Signal Processing - January 2016 - 102

ExAMPLE 3 (BEAMfORMING GAME [42])
Consider two N-antenna transmitters. Transmitter i ! {1, 2} has
to choose a beamforming vector w i ! C N , such that w iH w i = 1
(where the superscript H stands for Hermitian transpose). The signal observed by the single-antenna receiver i is given by
y i = h iiH w i x i + h Hji w j x j + z i, j = - i, h ji ! C N are fixed for all
(i, j), x i ! C, and z i ~C N (0, 1) is complex white Gaussian
noise. By choosing the utility function as u i = log (1 + SINR i)
with SINR i = (| h iiH w i | 2 E | x i | 2) / (1 +| h Hji w j | 2 E | x j | 2), it can
be shown that any point of the Pareto frontier can be reached by
beamforming vectors, which linearly combine the zero-forcing (ZF)
beamforming solution (w ZF
i ) and maximum ratio transmission
j [42]. Therefore, finding the
(MRT) beamforming solution ` w MRT
i
NBS amounts to finding the appropriate linear combination
coefficient a i, which is defined as w i = a i w ZF
+ (1 - a i) w MRT
.
i
i
The unique NE of the considered game corresponds to
NE
(a NE
1 , a 2 ) = (0, 0), i.e., each transmitter uses ZF beamforming. By
choosing the unique NE of the game under investigation to be the
NE
status quo point, i.e., m i = u i (a NE
i , a j ), the NBS is then given by
NBS
(a NBS
1 , a 2 ) = arg

max

[u 1 (a 1, a 2) - u 1 (0, 0)]

(a 1, a 2) ! [0, 1] 2

# [u 2 (a 1, a 2) - u 2 (0, 0)] .

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By construction, the obtained solution is necessarily more Paretoefficient than the NE. However, computing the NBS typically
requires more channel state information than what is required by
the NE [42].
SPECIAL CLASSES OF STRATEGIC-FORM GAMES
In this section, we review some special classes of strategic-form
games, which show a relevant share of the game-theoretic
approaches available in the SP literature. For the sake of brevity,
we list here only the distinguishing features of each class but also
provide a (nonexhaustive) list of relevant references that can be
used to gather more specific details on problem modeling and
solution tools. For other interesting classes of games (not reported
here due to space constraints), interested readers are referred to
specific literature on the topic (e.g., [22], [23], and [31]).
ZERO-SuM GAMES
One of the most common types of strategic-form games is the twoplayer zero-sum game. A two-player zero-sum game is a game in
which the sum of the utilities is zero or can be made zero by appropriate positive scaling and translation, which do not depend on the
players' actions or strategies. In other words, it is a game such that
K = {1, 2}, u 1 (s 1, s 2) + u 2 (s 1, s 2) = 0. In such a game, one
player is a maximizer, i.e., aims to maximize its gain, while the
other player is a minimizer, i.e., aims to minimize its losses (which
are the gains of the other player). In SP, zero-sum games are especially popular when modeling security games involving an attacker
and a defender. In such games, the attacker's gains are most often
equal to the defender's losses, yielding a zero-sum situation. An
example in this context can be found in [16], in which the interaction between a target and a MIMO radar-both smart-is modeled
as a two-player zero-sum game since the target and the radar are

completely hostile. The mutual information criterion is used in formulating the utility functions. In [43], the problem of polarimetric
waveform design for distributed MIMO radar from a game-theoretic
perspective is also formulated as a two-player zero-sum game
played between an opponent and the radar design engineer. In [2],
the authors use a two-player zero-sum game to model a watermarking problem where a source sequence (the cover text) needs
to be copyright-protected before it is distributed to the public.
Another example is given by a two-user communication channel
(such as the Gaussian multiple access channel) with a constraint
on the total sum-rate [44].
Despite being one of the most well-studied and analyzed
classes of strategic-form games in GT (in part because many
results can be derived), zero-sum games can be restrictive. In fact,
the majority of the studied problems in SP are better modeled as
nonzero-sum games.
CONtINuOuS quASI-CONCAvE GAMES
A game is said to be continuous if, for all k ! K, the utility function u k is continuous in the strategy profile s. It is said to be
quasi-concave if u k is quasi-concave with respect to s k for any
fixed s -k and S k is a compact and convex set. For such games,
we can take advantage of Theorem 2, which ensures the existence
of at least one pure-strategy NE. A flurry of research activity on
energy-efficient resource allocation in wireless communications
or sensor networks makes use of quasi-concave utility functions,
which aim at trading off the performance of network agents while
saving as much energy as possible. Since the performance usually
increases with the amount of resources employed, a useful modeling provides

u k (s) =

fk c

sk
1 + / j!k s j m
sk

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under the hypothesis of a one-dimensional strategy set S k =
[0, P max], with P max being the maximum transmit power. As long
as fk shows some desirable properties (such as sigmoidness),
which are often verified in many SP and communications scenarios, the ratio u k proves to be quasi-concave with respect to s k .
This is the case, for instance, when f (x) = (1 - e -x) M , M > 1 or
f (x) = e - (a/x), a 2 0.
CONtINuOuS CONCAvE GAMES
The same assumptions as for the previous special class of games are
made, except that u k is now a concave function of s k . The existence of a pure NE is guaranteed in such games since individual
concavity implies individual quasi-concavity. Interestingly, if we
make one more assumption, called the diagonally strict condition
(DSC), the uniqueness of the NE can also be guaranteed. This is
since sufficient conditions for ensuring uniqueness are quite rare
in the GT literature. The DSC is met if there exists a vector of
(strictly) positive components r = (r1, f, rK ), such that
6 (s, sl ) ! S 2, s ! sl : (s - sl ) ^c r (sl ) - c r (s) hT 2 0,

IEEE SIGNAL PROCESSING MAGAZINE [102] JANuARy 2016

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Table of Contents for the Digital Edition of Signal Processing - January 2016

Signal Processing - January 2016 - Cover1
Signal Processing - January 2016 - Cover2
Signal Processing - January 2016 - 1
Signal Processing - January 2016 - 2
Signal Processing - January 2016 - 3
Signal Processing - January 2016 - 4
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Signal Processing - January 2016 - 168
Signal Processing - January 2016 - Cover3
Signal Processing - January 2016 - Cover4
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