A Stackelberg game approach to distributed spectrum management.

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A STACKELBERG GAME APPROACH TO DISTRIBUTED SPECTRUM MANAGEMENT
Meisam Razaviyayn, Yao Morin and Zhi-Quan Luo
Department of Electrical and Computer Engineering
University of Minnesota, Minneapolis, MN 55455, USA
ABSTRACT
In this paper, we consider a cognitive radio system with one
primary (licensed) user and multiple secondary (unlicensed)
users. Considering the interference temperature constraints,
the secondary users compete for the available spectrum so as
to satisfy their need for communication. Borrowing the con-
cept of price from market theory, we develop a decentralized
Stackelberg game formulation for power allocation. In this
scheme, primary user (leader) announces prices for the avail-
able tones such that a system utility is maximized. Using the
announced prices, secondary users (followers) compete for
the available bandwidth to maximize their own utilities. We
show that this Stackelberg game is polynomial time solvable
under certain channel conditions. The proposed method is de-
composable across the tones and is more power efficient than
the Iterative Water-Filling Algorithm.
Index Terms— Cognitive Radio Network, Distributed
spectrum management, Stackelberg game, Mathematical Pro-
gram with Equilibrium Constraint (MPEC)
1. INTRODUCTION
The wireless communication spectrum is a valuable resource
that is currently under-utilized due to the usage of orthogonal
transmission schemes such as Frequency Division Multiple
Access (FDMA). In these static spectrum sharing policies,
the available frequency band is divided into multiple tones
(or bands) which are pre-assigned to all the users on a non-
overlapping basis. However, it is well known that such ‘static
orthogonal spectrum sharing’ approach can lead to low band-
width utilization. Spectrum-sensing cognitive radio technol-
ogy allows devices to dynamically and automatically seek out
and use the optimum frequencies and bandwidth. To take ad-
vantage of the unused spectrum capacity, the users dynami-
cally adapt to the spectral environment and change transmis-
sion or reception parameters on the fly. This allows for more
efficient wireless communications without causing harmful
interference with legacy systems or other devices using the
same frequency bands. From the optimization perspective,
the power allocation problem in the interference channel can
This research is supported in part by the National Science Foundation,
grant number CMMI-0726336, and by the AFOSR, grant number 00008547.
be formulated either as a cooperative utility maximization
problem [1] or as a noncooperative Nash game [2]. Although
finding the optimal power allocation is NP-hard [3], several
algorithms have been proposed to solve the cooperative opti-
mization problem suboptimally [4]. In the game theoretic for-
mulations, [2] and [5], users act selfishly in response to noise
and interference from other users. Besides, some pricing al-
gorithms [6] and Stackelberg game formulations [7] - [9] are
proposed to control the users’ behavior. However, unlike our
approach, these methods do not contain the soft interference
temperature constraint.
In this paper, we use the interference channel model and
the interference temperature constraint to introduce a pricing
mechanism for distributed power allocation in cognitive ra-
dio systems. In comparison to the pricing models in [10], a
Stackelberg game model is used in this paper and the price
defined in this paper is on a per-tone base. The per-tone pric-
ing scheme decomposes the problem across the tones which
makes the development of an efficient distributed algorithm
possible.
2. SYSTEM MODEL
Consider a cognitive radio system consists of K secondary
usersandoneprimaryuser. Eachuserconsistsofatransmitter-
receiver pair. We assume that spectrum sensing has been done
and there are N available frequency tones to be shared among
K users. Throughout, the term “user” refers to a secondary
user unless the word “primary” is present.
Let hn
mitter of User j and the receiver of User k on the n-th tone,
where n ∈ N
this notation hn
mitter of the k-th user and its intended receiver at tone n and
αn
jk
= |hn
crosstalk coefficient. Let sn
at the n-th tone. Assuming Gaussian signalling and treating
the interference as noise, the maximum rate that User k ∈ K
can achieve on the n-th tone is given by
jkbe the complex channel gain between the trans-
?
= {1,...,N} and j,k ∈ K
kkdenotes the channel gain between the trans-
?
= {1,...,K}. In
?
jk|2/|hn
kk|2denotes the normalized channel gain or
kbe the power allocated by User k
Rn
k= log
?
1 +
sn
k
j?=kαn
σn
k+?
jksn
j
?
,
(1)
3006 978-1-4244-4296-6/10/$25.00 ©2010 IEEE ICASSP 2010

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where σn
background noise plus the interference caused by the primary
user at User k on tone n. The total rate Rkthat can be reliably
communicated by User k is given by Rk=?N
ature constraint (defined later). Therefore, we need a scheme
to regulate the users’ power consumption so as to maximize a
network utility while respecting the interference temperature
constraint.
k
?
= Nn
0,k/|hn
kk|2and Nn
0,kis the variance of the
n=1Rn
k.
The allocated power may violate the interference temper-
3. STACKELBERG GAME FORMULATION
Since users are selfish, they act solely according to their best
interests. From the primary user’s point of view, these selfish
moves may lead to inefficient spectrum utilization and/or the
violation of interference temperature constraint. In what fol-
lows, we introduce a distributed pricing mechanism to guide
self users toward rational behaviors. Intuitively, “price” is an
indicator and a controller of the supplies and demands of a
product. With this motivation, we attach a price to the power
detected at the primary user’s receiver in each tone and use it
to rationalize power allocation for the users. For any per-tone
prices set by the primary user, the secondary users act as fol-
lowers and compete selfishly in a noncooperative Nash game
to maximize their individual utilities (defined below). The
primary user acts as a leader who sets the prices through the
maximization of its own utility function (to be defined later).
3.1. Secondary users’ noncooperative game
User k’s strategy set is {sk= {s1
pnbe the unit price of interference power over the primary
user on tone n and un
tone n, which is defined as follows:
k,...,sN
k}|sn
k≥ 0,∀n}. Let
kbe the utility function of User k on
un
k= wkRn
k(sn
k, ¯ sn
k) − pnβn
ksn
k,
(2)
where Rn
n-th tone defined in (1). The constant wk is the willing-
ness factor of purchase for User k and βn
coefficient from secondary User k to the primary user at
tone n. We assume that the interference temperature con-
straint,?K
user allocates the amount of power to maximize its utility
function. Therefore, the Nash equilibrium of the game can be
obtained by optimizing (2) simultaneously for all users in a
selfish manner, resulting in
⎡
j?=k
k(sn
k, ¯ sn
k) is the rate that User k can achieve on the
kis the channel
k=1βn
ksn
k≤ Qn, subsumes the users’ individual
power budget constraint. When the prices are fixed, each
sn∗
k =
⎣vn
k
pn− σn
k−
?
αn
jksn∗
j
⎤
⎦
+
,∀k,∀n,
(3)
where [x]+= max{x,0} and vn
Note that if the channel coefficient βn
we can consider βn
k=wk
pn.
kis not known at user k,
k= 1 in (2).
3.2. Primary user’s pricing mechanism
How should the primary user set the prices? For this purpose,
we can define a system utility function˜U which is the total
“revenue” of the system, i.e.,
˜U(p) =
N
?
n=1
(pn
K
?
k=1
βn
ksn∗
k),
(4)
where {sn∗
with the interference temperature constraint?K
at primary user on tone n. Therefore, the optimization prob-
lem of the primary user is:
k} is the Nash equilibrium that must comply
k=1βn
ksn∗
k
≤
Qn,∀n and Qnis the maximum tolerable interference power
max
p
N
?
n=1
(pn
K
?
k=1
βn
ksn∗
k) s.t. (3) and
K
?
k=1
βn
ksn∗
k ≤ Qn, ∀n
(5)
Notethat(5)isdecomposableacrossthetones. Hence, theop-
timization problem of the primary user is to find the best pn’s
such that the revenue is maximized based on the Nash equi-
librium solution of the game (3). This problem is non-convex
and in the form of a Mathematical Program with Equilibrium
Constraint (MPEC [13]). The following theorem simplifies
the optimization problem (5).
Theorem 1 Problem (5) is equivalent to the following prob-
lem for each tone n:
?
max
pn
min
pnQn,pn
K
?
2}, where
k=1
βn
ksn∗
k
?
s.t. (3)
(6)
Let us define Un? min{Un
1,Un
Un
1? pnQnand Un
2? pn
K
?
k=1
βn
ksn∗
k
(7)
According to Theorem 1, for each tone n, we need to find the
optimal price that maximizes Un. It is easy to see that Un
linear increasing function of pn. However, Un
plicated since it involves solving a parametric Linear Com-
plementarity Problem (LCP) with pnas the parameter. In the
following section, we analyze the behavior of Un
tion of pn.
1is a
2is more com-
2as a func-
4. BEHAVIOR OF THE REVENUE FUNCTION
Let In(pn) be the set of active users at tone n for the given
price pn, i.e., In(pn) = {k|sn∗
simplicity, we use Inin place of In(pn) in the rest of the
paper. Therefore, (3) is equivalent to
⎧
⎩
k(pn) > 0}. For notational
sn∗
k =
⎨
vn
k
pn− σn
k−
?
0
j∈In,j?=k
αn
jksn∗
j
k ∈ In
k / ∈ In
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Hence, we have
sn∗
In = Λ−1
In,In
?vIn
− σIn,c − ΛIn,c,InΛ−1
pn− σIn
?
, sn∗
In,c = [tIn,c]+= 0,
?vIn
(8)
tIn,c =vIn,c
pn
In,In
pn− σIn
?
where In,cis the complement of Inand sn∗
containing the allocated power (using the selfish approach)
for the active users.
ΛIn,In is the matrix containing the
crosstalk coefficients of the active users and ΛIn,c,In con-
tains the cross-talk coefficients from the active users to inac-
tive ones. σIn represents the noise power plus interference
from primary user for the active set at tone n. Hence, we have
In is a subvector
Un
2= βT
InΛ−1
In,In[vIn − pnσIn]
(9)
This implies Un
sequel, we analyze the maximum of Unbased on the behavior
of Un
2is a piece-wise linear function of pn. In the
2.
4.1. Monotone case
WhenUn
mean value theorem implies that the optimum price should
satisfy Un
timum price by using the bisection method over the interval
[0,pn
k
Theorem 2 provides sufficient condition for the monotonicity
of Un
2isanon-increasingfunctionofpn(seeFigure1), the
1= Un
2. In this case, we can easily find the op-
max], where pn
max= max
{vn
σn
k
k
} is the x-intercept of Un
2.
2.
Fig. 1: Unin tone n versus price pncase 1
Theorem 2 If?
The condition?
j?=kαn
jkβn
2is a non-increasing function of pn.
j≤ βn
k,∀k, there exits a unique
Nash equilibrium and Un
j?=kαn
jkβn
j≤ βn
kessentially says that the
interference among users should be weak.
4.2. Non-monotone case
In this case, the maximum of Unmay or may not be at the
intersection of Un
to calculate the break points in Un
2and Un
1(see Figure 2). Therefore, we need
2(the discontinuous points
in the derivative of Un
the optimum price for each tone. However, the number of
break points may increase exponentially with K. The follow-
ing theorem gives a sufficient condition to have at most K
break points.
2with respect to pn) in order to find
Theorem 3 Assuming the uniqueness of Nash equilibrium
for each price, and that any of the following two conditions is
satisfied.
•
ΛIn,c,InΛ−1
In,InvIn ≤ vIn,c
•
ΛIn,c,InΛ−1
In,InσIn ≤ σIn,c
then, if a user is active at a price level pn, it will remain active
for all prices less than pn.
Theorem 3 implies that under certain channel conditions (e.g.,
weak channel interferences), users that are priced out on a
tone will never return when pnincreases, and the cardinality
of active set Inis non-decreasing function of pn. This guar-
antees that the algorithm proposed in the next section has a
complexity that is polynomial in K.
Fig. 2: Unin tone n versus price pncase 2
5. FINDING THE OPTIMUM PRICE
In this section, we propose an algorithm to find pnthat max-
imizes Unfor each tone n. Although the algorithm may be
computationally expensive, its complexity is O(NK4) if the
condition in Theorem 3 is satisfied. As we see in simulations,
the algorithm could be much faster than IWFA, because we
keep track of the active set and we can calculate the resulting
power allocation according to (8).
For each tone n, the algorithm starts with the active user
set In,0= {kn
inactive user set In,c,0= {1,2,...,K}\{kn
tialize with Un
executes the following steps:
max} where kn
max= Argmaxk{vn
k
k} and the
σn
max} and we ini-
2,m= 0. At each iteration ν, the algorithm
• Step 1: Compute the next break point pn,ν+1in Un
pn,ν+1with discontinuity in the derivative of Un
pn,ν) and the corresponding user kν+1that is removed from or in-
serted to the active set at price pn,ν+1and update the active set to
In,ν+1
2(i.e., the point
2where pn,ν+1<
• Step 2: Compute Un,ν+1
In,ν+1in (8), and (7)
1
and Un,ν+1
2
by substituting pn,ν+1,
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• Step 3:
If Un,ν+1
2
< Un,ν+1
1
and Un,ν+1
2
> Un
2,m, then
- Un
2,m:= Un,ν+1
2
, pn
(m):= pn,ν+1
- set ν := ν + 1 and go to step 1
Else if Un,ν+1
2
< Un,ν+1
1
and Un,ν+1
2
≤ Un
2,m, then
- go to step 1
Else,
- Find pn
(i)such that Un
2(pn
(i)) and Un
1(pn
(i)) intersect at pn
(i)
- If Un
pn
1(pn
(i)) > Un
2,max, then pn∗:= pn
(i), else, pn∗:=
(m)and terminate
Note that the calculation of the next break point in step 1 can
be done by a simple ratio test. Indeed, we need to find the first
component of sn∗
Since the elements of sn∗
we can find the largest x-intercept of these lines by a simple
ratio test.
The proposed algorithm keeps track of the active set and
finds the maximum of Un. An important advantage of this
algorithm is that the resulting optimal power allocation may
not reach the interference temperature limit (see Figure 2) and
this leads to user power back off which is desirable.
In and tIn,c that goes to zero as pndecreases.
In and tIn,c are linear functions of
1
pn,
6. SIMULATION RESULTS
We consider a multiuser system with 32 secondary users and
32 tones and we compare our method with traditional IWFA
in terms of the total achievable rate (?
of wkand βn
tones. The crosstalk coefficients are drawn uniformly from
[0,1
of the total achievable rate, total exhausted power, and CPU
time. In contrast to IWFA, our method respects the interfer-
ence temperature constraints. Furthermore, simulations show
that the solution generated by our method uses less power
than that of the IWFA solution, and our method takes less
time, while the average achievable rates are almost the same.
Besides, the sum rate of IWFA solution saturates as SNR in-
creases, while Stackelberg solution does not.
kRk), the total ex-
hausted power (?
k
?
nsn
k), and the CPU time. The values
kare set to one and Qn’s are set equal across the
2]. We run the algorithms 50 times and plot the average
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197
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SNR(dB)
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