Communication NetworksA Second Course

Jean WalrandDepartment of EECSUniversity of California at Berkeley

Games in Networks: Overview

• Motivation

• Examples of Game

• In Networks

• Key Concepts

Motivation• Decisions made by people (agents)

• Selfish agents with conflicting interests

• Markets (interactions of agents) influence technological development and adoption

• Protocols limit or enable strategies

• Design protocols to create suitable incentives

• Pricing, service choice, contracts, regulations, cooperation, mergers and acquisitions, liability

• Game theory models interactions of selfish agents

Examples of Games• Matching Pennies:

• Alice and Bob both show a penny• If the faces match, Bob gives Alice $1.00• If the face do not match, Alice gives Bob $1.00

• Battle of the Sexes:• Alice and Bob choose to go to the opera or football• They prefer to go together, but Alice loves opera and hates

football; Bob loves football and hates opera• Prisoners’ Dilemma:

• Alice and Bob are accused of a crime and are interrogated separately

• If exactly one pleads guilty, that person gets 1 year and the other five years

• If both plead guilty, they get 3 years; if none pleads guilty, they get 2 years.

In Networks

• Physical

• MAC

• Routing

• Transport

• Interactions between providers

Games: Physical Layer

• Being Loud in a Restaurant• You are heard better if you talk louder• Everybody ends up being very loud and upset by the noise

• Power control in CDMA• Each user prefers to transmit at maximum power to improve

his capacity• However a user’s increased power reduces the capacity of

the other users because of increased interference

• Spectrum allocation• Each user prefers a wider spectrum• However, this decreases the capacity of the other users

Games: MAC

• Backup Window in WiFi• Each user prefers a smaller backup window size• However, this reduces the throughput of others

• Persistence in Aloha• Similar

• Priority Class in 11e• Incentive is to claim high priority for all traffic• However, this reduces the quality …

Games: Routing• Shortest Path

• Users may prefer shortest path• However, this my lead to congested paths

• Fastest Paths• Users may prefer fastest path• This may increase average delay

• Duplicated Paths• Users may prefer to duplicate their traffic across parallel paths• This may make everybody worse off

• Hot potato routing• Send packets off to competitor’s network asap

Games: Transport

• TCP• Users might want to be more aggressive• If everyone does it, everybody is worse off

• Proportional Fairness• Normal: user gets x = w/p• Strategic: user knows p = w + other weights• This leads strategic users to change their bids….

Games: Between Providers

• Interactions between network providers

• User ---[AT&T]----[Verizon]--- User• What are the incentives for Verizon to provide good

service to AT&T’s clients?• How should one allocate profits?

• User ---[AT&T]---[Google]• What are the incentives for AT&T to provide good

service to Google?• Should AT&T be allowed to charge Google more for

some services?

Key Concepts

• Game:• Multiple Agents• Agents act in some order, possibly multiple times• When she acts, agent knows some information

about previous actions, possibly incomplete• Ultimate reward of each agent depends on the

actions of all the agents• Agents may have incomplete information about

each other and the game evolution may be random• The choice of the actions may be randomized

Key Concepts

• Game Example: Matching Pennies

H T

H 1, -1 -1, 1

T -1, 1 1, -1

Actions of Alice

Actions of Bob

Reward of AliceReward of Bob

•One-Shot Game•Both players play simultaneously•Each player knows both reward functions

•One-Shot Game•Both players play simultaneously•Each player knows both reward functions

Key Concepts

• Game Example: Battle of the Sexes

•One-Shot Game•Both players play simultaneously•Each player knows both reward functions

•One-Shot Game•Both players play simultaneously•Each player knows both reward functions

O F

O 4, 1 2, 2

F 0, 0 1, 4 O = OperaF = Football

Key Concepts

• Game Example: Prisoners’ Dilemma

•One-Shot Game•Both players play simultaneously•Each player knows both reward functions

•One-Shot Game•Both players play simultaneously•Each player knows both reward functions

G = plead guiltyNG = plead not guilty

G NG

G 3, 3 1, 5

NG 5, 1 2, 2

Key Concepts

• Game Example: Dynamic

•2-player, 3-step game: P1, P2, P1•P1 does not see the action of P2•Each player knows both reward functions

•2-player, 3-step game: P1, P2, P1•P1 does not see the action of P2•Each player knows both reward functions

