# Robust remote estimation over the collision channel
in the presence of an intelligent jammer

Xu Zhang and Marcos M. Vasconcelos X. Zhang is with LSEC, Academy of
Mathematics and Systems Science, Chinese Academy of Sciences, Beijing, China.
X. Zhang was supported by China National Postdoctoral Program for Innovative
Talents under Grant No. BX2021346. E-mail<EMAIL_ADDRESS>M.
Vasconcelos is with the Commonwealth Cyber Initiative and the Bradley
Department of Electrical and Computer Engineering, Virginia Tech. M. M.
Vasconcelos was supported by funds from the Commonwealth Cyber Initiative
(CCI). E-mail<EMAIL_ADDRESS>
###### Abstract

We consider a sensor-receiver pair communicating over a wireless channel in
the presence of a jammer who may launch a denial-of-service attack. We
formulate a zero-sum game between a coordinator that jointly designs the
transmission and estimation policies, and the jammer. We consider two cases
depending on whether the jammer can sense the channel or not. We characterize
a saddle-point equilibrium for the class of symmetric and unimodal probability
density functions when the jammer cannot sense the channel. If the jammer can
sense if the channel is being used, we provide an efficient algorithm that
alternates between iterations of Projected Gradient Ascent and the Convex-
Concave Procedure to find approximate First-order Nash-Equilibria. Our
numerical results show that in certain cases the jammer may decide to launch a
denial-of-service attack with the goal of deceiving the receiver even when the
sensor decides not to transmit.

##  1 Introduction

Cyber-Physical Systems are characterized by the tight coupling between
physical, computing and communication components. Due to the confluence of
three distinct branches of systems science, many new security vulnerabilities
have emerged over the last decade as cyber-physical systems started to
dominate the technology landscape [1]. Many of the critical infrastructures
our society relies on are cyber-physical systems (e.g. industrial automation
systems, transportation networks, utility distribution networks, etc.). In
particular, remote sensing where one (or multiple) sensor(s) communicates its
measurements over a wireless channel to a non-collocated access point or base-
station is a fundamental building block of many cyber-physical systems [2].
The openness of the wireless medium creates a vulnerability to attacks that
could compromise the performance and safe operation of the entire system [3].

Denial-of-Service (DoS) is a class of cyber-attacks where a malicious agent,
often referred to as the jammer, may disrupt the communication link between
the legitimate transmitter-receiver pair. DoS attacks are widely studied at
different levels of modeling detail of the communication channel. For example
if the channel is assumed to be a physical layer model, the jammer may
introduce additional Gaussian noise to the transmitted signal. If the channel
is modeled at the network layer by a packet-drop channel, the jammer may
increase the probability of dropping a packet. We consider a medium access
control (MAC) layer model in which the jammer may decide to block the channel
by transmitting an interference signal that overwhelms the receiver, causing a
packet collision.

Figure 1: Block diagram for a game between a coordinator and a jammer. The
jammer may have access to the sensor’s decision to transmit. The coordinator
designs the policies for the sensor and the estimator.

We consider the remote estimation system depicted in Fig. 1, which is
comprised of a sensor and estimator pair communicating over a collision
channel in the presence of a jammer. The sensor makes a stochastic measurement
$X$ of a physical quantity according to a given distribution, and decides
whether to transmit it or not to the estimator. Communication is costly,
therefore, the sensor must transmit wisely. We consider two cases: 1. the
jammer cannot sense if the channel is being used by the sensor; 2. the jammer
can sense the channel, i.e., has access to $U$. Jamming is assumed to be
costly, therefore, the jammer must act strategically.

Finally, the estimator observes the channel output and declares an estimate
$\hat{X}$ for the sensor’s observation such as to minimize the expected
quadratic distortion between $X$ and $\hat{X}$. We study this problem as a
zero-sum game between a coordinator (system designer) and the jammer. Our goal
is to characterize equilibrium solutions and obtain efficient algorithms to
compute them. The main difference between our model and existing work in this
area is the presence of a virtual binary signaling channel that can be
exploited by the coordinator to guarantee a minimum level of performance of
the system in the presence of DoS attacks.

###  1.1 Related literature

There exists an extensive literature on strategic communication in the
presence of jammers. This class of problems seems to have started with the
work of Basar [4], which obtained a complete characterization of the saddle-
point equilibria when the sensor measurements and the channel are Gaussian.
Recently, an extension to the two-way additive Gaussian noise channel was
studied by McDonald et al. in [5]. A jamming problem where the transmitter and
estimator have different objectives was solved by Akyol et al. in [6] using a
hierarchical game approach. A jamming problem with and without common
randomness between the transmitter and estimator is studied Akyol in [7] and a
Stackelberg game formulation was considered by Gao et al. [8]. Another
interesting problem formulation is due to Shafiee and Ulukus in [9], where the
pay-off function is the mutual information between the channel input and
output. Jamming over fading channels was considered by Ray et al. in [10] and
subsequently by Altman et al. in [11]. An LTE network model was considered by
Aziz et al. in [12].

Another class of remote estimation problems focuses on the state estimation of
a linear time invariant system driven by Gaussian noise under DoS attacks. Li
et al. [13] studied a jamming game where the transmitter and jammer have
binary actions. A SINR-based model was considered by Li et al. in [14], where
the transmitter and jammer decide among multiple discrete power levels. The
case of continuum of power levels was studied by Ding et al. in [15]. A
jamming model over a channel with two modes (i.e., free mode and safe mode)
was analyzed by Wu et al. in [16]. A jamming problem with asymmetric feedback
information and multi-channel transmissions was considered by Ding et al. in
[17] and [18], respectively. A Stackelberg equilibrium approach to this
problem was considered by Feng et al. in [19]. The problem of optimizing the
attack scheduling policy from the jammer’s perspective was considered by Peng
et al. in [20].

The model described herein is closely related to the work of Gupta et al. [21,
22] and Vasconcelos and Martins [23, 24], where there is a clear distinction
between the channel being blocked vs. idle. As in [21], we assume that the
transmission decision $U$ may be available to the jammer, but not the full
input signal $X$. This assumption is realistic in the sense that the bits used
to encode $X$ may be encrypted. In the game considered in [21], it is assumed
that the receiver is fixed, and the game is played between the sensor and the
jammer. Instead, we follow Akyol [7] in which the sensor and estimator are
distinct agents implementing policies optimized by a coordinator [25].

