User-Centric Analysis on Jamming Game with Action
Detection Error
Jiniiang Liu, Liang Xiao*, Yan Li, Lianfen Huang
361000,
Dept. of Communication Engineering, Xiamen University,
China
*Email: [email protected]
Abstract-We formulate the interactions between a legitimate
transmitter and a smart jammer allocating power flexibly as a
power control Stackelberg anti-jamming game with the action
detection error from transmitter to jammer in a user-centric
view. More specifically, decision-making of the players follows
the subjective deviations specified by prospect theory instead
and Tversky to explain users' subjectivity in decision-making
under uncertainty environment
[5], [6].
Prospect theory is a descriptive and behavioral model of
decision making under risk for individuals. PT uses decision
weight deviating from probability weighting function in the
of the traditional objective assumption controlled by expected
value function instead of the stated objective probability
utility theory. The Stackelberg equilibrium of the game as well
directly
as the Nash equilibrium are analyzed and the impact of the
players' subjectivity and action detection error on the signal­
to-interference and noise ratio at the receiver is measured.
[5], [6].
The theory awarded the Nobel Prize for
its contribution on monetary transaction explains that people
would be risk seeking in losses and risk averse in gains.
Simulation results show that a subjective jammer which is less
Although the PT derived from monetary transaction, it is
likely to attack the transmitter results in the increasing of the
increasingly becoming a hotspot in a variety of areas
the signal-to-interference plus noise ratio and the behaviors of
the smart jammer in the Stackelberg game cause more damage
to the legal communications.
Index Terms-game theory, anti-jamming, Stackelberg, user­
centric, signal-to-interference plus noise ratio, power control.
[7]-[14].
To address smart jamming in real-life, PT is applied to
model the subjectivity of the players, i.e., januner and user in
corresponding jamming game
[7], [15],
which both proved the
existence of Nash Equilibria (NE) and computed it and eval­
uated the impact of the players' subjectivity on performance.
I. INT RODUCTION
However, assuming user and jammer behave simultaneously
Wireless conununication is highly vulnerable to jammer
attacks because of the broadcast nature and open access of
radio propagation. More specifically, jammer can put great
damage on the transmissions of users by sending malicious
signals. In particular, smart januner which is capable of
controlling over their strategies flexibly threatens the legal
communications effectively. Game theory has been studied as
an powerful tool to model and address such security issues
[1]-[4].
Traditional decision-making models in game theory are
based on expected utility theory from the perspective of the ra­
tional choice to quantify the players' perception of uncertainty
or error. However, an individual's behavior in reality is always
conducted by practical environment as well as its subjectivity
such as personality, psychological factor.
[5]
showed that the
assumption that an individual behaves rationally absolutely
doesn't hold in practical situation and the actual decision­
making process involving one's subjectivity is depart from
the prediction of expected utility theory. Motivated by such
a scenario, prospect theory (PT) was developed by Kahneman
The work of L. Xiao was partly supported by NSFC (61271242. 61001072.
61301097). NCETFJ. and Fundamental Research Funds for the Central
Universities (2012121028.2013121023).
The work of Huang was partially supported by 2011 National Natural
Science Foundation of China (Grant number 61172097) .2014 National
Natural Science Foundation of China (Grant number 61371081) 2013 Natural
Science Technology of Fujian (Grant number 2013H0048 ) and by 2013
Science Technology Project of Xiamen (3502Z20131155).
in NE does not conform to the reality and one player may not
have complete information of the other's identities.
In this paper,
we apply prospect theory to investigate
the anti-jamming communications in wireless network. More
specifically, we model the interaction between a legitimate
transmitter and a smart jammer as a power-controlled S­
tackelberg game (SG) based on PT, in which leader is the
transmitter that commits a power strategy in advance and
follower is the smart jammer that obtains the optimal strat­
egy by identifying the transmitter's strategy. However, given
the underlying behavior detection error and gossip channel
propagation error, we introduce the action detection error into
the channel from the transmitter to the jammer
[1], [16].
We
assume the smart jammer cannot obtain the actual transmitter's
strategy but knows the probability of action to be identified
successfully. Therefore, the decision weighting function was
developed in
[17]
to formulate subjectivity of the both players
for the action's identification probability.