L R L R L R L R

P1

P2

P1L

L R

R L R

0 2 2 0 0 1 3 00 3 2 0 0 1 2 0

R1R2

L R L R L R L R

P1

P2

P1L

L R

R L R

0 2 2 0 0 1 3 00 3 2 0 0 1 2 0

R1R2

Key Concepts

• Game Example: Cournot Duopoly

•One-shot game•Continuous action space•Each player knows both reward functions

•One-shot game•Continuous action space•Each player knows both reward functions

Two firms produce quantity q1 and q2 of a productThe price is A – q1 – q2

For i = 1, 2 the profit of firm i is qi(A – q1 – q2) - Cqi

Two firms produce quantity q1 and q2 of a productThe price is A – q1 – q2

For i = 1, 2 the profit of firm i is qi(A – q1 – q2) - Cqi

Key Concepts

• Game Example: Bayesian

•One-shot game•Continuous action space•Each player has an incomplete knowledge of some reward functions

•One-shot game•Continuous action space•Each player has an incomplete knowledge of some reward functions

Two firms produce quantity q1 and q2 of a productThe price is A – q1 – q2

For i = 1, 2, the profit of firm i is q i(A – q1 – q2) – Ciqi

For i = 1, 2, firm i knows Ci but only knows the distribution of C2 - i

Two firms produce quantity q1 and q2 of a productThe price is A – q1 – q2

For i = 1, 2, the profit of firm i is q i(A – q1 – q2) – Ciqi

For i = 1, 2, firm i knows Ci but only knows the distribution of C2 - i

Key Concepts

• Strategy:• Rule that specifies how to act at any time, as a

function (possibly randomized) of the information available

• Good Strategy?• Dominant• Iterated deletion of dominated strategies• Nash Equilibrium • Other concepts … later.

Key Concepts: Good Strategy

• Dominant Strategy:• Best, no matter what the other players do• Example: Being loud in a restaurant. (Hum, it all

depends on the reward function, of course.)• Counter-example: The best response generally

depends on the action of others. Think matching pennies – There is no dominant strategy

Key Concepts: Good Strategy• Iterated deletion of dominated strategies:

• Example: Guess 2/3 of average

•Let us all choose a number in {0, 1, …, 100}•The person who is closest to 2/3 of the average of all the numbers wins

•Let us all choose a number in {0, 1, …, 100}•The person who is closest to 2/3 of the average of all the numbers wins

•Average is at most 100•Nobody should guess more than 66•2/3 average is then at most 44•Nobody should guess more than 44•2/3 average is then at most 30•Nobody should guess more than 30•2/3 average is then at most 20 …•Only reasonable guess is 0

•Average is at most 100•Nobody should guess more than 66•2/3 average is then at most 44•Nobody should guess more than 44•2/3 average is then at most 30•Nobody should guess more than 30•2/3 average is then at most 20 …•Only reasonable guess is 0

Note: Assumes all players are very rational ….Note: Assumes all players are very rational ….

Key Concepts: Good Strategy

• Iterated deletion of dominated strategies:• Example:

L M RT 2, 4 2, 5 30, 3M 3, 5 40, 4 20, 2B 4, 3 10, 2 18, 1

L M RT 2, 4 2, 5 30, 3M 3, 5 40, 4 20, 2B 4, 3 10, 2 18, 1

Key Concepts: Good Strategy• Nash Equilibrium: No one wants to deviate unilaterally

• Examples:

G NG

G 3, 3 1, 5

NG 5, 1 2, 2

Prisoners’ Dilemma

H T

H 1, 0 0, 1

T 0, 1 1, 0Matching Pennies: 50/50

V W

V 1, 1 0, 0

W 0, 0 1, 1

O F

O 4, 1 2, 2

F 0, 0 1, 4

Coordination Game Battle of the Sexes

Key Concepts: Good Strategy• Nash Equilibrium: No one wants to deviate unilaterally

• Examples:

G NG

G 3, 3 1, 5

NG 5, 1 2, 2

Prisoners’ Dilemma

H T

H 1, 0 0, 1

T 0, 1 1, 0Matching Pennies: 50/50

V W

V 1, 1 0, 0

W 0, 0 1, 1

O F

O 4, 1 2, 2

F 0, 0 1, 4

Coordination Game Battle of the Sexes