Wireless Operators
in a Shared Spectrum
Mark Felegyhazi, Jean-Pierre Hubaux
EPFL, Switzerland
Infocom’06, Barcelona, Spain
April 26, 2006
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measured power [dB]
Spectrum utilization
frequency [GHz]
D. Cabric, S. M. Mishra, D. Willkomm, R. W. Brodersen, and A.
Wolisz, “A Cognitive Radio Approach for Usage of Virtual
Unlicensed Spectrum,” 14th IST Mobile and Wireless
Communications Summit, June 2005.
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Problem formulation
•
•
•
•
today, cellular operators own separate frequencies
operate wireless / cellular networks in an unlicensed spectrum
BUT, problem of interference
power control of the pilot signal
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System model (1/2)
• two operators: A and B
• set of base stations: BA and BB
• base stations are placed on the vertices of a
grid
• each base station of A has the same radio
range rA (relaxed later), same for B
• base stations emit pilot signals on the same
channel, with the radio ranges: rA, rB
• full coverage by combination of the two
operator’s coverage
• maximum power limit PMAX → RMAX
• if
2
rA  rB  RMIN 
2
d
• devices have omnidirectional antennas
4
System model (2/2)
• a set of users uniformly distributed in the
area
• free roaming
• users attach to the base station with the best
pilot signal


Pi  g iu


max 
Pi
N 0   Pj  g ju 


j


where the channel gain: g iu 
1
d iu2
• operators want to cover the largest area
with their pilot signal
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Power control game
•
•
•
•
static game G = (P, S, U)
operators → Players
pilot signal radio range → Strategy
Utility: coverage area of their own
pilot signal minus the interference
area
U i  Oi   i  Yi
where γi is the cooperation
parameter of player i:
• cooperativeness
• agreement
• power price
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Definitions
Let si , s j , si' S
Best response of player i to strategy sj of player j:


BRi ( s j )  si  S : U i ( si , s j )  U i ( si' , s j ), si'  S
Nash equilibrium:
sˆi  BRi (sˆ j ), i  A, B
Nash equilibrium strategies are mutual best responses to each other.
Pareto-superiority: A strategy profile ( si , s j ) is Pareto-superior to a
'
'
strategy profile ( s i , s j ) if for any player i we have:
Ui (si , s j )  Ui (si' , s'j )
with strict inequality for at least one player.
M. Felegyhazi and J.-P. Hubaux, “Game Theory in Wireless
Networks: A Tutorial,” EPFL Technical report LCA-REPORT2006-002, April 2006.
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 LIM 1
 LIM 2
 LIM 3
d2
2
  RMAX
 d2


1
 0.46
 1
2 4  2  2
8  8   2
  0.59
2
 1.75
 2
best response of player i
Best response values
radio range of player j (rj)
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 A   LIM 1
 B   LIM 1
radio range of player i (ri)
Mutual best responses = Nash equilibria
Nash equilibrium
radio range of player j (rj)
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Pareto-superior Nash equilibria
• rA  rB  RMAX (NEMAX)
• rA  rB  RMIN
(NEMIN)
2
2
rA2   2 d  rB    2 d   d 2  2drB  rB2
2
2 

 
(NEMIN,A,B)
• rA  d , rB  0
(NEMIN,A,B0)
 A   LIM 1
 LIM 1   A   LIM 2
 LIM 2   A   LIM 3
 LIM 3   A
NEMAX
no NE
NEMIN , B , A
NEMIN , B , A0
 LIM 1   B   LIM 2 no NE
no NE
NEMIN , B , A
NEMIN , B , A0
NEMIN , A, B
NEMIN , A, B
NEMIN , B , A
NEMIN , A, B 0
NEMIN , A, B 0
 B   LIM 1
 LIM 2   B   LIM 3 NEMIN , A, B
 LIM 3   B
NEMIN , A, B 0
NEMIN , B , A0
NEMIN , A, B NEMIN
NEMIN , B , A
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Repeated game (1/3)
Punisher strategy: Play RMIN in the first time step.
Then for each time step:
– play RMIN, if the other player played RMIN
– play RMAX for the next ki time steps, if the other
player played anything else
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Repeated game (2/3)
• cooperation utility: Ci  Ui  rA  rB  RMIN 
• deviation gain:
Di  Ui  rA  rB  RMAX 
utility of player j (Uj)
• defection utility:
Gi  U i  rA  BRi  RMIN  , rB  RMIN 
time steps (t)
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Repeated game (3/3)
Cooperation is enforceable: A Nash equilibrium based on RMIN is
enforceable using the Punisher strategy if:
Gj  Dj
 1     1 where ω is the discounting factor
C j  Dj
If the above condition holds, the punishment interval is defined by:
 Gj  Dj

ki  log 1 
 1    1
 C j  Dj 



Note: Similar result to the Folk-Theorem
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Generalization of the problem
→ base station might have different positions and ranges
Hardness result: Finding the maximum utility of a player
for general values of radio ranges is NP-complete.
Corollary: Finding Nash equilibria in the power control
game for general values of radio ranges is NP-complete.
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Conclusion
• Coexistence is a main problem in shared spectrum networks
• Power control of the pilot signal to cope with interference
• Single stage game:
– various Nash equilibria in the grid scenario, depending on γA and γB
• Repeated game:
– RMIN (cooperation) is enforceable with punishments
• General scenario = arbitrary ranges:
– the problem is NP-complete
http://winet-coop.epfl.ch
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