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
EPFL, Lausanne, Switzerland
Márk Félegyházi
Outline
• Intro to ad hoc networks• Problem formulation• Related work• Scenario – static case• Analysis• Simulation• Conclusion• Future work
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EPFL, Lausanne, Switzerland
Márk Félegyházi
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
EPFL, Lausanne, Switzerland
Márk Félegyházi
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]
EPFL, Lausanne, Switzerland
Márk Félegyházi
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
EPFL, Lausanne, Switzerland
Márk Félegyházi
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
EPFL, Lausanne, Switzerland
Márk Félegyházi
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)
EPFL, Lausanne, Switzerland
Márk Félegyházi
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)
EPFL, Lausanne, Switzerland
Márk Félegyházi
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)
EPFL, Lausanne, Switzerland
Márk Félegyházi
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
EPFL, Lausanne, Switzerland
Márk Félegyházi
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.
EPFL, Lausanne, Switzerland
Márk Félegyházi
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.
EPFL, Lausanne, Switzerland
Márk Félegyházi
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 %
EPFL, Lausanne, Switzerland
Márk Félegyházi
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
EPFL, Lausanne, Switzerland
Márk Félegyházi
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.
EPFL, Lausanne, Switzerland
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
EPFL, Lausanne, Switzerland
Márk Félegyházi
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.