Bargaining Dynamics in Exchange Networks Milan Vojnović Microsoft Research Joint work with Moez Draief llerton 2010, September 30, 2010
Feb 14, 2016
Bargaining Dynamics in Exchange NetworksMilan VojnovićMicrosoft Research
Joint work with Moez DraiefAllerton 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
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)
9
Edge-Balanced Dynamics
10
Natural Dynamics
11
Known FactsEdge-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
15
Path (cont’d)
16
Cycle
17
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 blossom23
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 analysis29
Simplified Dynamics
30
Path Bounding Process
31
Bounds
32
Bounds (cont’d)
33
Conclusion
34