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EPFL, Lausanne, Switzerland

Márk Félegyházi

Equilibrium Analysis of Packet Forwarding Strategiesin Wireless Ad Hoc Networks

– the Static Case

Márk Félegyházi

{mark.felegyhazi, jean-pierre.hubaux}@epfl.ch [email protected]

Levente ButtyánJean-Pierre Hubaux

Laboratory for computer Communications and Applications,Swiss Federal Institute of Technology (EPFL)

– Lausanne, Switzerland

TERMINODES Project (NCCR-MICS)http://www.terminodes.org

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Laboratory of Cryptography and System Security,

Budapest University of Technology and Economics


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Outline

• Intro to ad hoc networks• Problem formulation• Related work• Scenario – static case• Analysis• Simulation• Conclusion• Future work

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Ad Hoc Networks

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• self-organizing network – no infrastructure• each networking service is provided by the nodes themselves• we focus on packet forwarding


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Problem of cooperation

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Problem: If selfish nodes do not forward packets for others (do not cooperate with others), the network can be paralyzed.

Solution: Incentive for cooperation

• virtual currency (nuglets): Nodes pay if they use a service and get paid if they contribute to the service. [ButtyanH03]

• reputation system: Nodes maintain a belief about all nodes they have met. If a node is requesting a service, other nodes decide to provide it based on their belief about the requestor. [BucheggerLB02][MichiardiM03]


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Cooperation without incentives

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Question: Do we need these incentive mechanisms or can cooperation exist based on the self-interest of the nodes?

• Energy-efficient cooperation: Willingness to cooperate adapts to the energy class of the nodes. [SrinivasanNCR03]

S R3R1 R2 D

session:

energy class:

energy class of the session

two mechanisms: • class distribution mechanism• session acceptance mechanism


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Static network scenario

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• static network

• communication is based on

multi-hop relaying

• a communication chain is

called a route

• routes last for the whole

duration of the game

• each node is a source on only

one route

network configuration specific conditions for cooperation

s2

s1

s3


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Modeling packet forwarding as a game

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• time is slotted: nodes apply a decision for each time slot

• nodes apply a decision for each route where they are relays

• strategy is to define a cooperation level [0,1] for each time slot

• source benefits if packets arrive

• utility of the nodes is linear

• rationality of the players: goal is to maximize utility

Utility: G*(number of packets arrived) – C*(number of packets transmitted)

time0time slot: 1 t

cooperation level:

pi(0) pi(1) pi(t)


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Representation of the nodes as players

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node i is represented as a machine Mi

• Π is a multiplication gate corresponding the

multiplicative feature of packet forwarding

• σi represents the strategy of the node

node i is playing against the rest of the

network

(represented by the box denoted by A-i)


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Strategy of the nodes

))]1,(([)( )1( tSrii itrtp

0)( xi

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strategy function for node i:

example strategies:

Strategy Function

Initial cooperation

level

AllD (always defect)

AllC (always cooperate)

TFT (Tit-For-Tat)

0

1

1

1)( xi

xxi )(

non-reactive strategies: the output of the strategy function is independent of the input (example: AllD and AllC)

reactive strategies: the output of the strategy function depends on the input (example: TFT)


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Concept of dependency graph

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s2

s1

s3

s2

s1

s3

routes dependency graph

dependency: the benefit of each source is dependent on the behavior of its forwarders

dependency loop


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Analytical Results (1)

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Theorem 1: If a node does not have any dependency loops, then its best strategy is AllD.

s2

s1

s3

s2

s1

s3

Theorem 2: If a node has only non-reactive dependency loops, then its best strategy is AllD.

0)(1 xIf node s1 plays AllD:

Corollary 1: If every node plays AllD, it is a Nash-equilibrium.


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Analytical Results (2)

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Theorem 3: The best strategy for node i is TFT, if:

• Node i has a dependency loop with all of its

sources,

• the other nodes play TFT and

• (G + L) ¢ i > |Fi| ¢ C

where:

Δithe length of the longest dependency loop

G gain in one time slot if all traffic arrives at the destination

Cforwarding cost in one time slot if all traffic arrives at the destination

ω discounting factor

|Fi| number of sources for node i

Lloss in one time slot if no traffic arrives at the destination

s2

s1

s3

s2

s1

s3

routes dependency graph

Corollary 2: If Theorem 3 holds for every node, it is

a Nash-equilibrium.


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Simulation Scenario

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Number of nodes 100

Area type Torus

Area size 1500 m x 1500 m

Radio range 250 m

Route length 4 hops

Number of simulations

100

Confidence interval 95 %


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Simulation Results

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Theorem 3: The best

strategy for node i is TFT,

if:

• Node i has a dependency

loop with all of its sources,

• the other nodes play TFT

and

• (G + L) ¢ i > |Fi| ¢ C


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Conclusion

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• Model of packet forwarding in a static network using game theory

• Analytical results:

1. If everyone drops all packets, it is a Nash-equilibrium.

2. Given some conditions, there are Nash-equilibria, where all

nodes forward all packets (i.e., everyone cooperates in the

network).

• Simulation results: The conditions for cooperative Nash-equilibria

are very restrictive. In general, the likelihood that the conditions

for cooperation hold for every node is extremely small.


Márk Félegyházi

Future work

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• Quantify the probability that all nodes cooperate in the network

• The effect of the number of routes originating at each node

• Possible equilibria that involve only a part of the nodes (local

equilibria)

• Consider a mobile scenario – impact of mobility

• Emergence of cooperation


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Related work

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[Axelrod84] - R. Axelrod, The Evolution of Cooperation, Basic Books, New York, 1984.

[BucheggerLB02] – S. Buchegger, J-Y. Le Boudec, “Performance Analysis of the CONFIDANT Protocol (Cooperation Of Nodes--Fairness In Dynamic Ad-hoc NeTworks),” In Proc. 3rd ACM International Symposium on Mobile Ad Hoc Networking and Computing (MobiHoc'02), Lausanne, Switzerland, pp. 80-91, June 9-11, 2002.

[ButtyanH03] – L. Buttyán, J.-P. Hubaux, “Stimulating Cooperation in Self-Organizing Mobile Ad Hoc Networks,” to appear in ACM/Kluwer Mobile Networks and Applications (MONET) Special Issue on Mobile Ad Hoc Networks, Vol. 8 No. 5, October 2003.

[MichiardiM03] - P. Michiardi, R. Molva, “Core: A COllaborative REputation mechanism to enforce node cooperation in Mobile Ad Hoc Networks,” Communication and Multimedia Security 2002, Portoroz, Slovenia, September 26-27, 2002.

[SrinivasanNCR03] - V. Srinivasan, P. Nuggehalli, C. Chiasserini, R. Rao, “Cooperation in Wireless Ad Hoc Networks,” In Proceedings of IEEE Infocom ‘03, San Francisco, USA, March 30- April 3, 2003.

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