Bottleneck Routing Games on Grids

1. Bottleneck Routing Games on Grids Costas Busch RajgopalKannan Alfred Samman Department of Computer Science Louisiana State University

2. Talk Outline Introduction Basic Game Channel Game Extensions

3. 2-d Grid: nodes • Used in: • Multiprocessor architectures • Wireless mesh networks • can be extended to d-dimensions

4. Each player corresponds to a pair of source-destination Edge Congestion Bottleneck Congestion:

5. A player may selfishly choose an alternative path with better congestion Player Congestion Player Congestion: Maximum edge congestion along its path

6. Routing is a collection of paths, one path for each player Utility function for player : congestion of selected path Social cost for routing : bottleneck congestion

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. number of dimension changes plus source and destination Bends :

9. Basic congestion games on grids Price of Stability: Price of Anarchy: even with constant bends

10. Better bounds with bends Channel games: Path segments are separated according to length range Price of anarchy: Optimal solution uses at most bends

11. There is a (non-game) routing algorithm with bends and approximation ratio Optimal solution uses arbitrary number of bends Final price of anarchy:

12. Solution without channels: Split Games channels are implemented implicitly in space Similar poly-log price of anarchy bounds

13. Some related work: Price of Anarchy Definition Koutsoupias, Papadimitriou [STACS’99] Price of Anarchy for sum of congestion utilities [JACM’02] Arbitrary Bottleneck games [INFOCOM’06], [TCS’09]: Price of Anarchy NP-hardness

14. Talk Outline Introduction Basic Game Channel Game Extensions

15. Stability is proven through a potential function defined over routing vectors: number of players with congestion

16. In best response dynamics a player move improves lexicographically the routing vector Player Congestion

17. Before greedy move After greedy move

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

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

20. Price of Anarchy Optimal solution Nash Equilibrium High! Price of anarchy:

21. Talk Outline Introduction Basic Game Channel Game Extensions

22. channels Channel holds path segments of length in range: Row:

23. different channels same channel Congestion occurs only with path segments in same channel

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

25. In optimal routing : Optimal path of player must have a special edge with congestion Since otherwise:

26. In Nash Equilibrium social cost is:

27. First expansion Special Edges in optimal paths of

28. First expansion

29. Second expansion Special Edges in optimal paths of

30. Second expansion

31. In a similar way we can define: We obtain expansion sequences:

32. Redefine expansion:

34. If then constant k Contradiction

35. Therefore: Price of anarchy:

36. Tightness of Price of Anarchy Nash Equilibrium Optimal solution Price of anarchy:

37. Talk Outline Introduction Basic Game Channel Game Extensions

38. Split game Price of anarchy:

39. d-dimensional grid Channel game Price of anarchy: Split game Price of anarchy: