Game Theory and Mechanism Design Based Decentralized ...svandana/test_g000004.pdf · Introduction...
Transcript of Game Theory and Mechanism Design Based Decentralized ...svandana/test_g000004.pdf · Introduction...
Introduction Problem (P1) Problem (P2) Problem (P3)
Game Theory and Mechanism Design BasedDecentralized Algorithms for Power Allocation in Networks:
Cooperative and Non-cooperative Scenarios
Shruti Sharma
University of Michigan, Ann Arbor
Demos Teneketzis
University of Michigan, Ann Arbor
Asser Tantawi, Malgorzata Steinder, Michael Spreitzer
Service Management Middleware Group, IBM T. J. Watson Research Center
August 27, 2008, Yorktown, NY, USA
Introduction Problem (P1) Problem (P2) Problem (P3)
Outline
1 Introduction
2 Power allocation in the presence of interference: Cooperative networkModelOptimization problemDecentralized mechanism
3 Determination of CPU shares for clustered web servicesModelOptimization problemDecentralized algorithm
4 Power allocation in the presence of interference: Non-cooperative networkModelOptimization problemDecentralized mechanism
Introduction Problem (P1) Problem (P2) Problem (P3)
Resource allocation in computer and communication networks
Why resource management?Network services/applications are of numerous typesService users have different Quality of Service (QoS)requirementsService providers have limited resources
GoalGiven the available infrastructure,
satisfy maximum possible service demandbest meet the users’ QoS requirements
Achieving the goal: Two approachesCentralized optimizationDe-centralized optimization
Introduction Problem (P1) Problem (P2) Problem (P3)
Centralized vs. decentralized control
Difficulties with centralized controlRequires global information of the systemNot scalableOne failure can lead to the breakdown of entire systemCan not guarantee desired performance in the presence of selfishnetwork nodes with private information
Decentralized control is desirableEach controller would operate with little informationSystem size can be scaledSystem is less vulnerable to failure
Introduction Problem (P1) Problem (P2) Problem (P3)
Mechanism design: A Microeconomics approach
Analogy between networks in Economics and Engineering
Buisness,political, social
networks
Production goods,consumption goods,
private goods, public goods
Social welfare
Compuer,communication
networks
Technology / hardwareconstraints
Quality ofservice
Bandwidth, data rate, Power, CPU share, memory
Networks
Resources
Constraints
Objective
Technology constraints
Economics Engineering
Introduction Problem (P1) Problem (P2) Problem (P3)
Mechanism design framework
The goal of decentralized mechanism design
E A
M
π
µ h Outcomefunction
Messagecorrespondence
Goal
Message space
correspondneceEnvironment space Allocation space
Figure: Commuting diagram: The objectives of centralized and decentralized mechanisms
Realization:Users follow the rules as specified.
Decentralized mechanism consists of (M, µ, h).
Implementation:Users are selfish, desired behavior must be enforced indirectly.
Decentralized mechanism consists of (M, h).
Objective: h(µ(e)) ⊂ π(e) ∀e ∈ E
Introduction Problem (P1) Problem (P2) Problem (P3)
Three resource allocation problems
1) Transmission power allocation in a wireless network with interference where the usersare cooperative.
Obtained a decentralized algorithm based on externality formulation that obtainsglobally optimal power allocation.
3) CPU share determination for clustered web services on heterogenous nodes.
Obtained a decentralized algorithm that determines globally optimal CPU shares.
2) Transmission power allocation in a wireless cellular network with interference wherethe users are non-cooperative/selfish.
Obtained a power and tax determination mechanism based on public good formu-lation that obtains globally optimal power allocation at Nash equilibria.
Introduction Problem (P1) Problem (P2) Problem (P3)
Three resource allocation problems
1) Transmission power allocation in a wireless network with interference where the usersare cooperative.
Obtained a decentralized algorithm based on externality formulation that obtainsglobally optimal power allocation.
3) CPU share determination for clustered web services on heterogenous nodes.
Obtained a decentralized algorithm that determines globally optimal CPU shares.
2) Transmission power allocation in a wireless cellular network with interference wherethe users are non-cooperative/selfish.
Obtained a power and tax determination mechanism based on public good formu-lation that obtains globally optimal power allocation at Nash equilibria.
Introduction Problem (P1) Problem (P2) Problem (P3)
Three resource allocation problems
1) Transmission power allocation in a wireless network with interference where the usersare cooperative.
Obtained a decentralized algorithm based on externality formulation that obtainsglobally optimal power allocation.
3) CPU share determination for clustered web services on heterogenous nodes.
Obtained a decentralized algorithm that determines globally optimal CPU shares.
2) Transmission power allocation in a wireless cellular network with interference wherethe users are non-cooperative/selfish.
Obtained a power and tax determination mechanism based on public good formu-lation that obtains globally optimal power allocation at Nash equilibria.
Introduction Problem (P1) Problem (P2) Problem (P3) Model Optimization problem Decentralized mechanism
Power allocation in a cooperative network: Model (M1)
T1
R2
T2
T3
R3
R1
h21h11
h31
p1 ∈ P1 = [0,Pmax1 ]
p3 ∈ P3 = [0,Pmax3 ]
p2 ∈ P2 = [0,Pmax2 ]
N transmitter receiver pairs (Users), N := 1, 2, . . . , N
Transmissions of a user create interference to other usersInterference depends on the transmission powers
Performance determined by utilities: Ui (p) = Ui (p1, p2, . . . , pN), i ∈ N
Introduction Problem (P1) Problem (P2) Problem (P3) Model Optimization problem Decentralized mechanism
Model M1 (cont’)
T1
R2
T2
T3
R3
R1
h21h11
h31
p1 ∈ P1 = [0,Pmax1 ]
p3 ∈ P3 = [0,Pmax3 ]
p2 ∈ P2 = [0,Pmax2 ]
MC1h301
h101
h201
Interference Temperature (IT): Net radio frequency (RF) power measured at areceiving antenna per unit bandwidth
ITC: A measure to keep the RF noise floor below a safe threshold
NXi=1
pi hi01 ≤ P1
Multiple ITCs: C measurement centers (MCs)/ Users 0C := 01, 02, . . . , 0C
Introduction Problem (P1) Problem (P2) Problem (P3) Model Optimization problem Decentralized mechanism
Assumptions: Information available to the users
T1
R2
T2
T3
R3
R1
h21h11
h31
p1 ∈ P1 = [0,Pmax1 ]
p3 ∈ P3 = [0,Pmax3 ]
p2 ∈ P2 = [0,Pmax2 ]
MC1h301
h101
h201
User i ∈ N• Pi = [0,Pmax
i ]
• Utility Ui
User 0c, c ∈ C(MCc)
• Channel gainshj0c
, j ∈ N
Common knowledge
• P = [0,Pmax] ⊃ ∪i∈NPi
• # of active users NN remains constant
Introduction Problem (P1) Problem (P2) Problem (P3) Model Optimization problem Decentralized mechanism
The optimization problem
Problem (P1)
maxp
Xi∈N∪0C
Ui (p) = maxp
U(p)
subject to:
p ∈ S := p |NX
i=1
pi hi0c ≤ Pc , c ∈ C, pi ∈ Pi ∀ i ∈ N
∗ ∀ i ∈ N , Ui (p) : RN → R is a non-negative, strictly concave,
continuous function of p and Ui (p : pi = 0) = 0
∗ U0c (p) = 0, ∀ c ∈ C
Note: Problem (P1) has a unique optimum.
Objective
To develop a decentralized mechanism that obtains a solution to Problem (P1).
Introduction Problem (P1) Problem (P2) Problem (P3) Model Optimization problem Decentralized mechanism
The optimization problem
Problem (P1)
maxp
Xi∈N∪0C
Ui (p) = maxp
U(p)
subject to:
p ∈ S := p |NX
i=1
pi hi0c ≤ Pc , c ∈ C, pi ∈ Pi ∀ i ∈ N
∗ ∀ i ∈ N , Ui (p) : RN → R is a non-negative, strictly concave,
continuous function of p and Ui (p : pi = 0) = 0
∗ U0c (p) = 0, ∀ c ∈ C
Note: Problem (P1) has a unique optimum.
Objective
To develop a decentralized mechanism that obtains a solution to Problem (P1).
Introduction Problem (P1) Problem (P2) Problem (P3) Model Optimization problem Decentralized mechanism
Formulation as an externality problem
Introduction Problem (P1) Problem (P2) Problem (P3) Model Optimization problem Decentralized mechanism
Formulation as an externality problem
User feasible power profiles Si := p | pi ∈ Pi , p−i ∈ PN−1, i ∈ N
Introduction Problem (P1) Problem (P2) Problem (P3) Model Optimization problem Decentralized mechanism
Formulation as an externality problem
User feasible power profiles Si := p | pi ∈ Pi , p−i ∈ PN−1, i ∈ N
Introduction Problem (P1) Problem (P2) Problem (P3) Model Optimization problem Decentralized mechanism
Formulation as an externality problem
User feasible power profiles Si := p | pi ∈ Pi , p−i ∈ PN−1, i ∈ Nn – constraint-feasible power profiles
S0c := p |PN
i=1 pi hi0c ≤ Pc , pi ∈ P ∀ i ∈ N, c ∈ C
Feasible power profiles
S =T
i∈N∪0CSi = p |
PNi=1 pi hi0c ≤ Pc , c ∈ C, pi ∈ Pi ∀ i ∈ N
Introduction Problem (P1) Problem (P2) Problem (P3) Model Optimization problem Decentralized mechanism
Formulation as an externality problem
User feasible power profiles Si := p | pi ∈ Pi , p−i ∈ PN−1, i ∈ Nn – constraint-feasible power profiles
S0c := p |PN
i=1 pi hi0c ≤ Pc , pi ∈ P ∀ i ∈ N, c ∈ C
Feasible power profiles
S =T
i∈N∪0CSi = p |
PNi=1 pi hi0c ≤ Pc , c ∈ C, pi ∈ Pi ∀ i ∈ N
Introduction Problem (P1) Problem (P2) Problem (P3) Model Optimization problem Decentralized mechanism
A decentralized algorithm for transmission power allocation
0) Initialization:
Users (including users 0C) agree upon a common reference power profile
p(0) ∈ p | pi ∈ P ∀ i ∈ N (1)
Introduction Problem (P1) Problem (P2) Problem (P3) Model Optimization problem Decentralized mechanism
The decentralized algorithm (cont’)
0) Initialization (cont’):
A sequence of modification parameters τ (k)∞k=1 is chosen that satisfies,
0 < τ(k+1) ≤ τ
(k) ≤ 1, ∀ k ≥ 1 (2)
limk→∞
τ(k) = 0 (3)
limk→∞
σ(k) = lim
k→∞
kXt=1
τ(t) = ∞ (4)
The counter k is set to 0.
Example
τ (k) = 1kδ for δ ∈ (0, 1] satisfies (10) – (12).
Introduction Problem (P1) Problem (P2) Problem (P3) Model Optimization problem Decentralized mechanism
The decentralized algorithm (cont’)
1) k th iteration: (Individual optimization)User i , i = 1, 2, . . . , N, solves
p(k+1)i = argmaxp∈Si
Ui (p)−1
τ (k+1)‖p − p(k)‖2 (5)
MCi (user 0i ), i = 01, 02, . . . , 0C, solves
p(k+1)i = argmaxp∈S0c
0 −1
τ (k+1)‖p − p(k)‖2 (6)
Individual optimals p(k+1)i ∀ i are broadcast to all the users.
Introduction Problem (P1) Problem (P2) Problem (P3) Model Optimization problem Decentralized mechanism
The decentralized algorithm (cont’)
2) Calculation of user and time averages
Upon receiving p(k+1)i ∀ i , users compute for (k + 1)th iteration
p(k+1) =1
N + C
Xi∈N∪0C
p(k+1)i (7)
Introduction Problem (P1) Problem (P2) Problem (P3) Model Optimization problem Decentralized mechanism
The decentralized algorithm (cont’)
p(k+1) is used as a reference point in the (k + 1)th iteration.
Introduction Problem (P1) Problem (P2) Problem (P3) Model Optimization problem Decentralized mechanism
The decentralized algorithm (cont’)
2) (cont’)
User i, i ∈ N ∪ 0C , also computes the weighted averages
w (k+1)i =
1σ(k+1)
k+1Xt=1
τ (t)p(t)i , i ∈ N∪0C
=1
σ(k+1)
“σ(k)w (k)
i + τ (k+1)p(k+1)i
”, (8)
where σ(k+1) =k+1Xt=1
τ (t) = σ(k) + τ (k+1) (9)
The average calculated in (8) is stored in users’ memories.The counter k is increased to k + 1 and the process repeats from Step 1).
For (k + 1)th iteration, τ (k+2) ≤ τ (k+1) is chosen from the predefinedsequence in Step 0).
Introduction Problem (P1) Problem (P2) Problem (P3) Model Optimization problem Decentralized mechanism
Convergence to optimal solution
Theorem 1
The decentralized algorithm results in a power allocation which is the uniqueglobal optimum of Problem (P1).
The optimal power allocation is obtained as the limit of the sequences
w (k)i ∞k=1, i ∈ N ∪ 0C , all of which converge to the optimal allocation.
The above theorem has been proved using convex analysis.
Convergence to the optimum solution of Problem (P1) is guaranteed by thedecentralized algorithm.
Introduction Problem (P1) Problem (P2) Problem (P3) Model Optimization problem Decentralized algorithm
CPU share allocation: Model (M2)
Clusters Nodes
1,1ω
N,1ω
1,2ω
N,2ω
1,Dω
1
2
DN
1
Figure: CPU power allocation for clusters on heterogeneous nodes
ωd,n – CPU power for cluster d on node n, d ∈ 1, 2, . . . , D, n ∈ 1, 2, . . . , N.Each node has a CPU power capacity: Ωn, n ∈ 1, 2, . . . , N.Each cluster’s received QoS is represented by a utility: Ud (
PNn=1 ωd,n)
Introduction Problem (P1) Problem (P2) Problem (P3) Model Optimization problem Decentralized algorithm
The optimization problem
Problem (P2)
maxD∑
d=1
Ud
( N∑n=1
ωd,n
)
s.t.D∑
d=1
ωd,n ≤ Ωn, ∀ n ∈ 1, 2, . . . , N
0 ≤ ωd,n ≤ Ωd,n ∀ d ∈ 1, 2, . . . , D
The utility Ud (ω) is obtained from the queueing model analysis of each flowbelonging to the cluster.
The utility function thus constructed is concave in ω.
Introduction Problem (P1) Problem (P2) Problem (P3) Model Optimization problem Decentralized algorithm
Breaking up Problem (P2)
Nodes
Clusters 1
1
n
d
D
N
nd ,ω⎟⎠
⎞⎜⎝
⎛∑=
N
nnddU
1,ω
n
D
dnd Ω≤∑
=1,ω
cnω
rdω
Matrix W
Figure: Variables influencing the clusters and the nodes
W = [ωd,n]D×N , matrix of CPU power variables
R = [rd,n]D×N , row update matrix with rows r rd – updated by clusters
C = [cd,n]D×N , column update matrix with columns ccn – updated by nodes
Introduction Problem (P1) Problem (P2) Problem (P3) Model Optimization problem Decentralized algorithm
A decentralized algorithm for CPU power allocation
Step 0) Initialization: k = 0Sequence of penalty parameters τ (t)∞t=1 is chosen s.t.
