Application of Game Theory to Wireless NetworkingObjectives of this presentation Present a general...

Application ofGame Theory to

WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis

Stability andConvergence

Iterative Schemes

Simulations

Conclusion

{CongestionControl}

Application of Game Theory to WirelessNetworking

Tansu Alpcan

Deutsche Telekom Laboratories

1 / 44


WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}

Outline

Introduction

Power Control Games

Equilibrium Analysis

Stability and Convergence

Iterative Update Schemes

Simulations

Conclusion

{Congestion Control}

2 / 44


WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}

Objectives of this presentation

◮ Present a general game theoretic framework fordistributed control under limited informationexchange.

◮ Illustrate the game theoretic approach via a specificapplication: uplink power control in widebandwireless networks.

◮ Investigate existence and uniqueness of Nashequilibrium.

◮ Convergence and stability analysis ofcontinuous-time distributed algorithms.

◮ Study of relevant distributed iterative (update)algorithms and their convergence conditions to theequilibrium.

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}

Network Games

◮ Game theory (GT) involves multi-person decisionmaking.

◮ Autonomous parts of the networked systems (suchas mobiles, devices generating Internet traffic etc.)are modeled as players.

◮ Players interact and compete with each other on thesame system for limited and shared resources: e.g.quality of service, bandwidth...

◮ Players are associated with cost functions, whichthey minimize by choosing a strategy from a welldefined strategy space.

◮ Nash equilibrium (NE) provides an appropriatesolution concept, which is (approximately) optimalw.r.t. a global objective function.

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}

Why Game Theory

◮ The microprocessor revolution enabled production ofsystems with significant processing capacities →independent decision makers.

◮ These system are connected to each with a varietywired/wireless communication technologies resultingin networked systems → interaction betweendecision makers.

◮ The systems share various resources (but often haveonly local information) → competition for availableresources (resource allocation).

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}

Uplink Power Control in Wireless Networks

◮ Primary objective of (uplink) power control is toregulate the transmission power level of each mobilein order to obtain and maintain a satisfactory qualityof service or Signal-to-interference ratio (SIR) level.

◮ In wideband systems such as CDMA, signals of theusers interfere and affect each other’s service (SIR)level.

◮ In data networks, unlike in voice communication, SIRrequirements vary from one user to another.

◮ Emerging technologies such as cognitive radioempowers mobile users with independent decisionmaking capabilities.

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}

A Multicell Wireless Network

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}

Distributed Power Control

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}

Game Theoretic Formulation

◮ Game theory provides a natural framework for powercontrol in wireless systems, where mobiles (players)compete for service quality: e.g. cognitive radio.

◮ A mobile has no information on other player’s powerlevel or preferences. Therefore, use ofnoncooperative game theory is appropriate.

◮ Existence of a unique Nash equilibrium (NE) point isestablished in this multicell power control game.

◮ Convergence of continuous and discrete-timesynchronous and asynchronous update schemes aswell as of a stochastic update scheme isinvestigated.

◮ The power control game and the update algorithmsare demonstrated through numerical simulations.

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}

Network Model

◮ The system consists of L := {1, . . . , L} cells, with Ml

users in cell l .◮ Define 0 < hij < 1 as the channel gain. Let

secondary interference effects from neighboring cellsbe modeled as background noise, of variance σ2.

◮ The i th mobile transmits with an uplink power level ofpi ≤ pi ,max , which is received at the BS j asxij := hijpi . Then, SIR obtained by mobile i is givenby

γij :=Lhijpi

∑

k 6=i hkjpk + σ2

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}

Network Model

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}

Cost Function

◮ Each mobile is associated with a cost function:

Ji(xi , x−i , hi) = Pi(xi ) − Ui(γi(x))

◮ The benefit (utility) function, Ui(γi) quantifies theuser demand for quality of service or SIR level.

◮ The “pricing” function, Pi(pi) is imposed to limit theinterference, and hence, improve the systemperformance. It can also be interpreted as a cost onthe battery usage.

◮ Terminology clarification:

max Payoff = Benefit − “Cost ′′

min Cost = −Utility + Price

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}

Nash Equilibrium (NE)

DefinitionThe Nash equilibrium is defined as a set of strategies(and corresponding set of costs), with the property thatno player can benefit by modifying its own strategy whilethe other players keep theirs fixed.

