Distributed Power Allocation
Game for Uplink OFDM Systems
Gaoning He, Sophie Gault - MOTOROLA LABS
Mérouane Debbah - SUPÉLEC
Eitan Altman - INRIA INSTITUTE
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Outline
1.
System model
2.
Research background
3.
Distributed power allocation (DPA) game
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System model
• System settings
– The uplink of OFDM system in a single cell
– K users (Tx) are communicating with BS (Rx) using N subcarriers
– Transmitters (Tx) have partial (or incomplete) channel state information (CSI).
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System model
• A simplified model (MAC)
– K=2, N=1.
– gk(t) is the uplink channel gain of user k.
– Assume gk(t) has finite discrete channel states,
following a certain distribution.
g1  t 
g2  t 
MT1
BS
• Different channel state information (CSI) levels at Tx:
– Complete CSI
– Partial CSI
– Statistical CSI
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MT2
System model
• Individual rationality → optimization problem with selfish objective
– Channel gain matrix G  R K  N, where each element is gk, i
– Selfish objective of user k (average capacity)


N

pk ,i g k ,i


Ck  E G  log 2 1 
K
 i 1

2

    p j ,i g j ,i
j 1, j  k








– Average power constraint of user k
N
E
i 1
G
 pk ,i   Pk
pk ,i  0, i
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Research background
•
Complete CSI at Tx → The waterfilling game [1]
–
Optimization problem for (selfish) user 1 (which is similar for user 2):



p (g , g )g
max E g1 , g2 log 2 1  2 1 1 2 1  
p1 ( g1 , g 2 )
   p2 ( g1 , g 2 ) g 2  

s.t. E g1 , g2  p1  g1 , g 2    P1 & p1  g1 , g 2   0
–
Results in an interesting equilibrium (individual rationality=social rationality):


2





E g , g
2
 p1  g1 , g 2    1 
 , when g1  g 2
 1 2
g1 
1


where 1 and 2 




 
1 are decided from 
2 
 , when g 2  g1
 p2  g1 , g 2    2 
E g1 , g2
g
2

2 


•


2 
2 
 1 
 g1  g 2   P1
g1 
1 





1 
2 
 2 
 g 2  g1   P2
g2 
2 



Statistical CSI at Tx → Uniform power allocation (in the time domain)
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Research background
• Partial CSI at Tx → Distributed power allocation (DPA) game
– The optimization problem for user 1, maximizing its own average capacity C1
(and similarly for user 2):



p (g )g
maxC1  max E g1 , g2 log 2 1  2 1 1 1  
p1 ( g1 )
   p2 ( g 2 ) g 2  

s.t. E g1  p1  g1    P1 & p1  g1   0
– From Lagrangian duality, user 1’s best strategy p1(.) is given by equation:

 1
g1
E g2  2
  ,where 1 is chosen such that E g1  p1  g1    P1
   p1  g1  g1  p2  g 2  g 2  1
– Note that user 2’s strategy p2(.) is unknown to user 1, and user 1’s strategy can
not depend on the particular realization of g2.
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DPA game
• A non-cooperative game model

K , P uk kK , uk kK 
– Player set: K 1, 2 , where K K  2.
– Action set: P u1 , P u2  , where P u1  p1  a1  , p1  a2 , P u2  p2  a1  , p2  a2 .
– Utility (payoff) function set: u1 , u2  , denoted by u1 (P u1, P u2 ) and u2 (P u1, P u2 ).
• Intuitively, u1,u2 are defined as the average capacity of user 1 and 2
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DPA game
• What the game brings us?
– Each user can decide his “best strategy” relying on the assumption of
rationality of the other user.
– Results in Nash equilibrium (NE): a “stable” situation, where both (selfish)
users are satisfied with the choices of their strategies P u1 , P u2 , if given any other
strategies P u1 , P u2 
u1  P1 , P2   u1  P1 , P2 





u2  P1 , P2   u2  P1 , P2 
– The social performance of NE might be inefficient.
• Equilibrium analysis [2]-[5]
– Theorem 1: A NE exists in the game.
– Theorem 2: The game has a unique equilibrium. (proofs refer to the paper)
• Can we improve the performance of NE?
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DPA game
• Improve the performance of NE for the symmetric case
– Assume the channel state gk (k=1,2) can be a1 and a2 with probability ρ1 and ρ2
respectively.
– Assume both users coordinate to the same strategy set, i.e. P u1  P u2 P
More precisely, p1 (a )  p2 (a ) p(a ),  1, 2 .
– Assume both users have the same maximum power constraint, i.e. P1  P2 P
– Then the problem goes to find the optimal strategy set P  that maximizes the
average capacity for user k.
– The best strategy is given by:
 P
0,  ,
 p  a1  , p  a2    P 2 
 , 0 ,
 1 
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a2 1  a1 2
a2 1  a1 2
10
Numerical results
Left
Right
a1
1
1
a2
3
10
ρ1
0.5
0.5
ρ2
0.5
0.5
P
1
1
σ2
0.1
0.1
Table of parameters used in the simulation
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Numerical results
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Conclusions and future works
• We study the power allocation problem in the uplink of OFDM system with
partial knowledge at the transmitters. Most of the preview works on power
allocation game was achieved under assumption of complete CSI at the
transmitters.
• We introduced a theoretic approach to provide a natural setting for the
construction of efficient distributed algorithms.
• We establish the existence and uniqueness of Nash equilibrium
• For the symmetric case, we found the efficiency of NE can be improved by
forcing all users coordinating to the same optimal strategy set.
• Future work will consider this problem to the case of MIMO and correlated
equilibrium.
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Bibliography
•
•
•
•
•
[1] L. Lai and H. E. Gamal, “The Water-Filling Game in Fading Multiple Access Channels,”
IEEE Trans. on Information Theory, Dec. 2005.
[2] G. Debreu, “A Social Equilibrium Existence Theorem,” in Proc. Nat. Acad. Science, vol. 38,
pp. 886893, 1952.
[3] K. Fan, “Fixed Point and Minimax Theorems in Locally Convex Topological Linear Spaces,”
in Proc. Nat. Acad. Sciences, vol. 38, 1952, pp. 121126.
[4] I. L. Glicksberg, “A Further Generalization of the Kakutani Fixed Point Theorem with
Application to Nash Equilibrium Points,” in Proc. Amer. Math. Soc., vol. 3, 1952, pp. 170174.
[5] D. Fudenberg and J. Tirole, Game Theory. MIT Press, Cambridge, MASS, London,
England, 1991.
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Thank you !
Questions?
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