0 < τ(t + 1) ≤ τ(t), ∀ t ≥ 1 (10)
limt→∞
τ(t) = 0 (11)
limk→∞
σ(k) = lim
k→∞
kXt=1
τ(t) = ∞ (12)
W (k)= [0]D×N
Step 1) Row and Column updates:
r rd (k + 1) = arg max
ωrd : 0≤ωr
d≤Ωrd
Ud
“ NXn=1
ωd,n
”−
1τ (k+1)
‖ωrd − ωr
d (k)‖2 (13)
ccn(k + 1) = arg max
ωcn : 0≤ωc
n ,PD
d=1 ωd,n≤Ωn
0−1
τ (k+1)‖ωc
n − ωcn(k)‖2 (14)
(15)
Matrix R is sent to the nodes and matrix C is sent to the clusters.
Introduction Problem (P1) Problem (P2) Problem (P3) Model Optimization problem Decentralized algorithm
An example with two clusters and two nodes
R C
(R+C)2
W =rr2cc2
rw1
cc1
cw2cw1
rw2
rr1
Figure: Row and column updates by the clusters and the nodes
Introduction Problem (P1) Problem (P2) Problem (P3) Model Optimization problem Decentralized algorithm
Progression of decentralized algorithm
R(k)
C(k)
Satisfy individual cluster constraints
Satisfy node capacity constraints
Satisfy all constraints
Number of iterations
wei
ght
Cluster utility Penalty
(b)
(a)
Figure:
(a) Sequences R(k) and C(k) of row and column update matrices
(b) The weight of cluster utility and penalty terms in cluster optimization
Introduction Problem (P1) Problem (P2) Problem (P3) Model Optimization problem Decentralized algorithm
Decentralized algorithm for CPU power allocation (cont’)
Step 2) After receiving the updates from the nodes (respectively the clusters), the clusters(respectively the nodes) calculate the following averages.
W (k + 1) =12[R(k + 1) + C(k + 1)] (16)
bR(k + 1) =1Pk+1
t=1 τ(t)
k+1Xt=1
τ(t)R(t) (17)
bC(k + 1) =1Pk+1
t=1 τ(t)
k+1Xt=1
τ(t)C(t) (18)
cW (k + 1) =1Pk+1
t=1 τ(t)
kXt=0
τ(t + 1)W (t) (19)
Introduction Problem (P1) Problem (P2) Problem (P3) Model Optimization problem Decentralized algorithm
Time averaging of row and column update matrices
Satisfy individual cluster constraints
Satisfy node capacity constraints
++
+ ++
++
++
+
)(kτ
)(ˆ kC
)3(τ
Optimal solution of Problem (P)
)2(τ
)(ˆ kR
)1(τ )1( +kτ
Time average sequences lie in the respective convex and compact sets,therefore obey the respective constraints.
Time average sequences converge to the same matrix
The matrix of convergence is a feasible solution of Problem (P2).
Introduction Problem (P1) Problem (P2) Problem (P3) Model Optimization problem Decentralized algorithm
Convergence to optimal solution
Theorem 2
The sequences bR(k)∞k=1, bC(k)∞k=1, and cW (k)∞k=1 all converge to the optimumsolution of Problem (P2).
The above theorem has been proved using convex analysis.
Convergence to the optimum solution of Problem (P2) is guaranteed by thedecentralized algorithm.
