1
Informationally Efficient Multi‐user communication
Yi Su
Advisor: Professor Mihaela van der Schaar
Electrical Engineering, UCLA
2
• 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
3
Multi‐user communication networks
Distributed routing
Power control
Peer‐to‐peer system
etc…
4
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
6
Constraints in communication networks
• Resources– Bandwidth, power,spectrum, etc.
• Information– Real‐time
• Local observation• Exchanged message
– Non‐real‐time• A‐priori information aboutinter‐user coupling, protocols, etc.
7
Constraints in communication networks
• Resources– Bandwidth, power,spectrum, etc.
• Information– Real‐time
• Local observation• Exchanged message
– Non‐real‐time• A‐priori information aboutinter‐user coupling, protocols, etc.
Goal: multi‐user communication without information exchange
8
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
9
Existing approaches
• Local observation
Nash equilibrium
10
Existing approaches
• Local observation
Nash equilibrium
• Exchanged messages
Pareto optimality
Price!
11
Existing approaches
• Local observation
Nash equilibrium
• Exchanged messages
Pareto optimality
Existing results usually assume some
specific action and utility structures!
Price!
12
• Results with specific action and utility structures– Pure Nash equilibrium
• Concave gamesi) : 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
13
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 gamesi) : 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
14
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 gamesi) : 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
15
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 gamesi) : 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
16
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
Special
utility
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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
18
Existing approaches (cont’d)
u1
u2 Pareto boundaryGlobal information
Nash equilibriumDecentralized (limited) information
General modelse.g. concave/potential/supermodular games
Specific multi‐user communication applications
But in many communication systems, information is constrained and no message passing is allowed!
19
Our goals
u1
u2
Nash equilibriumDecentralized (limited) information
Pareto boundaryGlobal (exchanged) information
If information is constrained and
no message passing is allowed…
General modelse.g. concave/potential/supermodular games
Specific multi‐user communication applications
New classes of communication
games
20
When will it convergeto a NE ? And how fast ?
Our goals
u1
u2
Nash equilibriumDecentralized (limited) information
Pareto boundaryGlobal (exchanged) information
If information is constrained and
no message passing is allowed…
General modelse.g. concave/potential/supermodular games
Specific multi‐user communication applications
New classes of communication
games
21
When will it convergeto a NE ? And how fast ?
How to improve an inefficient NE without message passing ?
Our goals
u1
u2
Nash equilibriumDecentralized (limited) information
Pareto boundaryGlobal (exchanged) information
If information is constrained and
no message passing is allowed…
General modelse.g. concave/potential/supermodular games
Specific multi‐user communication applications
New classes of communication
games
22
When will it convergeto a NE ? And how fast ?
How to improve an inefficient NE without message passing ?
And can we still achieve Pareto optimality ?
Our goals
u1
u2
Nash equilibriumDecentralized (limited) information
Pareto boundaryGlobal (exchanged) information
If information is constrained and
no message passing is allowed…
General modelse.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
25
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 controlremaining 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
30
• 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
0kir
kimr
kijrk
iψ
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• Delay minimization in Jackson networks
Examples of ACSCG (cont’d)
i
j
m
0kir
kimr
kijrk
iψ
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• Delay minimization in Jackson networks
Examples of ACSCG (cont’d)
i
j
m
0kir
kimr
kijrk
iψ
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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]
43
• 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 dynamicssum constraint
additive coupling
state
A competition scenario in which every useraggressively 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 dynamicssum 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 overall 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 overall 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 overall 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 convergeto a NE ? And how 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
58
When will it convergeto a NE ? And how fast ?
Conclusion so far…
u1
u2
Nash equilibrium
Pareto boundary
If Information is constrained and
no message passing is available…
Sufficient conditions that guarantee linear convergence
Concave games
Power control,Flow control
ACSCG
59
• 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
knP
k
knσ
k kmn m
m nH P
≠∑
user n’s spectrum
max k kkRω∑
k
knσ
k kmn m
m nH P
≠∑
61
Performance comparison • Solutions without information exchange
