On Selfish Behavior in CSMA/CA Networks Mario Cagalj 1 Saurabh - - PowerPoint PPT Presentation

On Selfish Behavior in CSMA/CA Networks

Mario ˇ Cagalj1 Saurabh Ganeriwal2 Imad Aad1 Jean-Pierre Hubaux1

1 LCA-IC-EPFL 2 NESL-EE-UCLA

March 17, 2005

• IEEE Infocom 2005 -

Introduction

• CSMA/CA is the most popular MAC paradigm for wireless

networks

• CSMA/CA protocols rely on a (fair) random deferment of packet

transmission – where nodes control their own random delays

• CSMA/CA is efficient if nodes follow predefined rules, however,

nodes have a rational motive to cheat

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Cheating pays well

cheater well-behaved destination 1 2 3 4 5 6 7 8 2 4 6 8 10 12 14 16 Throughput(Mbits/s) ContentionwindowofCheater Cheater Well-behaved

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Our goal

• Study the coexistence of a population of greedy stations
• Derive the conditions for the stable and optimal functioning of a

population of greedy stations

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Model and assumptions

• Network of N nodes (transmitters) out of which C are cheaters

– IEEE 802.11 protocol – MAC layer authentication (no Sybil attack) – single collision domain (no hidden terminals) – nodes always have packets (of the same size) to transmit

• Contention window size-based cheating

– fix Wi = Wmin = Wmax (no exponential backoff) – delay transmissions for di ∈U {1, 2, . . . , Wi}

• Cheaters are rational (maximize their throughput ri)
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Cheaters utility function Ui

• Derived from Bianchi’s model of IEEE 802.11

– each cheater i controls its access probability τi = 2 Wi + 1 – cheater i receives the following throughput: Ui(τi, τ−i) ≡ ri(τi, τ−i) = τici

1(τ−i)

τici

2(τ−i) + ci 3(τ−i) ,

where τ−i ≡ (τ1, . . . , τi−1, τi+1, . . . , τN).

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Static CSMA/CA game

• Static games

– players make their moves independently of other players – players play the same move forever

• A move in the CSMA/CA game corresponds to setting the value
• f the cheater’s contention window Wi (that is, τi)
• Solution concept - Nash equilibrium (NE)

– W = (W1, W2, . . . , WC) is a NE point if ∀i = 1, . . . , C, we have: Ui(Wi, W−i) ≥ Ui( ˆ Wi, W−i) , ∀ ˆ Wi ∈ {1, 2, . . . , Wmax}

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Nash equilibria of the static game

Proposition 1. A vector W = (W1, . . . , WC) is a Nash equilibrium if and only if ∃i ∈ {1, 2, . . . , C} s.t. Wi = 1. Proposition 2. The static CSMA/CA game admits exactly W C

max − (Wmax − 1)C Nash equilibria.

Nash equilibria of the static game

Proposition 1. A vector W = (W1, . . . , WC) is a Nash equilibrium if and only if ∃i ∈ {1, 2, . . . , C} s.t. Wi = 1. Proposition 2. The static CSMA/CA game admits exactly W C

max − (Wmax − 1)C Nash equilibria.

• Two families of Nash equilibria

– define D ≡ {i : Wi = 1, i = 1, 2, . . . , C} – 1st family, |D| = 1, implying that only one cheater receives a non-null throughput (some allocations are Pareto-optimal!) – 2nd family, |D| > 1, implying ri = 0, for i = 1, . . . , C (known as the tragedy of the commons)

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How to avoid undesirable equilibria?

• Multiple Nash equilibria that are either highly unfair or highly

inefficient

How to avoid undesirable equilibria?

• Multiple Nash equilibria that are either highly unfair or highly

inefficient

• To derive a better solution we use Nash bargaining framework

– solve the following problem max C

i=1(ri − r0 i )

Π1 : s.t. r ∈ R r ≥ r0 . – where R is a finite set of feasible solutions, and r0

i ≡

maxi min−i ri, i = 1, 2, . . . , C is the disagreement point

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Uniqueness, fairness and optimality

• Π1 admits a unique, fair and Pareto-optimal solution W ∗

(Nash bargaining solution)

0.2 0.4 0.6 0.8 1 1.2 10 20 30 40 50 60 70 80 90 100 Throughput(Mbits/s) Contentionwindow(W)ofgreedystations Simulations Analytical Pareto-optimalpoint W* Nash-equilibriumpoint

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Dynamic CSMA/CA game

• W ∗ is a desirable solution, but is not a Nash equilibrium

– i.e., W ∗

i > 1, i = 1, . . . , C

Dynamic CSMA/CA game

• W ∗ is a desirable solution, but is not a Nash equilibrium

– i.e., W ∗

i > 1, i = 1, . . . , C

• In the model of dynamic games

– cheaters’ utility function changes to Ji = ri − Pi, where Pi is a penalty function – cheaters are reactive

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Penalty function Pi

Proposition 3. Define: Pi = pi, if τi > τ 0,

• therwise ,

where ∂pi/∂τi > ∂ri/∂τi and τi < 1, i = 1, 2, . . . , C. Then, Ji = ri − Pi has a unique maximizer τ.