###  1.2 Contributions

The main contributions of this article are:

  * •

We introduce a new class of signaling games among a sensor, an estimator and a
jammer with asymmetric information, over a collision channel model.

  * •

For a jammer without channel sensing capability, we obtain a sufficient
condition on the probability density function that guarantees the existence of
a saddle-point equilibrium despite the overall lack of convexity for the
coordinator’s sub-problem.

  * •

For a reactive jammer, we obtain an algorithm based on an interleaved
projected gradient ascent and a convex-concave procedure to efficiently obtain
an approximate first-order Nash-equilibrium.

##  2 System model

We consider the system in Fig. 1, which consists of a non-colocated sensor-
estimator pair communicating over a wireless network vulnerable to DoS
attacks. The sensor observes realizations of a random variable $X$, which is
distributed according to a continuous probability density function $f$. We
assume that $f$ is supported on the real line, i.e., $f(x)>0$,
$x\in\mathbb{R}$. However, the results herein can easily be extended to random
vectors. Upon observing $X=x$, the sensor decides whether to transmit its
observation or not according to a mixed policy
$\gamma:\mathbb{R}\rightarrow[0,1]$, such that

$\mathbf{P}(U=1\mid X=x)=\gamma(x),$ (1)

where the decision variable $U=1$ if the sensor transmits, and $U=0$ if the
sensor remains silent. Then, the channel input signal $S$ is determined as

$S=\begin{cases}X,&\text{if}\ \ U=1\\\ \varnothing,&\text{if}\ \
U=0,\end{cases}$ (2)

where $\varnothing$ is used to denote that the channel is idle.

The jammer operates strategically based on side information about the the
sensor’s decision to transmit or not. Upon observing $U=u$, the jammer blocks
the channel using a mixed policy $\varphi:\\{0,1\\}\rightarrow[0,1]$ such that

$\mathbf{P}(J=1\mid U=u)=\varphi(u),$ (3)

where $J=1$ denotes the jammer’s decision to block the channel, and $J=0$
denotes the decision not to block. Unlike [21], we are not precluding the
jammer to block an idle channel.

Given the input signal $S$ and the jammer’s decision $J$, the channel ouput
$Y$ is given by

$Y=\begin{cases}S&\text{if}\ \ J=0\\\ \mathsf{B}&\text{if}\ \ J=1,\end{cases}$
(4)

where $\mathsf{B}$ denotes that the channel has been blocked, and a DoS attack
has occurred.

Finally, the receiver uses an estimation policy
$\eta:\mathbb{R}\cup\\{\varnothing,\mathsf{B}\\}\to\mathbb{R}$ such that

$\eta(y)=\begin{cases}x&\text{if}\ \ y=x\\\ \hat{x}_{0}&\text{if}\ \
y=\varnothing\\\ \hat{x}_{1}&\text{if}\ \ y=\mathsf{B},\end{cases}$ (5)

where $\hat{x}_{0}$ and $\hat{x}_{1}$ are the so-called representation symbols
used by the receiver when the channel is idle and blocked, respectively. This
is another departure from the model in [21], which does not account for such
distinction. For the remainder of the paper, let
$\hat{x}\operatorname{\overset{def}{=}}(\hat{x}_{0},\hat{x}_{1})$.

We consider the same objective function of [21], which consists of three
terms: the estimation error, and the communication and jamming costs, as
follows

$\mathcal{J}\big{(}(\gamma,\eta),\varphi\big{)}=\mathbf{E}\big{[}(X-\hat{X})^{2}\big{]}+c\mathbf{P}(U=1)-d\mathbf{P}(J=1).$
(6)

Even though the sensor and the estimator act as independent agents, we adopt a
zero-sum game between a coordinator that seeks to minimize Eq. 6 by jointly
designing the pair $(\gamma,\eta)$, and the jammer whose goal is to maximize
Eq. 6 with respect to $\varphi$.

##  3 Jamming without sensing the channel

We start our analysis by considering the jammer without channel sensing, which
does not have access to the side information $U$. Since the jammer does not
have access to $U$, it randomly blocks the channel with some fixed probability
$\varphi\in[0,1]$, i.e., $\mathbf{P}(J=1)=\varphi.$ More importantly, the
random variable $J$ is independent of $U$, i.e., $J\perp\\!\\!\\!\perp U$.

We are interested in obtaining policies tuples
$(\gamma^{\star},\eta^{\star},\varphi^{\star})$ that constitute a saddle-point
equilibrium, i.e.,

$\mathcal{J}\big{(}(\gamma^{\star},\eta^{\star}),\varphi\big{)}\leq\mathcal{J}\big{(}(\gamma^{\star},\eta^{\star}),\varphi^{\star}\big{)}\leq\mathcal{J}\big{(}(\gamma,\eta),\varphi^{\star}\big{)},$
(7)

for all $\gamma,\eta,\varphi$ in their respective admissible policy spaces.

The first step is to obtain a structural result for the set of optimal
transmission strategies for the sensor.

######  Proposition 1 (Optimality of threshold policies)

For a system with a jammer with fixed jamming probability $\varphi\in[0,1]$,
and an arbitrary estimation policy indexed by representation symbols
$\hat{x}\in\mathbb{R}^{2}$, the optimal transmission strategy is111The
function $\mathbf{1}(\mathfrak{S})$ denotes the indicator function of the
Boolean statement $\mathfrak{S}$, i.e., $\mathbf{1}(\mathfrak{S})=1$ if
$\mathfrak{S}$ is true, and $\mathbf{1}(\mathfrak{S})=0$ if $\mathfrak{S}$ is
false.:

$\gamma_{\eta,\varphi}^{\star}(x)=\mathbf{1}\big{(}(1-\varphi)(x-\hat{x}_{0})^{2}\geq
c\big{)}.$ (8)

######  Proof 3.1.

Using the law of total expectation and the definition of the estimation policy
in Eq. 5, we rewrite Eq. 6 as follows:

