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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

Atomic Routing Games on Maximum Congestion

Costas Busch

Department of Computer Science

Louisiana State University

Collaborators: RajgopalKannan, LSU

MalikMagdon-Ismail, RPI

slide2

Outline of Talk

Introduction

Price of Stability

Price of Anarchy

Bicriteria Game

network routing
Network Routing

Each player corresponds to a pair of

source-destination

Objective is to select paths with small cost

slide4

Main objective of each player

is to minimize congestion:

minimize maximum utilized edge

slide5

Congestion Games:

A player may selfishly choose an alternative

path that minimizes congestion

slide6

Player cost function for routing :

Congestion

of selected path

Social cost function for routing :

Largest player cost

slide7

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

slide8

Results:

  • Price of Stability is 1
  • Price of Anarchy is

Maximum allowed path length

slide9

Outline of Talk

Introduction

Price of Stability

Price of Anarchy

Bicriteria Game

slide10

We show:

  • QoR games have Nash Equilibriums
  • (we define a potential function)
  • The price of stability is 1
slide11

Routing Vector

number of players with cost

slide13

Lemma:

If player performs a greedy move

transforming routing to then:

Proof Idea:

Show that the greedy move gives

a lower order routing vector

slide14

Player Cost

Before greedy move:

After greedy move:

Since player cost decreases:

slide15

Before greedy move

player was counted here

After greedy move

player is counted here

slide16

>

>

=

=

possible

increase

or decrease

possible

decrease

No change

Definite Decrease

Possible increase

END OF PROOF IDEA

slide17

Existence of Nash Equilibriums

Greedy moves give lower order routings

Eventually a local minimum for every player

is reached which is a Nash Equilibrium

slide18

Price of Stability

Lowest order routing :

  • Is a Nash Equilibrium
  • Achieves optimal social cost
slide19

Outline of Talk

Introduction

Price of Stability

Price of Anarchy

Bicriteria Game

slide21

Consider an arbitrary Nash Equilibrium

Path of player

maximum

congestion

in path

edge

slide22

In optimal routing :

Optimal path of player

must have an edge

with congestion

Since otherwise:

slide29

We obtain sequences:

There exist subsequence:

and

Where:

slide30

Maximum path length

Maximum edge utilization

Minimum edge utilization

Known relations

slide32

Outline of Talk

Introduction

Price of Stability

Price of Anarchy

Bicriteria Game

slide33

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

slide34

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
slide35

Player cost function for routing :

Congestion

of selected path

Cost of respective

routing class

slide37

Results:

  • Price of Stability is 1
  • Price of Anarchy is
slide38

We consider restricted QoR games

For any path :

Path length

Service Cost

of path

slide40

Consider an arbitrary Nash Equilibrium

Path of player

maximum

congestion

in path

edge

slide41

In optimal routing :

Optimal path of player

must have an edge

with congestion

Since otherwise:

slide48

We obtain sequences:

There exist subsequence:

and

Where:

slide49

Maximum path length

Maximum edge utilization

Minimum edge utilization

Known relations

slide50

We have:

By considering class service costs, we obtain:

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