Power Allocation Games on Interference Channels with Complete and Partial Information

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by  Krishna Chaitanya A

Power Allocation Games on Interference Channels
with Complete and Partial Information
Krishna Chaitanya A, Utpal Mukherji, Vinod Sharma
Department of ECE, Indian Institute of Science, Bangalore-560012
Email: {akc, utpal, vinod} @ece.iisc.ernet.in
Abstract—We consider a wireless channel shared by multiple the messages from strong and very strong interferers instead
transmitter-receiver pairs. Their transmissions interfere with of treating them as noise.
each other. Each transmitter-receiver pair aims to maximize its
long-termaverage transmissionratesubjecttoanaverage power All the above cited works consider a one shot non-
constraint. This scenario is modeled as a stochastic game under cooperativegame (or a Pareto point). As against that we con-
differentassumptions.Wefirstassumethateachtransmitterand sider a stochastic game over Gaussian interference channels,
5 receiverhasknowledgeofalldirectandcrosslinkchannelgains. wheretheuserswanttomaximizetheirlongtermaveragerate
1 We later relax the assumption to the knowledge of incident and have long term average power constraints (for potential
0 channel gains and then further relax to the knowledge of the advantagesofthisoveroneshotoptimization,see[8],[9]).For
2 direct link channel gains only. In all the cases, we formulate thissystem we obtainexistenceofNEanddevelopalgorithms
n tinheequparolibtylem(VIo)fpfirnobdlienmg tahnedNparsehseneqtuainlibarliguomritahsmatovasroilavteionthael to obtain NE via variational inequalities. Further more, the
a VI. above mentioned literature considers the problem when each
J user knows all the channel gains in the system while we also
9 Keywords—Interference channel, stochastic game, Nash equilib- consider the much more realistic situation when a user knows
1 rium, distributed algorithms, variational inequality. only its own channel gains.

] I. INTRODUCTION Thepaperisorganizedasfollows.InSectionII,wepresent
T the system model and formulate it as a stochastic game. In
I We consider a wireless channel which is being shared SectionIII,we studythisstochasticgameanddefinethe basic
.
s by multiple users to transmit their data to their respective terminology.In Section IV, we propose an algorithm to solve
c receivers. The transmissions of different users may cause theformulatedvariationalinequalityundergeneralconditions.
[
interference to other receivers. This is a typical scenario In Section V we use this algorithm to obtain NE when the

1 in many wireless networks. In particular, this can represent users have only partial information about the channel gains.
v inter-cell interference on a particular wireless channel in a In Section VI, we present numerical examples and Section
2 cellular network. The different users want to maximize their VII concludes the paper.
1 transmission rates. This system can be modeled in the game
4 theoretic framework and has been widely studied [1] - [6].
4 II. SYSTEMMODELAND NOTATION
0 In [1], the authors have considered parallel Gaussian
We consider a Gaussian wireless channel being shared
. interference channels. This setup is modeled as a strategic
1 by N transmitter-receiver pairs. The time axis is slotted and
formgameandexistenceanduniquenessofaNashequilibrium
0 all users’ slots are synchronized. The channel gains of each
5 (NE) is studied. The authors provide conditions under which
transmit-receive pair are constant during a slot and change
1 the water-filling function is a contraction and thus obtain
independentlyfromslottoslot.Theseassumptionsareusually
: conditions for uniqueness of NE and for convergence of
v made for this system [1], [9].
iterative water-filling. They extend these results to a multi-
i
X antenna system in [5] and consider an asynchronous version LetH (k)betherandomvariablethatrepresentschannel
ij
r of iterative water-filling in [6]. gainfromtransmitterj toreceiveri(fortransmitteri,receiver
a i is the intended receiver) in slot k. The direct channel power
An online algorithm to reach a NE for the parallel
Gaussian channels is presented in [2] when the channel gain gains |Hii(k)|2 ∈ Hd = {g1(d),g2(d),...,gn(d1)} and the cross
distributions are not known to the players. Its convergence is channel power gains |Hij(k)|2 ∈ Hc = {g1(c),g2(c),...,gn(c2)}.
also proved. In [4] authors describe some conditions under Let π and π be the probability distributions on H and H
d c d c
which parallel Gaussian interference channels have multiple respectively. We assume that, {H (k),k ≥ 0} is an i.i.d
ij
Nash equilibria. Using variational inequalities, they present sequence with distribution π where π = π if i = j and
ij ij d
an algorithm that converges to a Nash equilibrium which π = π if i 6= j. We also assume that these sequences are
ij c
minimizes the overall weighted interference. independentof each other.
