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

2 Talk Outline Introduction Basic Game Channel Game Extensions

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

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

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

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 Bends : number of dimension changes plus source and destination

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

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

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

14 Talk Outline Introduction Basic Game Channel Game Extensions

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

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

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 solutionNash Equilibrium Price of anarchy: High!

21 Talk Outline Introduction Basic Game Channel Game Extensions

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

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

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

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

26 In Nash Equilibrium social cost is:

27 Special Edges in optimal paths of First expansion

28 First expansion

29 Special Edges in optimal paths of Second expansion

30 Second expansion

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

32 Redefine expansion:

33

34 If then Contradiction constant k

35 Therefore: Price of anarchy:

36 Optimal solutionNash Equilibrium Price of anarchy: Tightness of Price of Anarchy

37 Talk Outline Introduction Basic Game Channel Game Extensions

38 Split game Price of anarchy:

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