Peter Bulychev Alexandre David Kim G. Larsen Marius Mikucionis TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAA A AA AA AA Computing Nash Equilibrium in Wireless Ad Hoc Networks Using Statistical Model Checking

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% Peter Bulychev [2] GASICS 2011 Master node

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=80% Peter Bulychev[3] GASICS 2011 Master node

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] GASICS 2011 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

Nash Eq in Wireless Ad Hoc Networks power=40% Peter Bulychev[5] GASICS 2011

Problem statement GASICS 2011 Peter Bulychev[6]  Input: 1.Each node is modeled by a parameterized Priced Timed Automata M(p), where p∈P and P is finite 2.System of N nodes is modeled by S(p 1, p 2, …, p N ) ≡ M(p 1 )||M(p 2 )||…||M(p N )||C 3.Each node k has a goal φ k (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 φ k is satisfied by a random run: U k (p 1, p 2, …, p k ) ≡ Pr[S(p 1, p 2, …, p k ) ⊨ φ k ]  Goal:  To find symmetric NE, i.e. to find p∈P s.t.: ∀ p’ ∈ P ⋅ U 1 (p, p, …, p)≥ U 1 (p’, p, …, p)

Problem statement GASICS 2011 Peter Bulychev[7]  Input: 1.Each node is modeled by a parameterized Priced Timed Automata M(p), where p∈P and P is finite 2.System of K nodes is modeled by S(p 1, p 2, …, p k ) ≡ M(p 1 )||M(p 2 )||…||M(p k )||C 3.Each node k has a goal φ k (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 φ k is satisfied by a random run: U k (p 1, p 2, …, p k ) ≡ Pr[S(p 1, p 2, …, p k ) ⊨ φ k ]  Goal:  To find symmetric NE, i.e. to find p∈P s.t.: ∀ p’ ∈ P ⋅ U 1 (p, p, …, p)≥ U 1 (p’, p, …, p) Nash Equilibrium might not exist in non-mixed strategies Thus, we will consider a relaxed definition of Nash Equilibrium

Problem statement GASICS 2011 Peter Bulychev[8]  Input: 1.Each node is modeled by a parameterized Priced Timed Automata M(p), where p∈P and P is finite 2.System of K nodes is modeled by S(p 1, p 2, …, p k ) ≡ M(p 1 )||M(p 2 )||…||M(p k )||C 3.Each node k has a goal φ k (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 φ k is satisfied by a random run: U k (p 1, p 2, …, p k ) ≡ Pr[S(p 1, p 2, …, p k ) ⊨ φ k ]  Goal:  To find symmetric δ-relaxed NE, i.e. to find p∈P s.t.: ∀ p’ ∈ P ⋅ U 1 (p, p, …, p)≥ δ* U 1 (p’, p, …, p)

Related work GASICS 2011 Peter Bulychev[9]  Pioneering work: “Game theory and the design of self-configuring, adaptive wireless networks”, MacKenzie et.al.,  Survey: “Using game theory to analyze wireless ad hoc networks”, Srivastava et.al.,  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?

Our SMC-based approach GASICS 2011 Peter Bulychev[10] SMC = Simulation + Statistics Scales to complex models Can provide confidence on its results

Our SMC-based approach GASICS 2011 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

SMC-based approach (Part I) Input: P – finite set of strategies, U(p i, p k ) – 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 := P 3.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 waiting 9. if ∀ p’ Ũ(p’, p) is already computed: 10. remove p from waiting 11. add p to candidates 12. return argmax p ∊ P min p’ ∊ P Ũ(p, p)/Ũ(p’, p) GASICS 2011 Peter Bulychev[12]

SMC-based approach (Part I) Input: P={p 1, p 2, …, p 10 } – finite set of strategies, U(p i, p k ) – utility function, d ∊ [0,1] - threshold Goal: find p ∊ P that maximizes min p’ ∊ P Ũ(p, p)/Ũ(p’, p) Ũ(p 1,p 1 )Ũ(p 10,p 1 ) Ũ(p 1,p 10 ) Ũ(p 10,p 10 ) Peter Bulychev[13] GASICS 2011

SMC-based approach (Part I) Peter Bulychev[14] GASICS 2011 Input: P={p 1, p 2, …, p 10 } – finite set of strategies, U(p i, p k ) – utility function, d ∊ [0,1] - threshold Goal: find p ∊ P that maximizes min p’ ∊ P Ũ(p, p)/Ũ(p’, p) Ũ(p 1,p 1 )Ũ(p 10,p 1 ) Ũ(p 1,p 10 ) Ũ(p 10,p 10 )

