Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
.
Uncoordinated Frequency Hopping under Nash
.
Equilibrium
Bingwen Zhang, Lifeng Lai
Worcester Polytechnic Institute
December 14, 2012
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
. Outline
1.
Introduction
Why Uncoordinated Frequency Hopping?
Why Nash Equilibrium
.2 Basic Model
Basic Model
.3 Equal Channel Quality Case
4. General Channel Quality Case
5.
One Access Multiple Jamming Case
6.
Multiple Access One Jamming Case
7.
Numerical Simulation
Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
Why Uncoordinated Frequency Hopping?
Why Nash Equilibrium
. Jamming Attack
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
Why Uncoordinated Frequency Hopping?
Why Nash Equilibrium
. Jamming Attack
The purpose of jamming attack is to interrupt the ongoing
wireless communications between legitimate users by injecting
signals into wireless media.
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
Why Uncoordinated Frequency Hopping?
Why Nash Equilibrium
. Jamming Attack
The purpose of jamming attack is to interrupt the ongoing
wireless communications between legitimate users by injecting
signals into wireless media.
Jamming attack is a kind of physical attack.
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
Why Uncoordinated Frequency Hopping?
Why Nash Equilibrium
. Jamming Attack
The purpose of jamming attack is to interrupt the ongoing
wireless communications between legitimate users by injecting
signals into wireless media.
Jamming attack is a kind of physical attack.
Jamming attack is easily launched cannot be fully addressed
by conventional cryptography.
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
Why Uncoordinated Frequency Hopping?
Why Nash Equilibrium
. Jamming Attack
The purpose of jamming attack is to interrupt the ongoing
wireless communications between legitimate users by injecting
signals into wireless media.
Jamming attack is a kind of physical attack.
Jamming attack is easily launched cannot be fully addressed
by conventional cryptography.
Can be mitigated by pread spectrum techniques, including
Direct Sequence Spread Spectrum (DSSS) and Frequency
Hopping (FH).
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
Why Uncoordinated Frequency Hopping?
Why Nash Equilibrium
. Coordinated Frequency Hopping
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
Why Uncoordinated Frequency Hopping?
Why Nash Equilibrium
. Coordinated Frequency Hopping
Need frequency hopping pattern
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
Why Uncoordinated Frequency Hopping?
Why Nash Equilibrium
. Coordinated Frequency Hopping
Need frequency hopping pattern
Transmitter and receiver hop according to this pattern
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
Why Uncoordinated Frequency Hopping?
Why Nash Equilibrium
. Coordinated Frequency Hopping
Need frequency hopping pattern
Transmitter and receiver hop according to this pattern
Pattern needs to be shared between transmitter and receiver
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
Why Uncoordinated Frequency Hopping?
Why Nash Equilibrium
. Coordinated Frequency Hopping
Need frequency hopping pattern
Transmitter and receiver hop according to this pattern
Pattern needs to be shared between transmitter and receiver
Pattern needs to be synchronized between transmitter and
receiver
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
Why Uncoordinated Frequency Hopping?
Why Nash Equilibrium
. Coordinated Frequency Hopping
Need frequency hopping pattern
Transmitter and receiver hop according to this pattern
Pattern needs to be shared between transmitter and receiver
Pattern needs to be synchronized between transmitter and
receiver
A third party does not this pattern cannot predict the
transmission
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
Why Uncoordinated Frequency Hopping?
Why Nash Equilibrium
. An example of a frequency hopping pattern
An example of Frequency Hopping pattern
5
channel
4
3
2
1
0
0
5
10
timeslot
15
20
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
Why Uncoordinated Frequency Hopping?
Why Nash Equilibrium
. Coordinated vs. Uncoordinated
Questions:
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
Why Uncoordinated Frequency Hopping?
Why Nash Equilibrium
. Coordinated vs. Uncoordinated
Questions:
For a network has N nodes, the total number of pre-shared
codes in this network is
N(N−1)
.
2
If N is large enough, how can
we store so many different patterns?
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
Why Uncoordinated Frequency Hopping?
Why Nash Equilibrium
. Coordinated vs. Uncoordinated
Questions:
For a network has N nodes, the total number of pre-shared
codes in this network is
N(N−1)
.
