1. Non-Cooperative Behavior in Wireless Networks Márk Félegyházi (EPFL) PhD. public defense July 9, 2007

2. 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

3. July 9, 2007Márk Félegyházi (EPFL) 3 Multi-Radio Channel Allocation Problem ► C orthogonal channels ► N communicating pairs of devices ► k radios at each device number of radios by sender i on channel x → Nash equilibrium: No player has an incentive to unilaterally deviate. Proposition: If S * is a NE in G MRCA, then d y,x ≤ 1, for any channel x and y. ► blabla, ► blabla, blabla

4. How to Share a Pie with Selfish Researchers Márk Félegyházi (EPFL) PhD. public defense July 9, 2007 Who Know Game Theory

5. July 9, 2007Márk Félegyházi (EPFL) 5 Problem Dining Game Theoreticians

6. July 9, 2007Márk Félegyházi (EPFL) 6 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? 

7. July 9, 2007Márk 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:

8. July 9, 2007Márk 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 total number of plates demanding pie x

9. July 9, 2007Márk Félegyházi (EPFL) 9 The pie-cut functions ► pies have all the same size and quality ► π t (k x ) – total size of the shares of any pie x ► π(k x ) – size of a share per plate 3 3

10. July 9, 2007Márk Félegyházi (EPFL) 10 Dining Game Theoreticians (DGT) game ► selfish (=hungry) researchers ► non-cooperative game G DGT – players → researchers – strategy → plate allocation – payoff → total amount of cookie ► payoff:

11. July 9, 2007Márk Félegyházi (EPFL) 11 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.

12. July 9, 2007Márk Félegyházi (EPFL) 12 The Question Where shall I put my plates?

13. July 9, 2007Márk Félegyházi (EPFL) 13 Recognition: In a stable state (NE), d y,x ≤ 1 for any two pies x and y. Cut the pies in (almost) the same number of pieces ► pick two pies x and y, where k x ≥ k y ► demand: d x,y = k x – k y

14. July 9, 2007Márk Félegyházi (EPFL) 14 Distribute your plates Truth 1: The researchers won’t change the position of their plates (NE), if: ► d x,y ≤ 1 and ► k i,x ≤ 1. Nash Equilibrium: ► pick two pies x and y, where k x ≥ k y ► demand: d x,y = k x – k y Put 1 plate per pie

15. July 9, 2007Márk Félegyházi (EPFL) 15 Put more plates to some pies Truth 2: The researchers won’t change the position of their plates (NE), if: ► d x,y ≤ 1, ► for any researcher i who has k i,x ≥ 2, x in C, ► for any researcher i who has k i,x ≥ 2 and x in C +, k i,y ≥ k i,x – 1, for all y in C – ► pick two pies x and y, where k x ≥ k y ► demand: d x,y = k x – k y ► more and less demanded pies C + and C – Nash Equilibrium: Put more plates to some pies

16. July 9, 2007Márk Félegyházi (EPFL) 16 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

17. July 9, 2007Márk Félegyházi (EPFL) 17 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

18. July 9, 2007Márk Félegyházi (EPFL) 18 Back to wireless networking ► C orthogonal channels – C pies ► N communicating pairs of devices – N researchers ► k radios at each device – k plates

19. July 9, 2007Márk Félegyházi (EPFL) 19 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