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 Introduction Price of Stability Price of Anarchy Outline of Talk 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 A player may selfishly choose an alternative path that minimizes congestion Congestion Games:

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

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

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

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

11 number of players with cost Routing Vector

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

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 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 > == No change Definite Decrease possible decrease possible increase or 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 Introduction Price of Stability Price of Anarchy Outline of Talk Bicriteria Game

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

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

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

23 In Nash Equilibrium social cost is:

24 Edges in optimal paths of

25

26 Edges in optimal paths of

27

28 In a similar way we can define:

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

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

31 Worst Case Scenario:

32 Introduction Price of Stability Price of Anarchy Outline of Talk 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 lengthService Cost of path

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

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

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

42 In Nash Equilibrium:

43 Edges in optimal paths of

44

45 Edges in optimal paths of

46

47 In a similar way we can define:

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

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

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