2
If N is large enough, how can
we store so many different patterns?
If two devices have not communicate before, how could they
share a common hopping pattern?
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
Why Uncoordinated Frequency Hopping?
Why Nash Equilibrium
. Cirlularly Dependency
Key establishment in
the presence of
jammer
Key establishment in
the presence of
jammer
Dependency circle
Dependency circle
FH
Frequency hopping
sequence
Frequency hopping
sequence
UFH
Figure: Cirlularly Dependency
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
Why Uncoordinated Frequency Hopping?
Why Nash Equilibrium
. UFH
To solve these problem, the basic scheme of UFH is proposed.
It is originally designed for jamming-resistant key establishment.
The authors design the UFH scheme as the transmitter and receiver
hop randomly between channels and the transmission is successful
when they are in the same channel and the same timeslot.
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
Why Uncoordinated Frequency Hopping?
Why Nash Equilibrium
. Nash Equilibrium
In game theory, the Nash Equilibrium is a solution concept of
a non-cooperative game involving two or more players, in which
each player is assumed to know the equilibrium strategies of the
other players, and no player has anything to gain by changing only
his own strategy unilaterally.
If each player has chosen a strategy and no player can benefit
by changing his or her strategy while the other players keep theirs
unchanged, then the current set of strategy choices and the
corresponding payoffs constitute a Nash Equilibrium.
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
Why Uncoordinated Frequency Hopping?
Why Nash Equilibrium
. Nash Equilibrium
Nash Equilibrium is a static point.
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
Why Uncoordinated Frequency Hopping?
Why Nash Equilibrium
. Nash Equilibrium
Nash Equilibrium is a static point.
Under Nash Equilibrium, changing one’s own strategy
unilaterally can not leads to greater reward.
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
Why Uncoordinated Frequency Hopping?
Why Nash Equilibrium
. Nash Equilibrium
Nash Equilibrium is a static point.
Under Nash Equilibrium, changing one’s own strategy
unilaterally can not leads to greater reward.
Everyone is taking his best strategy while taking into account
the decisions of the others.
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
Why Uncoordinated Frequency Hopping?
Why Nash Equilibrium
. Nash Equilibrium
Nash Equilibrium is a static point.
Under Nash Equilibrium, changing one’s own strategy
unilaterally can not leads to greater reward.
Everyone is taking his best strategy while taking into account
the decisions of the others.
Under Nash Equilibrium, the reward of trasmitter and receiver
in UFH is optimal.
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
Basic Model
. Basic Model
Alice: transmitter Bob: receiver Eve:jammer
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
Basic Model
. Basic Model
Alice: transmitter Bob: receiver Eve:jammer
Alice and Bob hop randomly between channels.
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
Basic Model
. Basic Model
Alice: transmitter Bob: receiver Eve:jammer
Alice and Bob hop randomly between channels.
The transmission between Alice and Bob is successful when
Alice and Bob use the same channel and at the same time Eve
is not jamming this channel.
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
Basic Model
. Basic Model
Channel
1
2
3
4
5
6
7
8
9
Alice
(Sender)
Bob
(Receiver)
Eve
(Jammer)
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Figure: BasicUncoordinated
Model Frequency Hopping under Nash Equilibrium
Bingwen Zhang, Lifeng Lai
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
Basic Model
. Basic Model
The average throughput of UFH is
R̄ =
N
∑
(
)
Ri pti pri 1 − pji ,
(1)
i=1
where N is the number of channels everyone can use in this game,
Ri is the data rate that can be achieved of channel i, pti ,pji are the
probabilities for Alice and Bob respectively to access channel i, pji
is the probability of Eve jams channel i, i ∈ {1, 2, ..., N}.
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
Basic Model
Alice and Bob are striving to make their average throughput
as large as possible. Eve, however, is trying to minimize the
average throughput at the same time. So we can model this
scenario as a zero-sum game. One party of the game is the
Alice and Bob, and the other is Eve. So we can view it as a
zero-sum game.