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Graphical interpretation

• Example: Pi = ki(τi − τ), with ki > ∂ri/∂τi

0.05 0.1 0.15 0.2

• 0.1

0.1 0.2 0.3 0.4

i

normalizedpayoff

Nashequilibriumpoint

ri

P

i

J

i = r i

P

i

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Making W ∗ a Nash equilibrium point

• Let

τ = min

i=1,...,C τi, (i.e., W =

max

i=1,...,C Wi)

• Each player j calculates a penalty pj

i to be inflicted on player i = j,

given that ri > rj, as follows: – pj

i = ri − rj

– note, ∂ri/∂τi > 0 and ∂rj/∂τi < 0 ⇒ ∂pj

i/∂τi > ∂ri/∂τi

– hence, Ji = ri − pj

i = rj has a unique maximizer τi = τj

Making W ∗ a Nash equilibrium point

• Let

τ = min

i=1,...,C τi, (i.e., W =

max

i=1,...,C Wi)

• Each player j calculates a penalty pj

i to be inflicted on player i = j,

given that ri > rj, as follows: – pj

i = ri − rj

– note, ∂ri/∂τi > 0 and ∂rj/∂τi < 0 ⇒ ∂pj

i/∂τi > ∂ri/∂τi

– hence, Ji = ri − pj

i = rj has a unique maximizer τi = τj

Then, τi = τ (i.e., Wi = W), i = 1, . . . , C, is a unique Nash equilibrium.

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Making W ∗ a Nash equilibrium point (cont.)

• Moving Nash Equilibrium

0.2 0.4 0.6 0.8 1 1.2 10 20 30 40 50 60 70 80 90 100 Throughput(Mbits/s) Contentionwindow(W)ofgreedystations Simulations Analytical Pareto-optimalNashequilibrium W* Nash-equilibria

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Implementation of the penalty function

• Achieved by selective jamming
• Penalty should result in ri = rj, therefore Tjam = (ri/rj −1)Tobs

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 50 100 150 200 250 Throughput(Mbits/s) Time(s) CheaterX Othercheaters 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 10 20 30 40 50 60 ThroughputofcheaterX(Mbits/s) Contentionwindow(WX)ofcheaterX UniquemaximizerforcheaterX Withjamming Withoutjamming

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Adaptive strategy

• Prescribes to a player what to do when the player is penalized

(jammed)

0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 50 100 150 200 250 300 350 400 450 500 Throughput(Mbits/s) Time(s) CheaterX Othercheaters 5 10 15 20 25 30 35 50 100 150 200 250 300 350 400 450 500 Contentionwindowsize Time(s) CheaterX Othercheaters

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Fully distributed algorithm

• Evolution of the contention windows and the aggregated cheaters’

throughput for N = 20, C = 7, the step size γ = 5

5 10 15 20 25 30 160 180 200 220 240 260 280 300 320 Contentionwindowsize Time(s) Ch.X:Distrib.prot.

• Ch. Y:Distrib.prot.

Ch.X, Y:ForcedCW 5 10 15 20 25 30 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 Contentionwindowsize Aggregatedthroughputofcheaters(Mbits/s) Ourprotocolstops atthispoint Ch.1-7:Distributedprot. Ch.1-7:ForcedCW

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Conclusions

• Static CSMA/CA game admits two families of Nash equilibria

– one family includes equilibria that are also Pareto-optimal

• Nash bargaining framework is appropriate to study fairness at the MAC layer

– using this framework we have identified the unique Pareto-optimal point exhibiting efficiency and fairness

• We have shown how to make this Pareto-optimal point a Nash equilibrium

point, by – providing a fully distributed algorithm that leads a population of greedy stations to the efficient Paret-optimal Nash equilibrium

Conclusions

• Static CSMA/CA game admits two families of Nash equilibria

– one family includes equilibria that are also Pareto-optimal

• Nash bargaining framework is appropriate to study fairness at the MAC layer

– using this framework we have identified the unique Pareto-optimal point exhibiting efficiency and fairness

• We have shown how to make this Pareto-optimal point a Nash equilibrium

point, by – providing a fully distributed algorithm that leads a population of greedy stations to the efficient Paret-optimal Nash equilibrium

• CSMA/CA game has a high educational value
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