Mitigating Self-interference among IEEE 802.22 Networks:
A Game Theoretic Perspective
Swastik Brahma and Mainak Chatterjee
Electrical Engineering and Computer Science
University of Central Florida
Orlando, FL 32816
Email: {sbrahma, mainak}@eecs.ucf.edu
Abstract—In this paper, we use game theory to mitigate selfinterference among cognitive radio based IEEE 802.22 networks
such that these networks can efficiently co-exist. When a network
experiences interference, it can adopt either one of two choicesswitch to a new band hoping to find a non-interfering one, or
stay with its current band hoping that the interfering network(s)
will move away to a new band. We model the spectrum band
switching process as an infinite horizon repeated game where
the aim of each network (player) is to find a channel void of
interference from it’s neighboring networks incurring minimal
cost. We investigate both pure and mixed strategy solution space
of the game and show that pure strategy solution of the game
is infeasible. Thus a mixed strategy is proposed which achieves
subgame-perfect Nash Equilibrium. Simulation results reveal that
using the proposed strategies, each network can find a clear
channel quickly and at the same time incur a low cost.
I. I NTRODUCTION
Recent experimental studies have shown that licensed radio
spectrum is highly under–utilized and that the usage is space
and time dependent [1],[2],[4], [7]. Based on these findings it
can be inferred that the radio spectrum has various spectrum
holes, i.e, band of frequencies assigned to a primary (licensed)
user, but, at a particular time and specific geographic location,
the band is not being utilized by that user. Utilization of
such unused bands can be significantly improved by making
it possible for a secondary user (who is not being serviced) to
access a spectrum hole at a given location and time.
Alongside, regulatory provisions are also being made to harness the under-utilized spectrum. For example, the cognitive
radio based IEEE 802.22 standard [5] has been defined to
make use of the sub900 MHz TV bands that have been opened
up due to the analog to digital transition of TV broadcast [8].
However, it is mandated that the unlicensed IEEE 802.22
devices detect and avoid interference with the licensed users
in a timely manner [6].
A typical 802.22 network consists of a base station (BS) and
consumer premise equipments (CPEs), just like a cellular architecture. The aim of these networks is to use spectrum bands
dynamically through incumbent sensing and avoidance. The
elements in these networks (i.e., cognitive radio enabled BSs
and CPEs) continuously perform spectrum sensing, dynamically identifying unused spectrum and operating on the same.
Upon detecting incumbents, these devices are required to
This research was sponsored by the Air Force Office of Scientific Research
(AFOSR) under the federal grant no. FA9550-07-1-0023.
switch to another channel or mode. Such networks, deployed
and operated by competing wireless service providers have to
self-coexist among themselves by accessing different parts of
the available spectrum. Since there is no coordination among
the networks in accessing the spectrum bands, each network
is prone to interference from its neighboring networks. Under
such circumstances even if appropriate measures are taken
at the air interface level, self-interference may drastically
degrade the performance of the network. Thus, different from
other IEEE 802 standards, where self-coexistence is not a
problem, it is required for IEEE 802.22 networks to take a
proactive approach and incorporate self-coexistence protocols
and algorithms as part of the standards definition.
In this paper, we examine the self-coexistence issues of
IEEE 802.22 networks from a game theoretic perspective. We
consider a system of multiple IEEE 802.22 networks (players),
operated by different service providers, some of which might
overlap with each other thereby becoming potential interferers.
If two or more overlapped networks operate using the same
spectrum band then interference will occur among them and
QoS of their users will degrade. Thus, each network tries
to access and use a spectrum band1 that is not in use by
any primary incumbent and also void of interference from its
neighboring2 802.22 networks. In this paper we extend our
previous work in [3], investigating subgame perfect equilibria
for the game and thereby generalize our analysis of the
problem. We model the process of allocating interference free
channels to each network as an infinite horizon simultaneous
move repeated game. When a player experiences interference
at any stage it can adopt either one of two strategies– ‘switch’
to another channel in the hope of finding a clear one or ‘stay’
with its current channel hoping that its interferers will move
away. We investigate both pure and mixed strategy solution
space of the game and show that pure strategy solution of the
game is infeasible. Thus, a mixed strategy subgame-perfect
Nash Equilibrium (SPNE) is proposed for the players. We
evaluate the convergence cost and convergence time of these
strategies through simulations.