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
Basic Model
In this game, the reward of Alice and Bob is R̄, the reward of
Eve is −R̄. So Alice and Bob want to maximize R̄, while Eve
wants to maximize −R̄. Then this basic setup forms a
zero-sum game. Thus Nash Equilibrium can be achieved
which means no player’s reward will increase by changing his
own strategy unilaterally. We are going to find out the Nash
Equilibrium in the following sections.
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
We first study the special case the qualities of all the N
channel are equal to illustrate the basic principle to obtain the
Nash Equilibrium in this game. Since R1 = R2 = ... = RN , we can
denote Ri = R, for ∀i ∈ {1, 2, ..., N}.
.
Proposition
.
In this case, the Nash Equilibrium is:
∗
∗
pti = pri ∗ = pji =
1
,
N
(2)
for
. ∀i ∈ {1, 2, ..., N}.
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
. Organization of Proof
Let A, B and E denote the support set for Alice, Bob and Eve
respectively. We first prove E = {1, 2, ..., N}. Then we prove
E ⊆ A , E ⊆ B, A = B. So A = B = E = {1, 2, ..., N}. In the end,
we determine the Nash Equilibrium.
Then we denote the strategy taken by Alice, Bob and Eve
achieving Nash Equilibrium as:
∗
∗
∗
∗
Pt = {pt1 , pt2 , ..., ptN },
P
t∗
= {pr1 ∗ , pr2 ∗ , ..., prN ∗ },
j∗
∗
∗
∗
{pj1 , pj2 , ..., pjN }.
P =
(3)
(4)
(5)
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
. Reward
The reward of Alice and Bob accessing channel i is:
)
)
(
(
Ri 1 − pji = R 1 − pji .
(6)
The reward of Eve jamming channel i is:
−
N
∑
Rj ptj prj = S + Ri pti pri ,
(7)
j=1,j̸=i
where S = −
∑N
t r
j=1 Rj pj pj .
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
. Prove: E = {1, 2, ..., N}.
If E ̸= {1, 2, ..., N}, then ∃i ∈ {1, 2, ..., N} s.t. pji = 0.
Case 1: pti = pri = 1, then it is obvious for Eve to increase his
reward by jamming channel i. So this is not a Nash
Equilibrium.
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
. Prove: E = {1, 2, ..., N}.
If E ̸= {1, 2, ..., N}, then ∃i ∈ {1, 2, ..., N} s.t. pji = 0.
Case 1: pti = pri = 1, then it is obvious for Eve to increase his
reward by jamming channel i. So this is not a Nash
Equilibrium.
Case 2: pti ̸= 1 and pri ̸= 1. Since Eve never jams channel i, so
Alice and Bob can always increase their reward by allocating
more probability into channel i. But they cannot achieve
maximum by setting pti = pri = 1. So Alice and Bob can
always increase their reward by changing their strategy
unilaterally. This is not a Nash Equilibrium.
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
. Prove: E ⊆ A , E ⊆ B, A = B.
If E * A, then Eve is jamming some channel that is never
used by the transmitter. So Eve can increase his reward by
jamming some other channel. So E ⊆ A 1 . The proof is the
same for E ⊆ B.
1
The ⊆ symbol does not mean proper set in this paper.
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
. Prove: E ⊆ A , E ⊆ B, A = B.
If A ̸= B, then Alice is transmitting on some channel that is
can never used by Bob or Bob is listening on some channel
that Alice’s message never comes from. This means Alice or
Bob is wasting her or his resources. So Alice and Bob can
increase their reward by allocating their probability on the
same set of channels. Thus, A = B.
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
. Prove: E ⊆ A , E ⊆ B, A = B.
If A ̸= B, then Alice is transmitting on some channel that is
can never used by Bob or Bob is listening on some channel
that Alice’s message never comes from. This means Alice or
Bob is wasting her or his resources. So Alice and Bob can
increase their reward by allocating their probability on the
same set of channels. Thus, A = B.
The above conclusion implies that if all the channel quality
are equal, then A = B = E = {1, 2, ..., N}.
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
∗
∗
∗
t
t
j
. Determine P , P and P .
First we will show that to achieve Nash Equilibrium,
1 − pji = C0 and pti pri = C1 , where C0 and C1 are constants
independent of i. If 1 − pji ̸= C0 , then ∃l1 , l2 ∈ {1, 2, ..., N} s.t.