$\mathcal{J}\big{(}(\gamma,\eta),\varphi\big{)}=\\\
\mathbf{E}\big{[}(X-\hat{x}_{0})\mid U=0,J=0\big{]}\mathbf{P}(U=0,J=0)\\\
+\mathbf{E}\big{[}(X-\hat{x}_{1})\mid U=0,J=1\big{]}\mathbf{P}(U=0,J=1)\\\
+\mathbf{E}\big{[}(X-\hat{x}_{1})\mid U=1,J=1\big{]}\mathbf{P}(U=1,J=1)\\\
+c\mathbf{P}(U=1)-d\mathbf{P}(J=1).$ (9)

From the fact that $(U,X)\perp\\!\\!\\!\perp J$ and $\mathbf{P}(J=1)=\varphi$,
we have:

$\mathcal{J}\big{(}(\gamma,\eta),\varphi\big{)}=\mathbf{E}\big{[}(X-\hat{x}_{0})\mid
U=0\big{]}\mathbf{P}(U=0)(1-\varphi)\\\
+\mathbf{E}\big{[}(X-\hat{x}_{1})\big{]}\varphi+c\mathbf{P}(U=1)-d\varphi,$
(10)

which is equivalent to

$\mathcal{J}\big{(}(\gamma,\eta),\varphi\big{)}=\int_{\mathbb{R}}(1-\varphi)(x-\hat{x}_{0})^{2}\big{(}(1-\gamma(x)\big{)}f(x)\mathrm{d}x\\\
+\int_{\mathbb{R}}c\gamma(x)f(x)\mathrm{d}x+\varphi\mathbf{E}\big{[}(X-\hat{x}_{1})\big{]}-d\varphi.$
(11)

Finally, when optimizing over $\gamma$ for fixed $\hat{x}$ and $\varphi$, we
have an infinite dimensional linear program with the following constraint:

$0\leq\gamma(x)\leq 1,\ \ x\in\mathbb{R}.$ (12)

The solution to this problem is obtained by comparing the arguments of the two
integrals, i.e., $x\in\\{\xi\mid\gamma^{\star}_{\eta,\varphi}(\xi)=0\\}$ if
and only if

$(1-\varphi)(x-\hat{x}_{0})^{2}\leq c.$ (13)

######  Remark 3.2.

Proposition 1 implies that the optimal transmission policy is always of the
threshold type. This threshold policy is symmetric only if $\hat{x}_{0}=0$.
The optimal policy will be characterized by finite lower and upper thresholds,
if $\varphi\in[0,1)$ or a degenerate policy called never transmit when
$\varphi=1$.

The structure of the optimal transmission policy in Proposition 1 implies that
the objective function assumes the following expression:

$\mathcal{J}\big{(}(\gamma_{\eta,\varphi}^{\star},\eta),\varphi\big{)}=\mathbf{E}\bigg{[}\min\Big{\\{}(1-\varphi)(X-\hat{x}_{0})^{2},c\Big{\\}}\bigg{]}\\\
+\varphi\Big{(}\mathbf{E}\big{[}(X-\hat{x}_{1})^{2}\big{]}-d\Big{)}.$ (14)

The second step in the analysis is to optimize over the estimation strategy,
which is a finite dimensional optimization problem over
$\hat{x}\in\mathbb{R}^{2}$. The pair
$(\hat{x}_{0}^{\star},\hat{x}_{1}^{\star})$ that minimizes Eq. 14 depends on
the pdf $f$, the constants $c,d$ and the strategy of the jammer. It is easy to
see that $\hat{x}_{1}^{\star}=\mathbf{E}[X]$. However, the optimal value of
$\hat{x}_{0}^{\star}$ does not admit a closed form expression and must be
found numerically, in general. However, in the following result we establish a
condition on the pdf $f$ such that the optimal
$\hat{x}_{0}^{\star}=\mathbf{E}[X]$ for any $c,d$ and $\varphi$.

######  Theorem 3.3.

If $f(x)$ is a symmetric and unimodal pdf around $\mathbf{E}[X]$, then

$\eta^{\star}(y)=\begin{cases}\mathbf{E}[X],&\text{if}\ \
y\in\\{\varnothing,\mathsf{B}\\}\\\ x,&\text{if}\ \ y=x.\end{cases}$ (15)

The proof of this result requires the following definitions.

######  Definition 3.4 (Symmetric rearrangement).

Let $\mathbb{D}\subset\mathbb{R}$ be a set of finite measure. Its symmetric
rearrangement $\mathbb{D}^{*}$ is defined as the open interval centered at the
origin whose measure is the same as $\mathbb{D}$.

######  Definition 3.5 (Symmetric decreasing rearrangement).

Let $f:\mathbb{R}\to\mathbb{R}$ be a nonnegative measurable function that
vanishes at infinity. Its symmetric decreasing rearrangement $f^{\downarrow}$
is

$f^{\downarrow}(x)\operatorname{\overset{def}{=}}\int_{0}^{\infty}\mathbf{1}\left(x\in\left\\{\xi\in\mathbb{R}\mid
f(\xi)>t\right\\}^{*}\right)\mathrm{d}t.$ (16)

######  Lemma 3.6 (Hardy-Littlewood Inequality [26]).

Let $f:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ be nonnegative
measurable functions that vanish at infinity. The following inequality holds:

$\int_{\mathbb{R}}f(x)g(x)\mathrm{d}x\leq\int_{\mathbb{R}}f^{\downarrow}(x)g^{\downarrow}(x)\mathrm{d}x,$
(17)

where $f^{\downarrow}$ and $g^{\downarrow}$ are the symmetric decreasing
rearrangements of $f$ and $g$, respectively.