We consider power allocation in a non-game-theoretic We denote (H (k),i,j = 1,...,N) by H(k) and its
ij
framework in [7] (see also other references in [7] for such realization vector by h(k) which takes values in H, the set of
a setup). In [7], we have proposed a centralized algorithm for allpossiblechannelstates.ThedistributionofH(k)isdenoted
finding the Pareto points that maximize sum rate when the by π. We call the channel gains (H (k),j =1,...,N) from
ij
receivers have knowledge of all the channelgains and decode all the transmitters to the receiver i an incident gain of user i
anddenotebyH (k)anditsrealizationvectorbyh (k)which We limit ourselves to stationary policies, i.e., the power
i i
takesvaluesinI,thesetofallpossibleincidentchannelgains. policy for every user in slot k depends only on the channel
The distribution of H (k) is denoted by π . state H(k) and not on k. In the current setup, it does not
i I
entail any loss in optimality. In fact now we can rewrite
Each user aims to operate at a power allocation that
the optimization problem in G to find policy P(H) such
maximizes its long term average rate under an average power
constraint. Since their transmissions interfere with each other, that ri = EH[log(1+Γi(P (H)))] is maximized subject to
affecting their transmission rates, we model this scenario as a EH[Pi(H)] ≤ Pi for all i. Similarly, we can rewrite the
optimizationproblemsingamesG andG .Weexpresspower
stochastic game. I D
policy of player i by P =(P (h),h∈H), where transmitter
i i
We first assume complete channel knowledgeat all trans- i transmits in channel state h with power P (h). We denote
i
mittersand receivers.If user i uses powerP (H(k)) in slot k, the power profile of all players by P=(P ,...,P ).
i 1 N
it gets rate log(1+Γ (P(H(k)))), where
i In the rest of the paper, we prove existence of a Nash
α |H (k)|2P (H(k)) equilibrium for each of these games and provide algorithm to
Γ (P(H(k)))= i ii i , (1)
i 1+ |H (k)|2P (H(k)) compute it.
j6=i ij j
P(H(k))=(P1(H(k)),...,PPN(H(k))) and αi is a constant III. GAME THEORETICREFORMULATION
thatdependsonthemodulationandcodingusedbytransmitter
Theoryofvariationalinequalitiesoffersvariousalgorithms
i andwe assume α =1 for alli. The aim of each user i is to
i to find NE of a given game [13]. A variational inequality
choose a power policy to maximize its long term average rate
problem denoted by VI(K,F) is defined as follows.
n
r (P ,P ),limsup 1 E[log(1+Γ (P (H(k))))], Definition 1. Let K ⊂ Rn be a closed and convex set, and
i i −i n→∞ n i F :K →K. The variationalinequality problem VI(K,F) is
k=1
X (2) defined as the problem of finding x∈K such that
subject to average power constraint F(x)T(y−x)≥0 for all y ∈K.
n
1
limsup E[P (H(k))]≤P , for each i, (3) We reformulate the Nash equilibrium problem at hand
i i
n
n→∞ to an affine variational inequality problem. We denote our
k=1
X game by G = (A )N ,(r )N , where r (P ,P ) =
where P denotes the power policies of all users except i. i i=1 i i=1 i i −i
We denot−eithis game by G . EH[log(1+Γi(P (H)))] and Ai = {Pi ∈ RN :
A EH[Pi(H)]≤Pi,(cid:0)Pi(h)≥0 for all(cid:1)h∈H}.