More specifically, the main contributions of this paper can
be summarized as follows.
1 Throughout this paper, we use the words “channel” and “band” interchangeably unless explicitly mentioned otherwise.
2 Two networks are said to be neighbors if they are in the interference range
of each other. Two neighboring networks are said to interfere with each other
if the operate using the same spectrum band.
1) We model the problem of mitigating self-interference
among IEEE 802.22 networks as an infinite horizon
repeated game, where the networks experiencing interference from each other always think that they may not
find a clear channel in current stage and the game might
be extended with some probability.
2) We propose a mixed strategy subgame-perfect Nash
equilibrium for the game, such that no player in no single
stage can change his strategy unilaterally and thereby
gain from it.
3) Simulation results suggest that using our proposed solution, a system of IEEE 802.22 networks converges
quickly with each network incurring a minimal cost.
II. P ROBLEM D ESCRIPTION AND G AME F ORMULATION
We assume that there are N IEEE 802.22 networks (operated by N different wireless service providers) competing for
one of M orthogonal spectrum bands not in use by primary
incumbents. The IEEE 802.22 networks can be partially or
completely overlapped geographically (i.e., coverage area)
with each other. If two or more networks are in the interference
range of each other (i.e, either partially or completely overlapped), they can not use the same spectrum band; otherwise
QoS of the users of all the networks will suffer. In this
scenario, we model the dynamic channel switching process
as an infinite horizon repeated game where the aim of each
network is to acquire a spectrum band free of interference
from its neighboring networks. We assume that the only
control information needed by a network for participating
successfully in this game is the number of neighbors it is
currently interfering with. This can be known from the beacons
broadcasted by each of the IEEE 802.22 networks in Foreign
Beacon Period (FBP) [5].
At the beginning of the game, each network dynamically
chooses one of the M allowable spectrum bands for its
operations. Time is divided into stages of the game. At each
stage of the game, a network can either experience interference
or has acquired a channel void of interference from it’s
neighbors. If a network perceives interference at any stage,
then the QoS of it’s users will degrade. Thus each network has
to pay a price when it experiences interference. Let this cost be
CI . Each of the networks experiencing interference will then
have to take decisions regarding whether to ‘stay’ with the
channel they currently have (hoping the interferers will move
away) or ‘switch’ to a new channel. It is assumed that this
decision is taken simultaneously by the networks experiencing
interference at each stage. When a network switches to a new
channel, it will have to reallocate the new spectrum among its
users. This also entails a cost. Let this cost be CS . We assume
that CS < CI , otherwise it does not make sense for a network
to consider switching to a new channel when interfered. The
game ends when all the networks are successful in acquiring
a clear spectrum band, and is re-initiated if primary users
(TV) starts transmission using IEEE 802.22 occupied band(s),
and thus the spectrum usage report changes for one or more
networks. In such a case, the IEEE 802.22 network(s) involved
will again try to access new band(s).
Next we formally present the stage game and the repeated
game. Without loss of generality, we focus our attention on
a particular network i ∈ N . Due to homogeneity of the
networks, the same reasoning applies to all other networks.
A. Stage Game
As discussed above, if interfered at any stage of the game,
network i ∈ N has the binary strategy set of ‘switching’ to
another channel or ‘staying’ with it’s current band. Using game
theoretic notation, the binary strategy set for network i can be
represented as:
Si = {switch, stay}
(1)
To generalize, we assume the existence of strategy sets
S1 , S2 , · · · , SN for the networks 1, 2, · · · , N . In this game, at
every stage, if network 1 chooses strategy s1 ∈ S1 , network
2 chooses strategy s2 ∈ S2 and so on, we can describe such
a set of strategies chosen by all N networks as one ordered
N -tuple (strategy profile), s = {s1 , s2 , · · · , sN }. The set of all
such strategy profiles is called the space of strategy profiles
S = S1 × S2 × · · · × SN .
Thus we can describe our stage game by the tuple g
= (k, S, c) where, k denotes the set of players (competing
networks), S denotes the space of strategy profiles and a vector
c of von Neumann-Morgenstern utility functions defined over
S. Thus, ci : S → R.
B. Repeated Game
When a set of networks (players)3 experiences interference
from each other, it is possible that using their {switch, stay}
strategy set, the networks do not find a clear channel in one
stage. Thus the stage game g might have to be played a number
of times before each of the players finds a clear channel.
Moreover, at each stage of the game, the players know that
they may not find a clear channel in the current stage and thus
the game may be extended with some probability. Also, in each
stage, all the players have been assumed to make their decision
simultaneously. Thus we model the process of finding a clear
channel for each of the networks experiencing interference as
an infinite horizon simultaneous move repeated game. We call
this repeated game G. The payoff or cost incurred by player
i ∈ N from playing the repeated game G is defined as the
sum of player i’s payoff at each individual stage of G until it
finds it’s clear channel. Let Ci be this payoff (cost). Thus,
Ci =
K
X
ci (st )
(2)
t=1
where, st ≡ (st1 , · · · , stN ) are the strategies that are played in
stage t. K is the stage at which player i obtains his clear
channel assignment with which he can continue operating
efficiently thereafter until unless the spectrum usage report
changes, i.e, M changes.
With the strategy set and cost functions defined, we now
seek to develop a mechanism of ‘switching’ or ‘staying’ such
that each network can find a channel void of interference
incurring as low a cost as possible.
3 We
use the term ‘network’ and ‘player’ interchangeably
III. S TAGE G AME A NALYSIS
In this section we analyze in detail the stage game g
defined in section II-A. In section IV we will investigate the
overall repeated game and also present the algorithm that the
networks invoke when experiencing interference. We assume
that all players are rational and pick their strategy keeping
only individual cost optimization in mind at every stage of
the game. We intend to find if there is a set of strategies
with the property that no network can benefit by changing
it’s strategy unilaterally while the other networks keep their
strategies unchanged (Nash equilibrium).
A. Exploring Pure Strategy Space
We start with the pure strategy space played by all the
networks. To simplify investigation of Nash equilibrium with
pure strategy, we consider the game with two players i and j
coexisting on one band. The game is represented in strategic
form in Table I. Each cell of the table corresponds to a possible
combination of the strategies of both players and contains a
pair representing the costs of players i and j, respectively.
Recall that CS is the cost incurred by a network when it
switches to a new band and CI in the price paid by a network
experiencing interference in terms of its reduced QoS. Also
CS < CI and M > 1 is the number of available channels.
i\j
Switch
Stay
Switch
(CS +
1
C , CS + M1−1 CI )
M −1 I
(0, CS )
Stay
(CS , 0)
(CI , CI )
TABLE I
PAYOFF M ATRIX FOR NETWORKS i AND j
For
we see that the cost
³ the strategy {switch, switch},
´
is CS + M1−1 CI , CS + M1−1 CI instead of (CS , CS ). This
is because when both players i and j switch4 , there is a
probability that both of them choose the same channel, thereby
continuing to interfere each other in the next stage of the
game. This probability is M1−1 . We thus give the expected
cost corresponding to the strategy {switch, switch} in table I
instead of an absolute cost.
As is evident from table I, the game has two pure strategy
Nash equilibriums – one corresponding to the strategy profile
{switch, stay} and the other corresponding to the profile
{stay, switch}. These two cases have been shown in boldface
in the table. In both of these cases neither player can reduce
his cost by playing a different strategy if the other player plays
his part.
Note, however, that neither player has a strictly dominating
strategy and the game is not solvable, i.e, NE’s are not
achievable, by iterated strict dominance. If everyone uses the
same pure strategy, then though the stage game has pure
strategy NE, having one of the NE strategy profiles as an
outcome of the game is not possible. Thus, to solve the game
at hand we must permit each player to use mixed strategy
where a choice is made with a particular probability. Next,
we investigate the mixed strategy solution space of the game.
4 When a player ‘switches’ it randomly selects a channel from M − 1
available channels - it cannot select the channel it is currently residing on.
B. Exploring Mixed Strategy Space
We now expand each players possible choices by including
mixed strategies. A player will ultimately execute exactly
one of his pure strategy choices which is determined randomly by a probability distribution over his finite strategy set,
{switch, stay}. Formally, we define the mixed strategy space
of player i as:
Simixed = {(switch = p), (stay = (1 − p))}
(3)
Here, player i chooses the strategy ‘switch’ with probability
p and chooses the strategy ‘stay’ with probability (1 − p).
We now need to find what values of (p, 1 − p) tuple will
allow each of the networks experiencing interference from
each other to find a clear channel, such that in no stage no
player has an incentive to change it’s strategy unilaterally
and at the same time incurring as low a cost as possible. For
simplicity of exposition we will first analyze the 2-player
case where network i experiences interference from only one
other network. We will then generalize it to the (n + 1)-player
scenario where network i faces interference from n other
networks (i.e, has n opponents) where n ∈ [1, N − 1].
2-Player Game: Let p be the probability that player j
‘switches’ (and 1 − p the probability that j ‘stays’). For player
i to play a mixed strategy, i must be indifferent between
‘switching’ and ‘staying’. In other words, the expected cost
of i when it chooses to ‘switch’ must be same with that of
] and E[cstay
when it chooses to ‘stay’. Let E[cswitch
] be the
i
i
expected cost of i choosing to ‘switch’ and ‘stay’ respectively.
From table I, E[cswitch
] can be written as:
i
·
¸
1
switch
E[ci
] = p CS +
CI + (1 − p)CS
(4)
M −1
Similarly, E[cstay
] is given by:
i
E[cstay
] = p × 0 + (1 − p)CI
i
(5)
] and E[cstay
As discussed above, E[cswitch
] must be same for
i
i
player i to randomize between his two pure strategies. Thus,
equating equations (4) and (5) we get:
·
¸
1
p CS +
CI + (1 − p)CS = (1 − p)CI
(6)
M −1
Solving the above equation for p, we get:
¶µ
¶
µ
1
CS
1−
p= 1−
CI
M
(7)
Thus, if player j uses the probability5 given by equation (7)
to ‘switch’, then player i does equally well if he plays any
one of his two pure strategies. In a similar way we can show
that player i has to ‘switch’ using the probability given by
equation 7 so that j becomes indifferent between ‘switching’
and ‘staying’. Thus the mixed strategy for
´ ¡ NE ¢is
³ achieving
1
S
1− M
.
that both players switch with probability 1 − C
CI
5 Note that, since we consider C < C and M > 1, the value of p given
S
I
by equation (7) lies in (0,1)
1
E[cswitch
]=
(8)
i
) #
"
(
µ
¶
¶
µ
n
k
X n
M −2
CI
pk (1 − p)n−k CS + 1 −
k
M −1
k=0
¡ ¢
where, nk pk (1 − p)n−k is the probability that k opponents
³
´k
M −2
of player i ‘switch’ from his n opponents. 1 − M
is
−1
the probability that, when k opponents switch, at least one
of them chooses (randomly) the same channel that player i
selects thereby continuing to interfere with i in the next stage.
Now, when player i chooses to ‘stay’, the strategies taken by
the opponents of i can again result in any k ∈ [0, n] opponents
of i choosing to ‘switch’. Thus, we can write E[cstay
] as:
i
µ
¶
n−1
X n
E[cstay
]
=
pk (1 − p)n−k CI
(9)
i
k
k=0
Note that in equation (9), k goes from 0 to n − 1 because
when all n opponents ‘switch’, the cost for player i is zero.
As discussed above, E[cswitch
] and E[cstay
] must be same for
i
i
player i to randomize between his two pure strategies. Thus,
equating equations (8) and (9) we get:
·
½
³
´k ¾ ¸
Pn ¡n¢ k
M −2
n−k
CS + 1 − M −1
CI
k=0 k p (1 − p)
Pn−1 ¡n¢ k
= k=0 k p (1 − p)n−k CI (10)
Simplifying equation (10) and using binomial theorem, we get:
µ
¶n
p
CS
1+
− pn −
=0
(11)
1−M
CI
In equation (11), since CS /CI > 0, we have:
M −1
<1
(12)
M
Equation (11) can be easily solved using numerical methods.
We use the bisection method to solve this equation. For any
values of N and M, we find that p has a finite value in the
range (0, 1)6 thus proving the existence of mixed strategy Nash
equilibrium point even in the generalized (n + 1)-player case.
p <
C. Discussion
Note that³in equation
´ ¡ (11), if¢ we put n = 1 (2 player case),
CS
1
we get p = 1 − CI 1 − M
, which tallies with the expression for the ‘switching’ probability derived in equation (7) for
the 2-player case.
6 Note
that p 6= 0 since CS /CI < 1
1
20 available channels
30 available channels
40 available channels
0.9
0.8
Switching Probabiltiy (p)
(n + 1)-Player Game: Let p be the ‘switching’ probability
of each one of player i’s n opponents (i.e, networks which
are interfering with i). Similar to the arguments for the 2player case, for player i to be indifferent between choosing
it’s two pure strategies, the expected cost when it chooses to
‘switch’ must be same with that of when it chooses to ‘stay’.
Let E[cswitch
] and E[cstay
] be the expected cost of i choosing
i
i
to ‘switch’ and ‘stay’ respectively. When player i chooses to
‘switch’, the strategies taken by the opponents of i can result
in any k ∈ [0, n] opponents of i choosing to ‘switch’. Thus,
we can write E[cswitch
] as:
i
Switching Probabiltiy (p)
0.9
0.7
0.6
0.5
0.7
0.6
0.5
10 Opponents
20 Opponents
30 Opponents
0.4
0.4
0
0.8
5
10
Number of Opponents (n)
15
20
0
50
100
Number of Channels (M)
150
200
(a)
(b)
Fig. 1. Switching probability p versus a) number of opponents n; b) number
of available channels M
Variation of p with n: Figure 1(a) shows how switching
probability p varies with the number of opponents (n) of
player i as per equation (11) for a fixed M . Here CS = 4
and CI = 6 (CS < CI ) for all three cases. As can be
seen from the graph, with increase in number of opponents
the ‘switching’ probability of i increases initially. This is
because, ‘staying’ for player i is justified only if all of
his opponents ‘switch’ to another channel. However, as the
number of opponents of i increases, the probability that all
of his opponents will ‘switch’ decreases. Thus player i thinks
that he is better off increasing his switching probability with
increasing n. As p increases with n, the number of players
that switches increases. For a fixed M , as the number of
players switching increases, the probability that if i switches
at least one of his opponents will choose the same channel as
i thereby continuing to interfere with him again in the next
stage also increases. Thus, as n approaches M , after a certain
point, i’s gain from using a higher ‘switching’ probability
when playing against a larger number of opponents starts
getting dominated by his fear of interfering with some of
his opponents in the next stage even after ‘switching’. Thus,
after crossing this threshold, player i starts decreasing his
‘switching’ probability. This trend can be seen in all the three
graphs in figure 1(a). When more channels are available, the
point where i starts decreasing p increases. Thus, for a given
n and M , the p given by equation (11), makes the optimal
number of players switch from among the (n + 1) players.
Variation of p with M : From figure 1(b) we can see that,
as the number of available channels (M ) is increased for
a given number of opponents, the ‘switching’ probability
increases. This is because as M increases, if two players
‘switch’, the probability that they choose the same channel
after switching decreases. Thus player i, goes on increasing
p with M . Note that when the number of available channels
is relatively less (for example, at M = 40), the switching
probability of i when playing against a larger number of
opponents is lesser. This is because player i’s gain from using
a higher switching probability when playing against a larger
number of opponents is dominated by his fear that even after
switching he might interfere with some of his opponents in
the next stage. The situation however gets reversed when
more channels are available (for example, at M = 120),
where player i uses a higher switching probability when
playing against larger number of opponents. In this case,
i’s gain from using a higher switching probability when
playing against a larger number of opponents dominates
his fear of interfering with some of his opponents in the
next stage even after ‘switching’. This ‘reversal’ nature of
In order to minimize E[ci ], we equate
p = (1 −
dE[ci ]
dp
to 0 and get
CS
1
)(1 −
)
2CI
M
(14)
Thus, if each player switches according to the ‘switching’
probability given by equation (14), he would minimize his
expected cost in a stage game. However, this is not the
same as the switching probability obtained from equation (7).
