Bargaining and Beamforming in
Interference Channels
Rami Mochaourab and Eduard Jorswieck
Communications Theory, Communications Laboratory
Dresden University of Technology, Dresden, Germany
e-mail: {Rami.Mochaourab, Eduard.Jorswieck}@tu-dresden.de
Zuleita K. M. Ho and David Gesbert
EURECOM, 2229 route des crˆetes
BP 193 F-06560 Sophia-Antipolis cedex
e-mail: {hokm, David.Gesbert}@eurecom.fr
Abstract—Utilizing the real-valued parametrization of each
transmitter’s efficient beamforming vectors, we propose a de-
centralized resource allocation scheme in the multiple-input
single-output interference channel. The scheme is motivated by
bargaining concepts in game theory. The aim of these concepts is
to improve the joint payoff of the users from the Nash equilibrium
outcome. In each bargaining-step, each user proposes a strategy.
A user accepts any proposal if it increases his payoff. Otherwise,
new proposals are made. When all proposals are accepted, a new
bargaining-stage begins. We prove the scheme’s convergence and
demonstrate its performance by simulations. In comparison to
previous approaches, our bargaining outcome is arbitrarily close
to the Pareto boundary of the achievable single-user rate region.
We further discuss the control overhead and complexity of this
scheme.
I. INTRODUCTION
We consider a setting in which two transmitter-receiver
pairs utilize the same spectral band simultaneously. Each
transmitter is equipped with N transmit antennas and each
receiver with a single antenna. This setting corresponds to
the multiple-input single-output (MISO) interference channel
(IFC) [1]. The performance of the systems in such a setting
is degraded by mutual interference, and their noncooperative
operation is usually not efficient [2]. Therefore, coordination
between the links is needed in order to improve the joint
outcome. In game theory, bargaining describes the process
in which the bargainers make use of an opportunity to gain
by coordinating their actions [3]. In this work, we consider
strategic bargaining between the links. The links decide on
their actions at each bargaining-step and bargain as long as
they experience improvement in their situation.
For the two-user MISO IFC, the efficient beamforming
vectors of each transmitter are parameterized by a single
real-valued parameter between zero and one in [4]. This
parametrization is valuable for designing efficient low com-
plexity distributed resource allocation schemes. In [5], this
parametrization is utilized and a bargaining algorithm is pro-
posed that requires two bit signaling between the transmitters.
In each bargaining-step, the transmitters reduce their beam-
forming parameters by an equal step-length leading to a joint
Part of this work has been performed in the framework of the European
research project SAPHYRE, which is partly funded by the European Union
under its FP7 ICT Objective 1.1 - The Network of the Future. This work is
also supported in part by the Deutsche Forschungsgemeinschaft (DFG) under
grant Jo 801/4-1.
increase in the links’ performance. In [6] a similar algorithm is
proposed. At each bargaining-step, each transmitter optimizes
its transmission to reduce a fixed amount of interference power
at unintended receivers. Both algorithms in [5] and [6] termi-
nate when at least one link experiences reduction in its out-
come. While both algorithms improve the joint performance of
the systems from the noncooperative state, these outcomes are
however not Pareto optimal. In [7], a distributed algorithm is
proposed that is performed between link pairs in a multi-link
system. The transmitters exchange interference levels (scalar
values) in each iteration and optimize their transmission such
that the interference levels are met at unintended receivers. The
interference levels are updated based on a necessary condition
for Pareto optimality. Although this condition is not proven to
be also sufficient, numerical evidence shows that the algorithm
converges to a Pareto optimal outcome almost surely.
In this work, we propose a bargaining process between
two links. The preference representation of the links in the
Edgeworth box [8] is utilized in order to describe this process.
We determine how each transmitter chooses its beamforming
vectors in each bargaining-step such that the bargaining out-
come lies arbitrarily close to the contract curve. The contract
curve in the Edgeworth box corresponds to all Pareto optimal
outcomes. Thus, our process is to converge to an outcome
which is arbitrarily close to the Pareto boundary. Our algorithm
requires four bit signaling between the transmitters in each
bargaining-step, and we show that only a few iterations are
necessary for the bargaining process to converge.
Notations: Column vectors and matrices are given in low-
ercase and uppercase boldface letters, respectively. a is the
Euclidean norm of a, a ∈ C
N
. |b| is the absolute value
of b, b ∈ C. (·)
H
denotes the Hermitian transpose. The
orthogonal projector onto the column space of Z is Π
Z
:=
Z(Z
H
Z)
−1
Z
H
. The orthogonal projector onto the orthogonal
complement of the column space of Z is Π
⊥
Z
:= I − Π
Z
,
where I is an identity matrix. Throughout the paper, the
subscripts k, are from the set {1, 2}.
II. P
RELIMINARIES
A. System and Channel Model
The quasi-static block flat-fading channel vector from trans-
mitter k to receiver is denoted by h
k
∈ C
N
.Weassume