the split of customers.

29
Example 3

• Analysis of equilibria

qf(T1)-f(T2)
p1-p2
?
Here, f is a concave function and strict-priority
scheduling is used.
30
Service Differentiation

• Model
• Examples
• Proposal ?

Joint work with Linhai He
31
Proposal

• Dynamic Pricing

• Fixed delay dynamic price
• Provider chooses target delays for both classes
• Adjust prices based on demand to guarantee the
  delays
• Users still choose the class which maximizes
  their net benefit

32
Multiprovider Network

• Model
• Nash Game
• Revenue Sharing

Joint work with Linhai He
33
Multiprovider Network

• Model ?
• Nash Game
• Revenue Sharing

Joint work with Linhai He
34
Model

• Pricing per packet

35
Multiprovider Network

• Model
• Nash Game ?
• Revenue Sharing

Joint work with Linhai He
36
Nash Game Formulation
p1
p2
Demand d(p1p2)
1
2
D

C1
C2
Provider 1
Provider 2

• A game between two providers
• Different solution concepts may apply, depend on
  actual implementation
• Nash game mostly suited for large networks

37
Nash Game Result

1. Bottleneck providers get larger share of revenues
   than others
2. Bottleneck providers may not have incentive to
   upgrade
3. Efficiency decreases quickly as network size gets
   larger (revenues/provider drop with size)

38
Multiprovider Network

• Model
• Nash Game
• Revenue Sharing ?

Joint work with Linhai He
39
Revenue Sharing

• Improving the game
• Model
• Optimal Prices
• Example

40
Revenue Sharing

• Improving the game ?
• Model
• Optimal Prices
• Example

41
Revenue Sharing- Improving the Game

• Possible Alternatives
• Centralized allocation
• Cooperative games
• Mechanism design
• Our approach design a protocol which
• overcomes drawbacks of non-cooperative pricing
• is in providers best interest to follow
• is suitable for scalable implementation

42
Revenue Sharing

• Improving the game
• Model ?
• Optimal Prices
• Example

43
Revenue Sharing- Model

• Providers agree to share the revenue equally, but
  still choose their prices independently

p1
p2
Demand d(p1p2)
D
2
1

C1
C2
Provider 1
Provider 2
44
Revenue Sharing

• Improving the game
• Model
• Optimal Prices ?
• Example

45
Revenue Sharing- Optimal Prices
? A system of equations on prices
46
Revenue Sharing- Optimal Prices (cont.)

• For any feasible set of mi, there is a unique
  solution
• On the link i with the largest m (, ?),
  pi N ? g(pi)
• On all other links, pj 0
• ? Only the most congested link on a route sets
  its total price

47
Revenue Sharing- Optimal Prices (cont.)

• Each provider solves its ?i based on local
  constraints

?i
pi
dr
? a Nash game with mi as the strategy
It can be shown that a Nash equilibrium exists
in this game.
48
Revenue Sharing- Optimal Prices (cont.)

• Comparison with social welfare maximization (TCP)

Social
Sharing
? A tradeoff between efficiency and fairness

• Incentive to upgrade
• Upgrade will always increase bottleneck
  providers revenue

49
Revenue Sharing- Optimal Prices (cont.)

• Efficient when capacities are adequate
• It is the same as that in centralized allocation
• Revenue per provider strictly dominates that in
  Nash game

50
Revenue Sharing- Optimal Prices (cont.)

• A local algorithm for computing mi
• that can be shown to converge to Nash
  equilibrium

51
Revenue Sharing- Optimal Prices (cont.)

• A possible scheme for distributed implementation

flows on route r
1
d
congestion price ?r0
hop count Nr0

?r max(?r, ?i)
NrNr1


No state info needs to be kept by transit
providers.
i
52
Revenue Sharing

• Improving the game
• Model
• Optimal Prices
• Example ?

53
Example
r2
r4
r1
r3
C12
C25
C33
demand 10 exp(-p2) on all routes
?i
prices
p2
link 1
p3
p1
link 3
p4
link 2
54
Wi-Fi Pricing

• Motivation
• Web-Browsing
• File Transfer

Joint work with John Musacchio
55
Wi-Fi Pricing

• Motivation ?
• Web-Browsing
• File Transfer

Joint work with John Musacchio
56
Motivation

• Path to Universal WiFi Access
• Massive Deployment of 802.11 base stations for
  private LANs
• Payment scheme might incentivize base station
  owners to allow public access.
• Direct Payments
• Avoid third party involvement.
• Transactions need to be self enforcing
• Payments
• Pay as you go In time slot n,- Base Station
  proposes price pn- Client either accepts or
  walks away
• What are good strategies?

57
Wi-Fi Pricing

• Motivation
• Web-Browsing ?
• File Transfer

Joint work with John Musacchio
58
Web Browsing

• Client Utility

• BS Utility

p1 p2 pN
59
Web Browsing

• Theorem Perfect Bayesian Equilibrium

60
Wi-Fi Pricing

• Motivation
• Web-Browsing
• File Transfer ?

Joint work with John Musacchio
61
File Transfer

• Client Utility

K.1K N

• BS Utility

p1 p2 pN
62
File Transfer

• Theorem Perfect Bayesian Equilibrium

63
Three Problems Conclusions
References Linhai He and Jean Walrand, Pricing
and Revenue Sharing for Internet Service
Providers.. To appear in JSAC 2006. John
Musacchio and Jean Walrand, Game Theoretic
Modeling of WiFi Pricing, Allerton, 10/1/2003