######  Proof 3.7.

(Proof of Theorem 1) From Eq. 14, it is easy to show that
$\hat{x}_{1}^{\star}=\mathbf{E}[X]$. Without loss of generality, we assume
that $\mathbf{E}[X]=0$. We focus on fixing $\varphi\in[0,1)$ and solving the
(equivalent) non-convex optimization problem

$\min_{\hat{x}_{0}\in\mathbb{R}}\ \
\int_{\mathbb{R}}\min\Big{\\{}(x-\hat{x}_{0})^{2},\frac{c}{1-\varphi}\Big{\\}}f(x)\mathrm{d}x.$
(18)

Our proof hinges on establishing the following inequality, under the symmetry
and unimodality assumption of $f$,

$\int_{\mathbb{R}}\min\Big{\\{}(x-\hat{x}_{0})^{2},\frac{c}{1-\varphi}\Big{\\}}f(x)\mathrm{d}x\\\
\geq\int_{\mathbb{R}}\min\Big{\\{}x^{2},\frac{c}{1-\varphi}\Big{\\}}f(x)\mathrm{d}x,\
\ \hat{x}_{0}\in\mathbb{R}.$ (19)

Therefore, implying that $\hat{x}_{0}^{\star}=0$. Next, we prove Eq. 19 by
considering the following equivalent inequality

$\frac{c}{1-\varphi}-\int_{\mathbb{R}}\min\Big{\\{}(x-\hat{x}_{0})^{2},\frac{c}{1-\varphi}\Big{\\}}f(x)\mathrm{d}x\\\
\leq\frac{c}{1-\varphi}-\int_{\mathbb{R}}\min\Big{\\{}x^{2},\frac{c}{1-\varphi}\Big{\\}}f(x)\mathrm{d}x,$
(20)

which can be represented by

$\int_{\mathbb{R}}\max\Big{\\{}\frac{c}{1-\varphi}-(x-\hat{x}_{0})^{2},0\Big{\\}}f(x)\mathrm{d}x\\\
\leq\int_{\mathbb{R}}\max\Big{\\{}\frac{c}{1-\varphi}-x^{2},0\Big{\\}}f(x)\mathrm{d}x.$
(21)

Define

$g(x;\hat{x}_{0})\operatorname{\overset{def}{=}}\max\Big{\\{}{c}/(1-\varphi)-(x-\hat{x}_{0})^{2},0\Big{\\}}$
(22)

and notice that $g^{\downarrow}(x;\hat{x}_{0})=g(x;0)$222 When a function is
symmetric and unimodal around a non-zero point (in this case $\hat{x}_{0}$),
its symmetric decreasing rearrangement corresponds to shifting the function to
the origin.. Moreover, from the assumption that $f$ is symmetric and unimodal
function implies that $f^{\downarrow}(x)=f(x)$. Since $g(x;\hat{x}_{0})$ and
$f(x)$ are nonnegative and vanish at infinity, we may use the Hardy-Littlewood
inequality, which implies in Eq. 21 and, equivalently, in Eq. 19.

The symmetry and unimodality assumptions on $f$ lead to closed form
characterizations for the optimal strategy of the coordinator. These
assumptions are common in the remote estimation literature (e.g. [21, 27, 23],
and references therein), and encompass a large class of distributions,
including Gaussian and Laplace. Without loss of generality, for the remainder
of the paper we assume that $\mathbf{E}[X]=0$.

The optimal transmitter and estimator’s strategies for a symmetric and
unimodal density, implies that the objective function for the jammer is given
by

$\mathcal{J}\big{(}(\gamma_{\eta^{\star},\varphi}^{\star},\eta^{\star}),\varphi\big{)}=\mathbf{E}\bigg{[}\min\Big{\\{}(1-\varphi)X^{2},c\Big{\\}}\bigg{]}\\\
+\varphi\Big{(}\mathbf{E}\big{[}X^{2}\big{]}-d\Big{)}.$ (23)

The objective function in Eq. 23 is concave with respect to $\varphi$, and we
can explicitly determine the optimal jamming probability $\varphi$ by
calculating its derivative and setting it to $0$.

######  Theorem 3.8.

If $f(x)$ is a symmetric and unimodal pdf such that $\mathbf{E}[X]=0$. The
optimal jamming probability  under the optimal transmission policy in
Proposition 1 and the optimal estimation policy in Theorem 3.3 is

$\varphi^{\star}=\left\\{\begin{array}[]{ll}{0}&{\text{ if
}2\int_{\sqrt{c}}^{+\infty}x^{2}f(x)\mathrm{d}x<d}\\\
{\tilde{\varphi}}&{\text{ if
}2\int_{\sqrt{c}}^{+\infty}x^{2}f(x)\mathrm{d}x\geq d}\end{array}\right.,$
(24)

where $\tilde{\varphi}$ is the unique constant in $[0,1)$ that satisfies

$2\int_{\sqrt{c/(1-\tilde{\varphi})}}^{+\infty}x^{2}f(x)\mathrm{d}x=d.$ (25)

######  Proof 3.9.

First, we represent Eq. 23 in integral form as

$\mathcal{J}\big{(}(\gamma_{\eta^{\star},\varphi}^{\star},\eta^{\star}),\varphi\big{)}=\int_{-\sqrt{c/(1-\varphi)}}^{\sqrt{c/(1-\varphi)}}(1-\varphi)x^{2}f(x)\mathrm{d}x\\\
+2\int_{\sqrt{c/(1-\varphi)}}^{+\infty}cf(x)\mathrm{d}x+\varphi\Big{(}\mathbf{E}\big{[}X^{2}\big{]}-d\Big{)}.$
(26)

Taking the derivative of the objective function with respect to $\varphi$, we
have rCl G(φ) & def= ∂∂φJ((γ_η^⋆,φ^⋆,η^⋆),φ)
= 2∫_c/(1-φ)^+∞ x^2f(x)d x -d.

Notice that $\mathcal{G}(\varphi)$ is a monotonically decreasing function with
respect to $\varphi$ and

$\displaystyle\mathcal{G}(0)$
$\displaystyle=2\int_{\sqrt{c}}^{+\infty}x^{2}f(x)\mathrm{d}x-d.$ (27)
$\displaystyle\lim_{\varphi\uparrow 1}\mathcal{G}(\varphi)$
$\displaystyle=-d.$ (28)

If $\mathcal{G}(0)\geq 0$, then the optimal $\varphi^{\star}=\tilde{\varphi}$
due to the fact that $\mathcal{G}(\tilde{\varphi})=0$. If $\mathcal{G}(0)<0$,
the objective function decreases with the increasing of $\varphi$. Therefore,
$\varphi^{\star}=0$.

Theorem 3.10 summarizes the saddle-point strategy for the game between a
coordinator jointly designing the transmission and estimation strategy against
the jammer.