We next assume that the ith transmitter-receiver pair has
Definition2. ApointP∗ is aNash Equilibrium(NE)ofgame
knowledgeofits incidentgainsH only.Thentherate of user
i G = (A )N ,(r )N if for each player i
i is i i=1 i i=1
r (P∗,P∗ )≥r (P ,P∗ ) for all P ∈A .
r (P ,P ), (cid:0) i i −i (cid:1) i i −i i i
i i −i
1 n ExistenceofapureNEforthestrategicgamesGA,GI and
limsupn EHi(k) EH−i(k)[log(1+Γi(Hi(k),H−i(k)))] , GD follows from the Debreu-Glicksberg-Fan Theorem ([10],
n→∞ Xk=1 (cid:2) (cid:3)(4)page no. 69), since in our game ri(Pi,P−i) is a continuous
function in the profile of strategies P = (P ,P ) ∈ A and
i −i
where Pi(H(k)) depends only on Hi(k) and EX denotes concave in Pi for GA,GI and GD.
expectation with respect to the distribution of X. Each user Definition3. Thebest-responseofplayeriisafunctionBR :
i
maximizes its rate subject to (3), we denote this game by GI. A−i → Ai such that BRi(P−i) maximizes ri(Pi,P−i),
We also consider a game assuming that each transmitter- subject to Pi ∈Ai.
receiver pair knows only its direct link gain H . This is the
ii
We see that the Nash equilibrium is a fixed point of the
most realistic assumption since each receiver i can estimate
best-responsefunction.Inthefollowingweprovidealgorithms
H and feed it back to transmitter i. In this case, the rate of
ii
user i is given by toobtainthisfixedpointforGA.InSectionVwewillconsider
G and G . Given other players’ power profile P , we use
I D −i
1 n Lagrangemethodtoevaluatethebestresponseofplayeri.The
ri(Pi,P−i),limsupn EHii(k) EH−ii(k) Lagrangian function is defined by
n→∞
[log(Xk1=+1 Γ (H ((cid:2)k),H (k)))]], (5) Li(Pi,P−i)=ri(Pi,P−i)+λi(Pi−EH[Pi(H)]).
i ii −ii
TomaximizeL (P ,P ),wesolveforP suchthat ∂Li =
where Pi(H(k)) is a function of Hii(k) only. Here, H−ii 0foreachh∈Hi .Tihus−,ithecomponentofithebestres∂pPoin(hse) of
denotesthe channelgains of all other links in the interference
playeri, BR (P ) correspondingto channelstate h is given
channelexceptH .Inthisgame,eachusermaximizesitsrate i −i
ii by
(5) under the average power constraint (3). We denote this
game by GD. BRi(P−i;h)=
We address these problems as stochastic games with the (1+ |h |2P (h))
set of feasible power policies of user i denoted by Ai and its max 0,λi(P−i)− j6=|hi |i2j j , (6)
utility by r . Let A=ΠN A . ( P ii )
i i=1 i
where λi(P−i) is chosen such that the average power con- IV. SOLVING THEVIFOR GENERALCHANNELS
straint is satisfied.
In [17], we proved that if H˜ = (I + Hˆ) is positive
It is easy to observe that the best-response of player i to semidefinite, then the fixed point iteration
a given strategy of other playersis water-filling on f (P )=
i −i P(n) =Π (P(n−1)−τF(P(n−1))), (11)
(f (P ;h),h∈H) where A
i −i
converges to a NE. This condition is much weaker than one
(1+ |h |2P (h))
f (P ;h)= j6=i ij j . (7) wouldobtainby usingthemethodsin [1]. Inthe currentsetup
i −i |h |2 we aim to find a NE even if H˜ is not positive semidefinite.
P ii
For this, we present an algorithm to solve the VI(A,F) in
For this reason, we represent the best-response of player i by general.
WF (P ). The notation used for the overall best-response
i −i
WF(P) = (WF(P(h)),h ∈ H), where WF(P(h)) = We note that a solution P∗ of VI(A,F) satisfies
(WF1(P−1;h),...,WFN(P−N;h)) and WFi(P−i;h) is as P∗ =Π (P∗−τF(P∗)). (12)
defined in (6). We use WF (P )=(WF (P ;h),h∈H). A
i −i i −i
Thus, P∗ is a fixed point of the mapping T(P) =
It is observed in [1] that the best-response WF (P ) is
i −i Π (P−F(P)). Using this fact, we reformulate the varia-
A
also the solution of the optimization problem
tional inequality problem as a non-convex optimization prob-
lem
minimize kP +f (P )k2, subject to P ∈A . (8)
i i −i i i
minimize kP−Π (P−F(P))k2,
A
Asaresultwecaninterpretthebest-responseastheprojection
subject to P∈A. (13)
of (−f (P ),...,−f (P )) on to A . We denote the
i,1 −i i,N −i i
projection of x on to A by Π (x). We consider (8), as The feasible region A of P, can be written as a Cartesian
i Ai
a game in which every player minimizes its cost function product of A , for each i, as the constraints of each player
i
kP +f (P )k2 with strategy set of player i being A . We aredecoupledin powervariables.As a result, we can splitthe
i i −i i
denote this game by G′. This game has the same set of NEs projection Π (.) into multiple projections Π (.) for each i,
A Ai
as G because the best responses of these two games are i.e., ΠA(x) = (ΠA1(x1),...,ΠAN(xN)). For each player i,
equal. We now formulate the variational inequality problem the projection operation Π (x ) takes the form
Ai i
corresponding to the game G′.
Π (x )=(max(0,x (h)−λ ),h∈H), (14)
Ai i i i
Observe that (8) is a convexoptimization problem. Given
where λ is chosen such that the average power constraint is
P , a necessary and sufficient condition for P∗ to be a i
−i i satisfied. Using (14), we rewrite the objective functionin (13)
solution of the convexoptimization problemof player i ([11],
as
page 210) is given by
kP−Π (P−F(P))k2 =
(P∗(h)+f (P ;h))(x (h)−P∗(h))≥0, (9) A
i i −i i i (P (h)−max{0,−f (P ;h)−λ })2
h∈H i i −i i
X
h∈H,i
for all xi ∈Ai. Thus, P∗ is a NE of the game G′ if (9) holds X 1+ |h |2P (h) 2
for each player i. We can rewrite the N inequalities in (9) in = min P (h), j ij j +λ
compact form as i |h |2 i
h∈H,i ( P ii )!
X
P∗+hˆ+HˆP∗ T (x−P∗)≥0 for all x∈A, (10) = (min{Pi(h),Pi(h)+fi(P−i;h)+λi})2. (15)
h∈H,i
(cid:16) (cid:17) X
where hˆ is a N -length block vector with N = |H|, the At a NE, the left side of equation (15) is zero and hence each
1 1
cardinality of H, each block hˆ(h),h ∈ H, is of length N minimum term on the right side of the equation must be zero
as well. This happens, only if
and is defined by hˆ(h) = 1 ,..., 1 and Hˆ is the
block diagonal matrix Hˆ =(cid:16)|hd1i1a|2g Hˆ(h|h)N,hN|∈2(cid:17)H with each P (h)= 0, if 1+Pj6=|ih|ihii|j2|2Pj(h) +λi >0,
block Hˆ(h) defined by n o i −1+Pj6=|ih|ihii|2j|2Pj(h) −λi, otherwise.
Here, the Lagrange multiplier λ can be negative, as the
0 if i=j,  i
[Hˆ(h)]ij = |hij|2, else. projectionsatisfies the averagepower constraintwith equality.
