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

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Nash Bargaining [Nash ’50] 2

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

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Nash Bargaining Solution Stable: Balanced: 4

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

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KT Procedure 6

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

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KT Elementary Graphs Path CycleBlossom Bicycle 8

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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

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Edge-Balanced Dynamics 10

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Natural Dynamics 11

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

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

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Linear Systems Refresher 14

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Path 15

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Path (cont’d) 16

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Cycle 17

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Cycle (cont’d) 18

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Blossom Non-linear system: 19

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Blossom (cont’d) 20

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Blossom (cont’d) path 21

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Blossom (cont’d) 22 Convergence time:

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

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

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Bicycle (cont’d) Convergence time: 25

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

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

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The Positive Gap Condition 28

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

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Simplified Dynamics 30

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Path Bounding Process 31

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Bounds 32

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Bounds (cont’d) 33

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Conclusion 34

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