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Introduction Model Decentralized mechanism Conclusion
A Game-Theoretic Approach toDecentralized Optimal Power Allocation
for Cellular Networks
Shruti SharmaPh.D. candidate, Electrical Engineering and Computer Science
and
Demos TeneketzisElectrical Engineering and Computer Science
University of Michigan, Ann Arbor
GameComm 2008, October 20, Athens, Greece
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Introduction Model Decentralized mechanism Conclusion
Outline
1 Introduction
2 Cellular network modelPower allocation problem
3 Decentralized mechanismSolution approach: Implementation theory frameworkA decentralized mechanism for power allocationResults
4 Conclusion
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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 otherusers’ information or strategies
Our workDesign of a decentralized power allocation mechanism that,
preserves private information of the users
makes the users willingly participate in the mechanism
is budget balanced
obtains optimal centralized allocations at all Nash equilibria
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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
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Introduction Model Decentralized mechanism Conclusion
Contribution
Developed decentralized power allocation mechanismfor cellular networks that,
preserves private information of the users
makes the users willingly participate in the mechanism
obtains 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
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Introduction Model Decentralized mechanism Conclusion Power allocation problem
The uplink model
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Introduction Model Decentralized mechanism Conclusion Power allocation problem
The uplink model
One base station (BS)
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Introduction Model Decentralized mechanism Conclusion Power allocation problem
The uplink model
One base station (BS)
N mobile users
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Introduction Model Decentralized mechanism Conclusion Power allocation problem
The uplink model
One base station (BS)
N mobile users
Transmission power of user i : pi
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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: pri = pi hi0
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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: pri = pi hi0
Signature codes not orthogonal
Causes interferenceQuality of Service (QoS) dependson: (pr
1, . . . , pri , . . . , p
rN)
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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: pri = pi hi0
Signature codes not orthogonal
Causes interferenceQuality of Service (QoS) dependson: (pr
1, . . . , pri , . . . , p
rN)
Multi User Detector (MUD) decoders
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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: pri = pi hi0
Signature codes not orthogonal
Causes interferenceQuality of Service (QoS) dependson: (pr
1, . . . , pri , . . . , p
rN)
Multi User Detector (MUD) decoders
Tax paid by i : ti (>, <, =) 0
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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: pri = pi hi0
Signature codes not orthogonal
Causes interferenceQuality of Service (QoS) dependson: (pr
1, . . . , pri , . . . , p
rN)
Multi User Detector (MUD) decoders
Tax paid by i : ti (>, <, =) 0
All users are self utility maximizers /behave strategically.
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Introduction Model Decentralized mechanism Conclusion Power allocation problem
Information available to the users
Private information of user i :
Maximum transmission power of i : Pmaxi
Channel gain from i to BS: hi0
Utility of user i : uAi (ti ,pr )
= −ti +ui (pr )−[
1− ISi(pr )
ISi(pr )
]
−tax paid + QoS received
Si := {pr | pri ∈ [0,Pmax
i hi0]; prj ∈R+,
j 6= i}
ui is concave in pr .(Sharma, Teneketzis (07))
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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 fixedUsers’ utilities are fixed
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Introduction Model Decentralized mechanism Conclusion Power allocation problem
The centralized power allocation problem
Problem (PC)max(t, pr )
N∑i=1
uAi (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 , pri ∈ [0,Pmax
i ]hi0}
(PC) obtains an allocation that is balanced in money transfers and maximizes thesum of utilities of all the users.
Solution of Problem (PC) = Ideal allocation
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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 theoryProvides guidelines for:
how the users should “communicate” with the BS, and
how “the information communicated by the users should be used by the BS todetermine allocations” so as to induce the selfish users to communicateinformation that results in optimal centralized allocations.