(|hii|2 At a NE Player i will not transmit if the ratio of total
interference plus noise to the direct link gain is more than
The characterization of Nash equilibrium in (10) corre-
some threshold.
sponds to solving for P in the variational inequality problem
VI(A,F), Wenowproposeaheuristicalgorithmtofindanoptimizer
of (13). This algorithm consists of two phases. In the first
F(P)T (x−P)≥0 for all x∈A, phase,itattemptstofindabetterestimateofapowerallocation
using the fixed point iteration on the mapping T(P) that is
where F(P)=(I+Hˆ)P+hˆ. close to a NE. For G this is algorithm (11) itself, which
A
converges to the NE when H˜ is positive semidefinite. When V. PARTIAL INFORMATION GAMES
this conditiondoes not hold, then we use it in Algorithm 1 to
Inpartialinformationgames,wecannotwritetheproblem
get a good initial point for the steepest descent algorithm of
offindinga NEas anaffinevariationalinequality,becausethe
Phase 2. We will show in Section VI that it indeed provides
best response is not water-filling and should be evaluated nu-
a very good initial point for Phase 2. For games G and G
I D merically.In this section, we show that we can use Algorithm
wewillprovidemorejustificationforPhase1byshowingthat
1 to find a NE even for these information games.
this corresponds to a better response dynamics. In the second
phase, using the estimate obtained from Phase 1 as the initial
point,thealgorithmrunsthesteepestdescentmethodtofinda A. Game GI
NE. Itis possiblethatthesteepestdescentalgorithmmaystop We first consider the game G and find its NE using
I
at a local minimum which is not a NE. This is because of the Algorithm 1. We follow on similar lines as in Sections III
non-convexnatureoftheoptimizationproblem.Ifthesteepest and IV. We write the variationalinequality formulationof the
descent method in Phase 2 terminates at a local minimum NE problem. For user i, the optimization at hand is
which is not a NE, we again invoke Phase 1 with this local
minimum as the initial point and then go over to Phase 2. We maximize r(I), subject to P ∈A , (16)
i i i
present the complete algorithm in Algorithm 1.
wherer(I) = π(h )E log 1+ |hii|2Pi(hi) .
In Section VI we providean example when H˜ is positive i hi∈I i 1+Pj6=i|hij|2Pj(Hj)
semidefinite and use the algorithmin [17] to obtain a NE. We The necessaryPand sufficienht op(cid:16)timality conditions for (cid:17)thie
convex optimization problem (16) are
also use Algorithm 1 and obtain the same NE (which will be
obtainedfromthefirstphaseitself).Nextweprovideexamples (x −P∗)T(−▽ r(I)(P∗,P ))≥0, for all x ∈A , (17)
where H˜ is not positive semidefinite. Thus the algorithm in i i i i i −i i i
[17] may not converge. The present algorithm provides the where ▽ r(I)(P∗,P ) is the gradient of r(I) with respect to
i i i −i i
NEs in just a few iterations of Phase 1 and Phase 2. power variables of user i. Then P∗ is a NE if and only if
(17) is satisfied for all i = 1,...,N. We can write the N
Algorithm 1 Heuristic algorithm to find a Nash equilibrium inequalities in (17) as
Fix ǫ>0,δ >0 and a positive integer MAX
(x−P∗)TF(P∗)≥0, for all x∈A, (18)
Phase 1 : Initialization phase
where F(P) = (−▽ r(I)(P),...,−▽ r(I)(P))T. Equation
Initialize P(i0) for all i=1,...,N. (18) is the required v1ar1iational inequalNityNcharacterization. A
for n=1→MAX do solution of the variational inequality is a fixed point of the
P(n) = T(P(n−1)) mapping T (P) = Π (P − τF(P)), for τ > 0. We use
I A
end for Algorithm1,tofindafixedpointofTI(P)byreplacingT(P)
go to Phase 2. in Algorithm 1 with TI(P).
Phase 2 : Optimization phase B. Better response
Inthissubsection,weinterpretT (P)asabetterresponse
Initialize t=1,P(t) =PMAX, I
foreachuser.Forthis,considertheoptimizationproblem(16).