1 − pjl1 = max{1 − pji }, 1 − pjl1 > 1 − pjl2 , where
i ∈ {1, 2, ..., N}, So Alice and Bob can always increase their
reward by moving their probability of accessing channel l2 into
accessing channel l1 . Thus this is not a Nash Equilibrium.
Then 1 − pji = C0 .
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
∗
∗
∗
t
t
j
. Determine P , P and P .
First we will show that to achieve Nash Equilibrium,
1 − pji = C0 and pti pri = C1 , where C0 and C1 are constants
independent of i. If 1 − pji ̸= C0 , then ∃l1 , l2 ∈ {1, 2, ..., N} s.t.
1 − pjl1 = max{1 − pji }, 1 − pjl1 > 1 − pjl2 , where
i ∈ {1, 2, ..., N}, So Alice and Bob can always increase their
reward by moving their probability of accessing channel l2 into
accessing channel l1 . Thus this is not a Nash Equilibrium.
Then 1 − pji = C0 .
The proof is same for pti pri = C1 .
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
∗
∗
∗
t
t
j
. Determine P , P and P .
First we will show that to achieve Nash Equilibrium,
1 − pji = C0 and pti pri = C1 , where C0 and C1 are constants
independent of i. If 1 − pji ̸= C0 , then ∃l1 , l2 ∈ {1, 2, ..., N} s.t.
1 − pjl1 = max{1 − pji }, 1 − pjl1 > 1 − pjl2 , where
i ∈ {1, 2, ..., N}, So Alice and Bob can always increase their
reward by moving their probability of accessing channel l2 into
accessing channel l1 . Thus this is not a Nash Equilibrium.
Then 1 − pji = C0 .
The proof is same for pti pri = C1 .
pji =
1
N
for ∀i ∈ {1, 2, ..., N}.
Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
Under Nash Equilibrium, R̄ =
∑N
i=1 C1
(
1−
1
N
)
= (N − 1) C1 .
Since Alice and Bob form one party of the game, so the we can
take the pair (Pt , Pr ) as the strategy of Alice and Bob. So Alice
and Bob want maximize R̄, that is, to maximize C1 . We build two
vectors
√ √
√
pt1 , pt2 , ..., ptN ],
√ √
√
A⃗2 = [ pr1 , pr2 , ..., prN ],
A⃗1 = [
(8)
(9)
A⃗1 and A⃗2 are two vector in RN . Then |A⃗1 | = 1 and |A⃗2 | = 1.
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
A⃗1 · A⃗2 =
N √
∑
pti ∗
√
i=1
pri ∗
=
N √
∑
√
C1 = N C1 ,
(10)
i=1
so we can convert the problem of maximizing C1 into maximizing
A⃗1 · A⃗2 . A⃗1 · A⃗2 = |A⃗1 ||A⃗2 | cos θ ≤ |A⃗1 ||A⃗2 | = 1, the equality holds
when θ = 0, where θ is the measure of angle between A⃗1 and A⃗2 .
So we can maximize C1 by setting θ = 0. The two vectors A⃗1 and
A⃗2 have the same length, and the same direction, so they are
√
equal. So pti ∗ = pri ∗ = C1 for ∀i ∈ {1, 2, ..., N}. Then we can
determine pti ∗ = pri ∗ =
1
N
for ∀i ∈ {1, 2, ..., N}. From the proof
above, Alice and Bob cannot increase their reward by changing to
another strategy unilaterally. Thus we have proved our proposition.
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
. General Channel Quality Case
In this section, we expand the case into general channel quality case, which is
R1 ≤ R2 ≤ ... ≤ RN .
.
Proposition
.
The Nash Equilibrium in this case is
pti
=
pri
√1
= ∑N
Ri
i=k
pji
=
for k ≤ i ≤ N, where k = min{k|Rk >
.
√1
N−k
Ri
1 − ∑N
1
i=k Ri
∑N−k
1
N
i=k Ri
(11)
,
Ri
(12)
,
}.
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
. Organization of Proof
Let A, B and E denote the support set for Alice, Bob and Eve
respectively. We first prove E = {k, k + 1, ..., N}, where k is a
certain integer that 1 ≤ k ≤ N. Then we prove E ⊆ A , E ⊆ B,
A = B, and A = B{k1 , k1 + 1, ..., N}. In the end, we prove k1 = k
and determine the Nash Equilibrium.
