Informationally Efficient Multi user communication Yi Su Advisor: - - PowerPoint PPT Presentation

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Informationally Efficient Multi‐user communication

Yi Su

Advisor: Professor Mihaela van der Schaar Electrical Engineering, UCLA

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• Motivation and existing approaches
• Informationally efficient multi‐user

communication

– Vector cases

• Convergence conditions with decentralized information
• Improve efficiency with decentralized information

– Scalar cases

• Achieve Pareto efficiency with decentralized information
• Conclusions

Outline

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Multi‐user communication networks

Distributed routing Power control Peer‐to‐peer system etc…

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Constraints in communication networks

• Resources

– Bandwidth, power,

spectrum, etc.

• Information

– Real‐time

• Local observation

5

Constraints in communication networks

• Resources

– Bandwidth, power,

spectrum, etc.

• Information

– Real‐time

• Local observation
• Exchanged message

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Constraints in communication networks

• Resources

– Bandwidth, power,

spectrum, etc.

• Information

– Real‐time

• Local observation
• Exchanged message

– Non‐real‐time

• A‐priori information about

inter‐user coupling, protocols, etc.

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Constraints in communication networks

• Resources

– Bandwidth, power,

spectrum, etc.

• Information

– Real‐time

• Local observation
• Exchanged message

– Non‐real‐time

• A‐priori information about

inter‐user coupling, protocols, etc. Goal: multi‐user communication without information exchange

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A standard strategic game formulation

• Consider a tuple

– The set of players : – The set of actions: and – Utility function: and – Utility region:

In communication networks, different operating points in can be chosen based on the information availability

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Existing approaches

• Local observation

Nash equilibrium

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Existing approaches

• Local observation

Nash equilibrium

• Exchanged messages

Pareto optimality

Price!

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Existing approaches

• Local observation

Nash equilibrium

• Exchanged messages

Pareto optimality

Existing results usually assume some specific action and utility structures! Price!

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• Results with specific action and utility structures

– Pure Nash equilibrium

• Concave games

i) : convex and compact; ii) : quasi‐concave in

• Potential games [Shapley]
• Super‐modular games [Topkis]

i) is a lattice; ii)

– Pareto optimality

• Network utility maximization [Kelly]
• Convexity is the watershed

Existing approaches (cont’d)

Use gradient play to find NE

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Existing approaches (cont’d)

Use gradient play to find NE Use best response to find NE

• Results with specific action and utility structures

– Pure Nash equilibrium

• Concave games

i) : convex and compact; ii) : quasi‐concave in

• Potential games [Shapley]
• Super‐modular games [Topkis]

i) is a lattice; ii)

– Pareto optimality

• Network utility maximization [Kelly]
• Convexity is the watershed

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Existing approaches (cont’d)

Use gradient play to find NE Use best response to find NE Use best response to find NE

• Results with specific action and utility structures

– Pure Nash equilibrium

• Concave games

i) : convex and compact; ii) : quasi‐concave in

• Potential games [Shapley]
• Super‐modular games [Topkis]

i) is a lattice; ii)

– Pareto optimality

• Network utility maximization [Kelly]
• Convexity is the watershed

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Existing approaches (cont’d)

Use gradient play to find NE Use best response to find NE Use best response to find NE

• Results with specific action and utility structures

– Pure Nash equilibrium

• Concave games

i) : convex and compact; ii) : quasi‐concave in

• Potential games [Shapley]
• Super‐modular games [Topkis]

i) is a lattice; ii)

– Pareto optimality

• Network utility maximization [Kelly]
• Convexity is the watershed

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Existing approaches (cont’d)

Researchers Applications Tools Altman CDMA uplink power control S‐modular games Berry Distributed interference compensation S‐modular games Barbarossa Power control Potential games Tse Spectrum sharing Repeated games Kelly End‐to‐end congestion control Pricing Goodman CDMA uplink power control Pricing Low End‐to‐end flow control Pricing Chiang Joint congestion and power control Pricing Poor Energy efficient power and rate control Equilibrium analysis Cioffi Power control in DSL systems Equilibrium analysis Yates Uplink power control for cellular radio Equilibrium analysis Wicker Selfish users in Aloha Equilibrium analysis Lazar Non‐cooperative optimal flow control Equilibrium analysis

S p e c i a l u t i l i t y

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Existing approaches (cont’d)

• Game theory

– Equilibrium characterization – Incentive design

• Optimization theory

– Computational complexity – Distributed algorithms

• Information theory

– Fundamental limits – Encoding and decoding schemes

Information is usually costless The focus is on strategic interactions among users Decentralization is not the focus

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Existing approaches (cont’d)

u1 u2

Pareto boundary Global information Nash equilibrium Decentralized (limited) information

General models

e.g. concave/potential /supermodular games

Specific multi‐user communication applications

But in many communication systems, information is constrained and no message passing is allowed!

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Our goals

u1 u2

Nash equilibrium Decentralized (limited) information Pareto boundary Global (exchanged) information

If information is constrained and no message passing is allowed… General models

e.g. concave/potential /supermodular games

Specific multi‐user communication applications New classes of communication games

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When will it c o nverge to a NE ? And ho w fast ?

Our goals

u1 u2

Nash equilibrium Decentralized (limited) information Pareto boundary Global (exchanged) information

If information is constrained and no message passing is allowed… General models

e.g. concave/potential /supermodular games

Specific multi‐user communication applications New classes of communication games

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When will it c o nverge to a NE ? And ho w fast ? H

• w to impro ve an ineffic ient

NE witho ut message passing ?

Our goals

u1 u2

Nash equilibrium Decentralized (limited) information Pareto boundary Global (exchanged) information

If information is constrained and no message passing is allowed… General models

e.g. concave/potential /supermodular games

Specific multi‐user communication applications New classes of communication games

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When will it c o nverge to a NE ? And ho w fast ? H

• w to impro ve an ineffic ient

NE witho ut message passing ? And c an we still ac hieve Pareto o ptimality ?

Our goals

u1 u2

Nash equilibrium Decentralized (limited) information Pareto boundary Global (exchanged) information

If information is constrained and no message passing is allowed… General models

e.g. concave/potential /supermodular games

Specific multi‐user communication applications New classes of communication games

23

• Motivation and existing approaches
• Informationally efficient multi‐user

communication

– Vector cases

• Convergence conditions with decentralized information
• Improve efficiency with decentralized information

– Scalar cases

• Achieve Pareto efficiency with decentralized information
• Conclusions

Outline

24

A reformulation of multi‐user interactions

• Consider a tuple

– The set of players: – The set of actions: – State space: – State determination function:

and

– Utility function:

and

In standard strategic game, It captures the structure of the coupling between action and state

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A reformulation of multi‐user interactions

• Consider a tuple

– The set of players: – The set of actions: – State space: – State determination function:

and

– Utility function:

and

In standard strategic game, Many communication networking applications have simple , which captures the aggregate effects of It captures the structure of the coupling between action and state

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• Power control

Communication games with simple states

aggregate interference

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Communication games with simple states

• Power control
• Flow control

remaining capacity aggregate interference

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Communication games with simple states

• Power control
• Flow control
• Random access

remaining capacity aggregate interference idle probability

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• Motivation and existing approaches
• Informationally efficient multi‐user

communication

– Vector cases

• Convergence conditions with decentralized information
• Improve efficiency with decentralized information

– Scalar cases

• Achieve Pareto efficiency with decentralized information
• Conclusions

Outline

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• Definition

– A multi‐user interaction

in which A1: action set is defined to be

Additively Coupled Sum Constrained Games

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• Definition

– A multi‐user interaction

in which A1: action set is defined to be

Additively Coupled Sum Constrained Games

Structure of the action set: resource is constrained

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• Definition

– A multi‐user interaction

in which A2: The utility function satisfies in which is an increasing and strictly concave function. Both and are twice differentiable.

Additively Coupled Sum Constrained Games

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• Definition

– A multi‐user interaction

in which A2: The utility function satisfies in which is an increasing and strictly concave function. Both and are twice differentiable.

Additively Coupled Sum Constrained Games

states cost

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• Definition

– A multi‐user interaction

in which A2: The utility function satisfies in which is an increasing and strictly concave function. Both and are twice differentiable.

Additively Coupled Sum Constrained Games

states Structure of the utility: additive coupling between action and state cost

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• Definition

– A multi‐user interaction

in which A2: The utility function satisfies in which is an increasing and strictly concave function. Both and are twice differentiable.

Additively Coupled Sum Constrained Games

diminishing return per invested action states Structure of the utility: additive coupling between action and state cost

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• Power control in interference channels

Examples of ACSCG

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• Power control in interference channels

Examples of ACSCG

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• Power control in interference channels

Examples of ACSCG

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• Delay minimization in Jackson networks

Examples of ACSCG (cont’d)

i j m

k i

r

k im

r

k ij

r

k i

ψ

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• Delay minimization in Jackson networks

Examples of ACSCG (cont’d)

i j m

k i

r

k im

r

k ij

r

k i

ψ

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• Delay minimization in Jackson networks

Examples of ACSCG (cont’d)

i j m

k i

r

k im

r

k ij

r

k i

ψ

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Nash equilibrium in ACSCG

• Existence of pure NE

– A subclass of concave games

• When is the NE unique? When does best

response converges to such a NE?

– Existing literatures are not immediately applicable

• Diagonal strict convexity condition [Rosen]
• Use gradient play and stepsizes need to be carefully chosen
• Super‐modular games [Topkis]
• Action space is not a lattice
• Sufficient conditions for specific and [Yu]

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• Best response iteration

Best response dynamics

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• Best response iteration

in which is chosen such that

• When does it converges?

– By intuition, the weaker the mutual coupling is, the

more likely it converges

– How to measure and quantify this coupling

strength?

Best response dynamics

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• Best response iteration

in which is chosen such that

• When does it converges?

– By intuition, the weaker the mutual coupling is, the

more likely it converges

– How to measure and quantify this coupling

strength?

Best response dynamics

sum constraint additive coupling state

A competition scenario in which every user aggressively uses up all his resources

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• Best response iteration

in which is chosen such that

• When does it converges?

– By intuition, the weaker the mutual coupling is, the

more likely it converges

– How to measure and quantify this coupling

strength?

Best response dynamics

sum constraint additive coupling state

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Define represents the maximum impact that user m’s action can make over user n’s state

A measure of the mutual coupling

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Convergence conditions

Theorem 1: If then best response dynamics converges linearly to a unique pure NE for any set of initial conditions.

• The contraction factor is a measure of the
• verall coupling strength
• If is affine, the condition in Theorem 1 is not

impacted by ; otherwise it may depend on .

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Convergence conditions

Contraction mapping

Theorem 1: If then best response dynamics converges linearly to a unique pure NE for any set of initial conditions.

• The contraction factor is a measure of the
• verall coupling strength
• If is affine, the condition in Theorem 1 is not

impacted by ; otherwise it may depend on .

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Theorem 1: If then best response dynamics converges linearly to a unique pure NE for any set of initial conditions.

• The contraction factor is a measure of the
• verall coupling strength
• If is affine, the condition in Theorem 1 is not

impacted by ; otherwise it may depend on .

Convergence conditions

Contraction mapping is a constant for affine

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• If have the same sign,

the condition in Theorem 1 can be relaxed to

• This is true in many communication scenarios

– Increasing power causes stronger interference – Increasing input rate congests the server

Convergence conditions

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• If have the same sign,

the condition in Theorem 1 can be relaxed to

• This is true in many communication scenarios

– Increasing power causes stronger interference – Increasing input rate congests the server

Convergence conditions

Strategic complements (or strategic substitutes)

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For , define [Walrand]

A special class of

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For , define [Walrand] Define

A special class of

A measure of the similarity between users’ parameters

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Convergence conditions

Theorem 2: If then best response dynamics converges linearly to a unique pure NE for any set of initial conditions.

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Theorem 1 Theorem 1 Theorem 2 Theorem 2

Convergence conditions

Theorem 2: If then best response dynamics converges linearly to a unique pure NE for any set of initial conditions.

Contraction mapping

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When will it c o nverge to a NE ? And ho w fast ?

Conclusion so far…

u1 u2

Nash equilibrium Pareto boundary If Information is constrained and no message passing is available…

Concave games

Power control, Flow control ACSCG

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When will it c o nverge to a NE ? And ho w fast ?

Conclusion so far…

u1 u2

Nash equilibrium Pareto boundary If Information is constrained and no message passing is available…

Suffic ient c o nditio ns that guarantee linear c o nvergenc e

Concave games

Power control, Flow control ACSCG

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• Power control in interference channels

Power control as an ACSCG

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Performance comparison

• Solutions without information exchange

– Iterative water‐filling algorithm [Yu]

• Solutions with information exchange

k n

P

k

k n

σ

k k mn m m n

H P

≠

∑

user n’s spectrum

max

k k k

R ω

∑

k

k n

σ

k k mn m m n

H P

≠

∑

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Performance comparison

• Solutions without information exchange

– Iterative water‐filling algorithm [Yu]

• Solutions with information exchange

k n

P

k

k n

σ

k k mn m m n

H P

≠

∑

user n’s spectrum

max

k k k

R ω

∑

k

k n

σ

k k mn m m n

H P

≠

∑

OSB = Optimal Spectrum Balancing ASB = Autonomous Spectrum Balancing

62

• Motivation and existing approaches
• Informationally efficient multi‐user

communication

– Vector cases

• Convergence conditions with decentralized information
• Improve efficiency with decentralized information

– Scalar cases

• Achieve Pareto efficiency with decentralized information
• Conclusions

Outline

63

How to model the mutual coupling

• A reformulation of the coupling

– State space – Utility function – State determination function – Belief function – Conjectural Equilibrium (CE) : a configuration of

belief functions and joint action satisfying and

n n ∈

=×

N

S S

:

n n n

u × → S A R

:

n n n

s

−

→ A S

:

n n n

s →

• A

S

1

( , , )

N

s s

∗ ∗

• 1

( , , )

N

a a a

∗ ∗ ∗

=

• (

)

( )

n n n n

s a s

∗ ∗ ∗ −

= a

• (

)

( )

arg max ,

n n

n n n n n a

a u s a a

∗ ∗ ∈

=

• A

64

How to model the mutual coupling

• A reformulation of the coupling

– State space – Utility function – State determination function – Belief function – Conjectural Equilibrium (CE) : a configuration of

belief functions and joint action satisfying and

n n ∈

=×

N

S S

:

n n n

u × → S A R

:

n n n

s

−

→ A S

:

n n n

s →

• A

S

1

( , , )

N

s s

∗ ∗

• 1

( , , )

N

a a a

∗ ∗ ∗

=

• (

)

( )

n n n n

s a s

∗ ∗ ∗ −

= a

• (

)

( )

arg max ,

n n

n n n n n a

a u s a a

∗ ∗ ∈

=

• A

it captures the aggregate effect of the other users’ actions it models the aggregate effect

• f the other users’ actions

65

How to model the mutual coupling

• A reformulation of the coupling

– State space – Utility function – State determination function – Belief function – Conjectural Equilibrium (CE) : a configuration of

belief functions and joint action satisfying and

n n ∈

=×

N

S S

:

n n n

u × → S A R

:

n n n

s

−

→ A S

:

n n n

s →

• A

S

1

( , , )

