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

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

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

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

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

6. 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 StabilityPrice of Anarchy is optimal coordinated routing with smallest social cost

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

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

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

11. 11 number of players with cost Routing Vector

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

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

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

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

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

17. 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. 18 Price of Stability Lowest order routing : Is a Nash Equilibrium Achieves optimal social cost

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

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

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

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

23. 23 In Nash Equilibrium social cost is:

24. 24 Edges in optimal paths of

25. 25

26. 26 Edges in optimal paths of

27. 27

28. 28 In a similar way we can define:

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

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

31. 31 Worst Case Scenario:

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

33. 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. 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. 35 Player cost function for routing : Congestion of selected path Cost of respective routing class

36. 36 Social cost function for routing : Largest player cost

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

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

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

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

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

42. 42 In Nash Equilibrium:

43. 43 Edges in optimal paths of

44. 44

45. 45 Edges in optimal paths of

46. 46

47. 47 In a similar way we can define:

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

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

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