– Iterative water‐filling algorithm [Yu]
• Solutions with information exchange
knP
k
knσ
k kmn m
m nH P
≠∑
user n’s spectrum
max k kkRω∑
k
knσ
k kmn m
m nH 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∈=× NS S:n n nu × →S A R
:n n ns − →A S:n n ns →A S
1( , , )Ns s∗ ∗1( , , )Na a a∗ ∗ ∗=
( ) ( )n n n ns a s∗ ∗ ∗−= a ( )( )arg max ,
n nn n n n n
aa 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∈=× NS S:n n nu × →S A R
:n n ns − →A S:n n ns →A S
1( , , )Ns s∗ ∗1( , , )Na a a∗ ∗ ∗=
( ) ( )n n n ns a s∗ ∗ ∗−= a ( )( )arg max ,
n nn n n n n
aa u s a a∗ ∗
∈=
A
it captures the
aggregate effect of
the other users’ actions
it models the aggregate effect
of 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∈=× NS S:n n nu × →S A R
:n n ns − →A S:n n ns →A S
1( , , )Ns s∗ ∗1( , , )Na a a∗ ∗ ∗=
( ) ( )n n n ns a s∗ ∗ ∗−= a ( )( )arg max ,
n nn n n n n
aa 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
of 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)
knI
21log 1
kKn
n k kn nk
PR
Iσ=
⎛ ⎞⎟⎜ ⎟= +⎜ ⎟⎜ ⎟⎜ +⎝ ⎠∑
1,Nk k k
n in ii i nI Pα
= ≠=∑
1 1k k k kI Pβ γ= −
actual play
conceived play
67
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
10,
k
jI
j kP
∂ = ≠∂
1
1
k
kIP
∂∂
68
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
10,
k
jI
j kP
∂ = ≠∂
1
1
k
kIP
∂∂
2fP
f2fσ
12 1f fH 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
10,
k
jI
j kP
∂ = ≠∂
1
1
k
kIP
∂∂
2fP
f2fσ
12 1f fH P
2fP
f2fσ
12 1f fH 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
10,
k
jI
j kP
∂ = ≠∂
1
1
k
kIP
∂∂
2fP
f2fσ
12 1f fH P
2fP
f2fσ
12 1f fH P
2fP
f2fσ
12 1f fH 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
10,
k
jI
j kP
∂ = ≠∂
1
1
k
kIP
∂∂
2fP
f2fσ
12 1f fH P
2fP
f2fσ
12 1f fH P
2fP
f2fσ
12 1f fH P
72
Main results• Stackelberg equilibrium
– Strategy profile that satisfies
• NE and SE are special CENE:
SE:
• Infinite set of CEOpen sets of CE that contain
NE and SE may exist
12
, 0N
k k k ki i
iPβ α γ
== =∑
γ
β
1R
•
•
1NER
1SER•
•
1 11 1
1 1, .
k kk k k k
k kI I
I PP 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
( )20.5,k
ijki jα = ≠∑
0.5 1 1.5 2 2.5 30
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
R1/R1NE
R2/R2NE
0.8 1 1.2 1.4 1.6 1.8 20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
R1/R1NE
R2/R2NE
R3/R3NE
( )20.33,k
ijki jα = ≠∑
76
Concave games
ACSCG
Conclusions so far…
u1
u2Pareto boundary
Nash equilibrium
How to improve an inefficient NE without message passing ?
If Information is constrained and
no message passing is allowed
Power control
77
Concave games
ACSCG
Overall efficiency may be improved!
Conclusions so far…
u1
u2Pareto boundary
Nash equilibrium
Build belief, learn, and adapt
How to improve an inefficient NE without 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
Rx1Tx2
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 of 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 5025
26
27
28
29
30
31
32
33
34
35
36
Number of nodes
Acc
umul
ativ
e th
roug
hput
(Mbp
s) Optimal throughputP-MACConjecture-based algorithmsIEEE 802.11 DCF
0 100 200 300 400 500 60031
31.5
32
32.5
33
33.5
34
34.5
35
35.5
36
Acc
umul
ativ
e th
roug
hput
(Mbp
s)
P-MACBest responseGradient 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 operating 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 observations 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 boundaryGlobal (exchanged) information
Nash equilibriumDecentralized (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 achieve Pareto optimality ?
Concave games
LCG
Random Access,Rate control
If Information is constrained and
no message passing is available…
97
Conclusions so far…
u1
u2 Pareto boundaryGlobal (exchanged) information
Nash equilibriumDecentralized (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 achieve Pareto optimality ?
Concave games
LCG
Random Access,Rate control
Pareto optimality can be achieved!
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
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• D. Topkis, Supermodularity and Complementarity. Princeton University Press, Princeton, 1998.
• F. Kelly, A. K. Maulloo, and D. K. H. Tan, “Rate control in communication networks: shadow prices, proportional fairness and stability,” Journal of the Operational Research Society, vol. 49, pp. 237‐252, 1998.
• M. Chiang, S. H. Low, A. R. Calderbank, and J. C. Doyle, “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.
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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.
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Linear convergence
• A sequence with limit is linearly convergent if there exists a constant such that
for k sufficiently large.
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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 na NE a
( )( ) ( )( )* *, , ,n n n n n n n nu 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 21
1 11
22
2 1 11
2 21
max ln 1 ( )
. . , 0, ( )
argmax ln 1 ( )
. . , 0. ( )
kK
k k kkK k kk
kK
k k kk
K k kk
Pa
N P
s t P P b
Pc
N P
s t P P d
α
α
=
=
′ =
=
⎛ ⎞⎟⎜ ⎟+⎜ ⎟⎜ ⎟⎜ +⎝ ⎠
≤ ≥
⎛ ⎞′ ⎟⎜ ⎟= +⎜ ⎟⎜ ⎟⎜ +⎝ ⎠
′ ′≤ ≥
∑
∑
∑
∑
max1
max2
P
PP
P
P
2 2 2 2 2 21 1 11 1 12 22 2 2 22 2 21 11, , ,k k k k k k k k k k k kN H H H N H H Hσ 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 kij iα σ max
iP
1 12,N k k k
n nnPα σ
=+∑ max
1P
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