loop
For this, using the gradientprojectionmethod,the updaterule
For each i, P(t+1) = Steepest Descent(P˜(t),i)
i i for power variables of user i is
where P˜(t) =(P(t+1),...,P(t+1),P(t),...,P(t)),
P(t+1) =i(P(t+1)1,...,P(t+1)i)−,1 i N Pi(n+1) =ΠAi(Pi(n)+τ▽iri(I)(P(n))). (19)
1 N
t=t+1, The gradientprojectionmethod ensures that for a given P(n),
Till kP(t)−T(P(t))k<ǫ −i
if kP(t)−P(t+1)k<δ and kP(t)−T(P(t))k>ǫ then ri(I)(Pi(n+1),P−(ni)) ≥ ri(I)(Pi(n),P−(ni)). Therefore, we can
Go to Phase 1 with P(0) =P(t) interpret P(n+1) as a better response to P(n) than P(n). As
i −i i
end if the feasible space A = ΠN A , we can combine the update
i=1 i
end loop rules of all players and write
function STEEPEST DESCENT(P(t),i)
P(n+1) =Π (P(n)−τF(P(n)))=T (P(n)). (20)
A I
▽f(P(t))=(∂∂Pfi((Ph))|P=P(t),h∈H) Thus,thePhase1ofAlgorithm1istheiteratedbetterresponse
where f(P)=kP−T(P)k2 algorithm.
for h∈H do
evaluate ∂∂Pfi((Ph))|P=P(t) using derivative approxima- ThenCoitnsimidpelrieasfitxheadt, pgoivinetnPP∗∗o,f Pth∗e bisetaterlorceaslpoonpsteimTuIm(Po)f.
tion −i i
(16) for all i. Since the optimization (16) is convex, P∗ is
end for i
P(t+1) =Π (P(t)−γ ▽f(P(t))) also a global optimum. Thus given P∗−i, P∗i is best response
i Ai i t for all i and hence NE is also a fixed point of the better
return P(t+1)
i response function. This gives further justification for Phase
end function
1 of Algorithm 1. We could not provide this justification for
G when H˜ is not positive semidefinite. Indeedwe will show
A
in the next section that in such a case Phase 1 often provides
5.5
a NE forG and G (forwhich also Phase 1 providesa better
I D
response dynamics; see Section V-D below) but not for GA. 5
4.5
C. Lower bound
e4
t
Inthe computationofNE,eachuseri isrequiredto know ra
3.5
the powerprofileP of all otherusers.We now givea lower m
−i u
bound on the utility r(I) of player i that does not depend on S3 NE with complete knowledge
i NE with incident gain knowledge
other players’ power profiles. 2.5 NE with direct link gain knowledge
s(Q) with incident gain knowledge
Wecaneasilyprovethatthefunctioninsidetheexpectation 2 s(Q) with direct link gain knowledge
in r(I) is a convex function of P (h ) for fixed P (h ) using
the ifact that ([16]) a function f :jKj⊆ Rn → R isi coinvex if 1.50 2 4 Aver6age tr8ansmit10 SNR c12onstr1a4int 16 18 20
and only if
Fig.1. Sumratecomparison atNashequilibrium pointsforExample1.
d2f(x+ty)
≥0, 5.5 NE with complete knowledge
dt2
NE with incident gain knowledge
for all x,y ∈K and t∈R is such that x+ty∈K. Then by 5 NE with direct link gain knowledge
Jensen’s inequality to the inner expectation in r(I), 4.5 s(Q) with incident gain knowledge
i s(Q) with direct link gain knowledge
e4
ri(I) = hXi∈Iπ(hi)E"log 1+ 1+ |hj6=iii|2|hPiij(|h2Pi)j(Hj)!# Sum rat3.35
|Ph |2P (h )
≥ π(hi)log 1+ 1+ ii|h |i2Ei[P (H )] 2.5
hXi∈I j6=i ij j j ! 2
|hP|2P (h )
= hXi∈Iπ(hi)log 1+ 1+ iij6=ii|hiji|2Pj!. (21) 1.50 2 4 Aver6age tr8ansmit10 SNR c12onstr1a4int 16 18 20
Fig.2. Sumratecomparison atNashequilibrium pointsforExample2.
