Peter BulychevAlexandre DavidKim G. Larsen

Marius Mikucionis

Computing Nash Equilibrium in Wireless Ad Hoc Networks

Using Statistical Model Checking

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

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

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

GASICS 2011

Nash Eq in Wireless Ad Hoc Networks

power=40%

power=40%

power=40%

Peter Bulychev[5]

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)

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

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)

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?

GASICS 2011

Our SMC-based approach

Peter Bulychev[10]

SMC = Simulation + Statistics

Scales to complex models

Can provide confidence on

its results

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

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]

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]

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)

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)

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)

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)

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:

…

…

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

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

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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Value of utility function for the cheater node

Results (3 nodes)

Peter Bulychev[22]

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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)

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

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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Results – 2 nodes

Peter Bulychev [26]

The Nash Equilibrium strategy here is trivial:UnitBackoffPeriod = 0

(transmit as soon as possible)

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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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Results – 2x2 nodes

Peter Bulychev [28]

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

GASICS 2011 Kim Larsen [30]

Questions?