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In Proceeding of IEEE Wireless Communications & Networking
Conference (WCNC), March, 2007
Dusit Niyato1 and Ekram Hossain2
1An
assistant professor in the School of Computer Engineering, at the
Nanyang Technological University, Singapore.
2An Associate Professor in the Department of Electrical and Computer
Engineering at University of Manitoba, Winnipeg, Canada.
Presented by: Ming-Lung
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Outline
 Introduction
 System Model
 Static Cournot Game
 Dynamic Cournot
 Local Stability Analysis
 Performance Evaluation
 Summary
 Comments
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Introduction
 The concepts of software defined radio and cognitive radio were
introduced to enhance the efficiency of frequency spectrum usage [1].
 Decision theory has been identified as one of the key techniques for
designing cognitive radio.
 Single –player decision making technique was used for this in the
literature.

For example, a partially observable Markov decision process (POMDP)
was used for dynamic spectrum access in an ad hoc network [4].
 On the other hand, game theory can be used for multi-player
optimization to achieve individual optimal solution.

In [5], a game-theoretic adaptive channel allocation scheme was
proposed for cognitive radio networks.
[1] J. Mitola, “Cognitive radio for flexible multimedia communications,” in Proc. MoMuC’99, pp. 3-10, 1999.
[4] Q. Zhao, L. Tong, and A. Swami, “Decentralized cognitive MAC for dynamic spectrum access,” in Proc. IEEE DySPAN’05, pp. 224-232,
Nov. 2005.
[5] N. Nie and C. Comaniciu, “Adaptive channel allocation spectrum etiquette for cognitive radio networks,” in Proc. IEEE DySPAN’05,
pp. 269-278, Nov. 2005.
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Problem formulation
 In this paper, we consider the problem of dynamic
spectrum sharing in a cognitive radio network.
 There is a primary user allocated with a licensed
radio spectrum the utilization of which could be
improved by sharing it with the secondary users.
 The secondary users are analogous to the firms who
compete for the spectrum offered by the primary user
and the cost of the spectrum is determined by using a
pricing function.
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Cournot game formulation
 A Cournot game is used to analyze this situation and
the Nash equilibrium is considered as the solution of
this game.
 A Cournot game is a game where players’ strategies are
about quantity, not price.
 That is, given the price, players compete for the quantity
they should demand (or produce, etc., depending on the
applied situation).
 The main objective of this Cournot game formulation
is to maximize the profit of all secondary users based
on the equilibrium adopted by all secondary users.
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Static Cournot game and dynamic
Cournot game
 Static Cournot game
 It is for the case where all of the secondary users can
completely observe the strategies adopted by each user and the
corresponding payoffs.
 The Nash equilibrium can be obtained in a centralized
fashion.
 However, the above assumption may not be valid. => Dynamic
Cournot game
 Dynamic Cournot game
 It is for the case where selection of the strategy by a secondary
user is based only on the pricing information from the primary
user.
 The stability condition for the Nash equilibrium is studied by
using local stability analysis.
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Primary and secondary users
 We consider a wireless system with a primary user and
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multiple secondary users.
The primary user is willing to share some portion of the
spectrum bi with secondary user i.
The primary user charges the secondary user for the
spectrum at a rate of c(b) per unit bandwidth, where b is
the amount of available bandwidth that can be shared.
After allocation, the secondary users transmit in the
allocated spectrum using adaptive modulation to enhance
the transmission performance.
The revenue of secondary user i is denoted by ri per unit of
achievable transmission rate.
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System model for spectrum sharing
For primary user’s
own usage
Shared with
secondary user
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Wireless transmission
 With adaptive modulation, the transmission rate can be
dynamically adjusted based on the channel quality.
 For uncoded quadrature amplitude modulation (QAM)
with square constellation (e.g., 4-QAM, 16-QAM) the biterror-rate (BER) in single-input single output Gaussian
noise channel can be well approximated as follows [6]:
 γ is the SNR at the receiver
 k is the spectral efficiency of the modulation scheme used.
 We assume that the spectral efficiency is a non-negative
real number
[6] A. J. Goldsmith and S.-G. Chua, “Variable rate variable power MQAM for fading channels,” IEEE Trans. Commun.,
vol. 45, no. 10, pp. 1218-1230, Oct. 1997.
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Wireless transmission (cont’d)
 To guarantee the quality of transmission, BER must be
maintained at the target level (i.e., BERitar).
 The spectral efficiency of the transmission for secondary
user i can be obtained from
 where
 In short, for secondary user i, given the received SNR γi,
target BERitar, and assigned spectrum bi, the transmission
rate (in bps) can be obtained.
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Oligopoly Market Competition and
Cournot Game
 The spectrum sharing problem can be modeled as an
oligopoly market competition in which the secondary
users (i.e., the firms) compete to share the bandwidth
offered by the primary user (i.e., the market).
 The firms compete with each other in terms of
requested spectrum size.
 The profit of a firm can be computed from
 the price charged by the primary user and
 the benefits gained from utilizing the allocated
spectrum.
 All the firms compete to achieve the highest profit.
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Static Cournot game
 Players: the secondary users.
 Strategy of each player: the allocated spectrum size
(denoted by bi for secondary user i) which is nonnegative.
 Payoff for each player: the profit of secondary user i
(denoted by pi) in sharing the spectrum with the
primary user and other secondary users.
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Primary user’s pricing function
 For primary user, we assume that the pricing function used
to charge the secondary users is given by
 where x and y are non-negative constants
 τ ≧ 1 (so that this pricing function is convex)
 B denotes the set of strategies of all secondary users.
 Let w denote the worth of the spectrum for the primary
user, the following condition is necessary to ensure that the
primary user is willing to share spectrum of size b with the
secondary users:

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Secondary user’s profit
 The revenue of the secondary user i: ri * ki * bi.
 The cost of spectrum allocation: bic(b).
 Therefore, the profit of the secondary user i is:
 Assume that the guard band used to separate the
spectrum allocated to different users is fixed and small,
the profit can be rewritten as follows:
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Nash equilibrium
 The optimal allocated spectrum size to one secondary user
depends on the strategies of other secondary users.
 Therefore, Nash equilibrium is considered as the solution
of the game to ensure that all secondary users are satisfied
with the solution.
 By definition, Nash equilibrium of a game is a strategy
profile (list of strategies, one for each player) with the
property that no player can increase his payoff by choosing
a different action, given the other players’ actions.
 The Nash equilibrium is obtained by using the best
response function which is the best strategy of one player
given others’ strategies.
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Nash equilibrium (cont’d)
 Best response function of secondary user i:
 The set B* = {b1*, …, bN*} denotes the Nash equilibrium
of this game if and only if
 where B-i* denotes the set of best responses for
secondary users j for j ≠ i.
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Nash equilibrium (cont’d)
 Mathematically, to obtained Nash equilibrium, we have to
solve the following set of equations:
 However, solving the above set of equations to obtain Nash
equilibrium is not straightforward and a numerical
method is required.
 In this paper, we use Nelder-Mean direct search method [9]
to obtain the Nash equilibrium.
 The Nelder-Mean method is a commonly used nonlinear
optimization technique for minimizing an objective function
without constraints in a many-dimensional space.
[9] J. A. Neldel and R. Mead, “A simplex method for function minimisation,” The Computer Journal 7, pp. 308-313,
1965
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Nash equilibrium (cont’d)
 We formulate an optimization problem with the objective
defined as follows:
 That is, we want to minimize the difference between
decision variables bi and the corresponding best response
function.
 Note that, the minimum value of the objective function is
zero if the algorithm reaches the Nash equilibrium.
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Dynamic Cournot game
 In a practical cognitive radio environment, not all
information is available:
 Available: Pricing function of primary user.
 Unavailable: Strategies and profits of other secondary
users.
 Therefore, we have to obtain Nash equilibrium of each
secondary user based on the interaction with the
primary user only.
 Since all secondary users are rational to maximize their
profits, they can adjust the spectrum size bi based on
the marginal profit function.
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Dynamic Cournot game (cont’d)
 The adjustment of the allocated spectrum size can be
modeled as a repeated Cournot game as follows:
 where bi(t) is the allocated spectrum size at time t.
 αi is the speed adjustment parameter (i.e., learning rate)
of secondary user i.
 The dynamic model of this Cournot game can be
expressed as follows:
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Local stability analysis
 Let us consider the case where τ = 1, that is, the pricing
function of the primary user is linear.
 The repeated Cournot game can be expressed in the
matrix form as follows:
 At the equilibrium, we have b(t+1) = b(t) = b, namely,
b = S(b), where S is the self-mapping function of fixed
point b.
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Local stability analysis (cont’d)
 The fixed point can be obtained by solving the set of
equations:
 With two secondary users, we have fixed points b0, b1,
b2, b3, which can be expressed as follows:
 where b3 is the Nash equilibrium.
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Local stability analysis (cont’d)
 We analyze local stability of this spectrum sharing based on
localization by considering the eigenvalues of the Jacobian
matrix of the mapping.
 Jacobian can be thought of as describing the amount of "stretching"
that a transformation imposes.
 Geometrically, if 0 < eigenvalue < 1, the eigenvector shrinks; if
eigenvalue is negative then eigenvector flips and points in the
opposite direction as well as being scaled by a factor equal to the
absolute value of eigenvalue.
Ax  x
 Thus, the fixed point is stable if and only if the eigenvalues are
all inside the unit circle of the complex plane (i.e., |λi| < 1 for i = 1
~ N).
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Local stability analysis (cont’d)
 The Jacobian matrix is given by:
 The stability condition for b0:
 These conditions imply that when the cost of the
spectrum is higher than the revenue gained from
allocated spectrum, a secondary user is willing to stay
out of the system.
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Local stability analysis (cont’d)
 After analysis, the fixed points b1 and b2 are never
stable.
 For the fixed point b3, the Jacobian matrix is given by:
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Local stability analysis (cont’d)
 The eigenvalues should satisfy the following equation:
 Basically, given r1, r2, k1, k2, x, and y, we can obtain the
relation of α1 and α2 so that the fixed point of Nash
equilibrium is stable.
 When the Nash equilibrium is stable, the profit of the
secondary users cannot be increased by altering the
allocated spectrum size (i.e., marginal profit is zero)
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Performance evaluation:
parameter setting
 We consider a cognitive radio environment with one
primary user and two secondary users sharing a
frequency spectrum of size 15 MHz.
 The target BER for both the users is BERitar = 10-4.
 For the pricing function of the primary user:
 x = 0 and y = 1
 τ is adjusted based on the evaluation scenario (e.g., τ =
1.0) => actually, τ is not changed in this paper
 The revenue of a secondary user per unit transmission
rate is ri = 10, i = 1, 2.
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Numerical result
 For Nash equilibrium
 Under different channel
quality, the Nash
equilibrium is located at
the different places.
 For the trajectory of
spectrum sharing (with α1
= α2 = 0.14)
 With the same speed
adjustment parameter,
better channel quality
results in more
fluctuations in the
trajectory to the Nash
equilibrium
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Numerical result (cont’d)
 Stability region: no obvious observation.
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Numerical result (cont’d)
 When the channel
quality becomes better,
spectrum size allocated
to the secondary users
becomes larger.
 The channel quality of
one secondary user
impacts the allocated
spectrum size for other
secondary user.
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Summary
 We have proposed a competitive spectrum sharing
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scheme for a cognitive radio network consisting of one
primary and multiple secondary users.
Competition occurs among the secondary users.
A static Cournot game has been used to analyze and
obtain the Nash equilibrium for the optimal allocated
spectrum size for the secondary users.
A dynamic Cournot game has been proposed to deal
with the problem of incomplete information.
Stability of the dynamic Cournot game has been
analytical investigated by using local stability theory.
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Comment
 This paper remind us the fact that the secondary users
may compete with each other rather than the
assumption that secondary users ignore the existence
of other secondary users.
 The pricing function in this paper reflects the total
allocated spectrum, that is, the more the allocated
spectrum, the more the primary user price the
spectrum.
 However, this paper didn’t focus much on the pricing
function and thus it gives the incentive to more studies.
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