On Selfish Behavior in CSMA/CA Networks Mario ˇ Cagalj 1 Saurabh Ganeriwal 2 Imad Aad 1 Jean-Pierre Hubaux 1 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 - IEEE Infocom 2005 - 1
Cheating pays well 8 Cheater Well-behaved 7 Throughput�(Mbits/s) 6 5 4 cheater 3 2 1 0 destination 0 2 4 6 8 10 12 14 16 well-behaved Contention�window�of�Cheater - IEEE Infocom 2005 - 2
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 - IEEE Infocom 2005 - 3
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 W i = W min = W max (no exponential backoff) – delay transmissions for d i ∈ U { 1 , 2 , . . . , W i } • Cheaters are rational (maximize their throughput r i ) - IEEE Infocom 2005 - 4
Cheaters utility function U i • Derived from Bianchi’s model of IEEE 802.11 – each cheater i controls its access probability 2 τ i = W i + 1 – cheater i receives the following throughput: τ i c i 1 ( τ − i ) U i ( τ i , τ − i ) ≡ r i ( τ i , τ − i ) = 3 ( τ − i ) , τ i c i 2 ( τ − i ) + c i where τ − i ≡ ( τ 1 , . . . , τ i − 1 , τ i +1 , . . . , τ N ) . - IEEE Infocom 2005 - 5
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 of the cheater’s contention window W i (that is, τ i ) • Solution concept - Nash equilibrium (NE) – W = ( W 1 , W 2 , . . . , W C ) is a NE point if ∀ i = 1 , . . . , C , we have: U i ( W i , W − i ) ≥ U i ( ˆ W i , W − i ) , ∀ ˆ W i ∈ { 1 , 2 , . . . , W max } - IEEE Infocom 2005 - 6
Nash equilibria of the static game Proposition 1. A vector W = ( W 1 , . . . , W C ) is a Nash equilibrium if and only if ∃ i ∈ { 1 , 2 , . . . , C } s.t. W i = 1 . Proposition 2. The static CSMA/CA game admits exactly max − ( W max − 1) C Nash equilibria. W C
Nash equilibria of the static game Proposition 1. A vector W = ( W 1 , . . . , W C ) is a Nash equilibrium if and only if ∃ i ∈ { 1 , 2 , . . . , C } s.t. W i = 1 . Proposition 2. The static CSMA/CA game admits exactly max − ( W max − 1) C Nash equilibria. W C • Two families of Nash equilibria – define D ≡ { i : W i = 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 r i = 0 , for i = 1 , . . . , C (known as the tragedy of the commons ) - IEEE Infocom 2005 - 7
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 ( r i − r 0 i ) Π 1 : s.t. r ∈ R r ≥ r 0 . – where R is a finite set of feasible solutions, and r 0 i ≡ max i min − i r i , i = 1 , 2 , . . . , C is the disagreement point - IEEE Infocom 2005 - 8
Uniqueness, fairness and optimality • Π 1 admits a unique , fair and Pareto-optimal solution W ∗ ( Nash bargaining solution ) 1.2 Pareto-optimal�point 1 Throughput�(Mbits/s) 0.8 0.6 0.4 Nash-equilibrium�point 0.2 Simulations Analytical 0 0 10 20 30 40 50 60 70 80 90 100 W * Contention�window�(W)�of�greedy�stations - IEEE Infocom 2005 - 9
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 J i = r i − P i , where P i is a penalty function – cheaters are reactive - IEEE Infocom 2005 - 10
