A Game-Theoretic Approach to Decentralized Optimal Power Allocation - - PowerPoint PPT Presentation

Introduction Model Decentralized mechanism Conclusion

A Game-Theoretic Approach to Decentralized Optimal Power Allocation for Cellular Networks

Shruti Sharma

Ph.D. candidate, Electrical Engineering and Computer Science

and

Demos Teneketzis

Electrical Engineering and Computer Science

University of Michigan, Ann Arbor GameComm 2008, October 20, Athens, Greece

Shrutivandana Sharma University of Michigan, Ann Arbor 1 / 34

Introduction Model Decentralized mechanism Conclusion

Outline

1

Introduction

2

Cellular network model Power allocation problem

3

Decentralized mechanism Solution approach: Implementation theory framework A decentralized mechanism for power allocation Results

4

Conclusion

Shrutivandana Sharma University of Michigan, Ann Arbor 2 / 34

Introduction Model Decentralized mechanism Conclusion

Overview

Set-up

Power allocation in cellular uplink and downlink networks Decentralized and asymmetric information Competitive/selfish/strategic users with no prior beliefs on other users’ information or strategies

Our work

Design of a decentralized power allocation mechanism that, preserves private information of the users makes the users willingly participate in the mechanism is budget balanced

• btains optimal centralized allocations at all Nash equilibria

Shrutivandana Sharma University of Michigan, Ann Arbor 3 / 34

Introduction Model Decentralized mechanism Conclusion

Literature survey

Uplink power control

User utility formulation: Ji, Huang (98); Famolari, Mandayam, Shah (99). Pricing (single cell): Alpcan, Basar, Srikant, Altman (02); Saraydar, Mandayam, Goodman (02). Pricing (Multi-cell networks): Saraydar, Mandayam, Goodman (01). Pricing (Interfernce Temperature Constraint): Huang, Berry, Honig. Equilibrium analysis: Do not achieve globally optimum allocation

Downlink power control

Common knowledge utilities: Liu, Honig, Jordan (00); Zhou, Honig, Jordan (01). Partial cooperation between base station and mobiles: Lee, Mazumdar, Shroff. Common knowledge/cooperation assumed to obtain optimum allocation

Shrutivandana Sharma University of Michigan, Ann Arbor 4 / 34

Introduction Model Decentralized mechanism Conclusion

Contribution

Developed decentralized power allocation mechanism for cellular networks that, preserves private information of the users makes the users willingly participate in the mechanism

• btains optimal centralized allocations at all Nash equilibria

balances the flow of money in the system Presented a method to characterize all Nash equilibria for a given system wide objective, and a given decentralized allocation mechanism

Shrutivandana Sharma University of Michigan, Ann Arbor 5 / 34

Introduction Model Decentralized mechanism Conclusion Power allocation problem

The uplink model

Shrutivandana Sharma University of Michigan, Ann Arbor 6 / 34

Introduction Model Decentralized mechanism Conclusion Power allocation problem

The uplink model

One base station (BS)

Shrutivandana Sharma University of Michigan, Ann Arbor 7 / 34

Introduction Model Decentralized mechanism Conclusion Power allocation problem

The uplink model

One base station (BS) N mobile users

Shrutivandana Sharma University of Michigan, Ann Arbor 8 / 34

Introduction Model Decentralized mechanism Conclusion Power allocation problem

The uplink model

One base station (BS) N mobile users Transmission power of user i: pi

Shrutivandana Sharma University of Michigan, Ann Arbor 9 / 34

Introduction Model Decentralized mechanism Conclusion Power allocation problem

The uplink model

One base station (BS) N mobile users Transmission power of user i: pi Channel gain from i to BS: hi0 Received power at BS: pr

i = pihi0

Shrutivandana Sharma University of Michigan, Ann Arbor 10 / 34

Introduction Model Decentralized mechanism Conclusion Power allocation problem

The uplink model

One base station (BS) N mobile users Transmission power of user i: pi Channel gain from i to BS: hi0 Received power at BS: pr

i = pihi0

Signature codes not orthogonal Causes interference Quality of Service (QoS) depends

• n: (pr

1, . . . , pr i , . . . , pr N)

Shrutivandana Sharma University of Michigan, Ann Arbor 11 / 34

Introduction Model Decentralized mechanism Conclusion Power allocation problem

The uplink model

One base station (BS) N mobile users Transmission power of user i: pi Channel gain from i to BS: hi0 Received power at BS: pr

i = pihi0

Signature codes not orthogonal Causes interference Quality of Service (QoS) depends

• n: (pr

1, . . . , pr i , . . . , pr N)

Multi User Detector (MUD) decoders

Shrutivandana Sharma University of Michigan, Ann Arbor 12 / 34

Introduction Model Decentralized mechanism Conclusion Power allocation problem

The uplink model

One base station (BS) N mobile users Transmission power of user i: pi Channel gain from i to BS: hi0 Received power at BS: pr

i = pihi0

Signature codes not orthogonal Causes interference Quality of Service (QoS) depends

• n: (pr

1, . . . , pr i , . . . , pr N)

Multi User Detector (MUD) decoders Tax paid by i: ti (>, <, =) 0

Shrutivandana Sharma University of Michigan, Ann Arbor 13 / 34

Introduction Model Decentralized mechanism Conclusion Power allocation problem

The uplink model

One base station (BS) N mobile users Transmission power of user i: pi Channel gain from i to BS: hi0 Received power at BS: pr

i = pihi0

Signature codes not orthogonal Causes interference Quality of Service (QoS) depends

• n: (pr

1, . . . , pr i , . . . , pr N)

Multi User Detector (MUD) decoders Tax paid by i: ti (>, <, =) 0 All users are self utility maximizers / behave strategically.

Shrutivandana Sharma University of Michigan, Ann Arbor 14 / 34

Introduction Model Decentralized mechanism Conclusion Power allocation problem

Information available to the users

Private information of user i :

Maximum transmission power of i: Pmax

i

Channel gain from i to BS: hi0 Utility of user i: uA

i (ti, pr)

= −ti +ui(pr)−

• 1 − ISi (pr)

ISi (pr)

• −tax paid + QoS received

Si := {pr | pr

i ∈[0, Pmax i

hi0]; pr

j ∈R+,

j = i} ui is concave in pr. (Sharma, Teneketzis (07))

Shrutivandana Sharma University of Michigan, Ann Arbor 15 / 34

Introduction Model Decentralized mechanism Conclusion Power allocation problem

Information available to the users

Common knowledge:

Number of users N System is static Channels gains are fixed Users’ utilities are fixed

Shrutivandana Sharma University of Michigan, Ann Arbor 16 / 34

Introduction Model Decentralized mechanism Conclusion Power allocation problem

The centralized power allocation problem

Problem (PC)

max

(t, pr ) N

• i=1

uA

i (ti, pr)

s.t.

N

• i=1

ti = 0 equivalently, max

(t, pr )∈SU N

• i=1

ui(pr) where SU = {(t, pr) |

N

• i=1

ti = 0, t ∈ RN, pr

i ∈ [0, Pmax i

]hi0} (PC) obtains an allocation that is balanced in money transfers and maximizes the sum of utilities of all the users.

Solution of Problem (PC) = Ideal allocation

Shrutivandana Sharma University of Michigan, Ann Arbor 17 / 34

Introduction Model Decentralized mechanism Conclusion Power allocation problem

How to obtain centralized solution

Characteristics of the uplink model

Decentralized information: Nobody has complete system information. Strategic users: The users are selfish.

Solution approach: Implementation theory

Provides guidelines for: how the users should “communicate” with the BS, and how “the information communicated by the users should be used by the BS to determine allocations” so as to induce the selfish users to communicate information that results in optimal centralized allocations. Reference: Implementation theory – Maskin (1985), Jackson (2001), Palfrey (2002), Stoenescu and Teneketzis (2005)

