Computing Nash Equilibrium in Wireless Ad Hoc Networks Using Statistical Model Checking
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Peter BulychevAlexandre DavidKim G. Larsen
Marius Mikucionis
Computing Nash Equilibrium in Wireless Ad Hoc Networks
Using Statistical Model Checking
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GASICS 2011
Nash Eq in Wireless Ad Hoc Networks
Consider a wireless network, where there is a master node that chooses the optimal parameters that should be used by other nodes
power=20%
power=20%
power=20%
Peter Bulychev [2]
Master node
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GASICS 2011
Nash Eq in Wireless Ad Hoc Networks
Now, if there are selfish nodes, they might want to change these parameters to achieve better performance
power=20%
power=20%
power=80%
Peter Bulychev[3]
Master node
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GASICS 2011
Nash Eq in Wireless Ad Hoc Networks
Now, if there are selfish nodes, they might want to change these parameters to achieve better performance
power=20%
power=90%
power=80%
Peter Bulychev [4]
We say that network configuration satisfies Nash equilibrium if it's not profitable for a node to alter its behavior to the detriment of other nodes
Master node
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GASICS 2011
Nash Eq in Wireless Ad Hoc Networks
power=40%
power=40%
power=40%
Peter Bulychev[5]
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GASICS 2011
Problem statement
Peter Bulychev[6]
Input:1. Each node is modeled by a parameterized Priced Timed
Automata M(p), where p∈P and P is finite2. System of N nodes is modeled by S(p1,
p2, …, pN) ≡ M(p1)||M(p2)||…||M(pN)||C3. Each node k has a goal φk (i.e. to transmit a message
within given timed and energy bounds)4. Utility function of a node k is defined as a probability
that φk is satisfied by a random run:Uk(p1, p2, …, pk) ≡ Pr[S(p1, p2, …, pk) ⊨ φk]
Goal: To find symmetric NE, i.e. to find p∈P s.t.:
∀p’ P U∈ ⋅ 1(p, p, …, p)≥ U1(p’, p, …, p)
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GASICS 2011
Problem statement
Peter Bulychev[7]
Input:1. Each node is modeled by a parameterized Priced Timed
Automata M(p), where p∈P and P is finite2. System of K nodes is modeled by S(p1,
p2, …, pk) ≡ M(p1)||M(p2)||…||M(pk)||C3. Each node k has a goal φk (i.e. to transmit a message
within given timed and energy bounds)4. Utility function of a node k is defined as a probability
that φk is satisfied by a random run:Uk(p1, p2, …, pk) ≡ Pr[S(p1, p2, …, pk) ⊨ φk]
Goal: To find symmetric NE, i.e. to find p∈P s.t.:
∀p’ P U∈ ⋅ 1(p, p, …, p)≥ U1(p’, p, …, p)
Nash Equilibrium might not exist in non-mixed strategiesThus, we will consider a relaxed definition of Nash Equilibrium
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GASICS 2011
Problem statement
Peter Bulychev[8]
Input:1. Each node is modeled by a parameterized Priced Timed
Automata M(p), where p∈P and P is finite2. System of K nodes is modeled by S(p1,
p2, …, pk) ≡ M(p1)||M(p2)||…||M(pk)||C3. Each node k has a goal φk (i.e. to transmit a message
within given timed and energy bounds)4. Utility function of a node k is defined as a probability
that φk is satisfied by a random run:Uk(p1, p2, …, pk) ≡ Pr[S(p1, p2, …, pk) ⊨ φk]
Goal: To find symmetric δ-relaxed NE, i.e. to find p∈P s.t.:
∀p’ P U∈ ⋅ 1(p, p, …, p)≥ δ*U1(p’, p, …, p)
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GASICS 2011
Related work
Peter Bulychev[9]
Pioneering work:“Game theory and the design of self-configuring, adaptive wireless networks”, MacKenzie et.al. , 2001.
Survey:“Using game theory to analyze wireless ad hoc networks”, Srivastava et.al., 2006.
Most of the papers use pure simulation(1) or analytical-based(2) approaches:(1) doesn’t provide confidence on its results(2) doesn’t scale to complex models
What can we propose?
