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Atomic Routing Games on Maximum Congestion. Costas Busch Department of Computer Science Louisiana State University. Collaborators: Rajgopal K annan , LSU Malik Magdon -Ismail, RPI. Outline of Talk. Introduction. Price of Stability. Price of Anarchy. Bicriteria Game.
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Atomic Routing Games on Maximum Congestion Costas Busch Department of Computer Science Louisiana State University Collaborators: RajgopalKannan, LSU MalikMagdon-Ismail, RPI
Outline of Talk Introduction Price of Stability Price of Anarchy Bicriteria Game
Network Routing Each player corresponds to a pair of source-destination Objective is to select paths with small cost
Main objective of each player is to minimize congestion: minimize maximum utilized edge
Congestion Games: A player may selfishly choose an alternative path that minimizes congestion
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 Stability Price of Anarchy is optimal coordinated routing with smallest social cost
Results: • Price of Stability is 1 • Price of Anarchy is Maximum allowed path length
Outline of Talk Introduction Price of Stability Price of Anarchy Bicriteria Game
We show: • QoR games have Nash Equilibriums • (we define a potential function) • The price of stability is 1
Routing Vector number of players with cost
Routing Vectors are ordered lexicographically = = = = < = < =
Lemma: If player performs a greedy move transforming routing to then: Proof Idea: Show that the greedy move gives a lower order routing vector
Player Cost Before greedy move: After greedy move: Since player cost decreases:
Before greedy move player was counted here After greedy move player is counted here
> > = = possible increase or decrease possible decrease No change Definite Decrease Possible increase END OF PROOF IDEA
Existence of Nash Equilibriums Greedy moves give lower order routings Eventually a local minimum for every player is reached which is a Nash Equilibrium
Price of Stability Lowest order routing : • Is a Nash Equilibrium • Achieves optimal social cost
Outline of Talk Introduction Price of Stability Price of Anarchy Bicriteria Game
We show for any restricted QoR game: Price of Anarchy =
Consider an arbitrary Nash Equilibrium Path of player maximum congestion in path edge
In optimal routing : Optimal path of player must have an edge with congestion Since otherwise:
We obtain sequences: There exist subsequence: and Where:
Maximum path length Maximum edge utilization Minimum edge utilization Known relations
Outline of Talk Introduction Price of Stability Price of Anarchy Bicriteria Game
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
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
Player cost function for routing : Congestion of selected path Cost of respective routing class
Social cost function for routing : Largest player cost
Results: • Price of Stability is 1 • Price of Anarchy is
We consider restricted QoR games For any path : Path length Service Cost of path
We show for any restricted QoR game: Price of Anarchy =
Consider an arbitrary Nash Equilibrium Path of player maximum congestion in path edge
In optimal routing : Optimal path of player must have an edge with congestion Since otherwise:
We obtain sequences: There exist subsequence: and Where:
Maximum path length Maximum edge utilization Minimum edge utilization Known relations
We have: By considering class service costs, we obtain: