1. Bargaining Dynamics in Exchange Networks Milan Vojnović Microsoft Research Joint work with Moez Draief Allerton 2010, September 30, 2010

2. Nash Bargaining [Nash ’50] 2

3. Nash Bargaining on Graphs [Kleinberg and Tardos ’08] 3

4. Nash Bargaining Solution Stable: Balanced: 4

5. Facts about Stable and Balanced [Kleinberg and Tardos ’08] 5

6. KT Procedure 6

7. Step 2: Max-Min-Slack 7 max sub. to

8. KT Elementary Graphs Path CycleBlossom Bicycle 8

9. Local Dynamics It is of interest to consider node-local dynamics for stable and balanced outcomes Two such local dynamics: – Edge-balanced dynamics (Azar et al ’09) – Natural dynamics (Kanoria et al ’10) 9

10. Edge-Balanced Dynamics 10

11. Natural Dynamics 11

12. Known Facts Edge-balanced dynamics Fixed points are balanced outcomes Convergence rate unknown 12

13. Outline Convergence rate of edge-balanced dynamics for KT elementary graphs A path bounding process of natural dynamics and convergence time Conclusion 13

14. Linear Systems Refresher 14

15. Path 15

16. Path (cont’d) 16

17. Cycle 17

18. Cycle (cont’d) 18

19. Blossom Non-linear system: 19

20. Blossom (cont’d) 20

21. Blossom (cont’d) path 21

22. Blossom (cont’d) 22 Convergence time:

23. Bicycle Non-linear dynamics: plus other updates as for blossom 23

24. Bicycle (cont’d) Similar but more complicated than for a blossom 24

25. Bicycle (cont’d) Convergence time: 25

26. Quadratic convergence time in the number of matched edges, for all elementary KT graphs 26

27. Outline Convergence rate of edge-balanced dynamics for KT elementary graphs A path bounding process of natural dynamics and convergence time Conclusion 27

28. The Positive Gap Condition 28

29. The Positive Gap Condition (cont’d) Enables decoupling for the convergence analysis 29

30. Simplified Dynamics 30

31. Path Bounding Process 31

32. Bounds 32

33. Bounds (cont’d) 33

34. Conclusion 34