Penalty function P i Proposition 3. Define: � p i , if τ i > τ P i = 0 , otherwise , where ∂p i /∂τ i > ∂r i /∂τ i and τ i < 1 , i = 1 , 2 , . . . , C . Then, J i = r i − P i has a unique maximizer τ . - IEEE Infocom 2005 - 11
Graphical interpretation • Example: P i = k i ( τ i − τ ) , with k i > ∂r i /∂τ i 0.4 P i r i 0.3 Nash�equilibrium�point normalized�payoff 0.2 0.1 J i = r P i i 0 -0.1 0 0.05 0.1 0.15 0.2 � � i - IEEE Infocom 2005 - 12
Making W ∗ a Nash equilibrium point • Let τ = i =1 ,...,C τ i , ( i.e. , W = min i =1 ,...,C W i ) max • Each player j calculates a penalty p j i to be inflicted on player i � = j , given that r i > r j , as follows: – p j i = r i − r j – note, ∂r i /∂τ i > 0 and ∂r j /∂τ i < 0 ⇒ ∂p j i /∂τ i > ∂r i /∂τ i – hence, J i = r i − p j i = r j has a unique maximizer τ i = τ j
Making W ∗ a Nash equilibrium point • Let τ = i =1 ,...,C τ i , ( i.e. , W = min i =1 ,...,C W i ) max • Each player j calculates a penalty p j i to be inflicted on player i � = j , given that r i > r j , as follows: – p j i = r i − r j – note, ∂r i /∂τ i > 0 and ∂r j /∂τ i < 0 ⇒ ∂p j i /∂τ i > ∂r i /∂τ i – hence, J i = r i − p j i = r j has a unique maximizer τ i = τ j Then, τ i = τ (i.e., W i = W ), i = 1 , . . . , C , is a unique Nash equilibrium. - IEEE Infocom 2005 - 13
Making W ∗ a Nash equilibrium point (cont.) • Moving Nash Equilibrium 1.2 Pareto-optimal�Nash�equilibrium 1 Throughput�(Mbits/s) 0.8 0.6 Nash-equilibria 0.4 0.2 Simulations Analytical 0 0 10 20 30 40 50 60 70 80 90 100 W * Contention�window�(W)�of�greedy�stations - IEEE Infocom 2005 - 14
Implementation of the penalty function • Achieved by selective jamming • Penalty should result in r i = r j , therefore T jam = ( r i /r j − 1) T obs 0.8 0.5 With�jamming Throughput�of�cheater�X�(Mbits/s) Cheater�X Without�jamming 0.45 0.7 Other�cheaters 0.4 0.6 Throughput�(Mbits/s) 0.35 0.5 0.3 0.4 0.25 0.2 Unique�maximizer�for�cheater�X 0.3 0.15 0.2 0.1 0.1 0.05 0 0 0 10 20 30 40 50 60 50 100 150 200 250 Contention�window�(W X )�of�cheater�X Time�(s) - IEEE Infocom 2005 - 15
Adaptive strategy • Prescribes to a player what to do when the player is penalized (jammed) 35 0.5 Cheater�X 0.45 Other�cheaters 30 Contention�window�size 0.4 Throughput�(Mbits/s) 0.35 25 0.3 20 0.25 0.2 15 0.15 0.1 10 Cheater�X 0.05 Other�cheaters 5 0 50 100 150 200 250 300 350 400 450 500 50 100 150 200 250 300 350 400 450 500 Time�(s) Time�(s) - IEEE Infocom 2005 - 16
Fully distributed algorithm • Evolution of the contention windows and the aggregated cheaters’ throughput for N = 20 , C = 7 , the step size γ = 5 30 30 25 25 Contention�window�size Contention�window�size 20 Our�protocol�stops 20 at�this�point 15 15 10 10 5 Ch.�X:�Distrib.�prot. 5 Ch.�1-7:�Distributed�prot. Ch. Y:�Distrib.�prot. Ch.�1-7:�Forced�CW Ch.�X, Y:�Forced�CW 0 0 0.6 0.65 0.7 0.75 0.8 0.85 0.9 0.95 1 160 180 200 220 240 260 280 300 320 Time�(s) Aggregated�throughput�of�cheaters�(Mbits/s) - IEEE Infocom 2005 - 17
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 - IEEE Infocom 2005 - 18
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