Reference: Implementation theory – Maskin (1985), Jackson (2001), Palfrey (2002),Stoenescu and Teneketzis (2005)
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Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results
The uplink problem in implementation theory framework
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Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results
The uplink problem in implementation theory framework
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Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results
The uplink problem in implementation theory framework
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Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results
The uplink problem in implementation theory framework
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Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results
The uplink problem in implementation theory framework
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Introduction Model Decentralized mechanism Conclusion Solution approach Game form Results
The uplink problem in implementation theory framework
Decentralized mechanism – Game form: (M, f )
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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 , {uAi }N
i=1)
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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,
uAi (f (m∗)) ≥ uA
i (f ((mi ,m∗/i))), ∀ mi ∈Mi , ∀ i ∈ {1, 2, . . . ,N}
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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 modelThe uplink model does not result in a game of completeinformation – 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 shouldhave properties of Nash equilibria.
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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 = 0
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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 = 0
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 = 0
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 utilitiesof all network users without getting divided among them
Each user obtains a different individual utility from theconsumption of the resource
Public good in uplink network
Power vector received at the base station: (pr1, pr
2, . . . , prN),
corresponding utilities: uAi (pr
1, pr2, . . . , 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
Public good analogy
Characteristics of public goods:
The presence of the resource simultaneously affects the utilitiesof all network users without getting divided among them
Each user obtains a different individual utility from theconsumption of the resource
Public good in uplink network
Power vector received at the base station: (pr1, pr
2, . . . , prN),
corresponding utilities: uAi (pr
1, pr2, . . . , 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
Outcome function:
p̂r (m) =1N
N∑i=1
pri , (2)
t̂i (m) = lTi (m)p̂r (m) + (pr
i − pri+1)
T diag(πi )(pri − pr
i+1)
−(pri+1 − pr
i+2)T diag(πi+1)(pr
i+1 − pri+2), i ∈ {1, 2, . . . ,N} (3)
where, (4)
l i (m) = πri+1 − πr
i+2 (5)
Equilibrium price does not depend on user’s own messageQuadratic 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
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:
p̂r (m) =1N
N∑i=1
pri , (2)
t̂i (m) = lTi (m)p̂r (m) + (pr
i − pri+1)
T diag(πi )(pri − pr
i+1)
−(pri+1 − pr
i+2)T diag(πi+1)(pr
i+1 − pri+2), i ∈ {1, 2, . . . ,N} (3)
where, (4)
l i (m) = πri+1 − πr
i+2 (5)
Equilibrium price does not depend on user’s own messageQuadratic 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 (t̂(m∗), p̂r (m∗)) be the allocation at m∗ determined by the gameform. Then,
(a) (t̂(m∗), p̂r (m∗)) is individually rational, and
(b) (t̂(m∗), p̂r (m∗)) is an optimal solution of Problem (PC).
Theorem 2:
Given an optimum received power vector p̂r∗ of Problem (PC ), there exists at leastone Nash equilibrium m∗ of the game corresponding to the proposed game form andthe users’ utility functions such that, p̂r (m∗) = p̂r∗.
Furthermore, given p̂r∗, the set of all Nash equilibria that result in p̂r∗ can be char-acterized.
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Introduction Model Decentralized mechanism Conclusion
Conclusion
Conclusion
Studied a power allocation problem for cellular uplink and downlink networksunder a game theoretic set up.
Developed a decentralized allocation mechanism that obtains optimal centralizedallocations at all Nash equilibria.
Presented a method to characterize all Nash equilibria corresponding to thedecentralized 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 resultinggame is not supermodular.
Developing algorithms or supermodular games that lead to the optimum centralizedtransactions.
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 networksunder a game theoretic set up.
Developed a decentralized allocation mechanism that obtains optimal centralizedallocations at all Nash equilibria.
Presented a method to characterize all Nash equilibria corresponding to thedecentralized 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 resultinggame is not supermodular.
Developing algorithms or supermodular games that lead to the optimum centralizedtransactions.
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 networksunder a game theoretic set up.
Developed a decentralized allocation mechanism that obtains optimal centralizedallocations at all Nash equilibria.
Presented a method to characterize all Nash equilibria corresponding to thedecentralized 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 resultinggame is not supermodular.
Developing algorithms or supermodular games that lead to the optimum centralizedtransactions.
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 Sharmaemail: [email protected]: http://www-personal.umich.edu/∼svandana
Demosthenis Teneketzisemail: [email protected]: http://www.eecs.umich.edu/∼teneket
Shrutivandana Sharma University of Michigan, Ann Arbor 34 / 34