######  Theorem 3.10 (Saddle-point equilibria).

Given a symmetric and unimodal pdf $f$ with $\mathbf{E}[X]=0$, communication
and jamming costs $c,d\geq 0$, the saddle-point strategy
$(\gamma^{\star},\eta^{\star},\varphi^{\star})$ for the game with jammer
without channel sensing is given by:

  * 1)

If $2\int_{\sqrt{c}}^{+\infty}x^{2}f(x)\mathrm{d}x<d$, the optimal policies
are

$\displaystyle\gamma^{\star}(x)=\mathbf{1}(x^{2}>c)$ (29)
$\displaystyle\eta^{\star}(y)=\begin{cases}0,&\text{if}\ \
y\in\\{\varnothing,\mathsf{B}\\}\\\ x,&\text{if}\ \ y=x,\end{cases}$ (30)
$\displaystyle\varphi^{\star}=0.$ (31)

  * 2)

If $2\int_{\sqrt{c}}^{+\infty}x^{2}f(x)\mathrm{d}x\geq d$, the optimal
policies are

$\displaystyle\gamma^{\star}(x)=\mathbf{1}\big{(}(1-\tilde{\varphi})x^{2}>c\big{)}$
(32) $\displaystyle\eta^{\star}(y)=\begin{cases}0,&\text{if}\ \
y\in\\{\varnothing,\mathsf{B}\\}\\\ x,&\text{if}\ \ y=x\end{cases}$ (33)
$\displaystyle\varphi^{\star}=\tilde{\varphi},$ (34)

where $\tilde{\varphi}$ is the unique solution of Eq. 25.

######  Proof 3.11.

We consider two cases:

Case 1 – Assume that $2\int_{\sqrt{c}}^{+\infty}x^{2}f(x)\mathrm{d}x<d$. If
the jammer chooses not to block the channel, i.e., $\varphi^{\star}=0$, using
Proposition 1 we have the corresponding optimal transmission strategy

$\gamma^{\star}(x)=\mathbf{1}(x^{2}>c).$ (35)

Under the jammer and transmitter’s policies above, Theorem 3.3 yields that
$\hat{x}^{\star}_{0}=0$ and $\hat{x}_{1}^{\star}=0$. In this case, the
policies satisfy
$\mathcal{J}\big{(}(\gamma^{\star},\eta^{\star}),\varphi^{\star}\big{)}\leq\mathcal{J}\big{(}(\gamma,\eta),\varphi^{\star}\big{)}.$

If the optimal transmission strategy is $\gamma^{\star}$ and the optimal
estimator is $\eta^{\star}$, using Theorem 3.8 and the assumption that
$2\int_{\sqrt{c}}^{+\infty}x^{2}f(x)\mathrm{d}x<d$, we get the optimal
jammer’s strategy is $\varphi^{\star}=0$. Therefore, we have
$\mathcal{J}\big{(}(\gamma^{\star},\eta^{\star}),\varphi\big{)}\leq\mathcal{J}\big{(}(\gamma^{\star},\eta^{\star}),\varphi^{\star}\big{)}.$

Case 2 – Assume that $2\int_{\sqrt{c}}^{+\infty}x^{2}f(x)\mathrm{d}x\geq d$.
If the jammer blocks the channel with probability $\tilde{\varphi}$, using
Proposition 1 gives the corresponding optimal transmission strategy

$\gamma^{\star}(x)=\mathbf{1}\big{(}(1-\tilde{\varphi})x^{2}>c\big{)}.$ (36)

Under the jammer and transmitter’s policies above, using Theorem 3.3 yields
that $\hat{x}_{0}=0$ and $\hat{x}_{1}=0$. Therefore, we have
$\mathcal{J}\big{(}(\gamma^{\star},\eta^{\star}),\varphi^{\star}\big{)}\leq\mathcal{J}\big{(}(\gamma,\eta),\varphi^{\star}\big{)}.$

If the optimal transmission strategy is $\gamma^{\star}$ and the optimal
estimator is $\eta^{\star}$, using Theorem 3.8 and the assumption that
$2\int_{\sqrt{c}}^{+\infty}x^{2}f(x)\mathrm{d}x\geq d$, we get the optimal
jammer’s strategy is $\varphi^{\star}=\tilde{\varphi}$. Therefore, we have
$\mathcal{J}\big{(}(\gamma^{\star},\eta^{\star}),\varphi\big{)}\leq\mathcal{J}\big{(}(\gamma^{\star},\eta^{\star}),\varphi^{\star}\big{)}.$

Figure 2: Optimal jamming probability for the jammer without channel sensing
$\varphi^{\star}$ as a function of $c$ and $d$. Here, $X\sim\mathcal{N}(0,1)$.

######  Example 3.12.

Consider $X\sim\mathcal{N}(0,1)$, and $c=d=1$. Since

$2\int_{1}^{\infty}x^{2}f(x)\mathrm{d}x=0.8012<1,$ (37)

the optimal jamming probability is $\varphi^{\star}=0,$ which implies that
$\gamma^{\star}$ and $\eta^{\star}$ are given by Eqs. 29 and 30. Let
$X\sim\mathcal{N}(0,2)$, and $c=d=1$. Since

$2\int_{1}^{\infty}x^{2}f(x)\mathrm{d}x=1.8378>1,$ (38)

the optimal jamming probability is $\varphi^{\star}=\tilde{\varphi}=0.7887$,
which implies that $\gamma^{\star}$ and $\eta^{\star}$ are given by Eqs. 32
and 33. Figure 2 shows the optimal jamming probability for the jammer without
channel sensing as a function of the communication and jamming costs, $c$ and
$d$, for $X\sim\mathcal{N}(0,1)$333The code used to obtain the examples in
this paper is available at GitHub
(https://github.com/mullervasconcelos/CDC22.git)..

##  4 Reactive jammer

When the jammer is able to sense if the channel is being used and uses this
knowledge to adjust its probability of blocking the channel, its policy
becomes:

$\varphi(0)\operatorname{\overset{def}{=}}\alpha\ \ \text{and}\ \
\varphi(1)\operatorname{\overset{def}{=}}\beta.$ (39)

For brevity, let $\theta\operatorname{\overset{def}{=}}(\alpha,\beta)$.

Notice that we allow the reactive jammer to block the channel even when the
sensor is not transmitting. To the best of our knowledge, the existing
literature on reactive jamming attacks precludes jamming when the channel is
not being used. There is a reason why the jammer may engage in such counter-
intuitive behavior: when the jammer only blocks a transmitted signal, it
creates a noiseless binary (signaling) channel between the transmitter and the
receiver, which may be exploited by the coordinator. If the jammer is allowed
to “block” the channel when the user is not transmitting, such binary
signaling channel is not noiseless anymore, because there will be uncertainty
if the decision variable at the transmitter is zero or one. This scenario is
illustrated in Fig. 3.

Figure 3: Signaling channel between the sensor and the receiver. The jammer
controls the transition probabilities $\alpha$ and $\beta$. When $\alpha=0$,
the channel is noiseless, i.e., the receiver can unequivocally decode whether
$U=1$ or $U=0$ from the ouput signal $Y$.

######  Proposition 4.13.

For a fixed jamming policy $\varphi$ parametrized by $\theta\in[0,1]^{2}$, and
a fixed estimation policy $\eta$ parametrized by $\hat{x}\in\mathbb{R}^{2}$,
the optimal transmission policy is:

$\gamma_{\eta,\varphi}^{\star}(x)=\mathbf{1}\Big{(}\beta(x-\hat{x}_{1})^{2}+c-d\beta\leq\\\
\alpha(x-\hat{x}_{1})^{2}+(1-\alpha)(x-\hat{x}_{0})^{2}-d\alpha\Big{)}.$ (40)

######  Proof 4.14.

For a reactive jammer, the random variables $X$ and $J$ are conditionally
independent given $U$. Using the law of total expectation, and employing the
estimation policy in Eq. 5, the cost function can be reformulated as

$\mathcal{J}\big{(}(\gamma,\eta),\varphi\big{)}=\mathbf{E}\big{[}(X-\hat{x}_{0})\mid
U=0\big{]}\mathbf{P}(U=0)(1-\alpha)\\\ +\mathbf{E}\big{[}(X-\hat{x}_{1})\mid
U=0\big{]}\mathbf{P}(U=0)\alpha-d\mathbf{P}(U=0)\alpha\\\
+\mathbf{E}\big{[}(X-\hat{x}_{1})\mid
U=1\big{]}\mathbf{P}(U=1)\beta+(c-d\beta)\mathbf{P}(U=1),$ (41)

which is equivalent to

$\mathcal{J}\big{(}(\gamma,\eta),\varphi\big{)}=\int_{\mathbb{R}}\big{[}\beta(x-\hat{x}_{1})^{2}+c-d\beta\big{]}\gamma(x)f(x)\mathrm{d}x\\\
+\int_{\mathbb{R}}\big{[}\alpha(x-\hat{x}_{1})^{2}+(1-\alpha)(x-\hat{x}_{0})^{2}-d\alpha\big{]}\big{(}(1-\gamma(x)\big{)}f(x)\mathrm{d}x.$
(42)

For fixed $\hat{x}\in\mathbb{R}^{2}$ and $\theta\in[0,1]^{2}$, the
transmission policy $\gamma$ that minimizes Eq. 42 is obtained by comparing
the arguments of the two integrals as follows:
$x\in\\{\xi\mid\gamma^{\star}_{\eta,\varphi}(\xi)=1\\}$ if and only if

$\beta(x-\hat{x}_{1})^{2}+c-d\beta\leq\\\
\alpha(x-\hat{x}_{1})^{2}+(1-\alpha)(x-\hat{x}_{0})^{2}-d\alpha.$ (43)

Given the optimal transmitter’s strategy in Proposition 4.13, the objective
function becomes

${}\mathcal{J}\big{(}(\gamma_{\varphi,\eta}^{\star},\eta),\varphi\big{)}=\mathbf{E}\Big{[}\min\big{\\{}\beta(X-\hat{x}_{1})^{2}+c-d\beta,\\\
\alpha(X-\hat{x}_{1})^{2}+(1-\alpha)(X-\hat{x}_{0})^{2}-d\alpha\big{\\}}\Big{]}\operatorname{\overset{def}{=}}\tilde{\mathcal{J}}(\hat{x},\theta).$
(44)

Notice that for fixed $\hat{x}\in\mathbb{R}^{2}$, $\mathcal{J}$ is a concave
function of $\theta$ for any pdf $f$. However, for fixed $\theta\in[0,1]^{2}$,
$\mathcal{J}$ is non-convex in $\hat{x}$. Therefore, the game between the
coordinator and the jammer reduces to the following minimax optimization
problem:

$\min_{\hat{x}\in\mathbb{R}^{2}}\max_{\theta\in[0,1]^{2}}\tilde{\mathcal{J}}(\hat{x},\theta),$
(45)

where $\tilde{\mathcal{J}}(\hat{x},\theta)$ is given by Eq. 44.

Unfortunately, the structure of Eq. 44 does not allow the same techniques we
used to find a saddle-point equilibrium for the jammer without channel
sensing. Instead, a useful alternative to the saddle-point (Nash-equilibrium)
are the solutions that satisfy the first-order stationarity conditions of the
minimization and the maximization problems, yielding in a larger class of
policies, called First-order Nash-equilibria (FNE) [28, 29, 30, 31].

######  Definition 4.15 (Approximate First-order Nash-equilibrium).

Let $\varepsilon>0$. A pair of policies
$(\hat{x}^{\star},\theta^{\star})\in\mathbb{R}^{2}\times[0,1]^{2}$ is an
approximate First-order Nash-equilibrium ($\varepsilon$-FNE) of the game if

$\|\nabla_{\hat{x}}\tilde{\mathcal{J}}(\hat{x}^{\star},\theta^{\star})\|_{2}\leq\varepsilon$
(46)

and

$\max_{\theta\in[0,1]^{2}}\langle\nabla_{\theta}\tilde{\mathcal{J}}(\hat{x}^{\star},\theta^{\star}),\theta-\theta^{\star}\rangle\leq\varepsilon.$
(47)

######  Proposition 4.16.

The function $\tilde{\mathcal{J}}(\hat{x},\theta)$ is differentiable in
$\hat{x}$ and $\theta$. Moreover, the partial gradients are

$\nabla_{\hat{x}}\tilde{\mathcal{J}}(\hat{x},\theta)=\textbf{E}\Bigg{[}\begin{bmatrix}0\\\
-2\beta(X-\hat{x}_{1})\end{bmatrix}\mathbf{1}(\gamma_{\eta,\varphi}^{\star}(X)=1)\\\
+\begin{bmatrix}-2(1-\alpha)(X-\hat{x}_{0})\\\
-2\alpha(X-\hat{x}_{1})\end{bmatrix}\mathbf{1}(\gamma_{\eta,\varphi}^{\star}(X)=0)\Bigg{]}$
(48)

and

$\nabla_{\theta}\tilde{\mathcal{J}}(\hat{x},\theta)=\textbf{E}\Bigg{[}\begin{bmatrix}0\\\
(X-\hat{x}_{1})^{2}-d\end{bmatrix}\cdot\mathbf{1}(\gamma_{\eta,\varphi}^{\star}(X)=1)\\\
+\begin{bmatrix}(X-\hat{x}_{1})^{2}-(X-\hat{x}_{0})^{2}-d\\\
0\end{bmatrix}\mathbf{1}(\gamma_{\eta,\varphi}^{\star}(X)=0)\Bigg{]}.$ (49)

######  Proof 4.17.

This result follows from the Leibniz rule and is omitted due to space
constraints.

###  4.1 Optimization algorithm

To obtain a pair of $\varepsilon$-FNE to the game in Eq. 45, we alternate
between a projected gradient ascent (PGA) step for the inner optimization
problem; and a convex-concave procedure (CCP) step for the outer optimization
problem.

We start with the description of the PGA step at a point
$(\hat{x}^{(k)},\theta^{(k)})$:

$\theta^{(k+1)}=\mathcal{P}_{[0,1]^{2}}\big{(}\theta^{(k)}+\lambda_{k}\,\nabla_{\theta}\tilde{\mathcal{J}}(\hat{x}^{(k)},\theta^{(k)}\big{)},$
(50)

where $\\{\lambda_{k}\\}$ is a step-size sequence (e.g.
$\lambda_{k}=0.1/\sqrt{k}$) and the projection operator is defined as
$\mathcal{P}_{[0,1]^{2}}(\theta):=\min_{\bar{\theta}\in[0,1]^{2}}\|\bar{\theta}-\theta\|_{2}$,
which is equal to

$\mathcal{P}_{[0,1]^{2}}\bigg{(}\begin{bmatrix}\alpha\\\
\beta\end{bmatrix}\bigg{)}=\bigg{[}\begin{array}[]{c}\max\big{\\{}0,\min\\{1,\alpha\\}\big{\\}}\\\
\max\big{\\{}0,\min\\{1,\beta\\}\big{\\}}\end{array}\bigg{]}.$ (51)

To update $\hat{x}^{(k)}$ for a fixed $\theta^{(k+1)}$, we use the property
that Eq. 44 can be decomposed as a difference of convex functions (DC
decomposition). Using the DC decomposition we obtain a specialized descent
algorithm [32], which is guaranteed to converge to stationary points of Eq. 44
for a fixed $\theta^{(k+1)}$[33]. CCP uses more information about the
structure of the objective function than standard Gradient Descent methods,
often leading to faster convergence [32].

Algorithm 1 PGA-CCP algorithm

0: PDF $f$, transmission cost $c$, jamming cost $d$

0: Estimated result $\hat{x}^{\star}$ and $\theta^{\star}$

1: Initialize $k\leftarrow 0,$ $\varepsilon,$ $\hat{x}^{(0)}$ and
$\theta^{(0)}$

2: repeat

3:
$\theta^{(k+1)}=\mathcal{P}_{[0,1]^{2}}\big{(}\theta^{(k)}+\lambda_{k}\,\nabla_{\theta}\tilde{\mathcal{J}}(\hat{x}^{(k)},\theta^{(k)}\big{)}$

4:
$\hat{x}^{(k+1)}=\mathcal{A}^{\dagger}(\theta^{(k+1)})\,g(\hat{x}^{(k)},\theta^{(k+1)})$

5: $k\leftarrow k+1$

6: until $\varepsilon$-FNE conditions (Eqs. 46 and 47) are satisfied

Notice that:

$\tilde{\mathcal{J}}(\hat{x},\theta)=\mathcal{F}(\hat{x},\theta)-\mathcal{G}(\hat{x},\theta),$
(52)

where

$\mathcal{F}(\hat{x},\theta)\operatorname{\overset{def}{=}}(1-\alpha)\hat{x}_{0}^{2}+(\alpha+\beta)\hat{x}_{1}^{2}+(1+\beta)\sigma_{X}^{2}+c-d(\alpha+\beta),$
(53)

and

$\mathcal{G}(\hat{x},\theta)\operatorname{\overset{def}{=}}\mathbf{E}\Big{[}\max\big{\\{}\beta(X-\hat{x}_{1})^{2}+c-d\beta,\\\
\alpha(X-\hat{x}_{1})^{2}+(1-\alpha)(X-\hat{x}_{0})^{2}-d\alpha\big{\\}}\Big{]}.$
(54)

The CCP for computing a local minima for the outer optimization problem is
given by

$\hat{x}^{(k+1)}=\arg\min_{\hat{x}}\left\\{\mathcal{F}(\hat{x},\theta^{(k+1)})-\mathcal{G}_{a}(\hat{x},\theta^{(k+1)};\hat{x}^{(k)})\right\\},$
(55)

where $\mathcal{G}_{a}(\hat{x},\theta^{(k+1)};\hat{x}^{(k)})$ is the affine
approximation of $\mathcal{G}(\hat{x},\theta^{(k+1)})$ with respect to
$\hat{x}$ at $\hat{x}^{(k)}$, while keeping $\theta^{(k+1)}$ fixed, i.e.,

$\mathcal{G}_{a}(\hat{x},\theta^{(k+1)};\hat{x}^{(k)})=\mathcal{G}(\hat{x}^{(k)},\theta^{(k+1)})\\\
+g^{T}(\hat{x}^{(k)},\theta^{(k+1)})(\hat{x}-\hat{x}^{(k)})$ (56)

and $g(\hat{x},\theta)$ is the gradient of $\mathcal{G}(\hat{x},\theta)$ with
respect to $\hat{x}$.

Because $\mathcal{F}$ is a quadratic function of $\hat{x}$ for a fixed
$\theta$, we may use the first-order necessary optimality condition of problem
Eq. 55 to find the recursion for $\hat{x}^{(k+1)}$ in closed form:

$\nabla_{\hat{x}}\mathcal{F}(\hat{x}^{(k+1)},\theta^{(k+1)})=g(\hat{x}^{(k)},\theta^{(k+1)}).$
(57)

The partial gradient of $\mathcal{F}(\hat{x},\theta)$ with respect to
$\hat{x}$ is

$\nabla_{\hat{x}}\mathcal{F}(\hat{x},\theta)=\left[\begin{array}[]{c}2(1-\alpha)\hat{x}_{0}\\\
2(\alpha+\beta)\hat{x}_{1}\end{array}\right].$ (58)

The partial gradient of $\mathcal{G}(\hat{x},\theta)$ with respect to
$\hat{x}$ is

$g(\hat{x},\theta)=\textbf{E}\Bigg{[}\begin{bmatrix}0\\\
-2\beta(X-\hat{x}_{1})\end{bmatrix}\mathbf{1}(\gamma_{\eta,\varphi}^{\star}(X)=0)\\\
+\begin{bmatrix}-2(1-\alpha)(X-\hat{x}_{0})\\\
-2\alpha(X-\hat{x}_{1})\end{bmatrix}\mathbf{1}(\gamma_{\eta,\varphi}^{\star}(X)=1)\Bigg{]}.$
(59)

Finally, define $\mathcal{A}:[0,1]^{2}\rightarrow\mathbb{R}^{2\times 2}$ as

$\mathcal{A}(\theta)=\left[\begin{array}[]{cc}2(1-\alpha)&0\\\
0&2(\alpha+\beta)\end{array}\right],$ (60)

and $\mathcal{A}^{\dagger}$ denotes its Moore-Penrose pseudo-inverse. Then,
the update of CCP can be compactly represented as

$\hat{x}^{(k+1)}=\mathcal{A}^{\dagger}\big{(}\theta^{(k+1)}\big{)}\,g\big{(}\hat{x}^{(k)},\theta^{(k+1)}\big{)}.$
(61)

The PGA-CCP algorithm is presented in Algorithm 1 and its empirical efficacy
is demonstrated in the following example.

######  Example 4.18.

Let $X\sim\mathcal{N}(0,\sigma^{2})$, and $c=d=1$. We set
$\varepsilon=10^{-5}$. Our algorithm results in the pairs of $\varepsilon$-FNE
in table 1 444The code we used to implement the PGA-CCP algorithm is available
at GitHub (https://github.com/mullervasconcelos/CDC22.git)..

Table 1: $\varepsilon$-FNE obtained using the PGA-CCP algorithm $\sigma^{2}$ | $\alpha^{\star}$ | $\beta^{\star}$ | $\hat{x}_{0}^{\star}$ | $\hat{x}_{1}^{\star}$
---|---|---|---|---
$1$ | $0.0760$ | $0.3172$ | $0.5169$ | $-0.4831$
$2$ | $0.0350$ | $0.1572$ | $0.7030$ | $-0.4338$
$3$ | $0.0136$ | $0.0937$ | $0.7618$ | $-0.4204$
$4$ | $0.0040$ | $0.0634$ | $0.7894$ | $-0.4148$
$5$ | $0$ | $0.0475$ | $0.8082$ | $-0.4039$

Figure 4 shows the optimal jamming probabilities $\alpha^{\star}$ and
$\beta^{\star}$ as a function of $\sigma^{2}$. Notice that the probability
that the jammer will block the channel even when the sensor does not transmit
is nonzero. Figure 5 shows that the optimal transmission policy is asymmetric
for $c=1,d=1$, and $X\sim\mathcal{N}(0,1)$. Figure 6 shows the convergence to
an $\varepsilon$-FNE for $c=1,d=1$, and $X\sim\mathcal{N}(0,1)$ using the PGA-
CCP (this paper) and the Gradient Descent Ascent (GDA) [31] algorithms. The
step size for PGA-CCP is set to be $\lambda=0.1$ and the step sizes for GA and
GD in GDA are set to be $\lambda_{\mathrm{GA}}=0.1$ and
$\lambda_{\mathrm{GD}}=0.01$, respectively555For the sake of fairness, the
step sizes for the GA update of both algorithms are set to be the same. Due to
the asymmetric nature of nonconvex-concave problems, GDA requires a time-scale
separation in the step-sizes to avoid convergence to limit cycles or even
divergence [31]. The reason for $\lambda_{\mathrm{GA}}>\lambda_{\mathrm{GD}}$
is that the inner maximization problem has better structure (the cost function
is concave) [31].. Our numerical results show that PGA-CCP converges with a
rate approximately $6$ times faster than the algorithm in [31].

Figure 4: Optimal jamming probabilities $\alpha^{\star}$ and $\beta^{\star}$
as a function of $\sigma^{2}$. Here, $c=1,d=1$, and
$X\sim\mathcal{N}(0,\sigma^{2})$. Figure 5: Optimal transmission policy for
$c=1,d=1$, and $X\sim\mathcal{N}(0,1)$. Figure 6: Convergence of PGA-CCP vs.
GDA for $c=1,d=1$, and $X\sim\mathcal{N}(0,1)$.

##  5 Conclusions and future work

We have studied a zero-sum signaling game with asymmetric information
involving a sensor, a jammer and an estimator. Two scenarios were considered:
jamming with and without channel sensing. For the jammer without channel
sensing, we have shown that under a symmetry and unimodality assumption of the
observation’s pdf, there exists a saddle-point equilibrium, where the optimal
transmission policy at the sensor is of the symmetric threshold type. For a
reactive jammer, the objective function does not admit a similar result.
Instead, we exploit the structure of outer and inner-optimization problem to
obtain an algorithm to find $\varepsilon$-FNE. There are many interesting
research directions for future work. The first possible extension is the case
when there are $n$ sensors sharing the network over a channel that can support
$k<n$ packets. Additionally, it is important to prove the convergence of the
PGA-CCP algorithm and the characterization of its convergence rate. Finally,
study how the coordinator and the jammer learn to play in equilibrium if they
do not have access to each other’s costs.

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