Introduction Problem (P1) Problem (P2) Problem (P3) Model Optimization problem Decentralized mechanism
Power allocation in a non-cooperative network: Model (M3)
BS
1
2
N
pNp1p2
h01h02
h0N
p1h01
p2h02
pNh0N
Pmax0
Figure: A downlink network with N mobiles and one base station
N mobile users, N := 1, 2, . . . , N, and one Base Station (BS)
Users experience interference due to the BS transmissions to other users,Quality of Service (QoS) received by user i depends on p := (p1, p2, . . . , pN)
Users are charged some tax t := (t1, t2, . . . , tN) by the BS for using the network;
NXi=1
ti = 0
Introduction Problem (P1) Problem (P2) Problem (P3) Model Optimization problem Decentralized mechanism
Model (M3) (cont’)
Utility function uAi : R1+N → R ∪ −∞ represents user i ’s satisfaction from
the tax payment ti and the QoS obtained from the BS transmisison p,
uAi (ti , p) := −ti + ui (p)−
"1− ISi
(p)
ISi(p)
#
where, Si := p | p ∈ [0, Pmax0 ]N
ISi(p) =
1, if p ∈ Si
0, otherwise
Assumptions:For each i ∈ N , ui (p) is strictly concave in p over Si .
Channel gain h0i and utility uAi is user i ’s private information.
Users are non-cooperative and selfish.
The number of users, their utilities and the channel gains from the BS to the usersremain fixed throughout a power allocation period.
N and Pmax0 are common knowledge.
Introduction Problem (P1) Problem (P2) Problem (P3) Model Optimization problem Decentralized mechanism
The optimization problem
Problem (P3)max(t,p)
Xi∈N
uAi (ti , p) (20)
s.t.Xi∈N
ti = 0 (21)
Problem (P3) is equivalent to Problem (P3.1)
Problem (P3.1)max
(t,p)∈ S
Xi∈N
ui (p) (22)
where, S := (t ,p) |Xi∈N
ti =0, t∈RN ; p∈ [0, Pmax0 ]N (23)
Note:
Problem (P3.1) has a unique optimal power profile p∗.
Optimal solution of Problem (P3.1) must be of the form (t , p∗), where t is anyfeasible tax profile satisfying (21).
Introduction Problem (P1) Problem (P2) Problem (P3) Model Optimization problem Decentralized mechanism
A decentralized mechanism for power and tax determination
Treating p = (p1, p2, . . . , pN) as a public good.
The message space:Each user i ∈ N sends a message mi ∈Mi := RN
+ × RN to the BS consisting ofthe power profile pi and price profile πi proposals.
mi := (πi , pi ); πi ∈ RN+, pi ∈ RN (24)
The outcome function:Based on the message profile m = (m1, m2, . . . , mN ), the BS sets the taxes andtransmission powers for the users,
p(m) =1N
NXi=1
pi , (25)
ti (m) = lTi (m)p(m) + (pi − pi+1)
T diag(πi )(pi − pi+1)
−(pi+1 − pi+2)T diag(πi+1)(pi+1 − pi+2), i ∈ N , (26)
where l i (m) = πi+1 − πi+2 (27)
In (26) and (27), i + 2 ≡ 1 for i = N − 1, and for i = N, i + 1 ≡ 1 and i + 2 ≡ 2.
Introduction Problem (P1) Problem (P2) Problem (P3) Model Optimization problem Decentralized mechanism
Optimality of the decentralized mechanism
Nash equilibrium:
uAi (t∗i (m∗), pi
∗(m∗)) ≥ uAi (t∗i ((mi , m∗/i)), pi
∗(mi , m∗/i))
∀ mi ∈ Mi := RN+ × RN , ∀ i ∈ N (28)
Theorem 3
The tax and power allocation (t(m∗), p(m∗)) at Nash equilibrium m∗ is,
(a) individually rational, i.e. all users weakly prefer (t(m∗), p(m∗)) to the initialallocation (0, 0), and
(b) an optimal solution of Problem (P3).
Theorem 4
Given the optimum power profile p∗ of Problem (P3), there exists at least one NE m∗of the game corresponding to the decentralized mechanism such that, p(m∗) = p∗.Furthermore, given p∗, the set of all NE that result in p∗ can be characterized.
Introduction Problem (P1) Problem (P2) Problem (P3) Model Optimization problem Decentralized mechanism
THANKS!
Introduction Problem (P1) Problem (P2) Problem (P3) Model Optimization problem Decentralized mechanism
Contact for further details:
email: [email protected]
Web: http ://www .umich.edu/∼svandana