If x is the strategy vector of players and X is the strategyspace such that x ∈ X ∀x, then x∗ is in NE when x∗

i ofany i th player satisfies

minxi

Ji(xi , x∗−i),

where Ji is the cost function of the i th player and x∗−i is

the equilibrium strategies of all other players.

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}

NE of a Generic Noncooperative Game

Assumptions:A1 The strategy space X of a noncooperative game, Θ isconvex, compact, and has a nonempty interior, X o 6= ∅.

A2 The cost function of the i th player, Ji (x), is twicecontinuously differentiable in all its arguments and strictlyconvex in xi , i.e. ∂2Ji(x)/∂x2

i ≥ 0.

Let ∇ be the pseudo-gradient operator:∇J :=

[

∇x1J1(x)T · · · ∇xM JM(x)T]T

.

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}

Let in addition G(x) be the Jacobian of ∇J with respect tox:

G(x) :=

b1 a12 · · · a1M...

. . ....

aM1 aM2 · · · bM

M×M

where bi := ∂2Ji (x)

∂x2i

and ai ,j := ∂2Ji(x)∂xi∂xj

.

We also define the symmetric matrixG(x) := G(x) + G(x)T .

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}

PropositionThe strategy vector x∗ ∈ X o is an inner NE solution of thegame Θ, if assumptions A1 and A2 hold, and∇J(x∗) = 0. In addition, if G(x) is positive definite for all xthen there can be at most one inner NE solution in thegame Θ. Furthermore, under A1, Θ admits a NE.

Notice that, this condition is quite similar to the strictconvexity condition where Hessian of a multivariablefunction f (x1, . . . , xn) is required to be positive definite:

H(f ) :=

∂2f∂x2

1

∂2f∂x1∂x2

· · · ∂2f∂x1∂xM

.... . .

...∂2f

∂xM∂x1

∂2f∂xM∂x2

· · · ∂2f∂x2

M

M×M

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}

A3 Let X := {x ∈ RM : hj(x) ≤ 0,∀j}, where

hj : RM → R,∀j , hj(x) is convex in its arguments for all j ,

and the set X is bounded and has a non-empty interior.In addition, the derivative of at least one of the constraintswith respect to xi , {dhj(x)/dxi ,∀j}, is nonzero fori = 1, 2, . . . M, ∀x ∈ X .

The Lagrangian function for player i in this game is givenby Li(x, µ) = Ji(x) +

∑rj=1 µi ,jhj(x).

TheoremThere exists a unique NE point in the M-playernoncooperative game Θ if A1, A2, and A3 hold.

TheoremUnder appropriate convexity conditions on the costfunctions J the multicell power control game definedadmits a unique inner Nash equilibrium solution.

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}

System Dynamics and Stability

◮ Each mobile uses a gradient algorithm to solve itsown optimization problem. The update scheme ofthe i th mobile is:

pi =dpi

dt= −λi

∂Ji

∂pi

◮ In terms of the received power level, xi , at the BS:

xi =dUi

dγi

Lλih2i

∑

j 6=i xj + σ2 − λihidPi

dpi:= φi(x).

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}

Mobile 1

.

.

.

h_1 p_1

h_2 p_2

h_M p_M

h_i p_ii

Mobile 2

Mobile M

Base Station

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}

Stability in a Cell

Define the quadratic and radially unbounded Lyapunovfunction

Vl :=∑

i∈Ml

φ2i (x)

A sufficient condition for Vl < 0, uniformly in the xi ’s, is

L > Ml − 1

in the symmetric case where Ui = Uj andxi = xj ∀i , j ∈ Ml , and for a large class of logarithmicutility functions of the form Ui = ui log(kγi + 1).

Then, the distributed power update scheme is globallyasymptotically stable!

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}

Lyapunov Function (representation)

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}

Outage Probability

◮ The outage probability of user i , denoted Oil , isdefined as the proportion of time that some SIRthreshold, γil , is not met for sufficient reception at thel th BS receiver

◮ By a careful choice of γil , a quality of service levelcan be established for each user. Assumeγi := γil = γik ∀l , k ∈ L as a simplification.

◮ The outage probability, Oi = Pr(γi ≤ γi), of the i th

mobile is

Oi(x, γi) = 1 − exp(−σ2γi

xi

)

Πj 6=i1

1 +γixjl

xi

.

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}

Outage-Based Cost Function

◮ Each mobile is associated with a cost function:

Ji(x) = Pi(xi) − Ui(Pri(γi(x) ≥ γi),

where Pri(γi(x) ≥ γi) = 1 − Oi(x, γi). Hence,Ui = ui log(1 − Oi(x, γi)).