SMC-based approach (Part I) Ũ(p 8,p 8 ) ≥ d*Ũ(s 6,s 8 ) Ũ(p 6,p 6 ) < d*Ũ(p 3,p 6 ) Peter Bulychev[15] GASICS 2011 Input: P={p 1, p 2, …, p 10 } – finite set of strategies, U(p i, p k ) – utility function, d ∊ [0,1] - threshold Goal: find p ∊ P that maximizes min p’ ∊ P Ũ(p, p)/Ũ(p’, p) Ũ(p 1,p 1 )Ũ(p 10,p 1 ) Ũ(p 1,p 10 ) Ũ(p 10,p 10 )

SMC-based approach (Part I) Peter Bulychev[16] GASICS 2011 Input: P={p 1, p 2, …, p 10 } – finite set of strategies, U(p i, p k ) – utility function, d ∊ [0,1] - threshold Goal: find p ∊ P that maximizes min p’ ∊ P Ũ(p, p)/Ũ(p’, p) Ũ(p 1,p 1 )Ũ(p 10,p 1 ) Ũ(p 1,p 10 ) Ũ(p 10,p 10 ) Ũ(p 8,p 8 ) ≥ d*Ũ(s 6,s 8 )

SMC-based approach (Part I) Peter Bulychev[17] GASICS 2011 “ Embarrassingly Parallelizable” argmax p ∊ P min p’ ∊ P Ũ(p, p)/Ũ(p’, p) Input: P={p 1, p 2, …, p 10 } – finite set of strategies, U(p i, p k ) – utility function, d ∊ [0,1] - threshold Goal: find p ∊ P that maximizes min p’ ∊ P Ũ(p, p)/Ũ(p’, p)

SMC-based approach (Part II) Peter Bulychev[18] GASICS 2011 Ũ(p k,p k ) Ũ(p k+1,p k ) Ũ(p n,p k ) Ũ(p k-1,p k ) … Ũ(p 1,p k ) … By definition p k satisfies δ-relaxed NE iff ∀ i ∈ [1,n] ⋅ U(p k, p k )≥ δ* U(p i, p k ) Now we: 1.Reestimate each Ũ(p i, p k ) using N SMC runs 2.Apply the following theorem: Theorem. W e can accept the hypothesis that p k satisfies δ-relaxed NE with a given significance level α, if: … …

Implementation details Peter Bulychev[19] GASICS 2011 SSH connection Python frontend node 1 node 2 node 3 node 4 UPPAAL backend

Case studies GASICS 2011 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 protocol 2.IEEE CSMA/CA protocol

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] GASICS 2011 Pr[Node.time (Node.Ok && Node.ntransmitted <= 5))

Value of utility function for the cheater node Results (3 nodes) Peter Bulychev[22] GASICS 2011

Results (Aloha) GASICS 2011 Peter Bulychev [23] N=2N=3N=4N=5N=6N=7N=8 Nash Eq The value of δ Ũ(s NE,s NE ) Opt Ũ(s opt, s opt ) Symmetric Nash Equilibrium and Optimal strategies for different number of network nodes #cores Time38m19m13m9m46s7m52s7m04s6m03s5m Time required to find Nash Equilibrium for N=3 100x100 parameter values (8xIntel Core2 2.66GHz CPU)

IEEE CSMA/CA protocol Peter Bulychev[24] GASICS 2011 nb:=0 be:=MinBE Delay for random(0..2 be ) UnitBackoffPeriod Channel is clear? nb:=nb+1 be:=min(be+1, MaxBE) nb>MaxNB? Failure Transmit Y N Y N Switch to transmitting IEEE CSMA/CA is based on the random backoff procedure We assume that a node can change its UnitBackoffPeriod parameter

IEEE CSMA/CA protocol Peter Bulychev [25] GASICS 2011 We tried to make our model realistic:  all the constant values have been taken from the ZigBee and IEEE standards  power consumption rates were taken from the specification of the real ZigBee chip (DACOM U- Power 500)

Results – 2 nodes Peter Bulychev [26] GASICS 2011 The Nash Equilibrium strategy here is trivial: UnitBackoffPeriod = 0 (transmit as soon as possible)

Coalitions GASICS 2011 Peter Bulychev [27]  No non-trivial NE strategy for the case 1xCheater 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

Results – 2x2 nodes Peter Bulychev [28] GASICS 2011

Peter Bulychev [29] Number of nodes in one coalition N=1N=2N=3N=4N=5 Nash Eq The value of δ Ũ(s NE,s NE ) Opt Ũ(s opt, s opt ) Computation time1m08s5m45s7m62s32m49s57m59s Symmetric Nash Equilibrium and Optimal strategies for different number of network nodes in CSMA/CA

GASICS 2011 Kim Larsen [30] Questions?