Atomic Routing Games on Maximum Congestion

1. Atomic Routing Games on Maximum Congestion Costas Busch Department of Computer Science Louisiana State University Collaborators: RajgopalKannan, LSU MalikMagdon-Ismail, RPI

2. Outline of Talk Introduction Price of Stability Price of Anarchy Bicriteria Game

3. Network Routing Each player corresponds to a pair of source-destination Objective is to select paths with small cost

4. Main objective of each player is to minimize congestion: minimize maximum utilized edge

5. Congestion Games: A player may selfishly choose an alternative path that minimizes congestion

6. Player cost function for routing : Congestion of selected path Social cost function for routing : Largest player cost

7. We are interested in Nash Equilibriums where every player is locally optimal Metrics of equilibrium quality: Price of Stability Price of Anarchy is optimal coordinated routing with smallest social cost

8. Results: • Price of Stability is 1 • Price of Anarchy is Maximum allowed path length

9. Outline of Talk Introduction Price of Stability Price of Anarchy Bicriteria Game

10. We show: • QoR games have Nash Equilibriums • (we define a potential function) • The price of stability is 1

11. Routing Vector number of players with cost

12. Routing Vectors are ordered lexicographically = = = = < = < =

13. Lemma: If player performs a greedy move transforming routing to then: Proof Idea: Show that the greedy move gives a lower order routing vector

14. Player Cost Before greedy move: After greedy move: Since player cost decreases:

15. Before greedy move player was counted here After greedy move player is counted here

16. > > = = possible increase or decrease possible decrease No change Definite Decrease Possible increase END OF PROOF IDEA

17. Existence of Nash Equilibriums Greedy moves give lower order routings Eventually a local minimum for every player is reached which is a Nash Equilibrium

18. Price of Stability Lowest order routing : • Is a Nash Equilibrium • Achieves optimal social cost

19. Outline of Talk Introduction Price of Stability Price of Anarchy Bicriteria Game

20. We show for any restricted QoR game: Price of Anarchy =

21. Consider an arbitrary Nash Equilibrium Path of player maximum congestion in path edge

22. In optimal routing : Optimal path of player must have an edge with congestion Since otherwise:

23. In Nash Equilibrium social cost is:

24. Edges in optimal paths of

26. Edges in optimal paths of

28. In a similar way we can define:

29. We obtain sequences: There exist subsequence: and Where:

30. Maximum path length Maximum edge utilization Minimum edge utilization Known relations

31. Worst Case Scenario:

32. Outline of Talk Introduction Price of Stability Price of Anarchy Bicriteria Game

33. We consider Quality of Routing (QoR) congestion games where the paths are partitioned into routing classes: With service costs: Only paths in same routing class can cause congestion to each other

34. An example: • We can have routing classes • Each routing class contains paths • with length in range • Service cost: • Each routing class uses a different • wireless frequency channel

35. Player cost function for routing : Congestion of selected path Cost of respective routing class

36. Social cost function for routing : Largest player cost

37. Results: • Price of Stability is 1 • Price of Anarchy is

38. We consider restricted QoR games For any path : Path length Service Cost of path

39. We show for any restricted QoR game: Price of Anarchy =

40. Consider an arbitrary Nash Equilibrium Path of player maximum congestion in path edge

41. In optimal routing : Optimal path of player must have an edge with congestion Since otherwise:

42. In Nash Equilibrium:

43. Edges in optimal paths of

45. Edges in optimal paths of

47. In a similar way we can define:

48. We obtain sequences: There exist subsequence: and Where:

49. Maximum path length Maximum edge utilization Minimum edge utilization Known relations

50. We have: By considering class service costs, we obtain: