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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


Atomic routing games on maximum congestion

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


Atomic routing games on maximum congestion

Main objective of each player

is to minimize congestion:

minimize maximum utilized edge


Atomic routing games on maximum congestion

Congestion Games:

A player may selfishly choose an alternative

path that minimizes congestion


Atomic routing games on maximum congestion

Player cost function for routing :

Congestion

of selected path

Social cost function for routing :

Largest player cost


Atomic routing games on maximum congestion

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


Atomic routing games on maximum congestion

Results:

  • Price of Stability is 1

  • Price of Anarchy is

Maximum allowed path length


Atomic routing games on maximum congestion

Outline of Talk

Introduction

Price of Stability

Price of Anarchy

Bicriteria Game


Atomic routing games on maximum congestion

We show:

  • QoR games have Nash Equilibriums

  • (we define a potential function)

  • The price of stability is 1


Atomic routing games on maximum congestion

Routing Vector

number of players with cost



Atomic routing games on maximum congestion

Lemma:

If player performs a greedy move

transforming routing to then:

Proof Idea:

Show that the greedy move gives

a lower order routing vector


Atomic routing games on maximum congestion

Player Cost

Before greedy move:

After greedy move:

Since player cost decreases:


Atomic routing games on maximum congestion

Before greedy move

player was counted here

After greedy move

player is counted here


Atomic routing games on maximum congestion

>

>

=

=

possible

increase

or decrease

possible

decrease

No change

Definite Decrease

Possible increase

END OF PROOF IDEA


Atomic routing games on maximum congestion

Existence of Nash Equilibriums

Greedy moves give lower order routings

Eventually a local minimum for every player

is reached which is a Nash Equilibrium


Atomic routing games on maximum congestion

Price of Stability

Lowest order routing :

  • Is a Nash Equilibrium

  • Achieves optimal social cost


Atomic routing games on maximum congestion

Outline of Talk

Introduction

Price of Stability

Price of Anarchy

Bicriteria Game


Atomic routing games on maximum congestion

We show for any restricted QoR game:

Price of Anarchy =


Atomic routing games on maximum congestion

Consider an arbitrary Nash Equilibrium

Path of player

maximum

congestion

in path

edge


Atomic routing games on maximum congestion

In optimal routing :

Optimal path of player

must have an edge

with congestion

Since otherwise:


Atomic routing games on maximum congestion

In Nash Equilibrium social cost is:





Atomic routing games on maximum congestion

We obtain sequences:

There exist subsequence:

and

Where:


Atomic routing games on maximum congestion

Maximum path length

Maximum edge utilization

Minimum edge utilization

Known relations



Atomic routing games on maximum congestion

Outline of Talk

Introduction

Price of Stability

Price of Anarchy

Bicriteria Game


Atomic routing games on maximum congestion

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


Atomic routing games on maximum congestion

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


Atomic routing games on maximum congestion

Player cost function for routing :

Congestion

of selected path

Cost of respective

routing class



Atomic routing games on maximum congestion

Results:

  • Price of Stability is 1

  • Price of Anarchy is


Atomic routing games on maximum congestion

We consider restricted QoR games

For any path :

Path length

Service Cost

of path


Atomic routing games on maximum congestion

We show for any restricted QoR game:

Price of Anarchy =


Atomic routing games on maximum congestion

Consider an arbitrary Nash Equilibrium

Path of player

maximum

congestion

in path

edge


Atomic routing games on maximum congestion

In optimal routing :

Optimal path of player

must have an edge

with congestion

Since otherwise:


Atomic routing games on maximum congestion

In Nash Equilibrium:





Atomic routing games on maximum congestion

We obtain sequences:

There exist subsequence:

and

Where:


Atomic routing games on maximum congestion

Maximum path length

Maximum edge utilization

Minimum edge utilization

Known relations


Atomic routing games on maximum congestion

We have:

By considering class service costs, we obtain: