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Bargaining Dynamics in Exchange Networks

Bargaining Dynamics in Exchange Networks. Milan Vojnović Microsoft Research Joint work with Moez Draief. Allerton 2010, September 30, 2010. Nash Bargaining [ Nash ’50]. Nash Bargaining on Graphs [Kleinberg and Tardos ’08]. Nash Bargaining Solution. Stable : . Balanced : .

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Bargaining Dynamics in Exchange Networks

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  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]

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

  4. Nash Bargaining Solution • Stable: • Balanced:

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

  6. KT Procedure

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

  8. KT Elementary Graphs Path Cycle Blossom Bicycle

  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)

  10. Edge-Balanced Dynamics

  11. Natural Dynamics

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

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

  14. Linear Systems Refresher

  15. Path

  16. Path (cont’d)

  17. Cycle

  18. Cycle (cont’d)

  19. Blossom • Non-linear system:

  20. Blossom (cont’d)

  21. Blossom (cont’d) path

  22. Blossom (cont’d) Convergence time:

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

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

  25. Bicycle (cont’d) Convergence time:

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

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

  28. The Positive Gap Condition

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

  30. Simplified Dynamics

  31. Path Bounding Process

  32. Bounds

  33. Bounds (cont’d)

  34. Conclusion

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