To the best of our knowledge, this paper first applies the
prospect theory to the Stackelberg game in anti-jamming wire­
less communications. We analyze the Stackelberg equilibria
(SE) of the PT-based power-controlled anti-januning game,
where the utility functions of both players are based on SINR
at the receiver and costs from power allocation. The NE
analysis of the PT-based anti-jamming game is provided as
a comparison.
The remainder of the paper is organized as follows. In
Section II, we briefly review the related work. In Section III,
(
t,j)
ax
pmax
x
ex
hx
Player
(Transmitter or jammer)
Action of Player x
Maximum power of Player x
Cost per unit Player x power
Channel gain between Player x and receiver
Background noise power
Action detection error matrix from transmitter to jammer
Prob. to successfully identify an action of transmitter
Objective weight of Player x
Subjective prob. weight func. of x
Instant payoff of Player x
EUT-based utility of Player x
PT-based utility of Player x
x =
The system model and PT-based Stackelberg game model are
presented. Next, the SE strategies in Section IV and the NE
strategies in Section are analyzed in V. In Section VI, we
(T
present the simulation results. Finally, conclusion is provided
1>
in Section VII.
c:
Qx
wx(C:)
ux(m,n)
II. RELATED WORK
U:Ul
Ux'"
Game theory-based approaches have been proposed by
x
TABLE I
many researchers to improve the communication security re­
SUMM ARY
cently. The game-theoretic randomized anti-jamming strategy
OF
S Y MBOLS AND NOTATION.
for power allocation and channel hopping is analyzed based
on learning algorithm in
[18].
Competing for dominating an
open spectrum access with jammer was investigated following
a stochastic game approach and using Q-Iearning to solve
for an optimal channel access strategy in
[19].
A januning
game for users without knowing the other users' identities was
formulated to model such uncertainties in
[20].
A dynamic
zero-sum game with asymmetric information was analyzed
against intelligent jammer that can detect transmission in
[21].
A power control game for cognitive radio network was
presented to formulated the SU with incomplete knowledge of
the jammer's location in
[22].
Due to the emerging smart jammers which are capable of
analyzing the opponents' strategy, wireless networks are under
serious threat. As a powerful method to describe the hierarchi­
cal behavior among players, Stackelberg game has recently at­
tracted a lot of attention. Power control against a smart jammer
was firstly investigated based on a Stackelberg game approach
and the corresponding Stackelberg equilibrium was presented
in
[23].
The defense strategy against an effective dynatnic
jamming attack was modeled as a Stackelberg game in wireless
personal area network in
[24].
In
[25],
a Stackelberg security
game with a cooperative jammer was analyzed to improve
physical layer security over multiuser OFDMA networks. The
decision making process of a correlated jammer for the cyber­
physical control system was modeled as a dynamic Stackelberg
game in
[26].
However, none of the above works takes the
subjectivity of the players into consideration; this is what we
focus on.
Prospect theory (PT) has emerged its promising to formulate
and illustrate the decision making process of humankind in
a variety of area such as communication networks
and smart energy management
[13], [14].
[7]-[12]
In addition, the
works of applying prospect theory into the anti-jamming
communication are still limited. Adjusting the selfish players
itself transmission probabilities over a collision channel was
considered as a random access game following the percepts
[7].
III. PT- BASED STACKELBERG ANTI-JAMMING GAME
A. Stackelberg Game
In this paper, we formulate the power allocation strategies of
nodes in anti-jamming wireless communication into Stackel­
berg game where the leader is a legitimate transmitter and the
follower is a smart jammer. The players aim to maximize their
individual utility ultimately via allocating translnission signal
power. More specifically, the transmitter commits a power
strategy for sending signal and then the jammer chooses a
best response for jamming correspondingly.
In the game, we consider finite discrete power action set
for transmitter and continuous power action set for jammer
at E
{Pm}, m 1, . . . , K, where translnission power is given by
Pm and 0 ::; at ::; Ptmax. Similarly, we denote the action of
jammer by aj and 0 ::; aj ::; prax. For simplicity, we also
mx
assume 0 ::; PI < P2 < . . . ::; pt a where p;,ax is the
to allocate. We denote the action of translnitter by
=
maximum power of the player
x.
We consider the interactions between two players as a non­
zero non-cooperative game. The translnitter with an action
at
=
Pm
aj q
Cx be the transmission cost per unit power of
behaves in advance and smart januner with
reacts latter. Let
the player
x, x =
t,j.
=
We take the signal-to-interference-plus
noise ratio (SINR) as a reward for the transmitter. We denote
Ux (m, n) as the instant utility of player with at Pm and
aj q. Thus the instantaneous payoffs are given by
htPm
(1)
- CtPm,
Ut(at Pm, aj q)
+hj q
htPm
(2)
Uj(at Pm, aj q) - +h q - Cj q,
j
whereht andhj are the channel gains between the players and
x
=
=
=
=
=
=
=
=
receiver, respectively and
B.
CJ
CJ
CJ
is the background noise power.
Action Detection Process
A prospect theory-based static januning game
However, due to the gossip channel propagation error and
was investigated to derivate the optimal strategy on channel
the behavior detection error, it is uncertainty for the smart
of PT in
access rate in
[15].
In this paper, we firstly apply PT into
jammer that the transmission action he knows is the actual
the Stackerberg game against a subjective smart jammer, in
behavior of transmitter. Therefore, misunderstanding to strate­
which there exists action detection error from the legitimate
gies of translnitter at jammer's view would influence the results
transmitter to the jammer.
of the SG.
IV. STACKELBERG EQUILIBRIUM IN THE PT-BASED
We can model the probability distribution matrix of the
action detection process as <I>
cm,n
[cm,nlo:C;m n:C;K'
,
denotes the probability action
and when
n
=
m = n,
cm,n
in which
indicates the probability for each
transmitter action to be accurately identified. For simplicity,
1-€
'
we assume cm n = c I'f m = n and otherWlse
cm n = K
-l'
,
,
We also assume the action identification matrix is known to
players in the SG.
In this section, we investigate the SE in the PT-based
anti-jamming game, where subjective smart jammer adjusts
strategy according to the behavior of the transmitter in the
case of action detection error from translnitter to jannner to
maximize its PT-based utility. To facilitate analysis, we assume
that the players in the same real-life has the same objective
weight, i.e.,
Prospect Theory
C.
ANTI-JAMMING GAME
to be taken as action
m
Although smart jammer has knowledge of the action i­
dentification matrix, the decision-making is guided with the
assumption that players are rational and uninfluenced by real­
at
=
aj.
The players choose their own power strategy to maximize
their individual PT-based utility to obtain the SE, thus the
optimal strategies in the game are given by
life perceptions. To close to the reality, prospect theory (PT)
(8)
has been proposed from a user-centric view to model the
(9)
subjective nature of human decision-making process.
Given the action detection error, we now apply the prospect
theory to model the Stackelberg anti-jatmning translnission
with Perlecs probability weight function. Moreover, subjective
Let Pm be a given strategy of the transmitter
and thus the optimal subjective jamming strategy is given by
Lemma IV.I.
players tend to under-weigh moderate and high probability
outcome and over-weigh low probability outcome. We denote
wx(c)
where
the probability weight function of player
ax
w x (c )
=
exp(-(-lnc)C>x),O
x
(3)
is defined as objective weight of players and
a;
pJmax '
=
=
Let
1, w x (c )
ulUT
=
c
denote the expected utility of translnitter aver­
at and thus the
expected utilities of the transmitter and jatnmer are given by
UE
t UT(at
UE
j UT(at
=
Pm , aj
=
q
htPi
) �
� cm, i(
- CtPi) ,
+hjq
=
Pm, aj
=
q
htPi
) �
- Cjq) .
� cm, i(+hJ. q
=
i=1
=
i=1
(Y
(4)
�
v
(5)
Therefore, facing the action detection error from the leader
to the follower, a subjective translnitter or jammer chooses
its transmission signal power to maximize its prospect theory­
based utility given by
U{'T(at
=
Pm, aj
K
=
q)
=
h
L Wt(Cm,i)( +t� q - CtPi),
i=1
(Y
hthj 2:: Wj(€m,;)Pi
_--i --" -'-'--- c=;
�
Cj 2:: Wj(€m,i)
�i=l
�j (
shows that player is objective.
aged over all the action realizations following
j
(6)
2
K
Cj 2:: Wj(€=,i)
----ic:;:--' h"'� -;hj:
a
­
K
decreases with the players subjectivity. In the special case of
ax
L Wj(c m,i)Pi <
i=1
0,
given by
< ax:S; 1,
K
Proof
0,
8U!T(at, aj)
8aj
We have82 U!T8
/ aj 2 < 0
U!T
is maximized by
for 0
:s;
aj :s; p rax.
Remark. As shown in
with respect to
indicating
a;vt
(10),
o.W..
),
(10)
U!T
We differentiate
the resulting equal to
- (Y
aj and
set
UPT is concave. Thus
K
=
�j (
hthj 2:: Wj(€=,i)Pi
_----"i7"-'--- �
cj 2:: Wj(€=,i)
i=l
- (Y
)
•
the smart jammer knows there is
not sufficient transmission if the power of the c Olmnunication
is lower than the cost for the jannner and thus the j atmner
chooses
a;
=
0 to ignore the ongoing transmission. On the
other hand, the jammer will apply the highest power to jatn
the current translnission if the jamming cost is negligible
compared with the the high transmission power. Otherwise,
We first consider the translnitter commits an action
which is identified as
Pi, 1 :s;
i
:s; K,
at
=
Pm
with weight probability
Wj(c m,i) by the subjective smart jatmner who weighs his own
the jatmning power is adjusted following
The probability
c
a;vt.
should not be too small because there is
no need to jamming or translnit when
c
is quite small, which
indicates channel condition is fairly bad. In the meantime, we
L�1 Wx(c 1,i)Pix inferring ax 0 and L�1 Wx(c K,i)Pi
ax p :,;'a , which can be proved. We assume / hj
overall utility. Since the subjective transmitter who aims to
have
minimize the reward for jammer knows the decision-making
indicating
process of the jammer, the reward part of transmitter utility in
is smaller enough otherwise the channel is terrible so as to
(6)
damage connnunications.
deviates from that of jannner's utility in
(7).
=
=
(Y
Lemma IV.2.
The optimal subjective transmission strategy of
the SG
K
K
K
htCj L Wj(c m,i)/ hj L Wt(Cm,i)Pi - Ct L Wt(Cm,i)Pi
i=1
i=1
i=1
and derivate it as:
0,
f l'
argmin
atE{Pm}m�1,
. .
h,Cj
argmin
a'E{Pm}m�1,
p max,
. .
dG
dat
f2 ,
,K
,K
K
2:: Wj(Em,;)
�h:c,2
K
- L Wt(Cm,i)Pi
i=1
t
f3,
o. w . ,
htCj
K
2:: W,(Em,')
i�
�j
Wt(c m,m)
--'----r=====
==-
2 JL�1 Wt(Cm,i)Pi
K
h
Wj(Cm,i)
tCj
L
1
i=l
d2 G
dat2
- CtWt(c m,m),
(13)
(14)
4
(11)
GO is concave. By (13), G is maximized
K ( )
htCj 2::t':-1 Wj(Em.i)
""
whI'ch can
L..,i=l Wt Cm,i Pi 4hjC,2
be calculated to obtain a;. Following the constraint,
m
o :s; at :s; pt ax , we have:
hi.
< C:t In the case, UtPT decreases with at for
1)
o :s; at :s; prax, thus a; = 0;
2<y2) ;� :s; Ct < �. : If L�1 Wt(CK,i)Pi <
c, ""K
.
. h at, thus a*t
UP
WIt
t T Increases
h,h,. L..,i=1 Wj( Cm,i),
2Cj
<Y
K
K
Ptmax - PK ,.·If h,hj ""
L..,i=l Wj( Cm,i) < ""
L..,i=l Wt( CI,i)Pi,
U[T decrease with at, thus a; = 0; Otherwise, a; is given
by (11) at f 2 ;
< Cj < 3) <Y+h�P!'wX < Ct < ;�: If L�l Wt(CK,i)Pi >
htCj 2::'-1 Wj(Em.i)
K ( ) p UPT is concave
""
>
L..,,
=l Wt C1,' " t
4h.Ct2
4h,c, 2 2:::"-1 W,(E1,')P'
K
h,e, 2::t':-1 Wj(Em,').
""
'
< CJ < and maxmnze
. . d wit
. h L..,i=l Wt( Cm,i)Pi hI. 2::{�1 Wj(Em.i)
,
4hjC, 2
.
4) 2(hj prax+ ) :s; Ct < <Y+h�prax: U[T(L�1 Wt(Cm,i)Pi)
increases with L�1 Wt(Cm,i)Pi which increases with at, for
>
htCj 2:::"-1 Wj(Em,')
K
K
""
L..,i=l Wt( Cm,i)Pi <
L..,i=l Wt( Cl,i)Pi < ""
(hjpmax+<y)2Cj K 4hjC,2
K
or Li=1 Wt(Cm,i)Pi >
Li=1 Wj(Cm,i) and
h,h,
We
find
WI'th
<y
_
K
.1
.
<
hi.
2<Y'
'
.
_
<y
_
1
.
decreases otherwise. The proofs about this case are similar
and omitted here because of the limitation of length.
5 ) Ct < 2(hjP';,:ax+ ) : In the case, U[T increases
0 -< at_< p'nw
t x, thus at* = ptmax =PK .
<y
for
Proof
Plugging
aj*
with
at
•
Remark. In conclusion, we have the SE of the PT-based
into
(6),
game given by
we have
(a;, aj)
in lennna 1 and lemma 2. The smart
j atmner allocates the optimal power according to the jamming
cost, channel state and the subjectivity for the action detection
U{'T(at =Pm, aj) =
error of the transmission power. Based on the jammer's subK
K
2 j K
L Wt(Cm,i)Pi(� - Ct), L Wj(Cm,i)Pi < �t�J L Wj(c m,i);jective strategy, the subjective transmission adjusts its strategy
i=l
i=l
i 1
in line with the transmission cost and the jannner's reward
J(
required to be minimized.
L Wt(Cm,i)Pi( <Y+h�p:nax - Ct),
i=l
K
(hjpmax+<y)2Cj K
>
Wj(Cm,i)Pi
Wj(Cm,i);v. NASH EQUILIBRIUM IN THE PT-BASED ANTI-JAMMING
h,hj
�
i;'
i 1
COMMUNI C ATION GAME
htCj 2:: Wj(Em,')
K
K
i=1
L Wt(Cm,i)Pi - Ct L Wt(Cm,i)Pi,
Let (at'E, afE ) denote the NE of the static game where
i=l
i=l
.1
1
.
----'.=.c
.:
--,,---,---K
o.w ..
(12)
According
to
(12),
we
define
G(at
jammer can't sense the ongoing transmission to adjust strategy
and two players behave at the same time. At a NE, given the
power levels of the other players, no player can improve its
UJT
utility by changing its power unilaterally.
UtPT(atNE, aNE) > UtPT(at, aNE)
UPT(aNE aNE) > UPT(aNE a. )
t
J
The
Lemma V.l.
'
NE
(afE, ayE)
(ptmax pmax )
( ptmax, 0) ,
J
-
J
-
J
t
J
'
J
'
(15)
'
(16)
of the game is given by
is concave and
afE
is obtained by setting
(19)
O. Similarly, the result from h can be obtained.
3)
is
h
If
(afE, afE) resulting
/ at
oUrT(at, afE)o
holds, the NE
obtained
by
setting
oUJT(afE, aj)o
/ at
=
equals
from
0
=
O.
•
=
'
J
(0 pmax )
(0,0) ,
'
(
J
ptmax,
VI. SIMULATION RESULTS
'
In this section, we evaluate the performance of the proposed
'
-.L
hj
ft W.j(€K,.;)P'
hjht.
1
_----"'''''-.=...
(
____
K
OJ 2:: Wj(€K,i)
_
(J
)
PT-based Stackelberg security game through simulations. We
)
investigate the impact of the j armner's objective weight aj on
the performance such as the average SINR and utilities showed
(1),(2)
by
Is,
(17)
i=l
(10),(11)
at
ht
=
=
hj
as
0 : 0.005 :
=
0.8
=
1
aj.
We set
Ct
=
0.4, Cj
with the noise power
=
=
0.5
receiver is presented in Fig.
0.8
SE and
argmin
to
0.62
the probability
c
0.33
and
for the game. The SINR at the legitimate
1,
showing that both that of the
0.65
from 0.5
For instance, the SINR decreases from about
Cj >
(J
SE and the NE decrease with jammer's objectivity (i.e.,
where
Ct <
under
In the simulation, for simplicity, we assume the users'
o.W.,
a,E{p=}==l, .. ,K
(17)
to the NE given by
different objective weights.
and
I2:
of both players. We also compare the performance
of the SE given by
objective weight
hI.
O'+h.p!nax,
J .1
h
and
at NE, as changes aj
=
0.7.
to
to
aj).
0.42 at
1, with
The reason is that a j armner with
subjectivity is less likely to jam the legal transmission as it
hjht. 2:: 1"-1 Wj(€K,i)Pi
.
(0'+hjP;nax)22::1"= 1 Wj(€K,i)'
K
hjh, 2:: Wj(€K,i)P,
overweighs its loss from wasteful jamming. Meanwhile, for a
fixed aj , a higher probability of action identified successfully
�i=l
(i.e.,
K
0'22:: Wj(€K,i)
c) results in a smaller SINR. This is because a subjective
c has more clear understanding
jammer with higher probability
�i=l
on action detection process. In addition, the smart jammer
can learn the ongoing transmission flexibly before decision
making.
Fig.
2
indicates the impacts of aj
both players with the probability
the transmitter decease with
aj,
c
=
on the utilities of
0.9.
The utilities of
while those of the jammer
increase with aj regardless of the scenarios. This is because
both subjective players are less likely to transmit or jam to
avoid highly unfavorable loss due to the action detection error.
One the other hand, the jammer in the SE strategy is more
intelligent than that in the NE strategy. Consequently, the
Proof
we differentiate
with respect to aj.
oUtPT
ht
( +h a
oat
jj
___
=
(J
UrT with respect to at and UJT
_
Ct)Wt(c m,m),
(18)
K
K
(J
=
J
J
"
'
=
proved similarly.
2)
afE
If
=
Is
holds, by
ptmax.
transmitter's utility is just opposite.
VII. CONCLUSION
Wj(c m,i)Pi
ouJPT hjht L
i=1
- Cj L Wj(c m,i). (19)
oaj
( +hjaj) 2
i=1
1) If h holds, by (18) we haveoUrTo
l at > 0, yielding
atNE ptmax · In this case by (19) we haveoUPTo
l a >0
ax
p
indicating afE
r . The cases (i.e., 12,h,I4) can be
__
jammer's utility at SE is higher than that at NE while the
(18)
oUrTo
l at
we have
However, differentiating
(19)
>
0,
In this paper, prospect theory is applied to formulate the
Stackelberg anti-jaming communication game and a static
jamming game, in which both the transmitter and the smart
jammer make decisions relying on their subjectivities and aim
to maximize the SINR-based utilities under power constraints
under the channel condition that there exists action detection
error from transmitter to jammer. We have derived the SE
strategies of both players and the NE of the jamming game
provided as a comparison to evaluate the impact of the
jammer's subjectivity about the probability of action identified
yielding
successfully on the jamming game, which is not considered
indicates that
in the traditional expected utility theory. Simulation results
0.9 1-----,--- --�-----,--- r=====il
....... NE,E=0.9
0.85
_SE,E=0.9
O.
-+ - NE,E=0.7
-*-SE,E=0.7
....----_.
0.5L_�_--c'=-----====:::!���
O.4
Objective weight of jammer,aj
Fig. 1.
Average SINR at the receiver: Performance comparison of a
transmitter and smart jammer vs. objective weights (Xj in an PT-based
Stackelberg game and a static game with action detection process,
0.2
o
-0.2
@
-0.4
::::l
-0.6
-0.8
-1
.;,
;.�*-
�'-,*--*- -*;",�-*.;..;.*.;. ...
-+-+--+-+-�- -+- -+
-
+
-
+--
- 1 . 2 L-___-'-___-'-___----'____-'---___--'
0.5
0.6
0.7
0.8
Objective weight of jammer,aj
0.9
Fig. 2,
Average transmitter and jammer utilities: Performance comparison
of a transmitter and smart jammer vs. objective weights (Xj in an PT-based
Stackelberg game and a static game with the probability E: = 0.9,
have shown that a subjective jammer with smaller probability
c
which is less likely to attack the transmitter leads to the
increasing of the SINR and the behaviors of the smart jammer
in SG does more harm to the legal communications.
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