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
. Reward
The reward of Alice and Bob accessing channel i is:
)
)
(
(
Ri 1 − pji = R 1 − pji .
(13)
The reward of Eve jamming channel i is:
−
N
∑
Rj ptj prj = S + Ri pti pri ,
(14)
j=1,j̸=i
where S = −
∑N
t r
j=1 Rj pj pj .
.
Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
Prove: E = {k, k + 1, ..., N}, where k is a certain integer
. that 1 ≤ k ≤ N.
If E ̸= {k, k + 1, ..., N}, then there exist some k2 , pjk2 = 0.
Case 1: If there does not exist a channel k1 < k2 , pjk1 > 0.
Then by setting k = k2 , we can express the support set of Eve in
the form of E = {k, k + 1, ..., N}, where k = k2 .
.
Bingwen Zhang, Lifeng Lai
.
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
Case 2: If there exist k1 < k2 , pjk1 > 0. That is, Eve gives up
jamming some channel that is not the worst in his support set.
The reward of Alice and Bob accessing channel k2 is
(
)
Rk2 1 − pjk2 = Rk2 . Then we have
)
(
Rk2 ≥ Rk1 > Rk1 1 − pjk1 ,
(15)
which means Alice and Bob can receive more reward if they put
more probability on using channel k2 . So this contradicts with
Nash Equilibrium. This completes the proof for
E = {k, k + 1, ..., N}, 1 ≤ k ≤ N.
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
. Prove: E ⊆ A, E ⊆ B, A = B
The proof is the same as it is in the previous section. So we
know A = B = {k1 , k1 + 1, ..., N}, where k1 ≤ k. This means Eve
never jams channel k1 to k − 1 if k1 ̸= k. For channel k1 to k − 1,
∑k−1
∑k−1 t r
∑k−1 t ∑k−1 r
t r
i=k1 Ri pi pi ≤ Rk−1
i=k1 pi pi < Rk−1 ( i=k1 pi )( i=k1 pi ). So
Alice and Bob should put all their probability from channel k1 to
k − 1 into channel k − 1. Then we can bound k1 by k − 1 ≤ k1 ≤ k.
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Bingwen Zhang, Lifeng Lai
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.
Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
∗
∗
t
t
j
. Determine P , P and P
∗
∗
We can solve Pj first. Ri (1 − pji ) = C0 for
∀i ∈ {k, k + 1, ..., N}. Then we have pji = 1 −
∀i ∈ {k, k + 1, ..., N}. Summing
C0 =
pji
C0
Ri
for
from k to N,
N−k
∑N 1 ,
pji = 1
i=k Ri
N−k
R
− ∑N i 1
i=k Ri
(16)
(17)
.
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
.
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
k = min{k|Rk >
N−k
∑N 1
i=k Ri
If Rm satisfy Rm >
}
∑N−m
N
1
, and then for m + 1,
i=m Ri
( N
)
∑ 1
> N − m,
Rm
Ri
i=m
( N
)
∑ 1
Rm+1
> N − (m + 1),
Ri
(18)
(19)
i=m+1
Rm+1 >
N − (m + 1)
,
∑N
1
(20)
i=m+1 Ri
.
Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
′
If we choose k instead of k, then we will show this is not a Nash Equilibrium.
′
′
If k > k, the reward of Alice and Bob accessing channel i ≥ k is
(
)
′
′
j
j
Ri 1 − pi = ∑N−k
1 . From above, for i < k , pi = 0. The reward of Alice and Bob
N
′
i=k Ri
′
′
accessing channel k − 1 is Rk′ −1 . We have k − 1 ≥ k, so
Rk′ −1

N
∑
1
1

Rk′ −1 
+
Ri
Rk′ −1
′
( ′
)
N− k −1
,
∑N
1
(21)
>
N − k + 1,
(22)
=
(
)
Rk′ 1 − pj ′ .
>
′
i=k −1 Ri

i=k
′
N−k
Rk′ −1 > ∑N
1
′
i=k
(23)
k
Ri
′
The right hand side term is the reward of Alice and Bob acessing channel k . So the
′
′
reward of Alice and Bob accessing channel k − 1 is better than accessing channel k ,
this contradicts with Nash Equilibrium.
Bingwen Zhang, Lifeng Lai
.
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
′
If k < k, then Rk′ ≤
′
∑N−k
N
1
′
i=k Ri
, so
′
pj ′ = 1 −
k
N−k
R
∑N i 1
′
i=k Ri
≤ 0.
(24)
′
′
This contradicts with our assumption that E = {k , k + 1, ..., N}
which means pj ′ > 0.
k
.
Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
Then we will find the strategy of on the side of Alice and Bob achieving Nash
Equilibrium.
If k1 = k − 1, then we have Rk−1 = C0 = ∑N−k
1 and Rk > Rk−1 . The total
N
i=k Ri
reward of Alice and Bob is
R̄
=
N
∑
∗
Ri pti pri (1 − pji )
(25)
i=k1
=
Rk−1 ptk−1 prk−1
+ C0
(26)

pti pri 
(27)
i=k+1
N
∑
C0 (ptk−1 + ptk )(prk−1 + prk ) +

=
N
∑
C0 ptk−1 prk−1 + ptk prk +

≤
)
pti pri
i=k

=
( N
∑
′
N
∑
′
C0 ptk prk +


pti pri 
(28)
i=k+1
pti pri  ,
(29)
i=k+1
.
Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
and the equality holds when ptk−1 = prk−1 = 0. So the reward
of Alice and Bob will increase if they transfer their effort of
accessing channel k − 1 to accessing channel k. So k1 = k − 1 is
not a Nash Equilibrium. So k1 = k = min{k|Rk >
.
Bingwen Zhang, Lifeng Lai
.
∑N−k
N
1
i=k Ri
.
}.
.
.
.
Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
. Determine Nash equilibrium of Alice and Bob
We can still use A⃗1 and A⃗2 . |A⃗1 | = |A⃗2 | = 1.
A⃗1 · A⃗2 =
N √
∑
i=k
=
N
∑
i=k
√
pti pri
(30)
C1
Ri
(31)
≤ |A⃗1 ||A⃗2 | = 1,
.
Bingwen Zhang, Lifeng Lai
(32)
.
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
the equality holds when A⃗1 and A⃗2 have the same direction.
A⃗1 and A⃗2 also have same length, so they are equal. Then
√
√1
Ri
C1 = (∑ 1 )2 . pti = pri = CR1i = ∑N √
1 .
N
i=k
√1
i=k
Ri
Ri
.
Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
Thus we have completed the proof for the Nash equilirium:
pti
=
√1
R
= ∑N i 1
√
i=k Ri
N−k
R
− ∑N i 1 ,
i=k Ri
pri
pji = 1
for k ≤ i ≤ N, where k = min{k|Rk >
∑N−k
N
1
i=k Ri
(34)
}.
.
Bingwen Zhang, Lifeng Lai
(33)
,
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
. Remark
In general channel quality case, Alice, Bob and Eve always
operate on the same set of channels.
Alice, Bob and Eve do not access or jam all channels. k is a
variable that is for Alice, Bob and Eve to decide on which
channels they should take actions.
When the channel quality is better, then Eve jams this
channel with a larger probability while Alice and Bob access
this channel with a smaller probability.
It is simple to verify that k ≥ 2, thus avoiding to be jammed
by Eve with probability 1.
Bingwen Zhang, Lifeng Lai
.
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
. One Access Multiple Jamming Case
In this section, we inherit the basic setting in the previous
section, the only difference is Eve can jam more than one channel
simultaneously. We assume Eve can jam M ≥ 2 channels
simultaneously and Alice and Bob can transmit and receive
through only one channel each time. Let Ω denote the set of
channels Eve jams, where Ω has M elements.
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Bingwen Zhang, Lifeng Lai
.
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.
Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
. One Access Multiple Jamming Case
.
Proposition
.
The Nash Equilibrium in this case is,
pti
∑
=
pj (Ω) = 1
Ω,i∈Ω
where k = min{k|Rk >
.
