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Computing Nash Equilibrium in Wireless Ad Hoc Networks Using Statistical Model Checking

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

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Computing Nash Equilibrium in Wireless Ad Hoc Networks Using Statistical Model Checking

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

  2. Nash Eq in Wireless Ad Hoc Networks power=20% power=20% Master node power=20% Consider a wireless network, where there is a master node that chooses the optimal parameters that should be used by other nodes GASICS 2011

  3. Nash Eq in Wireless Ad Hoc Networks power=20% power=20% Master node power=80% Now, if there are selfish nodes, they might want to change these parameters to achieve better performance GASICS 2011

  4. Nash Eq in Wireless Ad Hoc Networks 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 power=20% power=90% Master node power=80% Now, if there are selfish nodes, they might want to change these parameters to achieve better performance GASICS 2011

  5. Nash Eq in Wireless Ad Hoc Networks power=40% power=40% power=40% GASICS 2011

  6. Problem statement • Input: • Each node is modeled by a parameterized Priced Timed Automata M(p), where p∈P and P is finite • System of N nodes is modeled by S(p1, p2,…,pN) ≡ M(p1)||M(p2)||…||M(pN)||C • Each node k has a goal φk (i.e. to transmit a message within given timed and energy bounds) • 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∈Ps.t.: ∀p’∈P ⋅ U1(p, p, …, p)≥U1(p’, p, …, p) GASICS 2011

  7. Problem statement • Input: • Each node is modeled by a parameterized Priced Timed Automata M(p), where p∈P and P is finite • System of K nodes is modeled by S(p1, p2,…,pk) ≡ M(p1)||M(p2)||…||M(pk)||C • Each node k has a goal φk (i.e. to transmit a message within given timed and energy bounds) • 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∈Ps.t.: ∀p’∈P ⋅ U1(p, p, …, p)≥U1(p’, p, …, p) Nash Equilibrium might not exist in non-mixed strategies Thus, we will consider a relaxed definition of Nash Equilibrium GASICS 2011

  8. Problem statement • Input: • Each node is modeled by a parameterized Priced Timed Automata M(p), where p∈P and P is finite • System of K nodes is modeled by S(p1, p2,…,pk) ≡ M(p1)||M(p2)||…||M(pk)||C • Each node k has a goal φk (i.e. to transmit a message within given timed and energy bounds) • 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∈Ps.t.: ∀p’∈P ⋅ U1(p, p, …, p)≥δ*U1(p’, p, …, p) GASICS 2011

  9. Related work • Pioneering work: “Game theory and the design of self-configuring, adaptive wireless networks”, MacKenzieet.al. , 2001. • Survey: “Using game theory to analyze wireless ad hoc networks”, Srivastavaet.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

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

  11. Our SMC-based approach • 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 δs.t. we can accept the hypothesis that p is a δ-relaxed NE with a given significance level α GASICS 2011

  12. 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 minp’∊ PŨ(p, p)/Ũ(p’, p) Algorithm: • for everyp∊Pcompute estimation Ũ(p,p) • waiting := P • candidates := ∅ • whilelen(waiting)>1: • pick some unexplored pair (p’,p)∊P× waiting • computeestimation Ũ(p’, p) • if Ũ(p, p)/Ũ(p’, p) < d: • remove p from waiting • if ∀p’ Ũ(p’, p) is already computed: • remove p from waiting • add p to candidates • return argmaxp∊Pminp’∊ PŨ(p, p)/Ũ(p’, p) GASICS 2011

  13. SMC-based approach (Part I) Ũ(p10,p10) Ũ(p1,p10) Ũ(p1,p1) Ũ(p10,p1) Input: P={p1, p2, …, p10}– finite set of strategies, U(pi, pk) – utility function, d ∊ [0,1] - threshold Goal: find p∊P that maximizes minp’∊ PŨ(p, p)/Ũ(p’, p) GASICS 2011

  14. 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 minp’∊ PŨ(p, p)/Ũ(p’, p) Ũ(p10,p10) Ũ(p1,p10) Ũ(p1,p1) Ũ(p10,p1) GASICS 2011

  15. 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 minp’∊ PŨ(p, p)/Ũ(p’, p) Ũ(p10,p10) Ũ(p1,p10) Ũ(p8,p8) ≥d*Ũ(s6,s8) Ũ(p6,p6) < d*Ũ(p3,p6) Ũ(p1,p1) Ũ(p10,p1) GASICS 2011

  16. 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 minp’∊ PŨ(p, p)/Ũ(p’, p) Ũ(p10,p10) Ũ(p1,p10) Ũ(p8,p8) ≥d*Ũ(s6,s8) Ũ(p1,p1) Ũ(p10,p1) GASICS 2011

  17. 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 minp’∊ PŨ(p, p)/Ũ(p’, p) argmaxp∊Pminp’∊ P Ũ(p, p)/Ũ(p’, p) “Embarrassingly Parallelizable” GASICS 2011

  18. SMC-based approach (Part II) Ũ(pn,pk) By definition pksatisfies δ-relaxed NE iff ∀i∈[1,n] ⋅U(pk, pk)≥δ*U(pi, pk) Now we: • Reestimate each Ũ(pi, pk) using NSMC runs • Apply the following theorem: Theorem. We can accept the hypothesis that pk satisfies δ-relaxedNEwith a given significance level α, if: … … Ũ(pk+1,pk) Ũ(pk,pk) Ũ(pk-1,pk) … … Ũ(p1,pk) GASICS 2011

  19. Implementation details UPPAAL backend node 1 SSH connection SSH connection node 2 SSH connection node 3 Python frontend SSH connection node 4 GASICS 2011

  20. Case studies We used our tool to compute Nash Equilibrium for two CSMA (Carrier Sense Multiple Access) protocols: • k-persistent ALOHA CSMA/CD protocol • IEEE 802.15.4 CSMA/CA protocol GASICS 2011

  21. Aloha CSMA/CD protocol Pr[Node.time <= 3000](<>(Node.Ok&& Node.ntransmitted<= 5)) • 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 GASICS 2011

  22. Results (3 nodes) Value of utility function for the cheater node GASICS 2011

  23. Results (Aloha) Symmetric Nash Equilibrium and Optimal strategies for different number of network nodes Time required to find Nash Equilibrium for N=3 100x100 parameter values (8xIntel Core2 2.66GHz CPU) GASICS 2011

  24. IEEE 802.15.4 CSMA/CA protocol We assume that a node can change its UnitBackoffPeriod parameter nb:=0 be:=MinBE IEEE 802.15.4 CSMA/CA is based on the random backoff procedure Delay for random(0..2be) UnitBackoffPeriod Y Channel is clear? N nb:=nb+1 be:=min(be+1, MaxBE) Switch to transmitting N nb>MaxNB? Y Transmit Failure GASICS 2011

  25. IEEE 802.15.4 CSMA/CA protocol 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) GASICS 2011

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

  27. Coalitions • No non-trivial NE strategy for the case 1xCheater VS NxHonest • Let’s think about coalitions: NxCheaterVS 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 GASICS 2011

  28. Results – 2x2 nodes GASICS 2011

  29. Symmetric Nash Equilibrium and Optimal strategies for different number of network nodes in CSMA/CA GASICS 2011

  30. Questions? GASICS 2011

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