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Bargaining Dynamics in Exchange Networks Milan Vojnović Microsoft Research Joint work with Moez Draief Allerton 2010, September 30, 2010
Nash Bargaining [Nash ’50] 2
Nash Bargaining on Graphs [Kleinberg and Tardos ’08] 3
Nash Bargaining Solution Stable: Balanced: 4
Facts about Stable and Balanced [Kleinberg and Tardos ’08] 5
KT Procedure 6
Step 2: Max-Min-Slack 7 max sub. to
KT Elementary Graphs Path CycleBlossom Bicycle 8
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
Edge-Balanced Dynamics 10
Natural Dynamics 11
Known Facts Edge-balanced dynamics Fixed points are balanced outcomes Convergence rate unknown 12
Outline Convergence rate of edge-balanced dynamics for KT elementary graphs A path bounding process of natural dynamics and convergence time Conclusion 13
Linear Systems Refresher 14
Path (cont’d) 16
Cycle (cont’d) 18
Blossom Non-linear system: 19
Blossom (cont’d) 20
Blossom (cont’d) path 21
Blossom (cont’d) 22 Convergence time:
Bicycle Non-linear dynamics: plus other updates as for blossom 23
Bicycle (cont’d) Similar but more complicated than for a blossom 24
Bicycle (cont’d) Convergence time: 25
Quadratic convergence time in the number of matched edges, for all elementary KT graphs 26
Outline Convergence rate of edge-balanced dynamics for KT elementary graphs A path bounding process of natural dynamics and convergence time Conclusion 27
The Positive Gap Condition 28
The Positive Gap Condition (cont’d) Enables decoupling for the convergence analysis 29
Simplified Dynamics 30
Path Bounding Process 31
Bounds (cont’d) 33
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