博弈论视角下的多用户MIMO系统竞争设计综述
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本文档深入探讨了"Competitive Design of Multiuser MIMO Systems Based on Game Theory: A Unified View"，这是一篇发表在2008年9月《IEEE Journal on Selected Areas in Communications》上的经典论文。作者 Gesualdo Scutari、Daniel P. Palomar 和 Sergio Barbarossa 是 IEEE 会员，他们关注的是多用户MIMO（Multiple-Input Multiple-Output）系统中的非合作博弈理论应用，特别是在高斯干扰信道下的信息最大化问题。
在过去十年里，对于频选性信道的多用户MIMO系统，研究者们已经广泛探索了如何通过游戏理论来实现信息的最大化，以及如何确保非合作均衡（Nash Equilibrium, NE）的唯一性和分布式算法的收敛性。这些研究中提出了众多方法，尽管表面上看起来不同，但本文作者指出，这些技术其实可以被统一在一个视角下，即他们近期提出的将水填充操作解释为对一个适当的多面体集合进行投影的概念。
通过这种新颖的视角，作者提供了一个数学框架，它有助于提炼出一套通用的充分条件，这些条件能够确保游戏的稳定性和算法的有效运行。这标志着对多用户MIMO系统竞争设计中的游戏理论应用有了更深入和全面的理解。论文不仅回顾了现有的研究成果，还为后续的研究者提供了一个统一的分析工具，使得设计更加高效且易于扩展到不同的信道模型和应用场景。
在阅读这篇文章时，读者可以期待了解到如何利用博弈论解决多用户之间的竞争问题，理解信号处理和优化技术如何与游戏理论相结合，以及如何通过数学模型确保系统的公平性和效率。这对于通信工程领域，特别是多天线技术的发展具有重要的理论指导意义。
SCUTARI et al. : COMPETITIVE DESIGN OF MULTIUSER MIMO SYSTEMS BASED ON GAME THEORY: A UNIFIED VIEW 1091
systems of practical interest. What changes from one system
to the other is the structu re of the channel matrices. We may
have, in fact, as particular cases of (1)-(2): i) digital subscriber
lines [1], where the channel matrices are Toeplitz circulant,
the matrices Q
q
= W Diag(p
q
)W
H
incorporate the DFT
precoding W
H
, the vectors p
q
allocate the power across the
frequency bins, and the MUI is mainly caused by near-end
cross talk; ii) single (or multi) antenna CDMA cellular radio
systems, where the matrices Q
q
= F
q
F
H
q
contain in F
q
the
user codes within a given cell, and the MUI is essentially
intercell interferenc e [2]; iii) ad-hoc wireless MIMO networks,
where the channel matrices represent the MIMO channel of
each link [3].
Since our goal is to find distributed algorithms that do
not require neither a centralized control nor a coordination
among the links, we focus on transmission techniques where
no interference cancellation is performed and multiuser inter-
ference is treated as additive colored noise from each receiver.
Each channel is assumed to change sufficiently slowly to
be considered fixed during the whole transmission, so that
the information theoretical results are meaningful. Moreover,
perfect channel state information at both transmitter and
receiver sides of each link is assumed;
2
each receiver is also
assumed to measure with no errors the covariance matrix of
the noise plus MUI generated by the other users. Finally, we
assume that the channel matrices H
qq
are square nonsingular.
The more general case o f possibly rectangular nonfull rank
matrices is addressed in [21].
Under these assumptions, invoking the capacity expression
for the single user Gaussian MIMO channel−achievable us-
ing random Gaussian codes by all the users−the maximum
information rate on link q for a given set of users’ covariance
matrices Q
1
,...,Q
Q
is [33]
R
q
(Q
q
, Q
−q
)=logdet
I + H
H
qq
R
−1
−q
(Q
−q
)H
qq
Q
q
(3)
where R
−q
(Q
−q
) R
n
q
+
r=q
H
rq
Q
r
H
H
rq
is the MUI plus
noise covariance matrix observed by user q,andQ
−q
(Q
r
)
Q
r=1,r=q
is the set of all the users’ covariance matrices,
except the q-th one.
B. Game Theo retical Formulation
We formulate the system design within the framework of
game theory using as desirable criterion the concept of Nash
Equilibrium (NE) (cf. [5], [6]). Specifically, we consider
a strategic noncooperative game, in which the players are
the links and the payoff functions are the information rates
on each link: Each player q competes against the others by
choosing his transmit covariance matrix Q
q
(i.e., his strategy)
that maximizes his own information rate R
q
(Q
q
, Q
−q
) in (3),
subject to the transmit power constraint (2). A solution of the
game−aNE−is reached when each user, given the strategy
profiles of the others, does not get any rate increase by
unilaterally chang ing his own strategy. Stated in mathematical
2
Note that each user q is only required to known his own channel H
qq
,
but not the cross-channels {H
rq
}
r =q
from the other users.
terms, the game has the following structure:
(G ):
maximize
Q
q
R
q
(Q
q
, Q
−q
)
subject to Q
q
∈ Q
q
,
∀q ∈ Ω,
(4)
where Ω {1,...,Q} is the set of players (i.e., the links);
R
q
(Q
q
, Q
−q
) defined in (3) is the payoff function of player
q;andQ
q
is the set of admissible strategies (the covariance
matrices) for player q,defined as
3
Q
q
Q ∈ C
n
T
q
×n
T
q
: Q 0, Tr{Q} = P
q
. (5)
The solutions of game G are formally defined as follows.
4
Definition 1: A (pure) strategy profile Q
=
Q
q
q∈Ω
∈
Q
1
× ...× Q
Q
is a NE of game G if
R
q
(Q
q
, Q
−q
) ≥ R
q
(Q
q
, Q
−q
), ∀Q
q
∈ Q
q
, ∀q ∈ Ω. (6)
To write the Nash equilibria of game G in a convenient
form, we first introduce the following notations and defini-
tions. Given G , for each q ∈ Ω and Q
−q
∈ Q
−q
Q
1
×...×
Q
q−1
, Q
q+1
×...×Q
Q
, we write the eigendecomposition of
H
H
qq
R
−1
−q
(Q
−q
)H
qq
as:
H
H
qq
R
−1
−q
(Q
−q
)H
qq
=U
q
D
q
U
H
q
, (7)
where U
q
= U
q
(Q
−q
) ∈ C
n
T
q
×n
T
q
is a unitary matrix with
the eigenvectors, D
q
= D
q
(Q
−q
) ∈ R
n
T
q
×n
T
q
++
is a diagonal
matrix with the n
T
q
positive eigenvalues, and R
−q
(Q
−q
)=
R
n
q
+
r=q
H
rq
Q
r
H
H
rq
.
Given q ∈ Ω and Q
−q
∈ Q
−q
, the solution to problem (4)
is the well-known waterfilling solution (e.g., [33]):
Q
q
= WF
q
(Q
−q
),
(8)
with the waterfilling operator WF
q
(·) defined as
WF
q
(Q
−q
) U
q
μ
q
I −D
−1
q
+
U
H
q
, (9)
where U
q
= U
q
(Q
−q
), D
q
= D
q
(Q
−q
),andμ
q
is chosen to
satisfy Tr
(μ
q
I − D
−1
q
)
+
= P
q
, with (x)
+
max(0,x).
Using (8) and Definition 1, we can now c haracterize the
Nash Equilibria of the game G in a compact way as the
following waterfilling fixed-point equation:
Q
q
= WF
q
(Q
−q
)
, ∀q ∈ Ω. (10)
Remark 1 - Competitive maximization of transmission rates.
The choice of the objective function as in (3) requires the
use of ideal Gaussian codebooks with a proper covariance
matrix. In practice, Gaussian codes may b e substituted with
simple (suboptimal) finite-order signal constellations, such as
3
Observe that, in the definition of Q
q
in (5) the condition Q = Q
H
is
redundant, since any complex positive semidefinite matrix must be necessarily
Hermitian [41, Sec. 7.1]. Furthermore, we replaced, without loss of generality
(w.l.o.g.), the original inequality power constraint in (2) with equality, since,
at the optimum to each problem in (4), the constraint must be satisfied with
equality.
4
Observe that, for the payoff functions defined in (3), we can indeed limit
ourselves to adopt pure strategies w.l.o.g., as we did in (3), since every NE
of the game can be prove d to be achieva ble using pure strategies [18].
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