√1
R
= ∑N i 1 ,
√
i=k Ri
(N−k+1)−M
R
− ∑N i 1 ,
i=k Ri
pri
(35)
(36)
(N−k+1)−M
∑N 1
}.
i=k Ri
.
Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
. Multiple Access One Jamming Case
In this case, the sender and receiver can access multiple
channels in one time slot, but the attacker can only jam one
channel at a time. We assume Alice and Bob can access Mt and
Mr channels respectively, where 1 ≤ Mt ≤ N and 1 ≤ Mr ≤ N. The
strategy of Alice and Bob taken in a time slot is denoted by ΩA
and ΩB . Obviously, ΩA and ΩB are subset of A and B , and ΩA
and ΩB are Mt set and Mr set respectively. Let pti denote
∑
∑
r
ΩA PΩA , and pi denote
ΩB PΩB . And without loss of
i∈ΩA
i∈ΩB
generality, we can assume Mt ≥ Mr .
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
. Multiple Access One Jamming Case
.
Proposition
.
The Nash Equilirium in this case is given under different conditions: If k ≥ Mt ,
pti
Mt
√
∑N
=
Ri
i=k
pri
∑N
Ri
√1
(37)
,
(38)
Ri
N−k
Ri
1 − ∑N
=
for k ≤ i ≤ N, where k = min{k|Rk >
.
,
Ri
Mr
√
=
i=k
pji
√1
1
i=k Ri
∑N−k
1
N
i=k Ri
(39)
,
}.
.
Bingwen Zhang, Lifeng Lai
.
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.
.
Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
. Multiple Access One Jamming Case
.
Proposition
.
If Mt > k > Mr ,
pti
=
(40)
1,
for N − Mt + 1 ≤ i ≤ N,
pri
=
Mr
√
Ri
,
∑N
√1
i=k
(41)
Ri
pji
for k ≤ i ≤ N,
where k = min{k|Rk >
.
∑N−k
1
N
i=k Ri
=
}.
Bingwen Zhang, Lifeng Lai
N−k
Ri
1 − ∑N
1
i=k Ri
(42)
,
.
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
. Multiple Access One Jamming Case
.
Proposition
.
If k ≤ Mr ,
pti
=
1,
(43)
pri
=
1,
(44)
for N − Mt + 1 ≤ i ≤ N,
for N − Mr + 1 ≤ i ≤ N,
pji
for k ≤ i ≤ N, where k = min{k|Rk >
.
N−k
Ri
1 − ∑N
=
1
i=k Ri
∑N−k
1
N
i=k Ri
Bingwen Zhang, Lifeng Lai
(45)
,
}.
.
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
. Equal Channel Quality Case
average data rate
We first examine the case average throughput and set R = 100. In Figure below,
the average throughput is stabilized when time goes on. The average throughput is
R = 9. This is the same as the result calculation using R̄ = R N−1
.
N2
10
9
8
7
6
5
4
3
2
1
0
0
2
4
6
number of timeslot
8
10
4
x 10
Figure: Equal Channel Quality Case
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
. General Channel Quality Case
Ri are i.i.d drawn from exp(100). This basic setup is the same
for following simulation sections.
120
average data rate
100
80
60
40
20
0
0
2
4
6
8
number of timeslot
10
4
x 10
Figure: General Channel Quality Case
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
. General Channel Quality Case
R̄ =
(
∑NN−k1 2 .
√ )
i=k
From numerical simulation, we can see R̄ is
Ri
Gamma distributed.
0.015
0.01
0.005
0
0
10
20
30
Figure: Distribution of R̄
Bingwen Zhang, Lifeng Lai
40
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
. One Access Multiple Jamming Case
average data rate
20
15
10
5
0
0
2
4
6
number of timeslot
8
10
4
x 10
Figure: Multiple Jamming Case
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Bingwen Zhang, Lifeng Lai
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Uncoordinated Frequency Hopping under Nash Equilibrium
Introduction
Basic Model
Equal Channel Quality Case
General Channel Quality Case
One Access Multiple Jamming Case
Multiple Access One Jamming Case
Numerical Simulation
Thank you!
.
Bingwen Zhang, Lifeng Lai
.
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Uncoordinated Frequency Hopping under Nash Equilibrium