In many cases, an NE results from learning and evolution processes of all the game participants. Therefore, it is fundamental to predict and characterize the set of such points from the system design perspective of wireless networks. In the rest of the paper, we focus on characterizing the set of NEs. The following questions are addressed one by one.
-
(i)
Does an NE exist in our game?
-
(ii)
Is the NE unique or there exist multiple NE points?
-
(iii)
How to reach an NE if it exists?
-
(iv)
How does the system perform at NE?
Throughout this section we investigate the existence and uniqueness of a Nash equilibrium.
It is known that in general an NE point does not necessarily exist. In the following theorem we establish the existence of a Nash equilibrium in our game.
Theorem 1.
A Nash equilibrium exists in game
.
Proof.
Since
is convex, closed, and bounded for each
;
is continuous in both
and
; and
is concave in
for any set
, at least one Nash equilibrium point exists for
[12, 22].
Once existence is established, it is natural to consider the characterization of the equilibrium set. The uniqueness of an equilibrium is a rare but desirable property, if we wish to predict the network behavior. In fact, many game problems have more than one NE [12]. As an example of games with infinite NEs, we could consider a special case of our game
, namely, the symmetric waterfilling game [9] where the channel coefficients are assumed to be symmetric. Then, in general, our game
does not have a unique NE. But with the assumption of independent and identically distributed (i.i.d.) continuous entries in
, we will show that the probability of having a unique NE is equal to
.
For any player
, given all other players' strategy profile
, the best-response power strategy
can be found by solving the following maximization problem:
which is a convex optimization problem, since the objective function
is concave in
and the constraint set is convex. Therefore, the Karush-Kuhn-Tucker (KKT) conditions for optimization are sufficient and necessary for the optimality [5]. The KKT conditions are derived from the Lagrangian for each player
,
and are given by
where
for all
and for all
are dual variables associated with the power constraint and transmit power positivity, respectively. The solution to (11)–(13) is known as waterfilling [6]:
where
and
satisfies
In order to analyze the equilibrium set, we establish necessary and sufficient conditions for a point being an NE in the game 
Theorem 2.
A power strategy profile
is a Nash equilibrium of the game
if and only if each player's power
is the single-player waterfilling result
while treating other players' signals as noise. The corresponding necessary and sufficient conditions are:
The proof can be found in Appendix A.
From (16), it is easy to verify that necessarily
, since
and
for all
and for all
. Also, from (17), we have
This equation implies that, at the NE, all APs transmit at their maximum power by conveniently distributing the power over all the orthogonal channels.
However, it is still difficult to find an analytical solution from (16)–(18), since the system consisting of (14) and (15) is nonlinear. To simplify this problem, we could consider linear equations instead of nonlinear ones. The following lemma provides a key step in this direction.
Lemma 1.
For any realization of channel matrix
, there exist unique values of the Lagrange dual variables
and
for any Nash equilibrium of the game
. Furthermore, there is a unique vector
such that any vector
corresponding to a Nash equilibrium satisfies
The proof can be found in Appendix B.
Now, let
be the following
matrix:
where
is the
th column of
,
is the
identity matrix, and
is the zero vector of length
. Let
be the following vector of length
:
Then, (19) and (20) can be written in the form of linear matrix equation
Define the following sets:
and denote by
and
their cardinalities. From (18), if an index
we must have
. Without loss of generality, we assume that
for
. Let
be the
matrix formed from the first
rows and first
columns of
,
is formed from the first
elements of
, and
is formed from the first
elements of
. Then, any NE solution must satisfy
Let
be the
matrix formed from the columns of
that correspond to the elements of
. Similarly, let
be the vector of length
with entries
such that
(same order as they were in
). Then, any NE solution satisfies
Lemma 2.
For any realization of a random
channel gain matrix
with i.i.d. continuous entries, if
, the probability that
is equal to
.
Lemma 3.a
(
) If
and
, the probability that
is equal to
.
(
) If
, the probability that
is equal to
.
The proofs of Lemmas 2 and 3 can be found in Appendices C and D, respectively.
Based on Lemmas 1, 2, and 3, we derive the following theorem.
Theorem 3.
For any realization of a random
channel gain matrix
with statistically independent continuous entries, the probability that a unique Nash equilibrium exists in the game
is equal to
.
The proof can be found in Appendix E.
Thus, from Theorems 1 and 3, we have established the existence and uniqueness of NE in our game
.