N

s s

∗ ∗

• 1

( , , )

N

a a a

∗ ∗ ∗

=

• (

)

( )

n n n n

s a s

∗ ∗ ∗ −

= a

• (

)

( )

arg max ,

n n

n n n n n a

a u s a a

∗ ∗ ∈

=

• A

beliefs are realized each user behaves optimally according to its expectation it captures the aggregate effect of the other users’ actions it models the aggregate effect

• f the other users’ actions

66

CE in power control games [SuTSP’09]

• One leader and multiple followers
• State space

–

: the interference caused to user n in channel k

• Utility function
• State determination function
• Belief function (linear form)

k n

I

2 1

log 1

k K n n k k n n k

P R I σ

=

⎛ ⎞ ⎟ ⎜ ⎟ = + ⎜ ⎟ ⎜ ⎟ ⎜ + ⎝ ⎠

∑

1, N k k k n in i i i n

I P α

= ≠

= ∑

1 1 k k k k

I P β γ = −

• actual play

conceived play

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Why Linear belief?

is piece‐wise linear; , if the number of frequency bins is sufficiently large.

Linear belief is sufficient to capture the interference coupling!

1 1

0,

k j

I j k P ∂ = ≠ ∂

1 1 k k

I P ∂ ∂

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Why Linear belief?

is piece‐wise linear; , if the number of frequency bins is sufficiently large.

Linear belief is sufficient to capture the interference coupling!

1 1

0,

k j

I j k P ∂ = ≠ ∂

1 1 k k

I P ∂ ∂

2 f

P

f

2 f

σ

12 1 f f

H P

69

Why Linear belief?

is piece‐wise linear; , if the number of frequency bins is sufficiently large.

Linear belief is sufficient to capture the interference coupling!

1 1

0,

k j

I j k P ∂ = ≠ ∂

1 1 k k

I P ∂ ∂

2 f

P

f

2 f

σ

12 1 f f

H P

2 f

P

f

2 f

σ

12 1 f f

H P

70

Why Linear belief?

is piece‐wise linear; , if the number of frequency bins is sufficiently large.

Linear belief is sufficient to capture the interference coupling!

1 1

0,

k j

I j k P ∂ = ≠ ∂

1 1 k k

I P ∂ ∂

2 f

P

f

2 f

σ

12 1 f f

H P

2 f

P

f

2 f

σ

12 1 f f

H P

2 f

P

f

2 f

σ

12 1 f f

H P

71

Why Linear belief?

is piece‐wise linear; , if the number of frequency bins is sufficiently large.

Linear belief is sufficient to capture the interference coupling!

1 1

0,

k j

I j k P ∂ = ≠ ∂

1 1 k k

I P ∂ ∂

2 f

P

f

2 f

σ

12 1 f f

H P

2 f

P

f

2 f

σ

12 1 f f

H P

2 f

P

f

2 f

σ

12 1 f f

H P

72

Main results

• Stackelberg equilibrium

– Strategy profile that satisfies

• NE and SE are special CE

NE: SE:

• Infinite set of CE

Open sets of CE that contain NE and SE may exist

1 2

,

N k k k k i i i

P β α γ

=

= =

∑

γ

β

1

R

• 1

NE

R

1 SE

R

• 1

1 1 1 1 1

, .

k k k k k k k k

I I I P P P β γ ∂ ∂ = − ⋅ = − ∂ ∂

( )

( )

* * 1 1

, a NE a

( )

( )

( )

( )

* * 1 1 1 1 1 1 1 1

, , , u a NE a u a NE a a ≥ ∀ ∈ A

73

Achieving the desired CE

• Conjecture‐based rate maximization (CRM)

solvable using dual method

leader followers

74

Discussion about CRM

• Essence of CRM

– local approximation of the computation of SE

• Advantages

– the structure of the utility function is explored – only local information is required – it can be applied in the cases where N>2 – if it converges, the outcome is a CE