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GASICS 2011
Our SMC-based approach
Peter Bulychev[10]
SMC = Simulation + Statistics
Scales to complex models
Can provide confidence on
its results
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GASICS 2011
Our SMC-based approach
Peter Bulychev11]
First, we use simulation-based algorithm to find a strategy p that is a good candidate for δ-relaxed NE for as large δ as it is possible
Then we apply statistics to compute δ s.t. we can accept the hypothesis that p is a δ-relaxed NE with a given significance level α
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GASICS 2011
SMC-based approach (Part I)Input: P – finite set of strategies, U(pi, pk) – utility function, d [0,1] - threshold∊Goal: find p P that maximizes min∊ p’ P∊ Ũ(p, p)/Ũ(p’, p)Algorithm:
1. for every p P ∊ compute estimation Ũ(p,p)2. waiting := P3. candidates := ∅4. while len(waiting)>1:5. pick some unexplored pair (p’,p) P × waiting∊6. compute estimation Ũ(p’, p)7. if Ũ(p, p)/Ũ(p’, p) < d:8. remove p from waiting9. if p’ Ũ(p’, p) is already computed:∀10. remove p from waiting11. add p to candidates12. return argmaxp P∊ minp’ P∊ Ũ(p, p)/Ũ(p’, p)
Peter Bulychev[12]
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GASICS 2011
SMC-based approach (Part I)Input: P={p1, p2, …, p10} – finite set of strategies, U(pi, pk) – utility function, d [0,1] - threshold∊Goal: find p P that maximizes min∊ p’ P∊ Ũ(p, p)/Ũ(p’, p)
Ũ(p1,p1) Ũ(p10,p1)
Ũ(p1,p10)Ũ(p10,p10)
Peter Bulychev[13]
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GASICS 2011
SMC-based approach (Part I)
Peter Bulychev[14]
Input: P={p1, p2, …, p10} – finite set of strategies, U(pi, pk) – utility function, d [0,1] - ∊thresholdGoal: find p P that maximizes min∊ p’ P∊ Ũ(p, p)/Ũ(p’, p)
Ũ(p1,p1) Ũ(p10,p1)
Ũ(p1,p10)Ũ(p10,p10)
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GASICS 2011
SMC-based approach (Part I)
Ũ(p8,p8) ≥ d*Ũ(s6,s8)
Ũ(p6,p6) < d*Ũ(p3,p6)
Peter Bulychev[15]
Input: P={p1, p2, …, p10} – finite set of strategies, U(pi, pk) – utility function, d [0,1] - ∊thresholdGoal: find p P that maximizes min∊ p’ P∊ Ũ(p, p)/Ũ(p’, p)
Ũ(p1,p1) Ũ(p10,p1)
Ũ(p1,p10)Ũ(p10,p10)
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GASICS 2011
SMC-based approach (Part I)
Peter Bulychev[16]
Input: P={p1, p2, …, p10} – finite set of strategies, U(pi, pk) – utility function, d [0,1] - ∊thresholdGoal: find p P that maximizes min∊ p’ P∊ Ũ(p, p)/Ũ(p’, p)
Ũ(p1,p1) Ũ(p10,p1)
Ũ(p1,p10)Ũ(p10,p10)
Ũ(p8,p8) ≥ d*Ũ(s6,s8)
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GASICS 2011
SMC-based approach (Part I)
Peter Bulychev[17]
“EmbarrassinglyParallelizable”
argmaxp P∊ minp’ P∊ Ũ(p, p)/Ũ(p’, p)
Input: P={p1, p2, …, p10} – finite set of strategies, U(pi, pk) – utility function, d [0,1] - ∊thresholdGoal: find p P that maximizes min∊ p’ P∊ Ũ(p, p)/Ũ(p’, p)
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GASICS 2011
SMC-based approach (Part II)
Peter Bulychev[18]
Ũ(pk,pk)
Ũ(pk+1,pk)
Ũ(pn,pk)
Ũ(pk-1,pk) …
Ũ(p1,pk)
…
By definition pk satisfies δ-relaxed NE iff∀i [1,n] U(p∈ ⋅ k, pk)≥ δ*U(pi, pk)
Now we:1. Reestimate each Ũ(pi, pk) using N SMC runs2. Apply the following theorem:Theorem. We can accept the hypothesis that pk satisfies δ-relaxed NE with a given significance level α, if:
…
…
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GASICS 2011
Implementation details
Peter Bulychev[19]
SSH connection
SSH connectionSSH connection
SSH connection
Python frontend
node 1
node 2
node 3
node 4
UPPAAL backend
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GASICS 2011
Case studies
Peter Bulychev[20]
We used our tool to compute Nash Equilibrium for two CSMA (Carrier Sense Multiple Access) protocols:1. k-persistent ALOHA CSMA/CD protocol2. IEEE 802.15.4 CSMA/CA protocol
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GASICS 2011
Aloha CSMA/CD protocol Simple random access
protocol (based on p-persistent ALOHA) several nodes sharing the
same wireless medium each node has always data to
send, and it sends data after a random delay
in case of collision both stations wait for a random delay
delay has a geometrical distribution with parameter p=TransmitProb
Peter Bulychev[21]
Pr[Node.time <= 3000](<>(Node.Ok && Node.ntransmitted <= 5))
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GASICS 2011
Value of utility function for the cheater node
Results (3 nodes)
Peter Bulychev[22]
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GASICS 2011
Results (Aloha)
Peter Bulychev [23]
N=2 N=3 N=4 N=5 N=6 N=7 N=8Nash Eq 0.37 0.40 0.35 0.35 0.41 0.42 0.41The value of δ
0.992 0.993
0.992
0.990 0.993 0.992
0.987
Ũ(sNE,sNE) 0.99 0.98 0.95 0.89 0.75 0.61 0.50Opt 0.37 0.30 0.26 0.22 0.19 0.15 0.14Ũ(sopt, sopt) 0.99 0.98 0.96 0.90 0.87 0.81 0.76Symmetric Nash Equilibrium and Optimal strategies for
different number of network nodes
#cores 4 8 12 16 20 24 28 32Time 38m 19m 13m 9m46s 7m52s 7m04s 6m03s 5m
Time required to find Nash Equilibrium for N=3 100x100 parameter values
(8xIntel Core2 2.66GHz CPU)
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GASICS 2011
IEEE 802.15.4 CSMA/CA protocol
Peter Bulychev[24]
nb:=0be:=MinBE
Delay for random(0..2be)UnitBackoffPeriod
Channel is clear?
nb:=nb+1be:=min(be+1, MaxBE)
nb>MaxNB?
Failure Transmit
Y
N
Y
N
Switch to transmitting
IEEE 802.15.4 CSMA/CA is based on the random backoff procedure
We assume that a node can change its UnitBackoffPeriod
parameter
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GASICS 2011
IEEE 802.15.4 CSMA/CA protocol
Peter Bulychev [25]
We tried to make our model realistic: all the constant values have been taken from the
ZigBee and IEEE 802.15.4 standards power consumption rates were taken from the
specification of the real ZigBee chip (DACOM U-Power 500)
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GASICS 2011
Results – 2 nodes
Peter Bulychev [26]
The Nash Equilibrium strategy here is trivial:UnitBackoffPeriod = 0
(transmit as soon as possible)
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GASICS 2011
Coalitions
Peter Bulychev [27]
No non-trivial NE strategy for the case1xCheater VS NxHonest
Let’s think about coalitions:NxCheater VS NxHonest
This can correspond to the situation when several wireless devices belong to the same user. In this case it’s not profitable for a user if these devices compete with each other
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GASICS 2011
Results – 2x2 nodes
Peter Bulychev [28]
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GASICS 2011 Peter Bulychev [29]
Number of nodes in one coalition
N=1 N=2 N=3 N=4 N=5
Nash Eq 11 8 15 25 28The value of δ 0.900 0.985 0.986 0.990 0.990 Ũ(sNE,sNE) 0.86 0.76 0.81 0.85 0.83 Opt 13 23 31 34 48 Ũ(sopt, sopt) 0.87 0.85 0.87 0.87 0.86 Computation time
1m08s 5m45s 7m62s 32m49s 57m59s
Symmetric Nash Equilibrium and Optimal strategies for different number of network nodes in CSMA/CA
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GASICS 2011 Kim Larsen [30]
Questions?