Non-Cooperative Behavior

in Wireless Networks

Márk Félegyházi (EPFL)PhD. public defense

July 9, 2007

July 9, 2007 Márk Félegyházi (EPFL) 2

Summary of my research

► Ch 1: A tutorial on game theory► Ch. 2: Multi-radio channel allocation in wireless networks► Ch. 3: Packet forwarding in static ad-hoc networks► Ch. 4: Packet forwarding in dynamic ad-hoc networks► Ch. 5: Packet forwarding in multi-domain sensor networks► Ch. 6: Cellular operators in a shared spectrum► Ch. 7: Border games in cellular networks

Part II: Non-cooperative users

Part III: Non-cooperative network operators

Part I: Introduction to game theory

July 9, 2007 Márk Félegyházi (EPFL) 3

Multi-Radio Channel Allocation Problem

► C orthogonal channels► N communicating pairs of devices► k radios at each device

,i xknumber of radios

by sender i on channel x

→

,i i xx C

k k

,x i xi N

k k

, ( )i i i x xx C

u k k

Nash equilibrium: No player has an incentive to unilaterally deviate.

* * *( , ) ( , ),i i i i i i iu s s u s s s S

Proposition: If S* is a NE in GMRCA, then dy,x ≤ 1, for any channel x and y.

► blabla, ► blabla, blabla

How to Share a Pie with Selfish Researchers

Márk Félegyházi (EPFL)PhD. public defense

July 9, 2007

Who Know Game Theory

July 9, 2007 Márk Félegyházi (EPFL) 5

Problem

Dining Game Theoreticians

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Motivation

► pies were controlled by a trusted central authority– “Mark, I would strongly encourage you share the pie

with Panos”

► it was difficult to get enough plates

► no central control how to cut the pies► it is easy to get more plates to get a bigger share

BEFORE

NOW

What is the effect of selfish behavior in pie sharing?

July 9, 2007 Márk Félegyházi (EPFL) 7

System model

► C pies► N selfish and rational (= hungry)

researchers► k plates for each researcher

SYSTEM:

► the central authority does not help to share the pies

► pies have the same size and quality (strawberry)

► each researcher can reach any pie (by allocating a plate there)

► pies are fairly shared► one slice on one plate

ASSUMPTIONS:

July 9, 2007 Márk Félegyházi (EPFL) 8

total number of plates by researcher i

number of plates by researcher i at pie x

Example

► C = 6 pies► N = 4 hungry researchers► k = 4 plates for each researcher

,i xk

,i i xx C

k k

,x i xi N

k k

total number of plates demanding pie

x

July 9, 2007 Márk Félegyházi (EPFL) 9

The pie-cut functions► pies have all the same size and quality ► π t(kx) – total size of the shares of any pie x

► π(kx) – size of a share per plate

33

July 9, 2007 Márk Félegyházi (EPFL) 10

Dining Game Theoreticians (DGT) game

► selfish (=hungry) researchers► non-cooperative game GDGT

– players → researchers– strategy → plate allocation – payoff → total amount of cookie

► payoff:

, ( )i i i x xx C

u k k

(3) (4) 2 (4)Maxim iu

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Stability: Nash equilibrium

Nash equilibrium: No researcher changes if the others keep their plates.

Best response: Best strategy of a researcher given the strategies of others.

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The Question

Where shall I put my plates?

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Recognition: In a stable state (NE), dy,x ≤ 1 for any two pies x and y.

Cut the pies in (almost) the same number of pieces

x

► pick two pies x and y, where kx ≥ ky► demand: dx,y = kx – ky

y

July 9, 2007 Márk Félegyházi (EPFL) 14

Distribute your plates

Truth 1: The researchers won’t change the position of their plates (NE), if:

► dx,y ≤ 1 and

► ki,x ≤ 1.

Nash Equilibrium:

► pick two pies x and y, where kx ≥ ky► demand: dx,y = kx – ky

Put 1 plate per pie

July 9, 2007 Márk Félegyházi (EPFL) 15

Put more plates to some pies

Truth 2: The researchers won’t change the position of their plates (NE), if: ► dx,y ≤ 1,

► for any researcher i who has ki,x ≥ 2, x in C,

► for any researcher i who has ki,x ≥ 2 and x in C+, ki,y ≥ ki,x – 1, for all y in C–

,

( 1) ( 1)

( 1) ( )x x

i xx x

k kk

k k

► pick two pies x and y, where kx ≥ ky► demand: dx,y = kx – ky► more and less demanded pies C+ and C–

Nash Equilibrium:

Put more platesto some pies

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Convergence to stable states

Algorithm with imperfect info:► researchers don’t know the

demand for pies they are not demanding themselves

► move plates from demanded pies to other randomly chosen pies

► desynchronize the changes► convergence is not ensured

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Summary

► hungry researchers having several plates► Dining Game Theoreticians game► results for a stable pie sharing (NE):

– researchers should use all their plates– similar demand for each pie– two types of stable states– NE are efficient both in theory and practice

► fairness issues► equilibria for coalitions► algorithms to achieve efficient NE:

– centralized algorithm with perfect information– distributed algorithm with imperfect information

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Back to wireless networking

► C orthogonal channels – C pies► N communicating pairs of devices – N researchers► k radios at each device – k plates

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Some contributions

► Stability and convergence of multi-radio channel allocation in wireless networks

► Cooperation conditions for packet forwarding in ad hoc networks► Spectrum sharing strategies of wireless network (cellular) operators