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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

Network Routing

Each player corresponds to a pair of

source-destination

Objective is to select paths with small cost

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

- QoR games have Nash Equilibriums
- (we define a potential function)

- The price of stability is 1

number of players with cost

If player performs a greedy move

transforming routing to then:

Proof Idea:

Show that the greedy move gives

a lower order routing vector

>

=

=

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

We show for any restricted QoR game:

Price of Anarchy =

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

- 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

Social cost function for routing :

Largest player cost

We show for any restricted QoR game:

Price of Anarchy =

By considering class service costs, we obtain:

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