Bargaining Dynamics in Exchange NetworksMilan VojnovićMicrosoft Research

Joint work with Moez DraiefAllerton 2010, September 30, 2010

Nash Bargaining[Nash ’50]

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Nash Bargaining on Graphs[Kleinberg and Tardos ’08]

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Nash Bargaining Solution• Stable: • Balanced:

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

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KT Procedure

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Step 2: Max-Min-Slack

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maxsub. to

KT Elementary Graphs

Path Cycle Blossom 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)

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Edge-Balanced Dynamics

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Natural Dynamics

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Known FactsEdge-balanced dynamics•Fixed points are balanced outcomes•Convergence rate unknown

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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

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Linear Systems Refresher

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Path

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Path (cont’d)

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Cycle

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Cycle (cont’d)

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Blossom• Non-linear system:

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Blossom (cont’d)

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Blossom (cont’d) path

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Blossom (cont’d)

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Convergence time:

Bicycle• Non-linear dynamics:

plus other updates as for blossom23

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

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Bicycle (cont’d)Convergence time:

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

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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

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The Positive Gap Condition

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The Positive Gap Condition (cont’d)

• Enables decoupling for the convergence analysis29

Simplified Dynamics

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Path Bounding Process

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Bounds

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Bounds (cont’d)

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Conclusion

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