This means that the Nash Equilibrium obtained by using
the switching probability given by equation (7) is not Pareto
optimal. However, note that the strategy– each player chooses
his ‘switching’ probability p such that his expected cost in the
stage game is minimized– does not constitute a NE. This is
because the expected cost of player i choosing to ‘switch’ is
not the same with the expected cost of i opting to ‘stay’ if the
other player ‘switches’ according to the probability given by
equation (14). Thus player i will have an incentive to change
his strategy and play the one that gives him minimum expected
cost.
IV. I NFINITE H ORIZON R EPEATED G AME
We now investigate the overall repeated game G, which the
networks experiencing interference from each other play in
order to find a clear channel. For this overall game G, the
following strategy comprises a subgame-perfect equilibrium–
each player in each stage ‘switches’ according to the probability given by equation (11) until they find a clear channel.
This is because, in no stage no player can gain by deviating
from the specified strategy and then conforming, given that
the other players stick to their strategy, and so from the one
stage deviation principle [10] these strategies form a subgameperfect equilibrium.
Based on the above argument, algorithm 1 presents the
procedure that will be invoked by network i, if it experiences
interference in any stage. Note that algorithm 1 is distributed
in nature since it is invoked independently by each network
experiencing interference.
V. P ERFORMANCE E VALUATION
We conducted simulations to evaluate the improvements
achieved by the proposed mixed strategy subgame-perfect
equilibrium. Source code for the experiment has been written
in Java. We assume N IEEE 802.22 networks compete for
one of M available spectrum bands. Each of the networks
is associated with a mixed strategy space of ‘switch’ and
‘stay’. The system converges when all the networks acquire a
spectrum band void of interference from it’s neighboring IEEE
802.22 networks. Also, network i is said to have converged
when i has obtained a clear channel with which it can operate
efficiently thereafter. The cost incurred by network i in finding
an interference free channel is calculated using equation (2).
We assumed CS = 4 and CI = 6 (CS < CI ).
200
100
90
80
55 available channels
60 available channels
65 available channels
70 available channels
70
60
50
40
30
20
160
140
120
100
80
60
40
20
10
0
5
30 networks
35 networks
40 networks
180
Average Convergence Cost per Network
Pareto Optimality: Let us analyze the following strategy of
the players– each player chooses his ‘switching’ probability p
such that his expected cost in the stage game is minimized.
For simplicity, we investigate this using the 2-player case. Let
E[ci ] be the expected cost of player i in the stage game. Since
the payoff matrix is symmetric, the p chosen by each player
will be same. E[ci ] can be written from the payoff matrix in
table I as
µ
¶
1
2
CI + p(1 − p)CS +
E[ci ] = p CS +
M −1
(1 − p)p × 0 + (1 − p)(1 − p)CI
1
= p2
CI + pCS + (1 − p)2 CI
(13)
M −1
Algorithm 1 Mitigate Self-Interference
Require: M > 1, CS < CI
1: Estimate the number of interfering networks n
2: Calculate switching probability p from the root of the
following equation
in (0,1):
³
´n
p
S
1 + 1−M
− pn − C
CI = 0
3: Generate random number r ∈ [0, 1]
4: if r ≤ p then
5:
Select a channel randomly from (M − 1) available
channels.
6: else
7:
Stay with current channel.
8: end if
Average Convergence Cost per Network
the ‘switching’ probability of i for playing against a larger
number of opponents when more channels become available
corroborates with the argument given for the variation of p
with n in figure 1(a).
10
15
20
25
30
35
40
Number of Competing Networks (N)
45
50
0
40
50
60
70
80
Number of Available Channels (M)
90
100
(a)
(b)
Fig. 2. Convergence cost per network a) with number of competing networks;
b) with number of available channels
A. Convergence Cost
During network bootup, we allow each network to randomly
choose a channel from M available channels. Figure 2(a)
shows the average cost incurred by a network to find a clear
channel with varying number of competing IEEE 802.22 networks. As can be seen from the graph, the average convergence
cost of a network increases with the number of competing
networks N , for a fixed number of channels M . Also note
that the cost does not increase linearly with N . As the ratio of
‘N/M’ increases, the rate of increase of cost also increases. As
expected, when the number of available channels is increased,
the average convergence cost of a network decreases.
