Game Theory and Mechanism Design Based Decentralized ...svandana/test_g000004.pdf · Introduction Problem (P1) Problem (P2) Problem (P3) Game Theory and Mechanism Design Based Decentralized

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


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


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


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


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


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


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


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


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


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).



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).


Formulation as an externality problem



User feasible power profiles Si := p | pi ∈ Pi , p−i ∈ PN−1, i ∈ N



User feasible power profiles Si := p | pi ∈ Pi , p−i ∈ PN−1, i ∈ N



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



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


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)


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).



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.



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)



p(k+1) is used as a reference point in the (k + 1)th iteration.



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).


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)



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.


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


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.


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


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


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)


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).


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.


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


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.



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).


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.


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.


THANKS!


Contact for further details:

email: [email protected]

Web: http ://www .umich.edu/∼svandana

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