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 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 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 32

Bounds (cont’d) 33

Conclusion 34