Figure 2(b) shows the average cost incurred by a network
to find a clear channel with varying M . As can be seen from
the graph, as the number of available channels is increased the
average cost incurred by a network decreases. Again note that
the cost does not decrease linearly with M . As the ratio of
1
Pure Strategy
Mixed Strategy NE
500
0.9
Percentage of Networks Converged
Average Convergence Cost per Network
600
400
300
200
100
0.8
0.7
0.6
0.5
0.4
0.3
0.2
25 available channels
30 available channels
35 available channels
0.1
0
50
55
60
65
70
Number of Available Channels (M)
Fig. 3.
Convergence cost per network using pure and mixed strategies.
‘N/M’ decreases, the rate of decrease of cost also decreases.
Also, as expected, for larger number of competing networks,
the cost incurred by a network to find a clear channel increases.
B. Mixed versus Pure Strategy
Figure 3 shows the average convergence cost of a network
when they adopt pure strategies7 against the average cost incurred by a network when playing mixed subgame-perfect NE
strategies. Both costs have been plotted with varying number
of available channels. For this experiment, all 35 networks
were allocated the same channel at the beginning of the game.
This was done to make the average convergence cost of the
networks when using pure strategy to be comparable with
the average cost incurred by the networks when they adopt
mixed strategies. As can be seen from the graph, the average
convergence cost of a network when playing the proposed
mixed strategy NE is lower than the cost incurred by the same
when the networks adopt pure strategy. The difference in the
two costs is particularly more contrasting when the ‘N/M’ ratio
is high. In this case, due to the ‘scarcity’ of available channels,
when two players switch, the probability that they choose the
same channel is large. However, as the ‘N/M’ ratio decreases,
the performance of pure strategy improves and approaches
that of the proposed mixed strategy NE. This is because as
M increases, when two players switch, the probability that
they choose the same channel after switching decreases. Also
recall that in the mixed strategy solution space, as shown
in Figure 1(b), with increasing M , the players’ switching
probabilities start tending to 1. Thus when large number of
channels are available, the performance given by pure strategy
starts approaching that obtained from mixed strategy.
C. Convergence Time
Figure 4 shows the cumulative distribution function (CDF)
of the percentage of nodes that converged versus the number
of stages required by them to do so. For this experiment, all
20 IEEE 802.22 networks were allocated the same channel at
the beginning of the game to make the plots corresponding
to the different cases in the figure comparable. This graph
highlights the fact that while a system of IEEE 802.22 may
take longer to converge, many of it’s individual networks may
have found a clear channel within only a few stages. For
example, in figure 4 though the system of 20 networks with
7A
network always ‘switches’ when experiencing interference.
0
0
1
2
3
4
5
6
7
8
9
10
Number of Stages Played
11
12
13
14
Fig. 4. Percentage of networks converged versus number of stages required.
25 available channels takes 14 stages to converge, more than
50% of it’s networks has converged within stage 6, around
75% has converged within stage 9 and so on. As the number of
channels is increased more and more networks start converging
faster. For example, when M becomes 30, around 70% of
the networks has found a clear channel within only 4 stages
(versus 30% converged networks at stage 4 for M = 25).
VI. C ONCLUSIONS
In this paper we investigate the problem of mitigating selfinterference among a system of IEEE 802.22 networks where
N networks compete for one of M spectrum bands. Using
game theory, we model the process of allocating interference
free channels to each network as an infinite horizon simultaneous move repeated game. When a network experiences
interference in any stage it can adopt either one of two
strategies– ‘switch’ to another channel or ‘stay’ with it’s one.
We investigate the stage game and show that there does not
exist any pure strategy solution for the game. Thus a mixed
strategy NE is proposed for the players when they experience
interference at any stage. Furthermore, for the overall repeated
game, it is shown that if in each stage the players experiencing
interference play their mixed strategy NE, then the overall
game is subgame-perfect. Simulation results suggest that using
the proposed strategies, each network can find a clear channel
quickly at the same time incurring a very low cost. Moreover,
the proposed algorithm works in a distributed manner, thus
making the system scalable.
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