Shrutivandana Sharma University of Michigan, Ann Arbor 18 / 34

Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results

The uplink problem in implementation theory framework

Shrutivandana Sharma University of Michigan, Ann Arbor 19 / 34

Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results

The uplink problem in implementation theory framework

Shrutivandana Sharma University of Michigan, Ann Arbor 20 / 34

Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results

The uplink problem in implementation theory framework

Shrutivandana Sharma University of Michigan, Ann Arbor 21 / 34

Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results

The uplink problem in implementation theory framework

Shrutivandana Sharma University of Michigan, Ann Arbor 22 / 34

Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results

The uplink problem in implementation theory framework

Shrutivandana Sharma University of Michigan, Ann Arbor 23 / 34

Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results

The uplink problem in implementation theory framework

Decentralized mechanism – Game form: (M, f)

Shrutivandana Sharma University of Michigan, Ann Arbor 24 / 34

Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results

The uplink problem in implementation theory framework

Decentralized mechanism – Game form: (M, f) Induced game: (M, f, {uA

i }N i=1)

Shrutivandana Sharma University of Michigan, Ann Arbor 25 / 34

Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results

The uplink problem in implementation theory framework

Nash equilibrium: A message profile m∗ is a NE if,

uA

i (f(m∗))

≥ uA

i (f((mi, m∗/i))),

∀ mi ∈ Mi, ∀ i ∈ {1, 2, . . . , N}

Shrutivandana Sharma University of Michigan, Ann Arbor 26 / 34

Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results

Interpretation of Nash equilibria

Traditional definition of Nash equilibria

– for games of complete information

Difference in the uplink model

The uplink model does not result in a game of complete information – Users’ utilities/channel gains are private information Users are involved in a message exchange process with the BS Interpretation The stationary points of the message exchange process should have properties of Nash equilibria.

Shrutivandana Sharma University of Michigan, Ann Arbor 27 / 34

Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results

Desirable properties of a decentralized mechanism

Implementation in Nash equilibria:

A game form (M, f) “fully implements the goal correspondence π in Nash equilibria” if, for all problem environments, Set of allocations at all Nash equilibria = Set of optimal centralized allocations

Shrutivandana Sharma University of Michigan, Ann Arbor 28 / 34

Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results

Desirable properties of a decentralized mechanism

Implementation in Nash equilibria:

A game form (M, f) “fully implements the goal correspondence π in Nash equilibria” if, for all problem environments, Set of allocations at all Nash equilibria = Set of optimal centralized allocations

Individual rationality:

A game form (M, f) is individually rational if, for all users, Utility at all Nash equilibria ≥ Utility before/without participating in the allocation process specified by the game form

Shrutivandana Sharma University of Michigan, Ann Arbor 28 / 34

Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results

Desirable properties of a decentralized mechanism

Implementation in Nash equilibria:

A game form (M, f) “fully implements the goal correspondence π in Nash equilibria” if, for all problem environments, Set of allocations at all Nash equilibria = Set of optimal centralized allocations

Individual rationality:

A game form (M, f) is individually rational if, for all users, Utility at all Nash equilibria ≥ Utility before/without participating in the allocation process specified by the game form

Budget balance:

A game form (M, f) is budget balanced if, Net money transfer in the system =

Shrutivandana Sharma University of Michigan, Ann Arbor 28 / 34

Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results

Public good analogy

Characteristics of public goods:

The presence of the resource simultaneously affects the utilities

• f all network users without getting divided among them

Each user obtains a different individual utility from the consumption of the resource

Shrutivandana Sharma University of Michigan, Ann Arbor 29 / 34

Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results

Public good analogy

Characteristics of public goods:

The presence of the resource simultaneously affects the utilities

• f all network users without getting divided among them

Each user obtains a different individual utility from the consumption of the resource Public good in uplink network Power vector received at the base station: (pr

1, pr 2, . . . , pr N),

corresponding utilities: uA

i (pr 1, pr 2, . . . , pr N), i ∈ {1, 2, . . . , N}

Reference: Nash implementation mechanisms – Groves, Ledyard (77); Hurwicz (79); Walker (81)

Shrutivandana Sharma University of Michigan, Ann Arbor 29 / 34

Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results

A game form for the uplink power allocation problem

Message space:

mi := (πi, pr

i );

πi ∈ RN

+, pr i ∈ RN,

i ∈ {1, 2, . . . , N} (1) (Price vector, Power vector) proposal for N users

Shrutivandana Sharma University of Michigan, Ann Arbor 30 / 34

Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results

A game form for the uplink power allocation problem

Message space:

mi := (πi, pr

i );

πi ∈ RN

+, pr i ∈ RN,

i ∈ {1, 2, . . . , N} (1) (Price vector, Power vector) proposal for N users

Outcome function:

ˆ pr(m) = 1 N

N

• i=1

pr

i ,

(2) ˆ ti(m) = lT

i (m)ˆ

pr(m) + (pr

i − pr i+1)T diag(πi)(pr i − pr i+1)

−(pr

i+1 − pr i+2)T diag(πi+1)(pr i+1 − pr i+2),

i ∈ {1, 2, . . . , N} (3) where, (4) li(m) = πr

i+1 − πr i+2

(5) Equilibrium price does not depend on user’s own message Quadratic penalty term forces the users to agree on one power allocation

Shrutivandana Sharma University of Michigan, Ann Arbor 30 / 34

Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results

Results

Theorem 1: Let m∗ be a Nash equilibrium of the game specified by the game form and the users’ utility functions. Let (ˆ t(m∗), ˆ pr(m∗)) be the allocation at m∗ determined by the game

• form. Then,

(a) (ˆ t(m∗), ˆ pr(m∗)) is individually rational, and (b) (ˆ t(m∗), ˆ pr(m∗)) is an optimal solution of Problem (PC). Theorem 2: Given an optimum received power vector ˆ pr∗ of Problem (PC), there exists at least

• ne Nash equilibrium m∗ of the game corresponding to the proposed game form and

the users’ utility functions such that, ˆ pr(m∗) = ˆ pr∗. Furthermore, given ˆ pr∗, the set of all Nash equilibria that result in ˆ pr∗ can be char- acterized.

Shrutivandana Sharma University of Michigan, Ann Arbor 31 / 34

Introduction Model Decentralized mechanism Conclusion

Conclusion

Conclusion

Studied a power allocation problem for cellular uplink and downlink networks under a game theoretic set up. Developed a decentralized allocation mechanism that obtains optimal centralized allocations at all Nash equilibria. Presented a method to characterize all Nash equilibria corresponding to the decentralized mechanism.

Shrutivandana Sharma University of Michigan, Ann Arbor 32 / 34

Introduction Model Decentralized mechanism Conclusion

Conclusion

Conclusion

Studied a power allocation problem for cellular uplink and downlink networks under a game theoretic set up. Developed a decentralized allocation mechanism that obtains optimal centralized allocations at all Nash equilibria. Presented a method to characterize all Nash equilibria corresponding to the decentralized mechanism.

Future scope

We have a constructive proof for the existence of Nash equilibria. We do not have an algorithm to show how to converge to the Nash equilibria. Orthogonal/greedy search is not guaranteed to converge because the resulting game is not supermodular.

Shrutivandana Sharma University of Michigan, Ann Arbor 32 / 34

Introduction Model Decentralized mechanism Conclusion

Conclusion

Conclusion

Studied a power allocation problem for cellular uplink and downlink networks under a game theoretic set up. Developed a decentralized allocation mechanism that obtains optimal centralized allocations at all Nash equilibria. Presented a method to characterize all Nash equilibria corresponding to the decentralized mechanism.

Future scope

We have a constructive proof for the existence of Nash equilibria. We do not have an algorithm to show how to converge to the Nash equilibria. Orthogonal/greedy search is not guaranteed to converge because the resulting game is not supermodular. Developing algorithms or supermodular games that lead to the optimum centralized transactions.

Shrutivandana Sharma University of Michigan, Ann Arbor 32 / 34

Introduction Model Decentralized mechanism Conclusion

Thank You!

Shrutivandana Sharma University of Michigan, Ann Arbor 33 / 34

Introduction Model Decentralized mechanism Conclusion

Questions?

Contact:

Shrutivandana Sharma

email: svandana@umich.edu web: http://www-personal.umich.edu/∼svandana

Demosthenis Teneketzis

email: teneket@eecs.umich.edu web: http://www.eecs.umich.edu/∼teneket

Shrutivandana Sharma University of Michigan, Ann Arbor 34 / 34