◮ The utility function, Ui(Pri(γi(x) ≥ γi) quantifies theuser demand for a certain level of service or outageprobability.

TheoremUnder certain convexity assumptions, the multicell powercontrol game defined admits a unique inner Nashequilibrium solution.

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}

Synchronous Update

Consider a discrete-time update scheme in a system withM mobiles where each mobile uses a discretized gradientalgorithm to solve its optimization problem:

pi(n + 1) = pi(n) − λi∂Ji

∂pi∀i ∈ M ,

where n = 1, 2, . . ., denotes the update instances and λi

is the user-specific step size constant.This can also be defined as

xi(n + 1) = Ti(x(n)) := xi(n) − λ∂Ji

∂xi∀i ∈ M .

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}

Synchronous Update

TheoremLet xmax = αxmin for some α > 0 andX := {x ∈ R

Mx : xmin ≤ xil ≤ xmax ∀i , l}. Thesynchronous power update algorithm

pi(n + 1) = pi(n) − λi∂Ji

∂pi∀i ∈ M

converges to the unique NE point of the powercontrol game,p∗ := [x∗

1/h1, . . . , x∗M/hM ], on the set X if

λKsynch < 1,

andα < 1 +

√

1 + γmin,

where Ksynch is a function of the system parameters andconstant.

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}

Asynchronous Power Update

◮ A natural generalization of the synchronous updateis the asynchronous update scheme.

◮ It is more realistic since it is difficult for the mobiles tosynchronize their exact power update instances in apractical implementation.

◮ In this particular case, the convergence analysisabove also applies to the asynchronous updatealgorithm.

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}


Define a sequence of nonempty, convex, and compactsets

X (k) := [x∗1 − δ(k), x∗

1 − δ(k)] × [x∗2 − δ(k), x∗

2 − δ(k)]

× . . . [x∗M − δ(k), x∗

M − δ(k)],

where δ(k) := ‖x(k) − x∗‖. By the previous Theorem,δ(k + 1) < δ(k), we have

. . . ⊂ X (k + 1) ⊂ X (k) ⊂ . . . X .

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}


Definition 1 [Synchronous Convergence Condition] For asequence of nonempty sets {X (k)} with. . . ⊂ X (k + 1) ⊂ X (k) ⊂ . . . X , we haveT (x) ∈ X (k + 1), ∀k , and x ∈ X (k). Furthermore, if {yk}is a sequence such that yk ∈ X (k) for every k , then everylimit point of {yk} is a fixed point of T .

Definition 2 [Box Condition] For every k , there exist setsXi(k) ⊂ Xi such that

X (k) := X1(k) × X2(k) × · · · × XM(k).

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}


Both of these conditions are satisfied in this case bydefinition of X (k) and synchronous convergencetheorem.

Therefore, it immediately follows from asynchronousconvergence theorem [Bertsekas] that theasynchronous power update algorithm converges tothe unique NE point of the power control game.

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}

Stochastic Power Update

◮ In a real life implementation, communicationconstraints, approximations, estimation andquantization errors are not negligible.

◮ Hence, a mobile does not have access to the exactvalues of the system parameters such as its ownchannel gain or the feedback terms provided by theBS.

◮ These uncertainties can be captured by defining astochastic update algorithm for analysis purposes.

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}

Communication Constraints

.

.

.

h_1 p_1

h_2 p_2

h_M p_M

Mobile1

Mobile2

MobileM

h_i p_ii

Q( h_i p_i)i

Quantizer (Q)

Base Station

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}


For each i ∈ M, let ξi(n) n = 1, 2, . . . be a sequence ofindependent identically distributed (iid) random variablesdefined on the common support set [1 − ε, 1 + ε], where0 < ε < 1.

We further assume the sequence ξi is independent of thepast of ξj , j 6= i .

Using these random sequences, we model the aggregateuncertainty in the term ∂Ji/∂pi due to quantization,estimation, and multiplicatively approximation errors.The stochastic counterpart of the synchronous updatealgorithm is given by

pi(n + 1) = pi(n) − λiξi(n) ∂Ji∂pi

∀i ∈ M.

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}


This can also be described in terms of received powerlevels at the base station as

xi(n + 1) = xi(n) − λξi(n)∂Ji

∂xi

=: Ti(x(n); ξi(n)) ∀i ∈ M.

