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

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

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

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

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

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

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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

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

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

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

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11 number of players with cost Routing Vector

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

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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

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

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

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

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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

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

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

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

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

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

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23 In Nash Equilibrium social cost is:

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24 Edges in optimal paths of

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26 Edges in optimal paths of

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27

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28 In a similar way we can define:

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

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

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31 Worst Case Scenario:

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

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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

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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

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

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36 Social cost function for routing : Largest player cost

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

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

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

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

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

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42 In Nash Equilibrium:

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43 Edges in optimal paths of

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44

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45 Edges in optimal paths of

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46

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47 In a similar way we can define:

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

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

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

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