75

Simulation results

Average rate improvements: 2‐user case: 24.4% for user 1; 33.6% for user 2 3‐user case: 26.3% for user 1; 9.7% for user 2&3

( )

2

0.5,

k ij k

i j α = ≠

∑

0.5 1 1.5 2 2.5 3 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 R1/R1

NE

R2/R2

NE

0.8 1 1.2 1.4 1.6 1.8 2 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 R1/R1

NE

R2/R2

NE

R3/R3

NE

( )

2

0.33,

k ij k

i j α = ≠

∑

76

Concave games

ACSCG

Conclusions so far…

u1 u2

Pareto boundary Nash equilibrium

H

• w to impro ve an ineffic ient

NE witho ut message passing ?

If Information is constrained and no message passing is allowed

Power control

77

Concave games

ACSCG

Overall effic ienc y may be impro ve d!

Conclusions so far…

u1 u2

Pareto boundary Nash equilibrium Build belief, learn, and adapt

H

• w to impro ve an ineffic ient

NE witho ut message passing ?

If Information is constrained and no message passing is allowed

Power control

78

• Motivation and existing approaches
• Informationally efficient multi‐user

communication

– Vector cases

• Convergence conditions with decentralized information
• Improve efficiency with decentralized information

– Scalar cases

• Achieve Pareto efficiency with decentralized information
• Conclusions

Outline

79

Linearly coupled games

• A non‐cooperative game model
• Users’ states are linearly impacted by their

competitor’s actions

• Contributions

– Characterize the structures of the utility functions – Explicitly compute Nash equilibrium and Pareto

boundary

– A conjectural equilibrium approach to achieve

Pareto boundary without real‐time information exchange

80

A multi‐user interaction is considered a linearly coupled game if the action set is convex and the utility function satisfies in which . In particular, the basic assumptions about include: A1: is non‐negative; A2: is strictly linearly decreasing in ; is non‐increasing and linear in .

Definition

States are linearly impacted by actions

81

Denote . A3: is an affine function, A4:

Definition (cont’d)

Actions are linearly coupled at NE and PB

82

• For the games satisfying A1‐A4, the utility

functions can take two types of form:

– Type I [SuJSAC’10]

• e.g. random access

– Type II [SuTR’09]

• e.g. rate control

Two basic types

83

• For the games satisfying A1‐A4, the utility

functions can take two types of form:

– Type I [SuJSAC’10]

• e.g. random access

– Type II [SuTR’09]

• e.g. rate control

Two basic types

84

• Player set:

– nodes in a single cell

• Action set:

– transmission probability

• Payoff:

– throughput

• Key issues

– stability, convergence, throughput, and fairness

Type I games: wireless random access

Tx1 Rx1 Tx2 TxK Rx2 RxK

85

• Individual conjectures

– state: – linear belief:

• Two update mechanisms

– Best response – Gradient play

Conjecture‐based Random Access

actual play conceived play

86

Main results

• Existence of CE

– all operating points in action space are CE

• Stability and convergence

– sufficient conditions

• Throughput performance

– the entire throughput region can

be achieved with stable CE

• Fairness issue

– conjecture‐based approaches

attain weighted fairness

Protocol design: how to achieve efficient outcomes?

87

How to select suitable ak?

• Adaptively alter ak when the network size

changes

• Adopt aggregated throughput or “idle

interval” as the indicator of the system efficiency

• Advantages

– No need of a centralized solver – Throughput efficient with fairness guarantee – Stable equilibrium – Autonomously adapt to traffic fluctuation

88

Engineering interpretation

• DCF vs. the best response update

– re‐design the random access protocol

89

Engineering interpretation

• DCF vs. the best response update

– re‐design the random access protocol

similar different

90

Engineering interpretation

• DCF vs. the best response update

– re‐design the random access protocol

similar different CBRA makes use

• f 4-bit information,

while DCF only uses 2 bits

91

Simulation results

• Throughput
• Stability and convergence

5 10 15 20 25 30 35 40 45 50 25 26 27 28 29 30 31 32 33 34 35 36 Number of nodes Accumulative throughput (Mbps) Optimal throughput P-MAC Conjecture-based algorithms IEEE 802.11 DCF 100 200 300 400 500 600 31 31.5 32 32.5 33 33.5 34 34.5 35 35.5 36 Accumulative throughput (Mbps) P-MAC Best response Gradient play

DCF: low throughput; P‐MAC: needs to know the number of nodes P‐MAC: instability due to the online estimation

92

• Utility function
• Nash equilibrium
• Pareto boundary
• Efficiency loss

Conventional solutions in Type II games

93

• At stage t,
• Theorem 5: A necessary and sufficient

condition for the best response dynamics to converge is

Best response dynamics in Type II games

Determine the eigenvalues of the Jacobian matrix Observed state Linear belief

94

• Theorem 6: All the operating points on the

Pareto boundary are globally convergent CE under the best response dynamics. The belief configurations lead to Pareto‐optimal

• perating points if and only if

–

: the ratio between the immediate performance degradation and the conjectured long‐term effect

Stability of the Pareto boundary

Theorem 5 and expressions of Pareto boundary and CE

95

Pricing vs. conjectural equilibrium

• Pricing mechanism in communication networks

[Kelly][Chiang]

– Users repeatedly exchange coordination signals

• Conjectural equilibrium for linearly coupled games

– Coordination is implicitly implemented when the

participating users initialize their belief parameters

– Pareto‐optimality can be achieved solely based on local

• bservations on the states

– No message passing is needed during the convergence

process

– The key problem is how to design belief functions

96

Conclusions so far…

u1 u2

Pareto boundary Global (exchanged) information Nash equilibrium Decentralized (limited) information Decentralized (insufficient) information

The optimal way of designing the beliefs and updating the actions based on conjectural equilibrium is addressed

Can we still ac hieve Pareto o ptimality ?

Concave games

LCG Random Access, Rate control

If Information is constrained and no message passing is available…

97

Conclusions so far…

u1 u2

Pareto boundary Global (exchanged) information Nash equilibrium Decentralized (limited) information Conjectural equilibrium Decentralized (insufficient) information

The optimal way of designing the beliefs and updating the actions based on conjectural equilibrium is addressed

Can we still ac hieve Pareto o ptimality ?

Concave games

LCG Random Access, Rate control

Pareto o ptimality c an be ac hieved!

If Information is constrained and no message passing is available…

98

Conclusions

• We define new classes of games emerging in

multi‐user communication networks and investigate the information and efficiency trade‐off

– Provide sufficient convergence conditions to NE – Suggest a conjectural equilibrium based approach

to improve efficiency

– Quantify the performance improvement

99

References

• J. Rosen, “Existence and uniqueness of equilibrium points for

concave n‐person games,” Econometrica, vol. 33, no. 3, pp. 520‐534, Jul. 1965.

• D. Monderer and L. S. Shapley, “Potential games,” Games
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University Press, Princeton, 1998.

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communication networks: shadow prices, proportional fairness and stability,” Journal of the Operational Research Society, vol. 49, pp. 237‐252, 1998.

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“Layering as optimization decomposition: A mathematical theory of network architectures,” Proc. of the IEEE, vol. 95,

• no. 1, pp. 255‐312, January 2007.

100

References (cont’d)

• W. Yu, G. Ginis, and J. Cioffi, “Distributed multiuser power

control for digital subscriber lines,” IEEE J. Sel. Areas Commun., vol. 20, no. 5, pp. 1105‐1115, June 2002.

• J. Mo and J. Walrand, “Fair end‐to‐end window‐based

congestion control,” IEEE Trans. on Networking, vol. 8, no. 5,

• pp. 556‐567, Oct. 2000.

101

References (cont’d)

• Y. Su and M. van der Schaar, “Structural solutions for

additively coupled sum constrained games,” UCLA technical Report, 2010.

• Y. Su and M. van der Schaar, “Conjectural equilibrium in

multiuser power control games,” IEEE Trans. Signal Processing, vol. 57, no. 9, pp. 3638‐3650, Sep. 2009.

• Y. Su and M. van der Schaar, “A new perspective on multi‐

user power control games in interference channels,” IEEE

• Trans. Wireless Communications, vol. 8, no. 6, pp. 2910‐2919,

June 2009.

• Y. Su and M. van der Schaar, “Linearly coupled

communication games,” UCLA technical Report, 2009.

• Y. Su and M. van der Schaar, “Dynamic conjectures in random

access networks using bio‐inspired learning,” IEEE JSAC special issue on Bio‐Inspired Networking, May 2010.

102

Linear convergence

• A sequence

with limit is linearly convergent if there exists a constant such that for k sufficiently large.

103

Solutions with information exchange

• Users aim to solve
• They can pass coordination messages

and user n behaves according to

user n’s impact over user m’s utility

104

Solutions with information exchange

• Gradient play

Theorem 3: If gradient play converges for a small enough stepsize.

Lipschitz continuity and gradient projection algorithm

105

Solutions with information exchange

• Jacobi update

Theorem 4: If Jacobi update converges for a small enough stepsize.

Lipschitz continuity, descent lemma, and mean value theorem

106

Solutions with information exchange

• Convergence to an operating point that satisfies the

KKT conditions is guaranteed

• Total utility is monotonically increasing
• Global optimality is guaranteed if the original

problem is convex, otherwise not

• Developed for general non‐convex problem in which

convex NUM solutions may not apply in general

107

Stackelberg equilibrium

• Definition

– Leader (foresighted): only one – Follower (myopic):

the remaining ones

– Strategy profile that satisfies

• Existence and computation of SE in the

power control games [SuTWC’09]

( )

( )

* *

,

n n

a NE a

( )

( )

( )

( )

* *

, , ,

n n n n n n n n

u a NE a u a NE a a ≥ ∀ ∈ A

108

A two‐user formulation

• Bi‐level Programming

where

upper level problem ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ lower level problem ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩

1 2

1 1 2 2 1 1 1 1 2 2 2 1 1 1 2 2 1

max ln 1 ( ) . . , 0, ( ) arg max ln 1 ( ) . . , 0. ( )

k K k k k k K k k k k K k k k k K k k k

P a N P s t P P b P c N P s t P P d α α

= = ′ = =

⎛ ⎞ ⎟ ⎜ ⎟ + ⎜ ⎟ ⎜ ⎟ ⎜ + ⎝ ⎠ ≤ ≥ ⎛ ⎞ ′ ⎟ ⎜ ⎟ = + ⎜ ⎟ ⎜ ⎟ ⎜ + ⎝ ⎠ ′ ′ ≤ ≥

∑ ∑ ∑ ∑

max 1 max 2 P P

P P P

2 2 2 2 2 2 1 1 11 1 12 22 2 2 22 2 21 11

, , ,

k k k k k k k k k k k k

N H H H N H H H σ α σ α = = = =

109

Problems with the SE formulation

• Computational complexity

– intrinsically hard to compute

• Information required for playing SE

– Global information

• Realistic assumption

– Local information – Any appropriate solutions other than SE and NE?

{ } { } {

}

, ,

k k ij i

α σ

max i

P

1 1 2

,

N k k k n n n

P α σ

=

+

∑

max 1

P

110

• Priority‐based fair medium access control

– Traffic classes with positive weights

• Conjecture‐based protocol

Weighted Fairness

111

Some distributed iterative algorithms

• Best response
• Jacobi update
• Gradient play

stepsize