Let xi(n) (ξi(n)) be random (random iid) sequences for alli , where ξi is associated with the probability densityfunction fξi (ξi) defined on the support set [1 − ε, 1 + ε],0 < ε < 1, and the random vector x takes its values onthe set X := {x ∈ R

Mx : xmin ≤ xil ≤ xmax ∀i , l}.Furthermore, let α > 0 be defined as α := xmax/xmin.

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}


TheoremThe stochastic power update algorithm convergesalmost surely to the unique NE point of the powercontrol game, p∗, if

α <12√

γmin +14

andλ(1 + ε)Ksto < 1

hold. Here Ksto is a function of the system parametersand constant.

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}

0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

X Coordinate

Y C

oord

inat

eLocations of Base Stations and Mobiles

Base StationMobile

Locations of base stations and the paths of mobiles.

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}

0 100 200 300 400 500 600 700 800 900 10000

100

200

300

400

500

600

700

800

900

1000

time steps

pow

er le

vels

Power Levels of Mobiles

Power levels of selected mobiles with respect to time.

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}

100 200 300 400 500 600 700 800 900 100015

20

25

30

35

40

45

time steps

SIR

(dB

)SIR Values of Mobiles

SIR values of selected mobiles (in dB) versus time.

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}

Conclusion

◮ We have considered a noncooperative power controlgame with a utility defined as the function of the theSIR level or outage probability.

◮ We have proved that this game admits a uniqueNash equilibrium for uniformly strictly convex pricingfunctions and/or under some technical assumptionson the SIR threshold levels.

◮ We have established the global convergence ofcontinuous-time as well as discrete-timesynchronous, asynchronous, and stochastic iterativepower update algorithms to the unique NE of thegame under some conditions.

◮ Finally, through simulation studies we havedemonstrated the convergence and robustnessproperties of power update schemes developed.

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}

Conclusion

◮ We have presented a game theoretic framework fordistributed control where individual parties (clients,mobile devices, etc.) compete for resources andhave limited information.

◮ We have established conditions for existence anduniqueness of Nash equilibrium in the resultinggame.

◮ We have studied convergence and stabilityproperties of continuous-time as well asdiscrete-time distributed algorithms.

◮ Using two example power control games in thecontext of wireless networks, we have illustrated theframework presented.

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}

Congestion Control Problem

◮ The userscommunicate witheach other on thenetwork by sharingthe availablebandwidth.

◮ The bandwidthbecomes congestedas a resource whenthe total demandexceeds thecapacity.

The problem is complicated by communicationconstraints such as communication delays, distributednature of the system, and users requesting as muchbandwidth as possible.

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}

Congestion Control Game

The Network Fluid approximation model. N nodes and Llinks with capacities Cl . M users, eachassociated with a (unique) connection.Routes are fixed and described by therouting matrix A.

Flow Rates: User i has a nonnegative flow rate xi . Flowssatisfy the capacity constraint Ax ≤ C.

Cost Function: Each user is associated with a costfunction

Ji(x; C, A) = Pi(x; C, A) − Ui(xi ), i ∈ M.

⋄ The function P acts as a “feedback” termindicating the state of the network.⋄ The function U models the user’s demandfor bandwidth.

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}

Congestion Control: Overview of Results

◮ Developed a general framework for study of networkcongestion control based on game theory.

◮ Developed distributed, end-to-end congestion controlalgorithms and analyzed their stability and delayrobustness properties both theoretically andnumerically.

◮ Utilized randomized algorithms to investigate stabilityof discrete-time nonlinear algorithms in cases whereanalytical models are intractable.

◮ Verified theoretical results obtained through bothnumerical and realistic packet level simulations.

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}

Simulation Results

A Nam screenshot of thegeneral (arbitrary) topology

network.

0 5 10 150

5

10

15x 10

5

Time (seconds)F

low

Rat

e (b

ps)

3 Selected Flows in General Network Topology

User 1User 2User 3

Three flows from nodes 7, 8,and 9 to node 6 are shown

where these users aresymmetric.

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WirelessNetworking

Tansu Alpcan

Introduction

Power ControlGames

EquilibriumAnalysis


Iterative Schemes

Simulations

Conclusion

{CongestionControl}

Merci!

My publications are available for download on my website(research section) at:

http://decision.csl.uiuc.edu/˜alpcan/

or

http://deutsche-telekom-laboratories.de/˜alpcan/

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Application of Game Theory to Wireless NetworkingObjectives of this presentation Present a general...

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