Bargaining Dynamics in Exchange Networks
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Bargaining Dynamics in Exchange NetworksMilan VojnovićMicrosoft Research
Joint work with Moez DraiefAllerton 2010, September 30, 2010
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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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KT Elementary Graphs
Path Cycle Blossom 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)
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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:
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Bicycle• Non-linear dynamics:
plus other updates as for blossom23
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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
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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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