The above lower bound r(I) (P )Pof r(I)(P ,P ) does not
i,LB i i i −i and Jensen’s inequality as in (21). The optimal solution for
depend on the power profile of players other than i. We can
the lower bound is the water-filling solution
choose a power allocation P of player i that maximizes
i
ri(,IL)B(Pi). It is the water-filling solution given by P (h )= max 0,λ − 1+ j6=iE[|Hij|2]Pj .
i ii i |h |2
1+ |h |2P ( P ii )
P (h )= max 0,λ − j6=i ij j .
i i i |h |2
( P ii ) VI. NUMERICAL EXAMPLES
Let P∗ = (P∗,P∗ ) be a NE, and let P† be the maxi- InthissectionwecomparethesumrateachievedataNash
i −i i
mizer for the lower bound r(I) (P ). Then, r(I)(P∗,P∗ )≥ equilibrium under the different assumptions on the channel
i,LB i i i −i
r(I)(P ,P∗ ) for all P ∈ A , in particular for P = P†. gainknowledge,obtainedusingthealgorithmsprovidedabove.
i i −i i i i i In all the numerical examples, we have chosen τ = 0.1 and
Thus, ri(I)(P∗i,P∗−i) ≥ ri(I)(P†i,P∗−i). But, ri(I)(P†i,P∗−i) ≥ thestepsizeinthesteepestdescentmethodγt =0.5 for t=1
r(I) (P†). Therefore, r(I)(P∗,P∗ ) ≥ r(I) (P†). But, in andisupdatedafter10iterationsasγ = γt . We choose
i,LB i i i −i i,LB i t+10 1+γt
general it may not hold that r(I)(P∗,P∗ )≥r(I)(P†,P† ). a 3-user interference channel for Examples 1 and 2 below.
i i −i i i −i
For Example 1, we take H = {0.3,1} and H =
d c
D. Game GD {0.2,0.1}.We assume that all elements of Hd,Hc occur with
equalprobability,i.e.,withprobability0.5.Now,theH˜ matrix
We now consider the game GD where each user i has is positive definite and there exists a unique NE. Thus, the
knowledge of only the corresponding direct link gain Hii. fixed point iteration (11) converge to the unique NE for GA.
In this case also we can formulate the variational inequality Algorithm 1 also converges to this NE not only for G but
A
characterization. The variational inequality becomes also for G and G .
I D
(x−P∗)TFD(P∗)≥0, for all x∈A, (22) We compare the sum rates for the NE under different
assumptions in Figure 1. We have also computed Q = P†
where FD(P)=(−▽1r1(D)(P),...,−▽NrN(D)(P))T. We use thatmaximizesthecorrespondinglowerbounds(21),evaluated
Algorithm1 to solvethe variationalinequality(22) by finding thesum rate s(Q)andcomparedtothe sum rateat a NE.The
fixed points of TD(P) = ΠA(P−τFD(P)). Also, one can sum rates at Nash equilibria for GI and GD are close. This is
show that as for TI, TD provides a better response strategy. becausethevaluesofthecrosslinkchannelgainsarecloseand
We can also derive a lower bound on r(D) using convexity hence knowing the cross link channel gains has less impact.
i
allocation.
7
NE with complete knowledge For all the three examples, for GI and GD, Phase 1 itself
6 NE with incident gain knowledge provides the NE.
NE with direct link gain knowledge
s(Q) with incident gain knowledge We have run Algorithm 1 on many more examples and
5 s(Q) with direct link gain knowledge
e found that it computed the NE, and for G and G the Phase
t I D
ra 1 itself provided the NE.
4
m
u
S
3 VII. CONCLUSIONS
We have considered a channel shared by multiple
2
transmitter-receiver pairs causing interference to each other.
We have modeled this system as a non-cooperativestochastic
10 2 4 Aver6age tr8ansmit10 SNR c12onstr1a4int 16 18 20 game.Differenttransmitter-receiverpairsmayormaynothave
channelgaininformationaboutotherpairs’channelgains.Ex-
Fig.3. SumratecomparisonatNashequilibrium points forExample3.
ploiting variational inequalities, we provide an algorithm that
We now give a couple of examples in which H˜ is not obtains NE in the various examples studied quite efficiently.
positive semidefinite and hence fixed point iteration (11) fails
to converge to a NE but Algorithm 1 converges to a NE for REFERENCES
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