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Atomic Routing Games on Maximum Congestion

Costas BuschDepartment of Computer Science

Louisiana State University

Collaborators: Rajgopal Kannan, LSUMalik Magdon-Ismail, RPI

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Introduction

Price of Stability

Price of Anarchy

Outline of Talk

Bicriteria Game

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Network RoutingEach player corresponds to a pair of source-destination

Objective is to select paths with small cost

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

3 congestion C

iplayer

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A player may selfishly choose an alternativepath that minimizes congestion

CC 31 congestion

Congestion Games:

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Player cost function for routing :i

ii Cppc )(

p

Congestionof selected path

Social cost function for routing :

CpSC )(

p

Largest player cost

We are interested in Nash Equilibriumswhere every player is locally optimal

Metrics of equilibrium quality:

p

Price of Stability

)()(min *pSCpSC

p

Price of Anarchy

)()(max *pSCpSC

p

*p is optimal coordinated routing with smallest social cost

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Results:• Price of Stability is 1

• Price of Anarchy is )log( nLO

Maximum allowed path length

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Introduction

Price of Stability

Price of Anarchy

Outline of Talk

Bicriteria Game

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We show:

• QoR games have Nash Equilibriums

(we define a potential function)

• The price of stability is 1

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],,,,,[)( 21 Nk mmmmpM

number of players with cost km k

Routing Vector

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Routing Vectors are ordered lexicographically

],,,[)( 21 NmmmpM

],,,[)( 21 NmmmpM

= = = =

],,,,,[)( 11 Nkk mmmmpM

],,,,,[)( 11 Nkk mmmmpM

< < = =

)()( pMpM

)()( pMpM )( pp

)( pp

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If player performs a greedy movetransforming routing to then:p p pp

iLemma:

Proof Idea:Show that the greedy move gives a lower order routing vector

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kk

ii Cppck )(ii Cppck )(

Player CostiBefore greedy move:After greedy move:

Since player cost decreases:

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],,,,,,,[)( 11 Nkkk mmmmmpM

Before greedy move player was counted herei

],,,,,,,[)( 11 Nkkk mmmmmpM

After greedy moveplayer is counted herei

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],,,,,,,[)( 11 rkkk mmmmmpM

],,,,,,,[)( 11 rkkk mmmmmpM

> ==No change

Definite Decrease

possibledecrease

possibleincreaseor decrease

Possible increase

>

END OF PROOF IDEA

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Existence of Nash Equilibriums

Greedy moves give lower order routings

Eventually a local minimum for every playeris reached which is a Nash Equilibrium

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minp

Price of Stability

Lowest order routing :

*min )( SCpSC

• Is a Nash Equilibrium

• Achieves optimal social cost

1)(Stability of Price *min

SCpSC

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Introduction

Price of Stability

Price of Anarchy

Outline of Talk

Bicriteria Game

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We show for any restricted QoR game:

Price of Anarchy = )log( nLO

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Path of player

Consider an arbitrary Nash Equilibriump

i

iCedgemaximum congestionin path

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must have an edge with congestion

Optimal path of player

In optimal routing :*p

i

iC1 iCC

)(111*)( ppcCCCppc iiii

**)( CpSC

Since otherwise:

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C

00

0

edges use that Paths: Congestion of Edges :ECE

In Nash Equilibrium social cost is: CSC

0 0

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C 1C1C

0 0

Edges in optimal paths of 0

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C 1C1C

0 01 1

11

1

edges use that Players:1 least at Congestion of Edges :

ECE

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C 1C1C 2C 2C2C2C

0 01 1

Edges in optimal paths of 1

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C *SC *SC 2C 2C2C

0 01 1

2C

2 2

22

2

edges use that Players:2 least at Congestion of Edges :

ECE

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In a similar way we can define:

jj

j

EjCE

edges use that Players: least at Congestion of Edges :

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,,,,

,,,,

3210

3210

EEEEWe obtain sequences:

There exist subsequence:110

110

,,,,,,,

s

ss EEEE

||2|| 1 jj EEWhere: ||2|| 1 ss EEand1sj

ns log

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||))1((|| 11 ss EsCL

|||| 1*

s

s

EC

Maximum edge utilization

Minimum edge utilization

LMaximum path length

)log(* nLOCC

ns log ||2|| 1 ss EEKnown relations

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Worst Case Scenario:

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Introduction

Price of Stability

Price of Anarchy

Outline of Talk

Bicriteria Game

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We consider Quality of Routing (QoR) congestion games where the pathsare partitioned into routing classes:

QQQ ,,, 21

)()()( 21 QSQSQS

With service costs:

Only paths in same routing class can causecongestion to each other

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An example:

•We can have routing classes)(lognO

•Each routing class contains paths with length in range

jQ]2,2( 1jj

12)( jjQS•Service cost:

•Each routing class uses a different wireless frequency channel

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Player cost function for routing :i

iii SCppc )(

p

Congestionof selected path

Cost of respectiverouting class

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Social cost function for routing :

SCpSC )(

p

Largest player cost

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Results:• Price of Stability is 1

• Price of Anarchy is

)log),(min( ** nSCO

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We consider restricted QoR games

For any path :p )(|| pSp

Path length Service Cost of path

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We show for any restricted QoR game:

Price of Anarchy = )log),(min( ** nSCO

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Path of player

Consider an arbitrary Nash Equilibriump

i

iCedgemaximum congestionin path

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must have an edge with congestion

Optimal path of player

In optimal routing :*p

i

iC

*SCC i

)(111 *** ppcSCCSSCSCcp iiiiiiii

***)( SCpSC

Since otherwise:

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C

00

0

edges use that Paths: Congestion of Edges :ECE

In Nash Equilibrium: SCSC

0 0

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C *SC *SC

0 0

Edges in optimal paths of 0

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C *SC *SC

0 01 1

11

*1

edges use that Players:least at Congestion of Edges :E

SCE

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C *SC *SC *2SC *2SC *2SC *2SC

0 01 1

Edges in optimal paths of 1

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C *SC *SC *2SC *2SC *2SC

0 01 1

*2SC

2 2

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*2

edges use that Players:2least at Congestion of Edges :

ESCE

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In a similar way we can define:

jj

j

E

jSCE

edges use that Players:

least at Congestion of Edges : *

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,,,,

,,,,

3210

3210

EEEEWe obtain sequences:

There exist subsequence:110

110

,,,,,,,

s

ss EEEE

||2|| 1 jj EEWhere: ||2|| 1 ss EEand1sj

ns log

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||))1((|| 1*

1 ss ESsCL

|||| 1*

s

s

EC

Maximum edge utilization

Minimum edge utilization

*SLMaximum path length

)log( ** nSOCC

ns log ||2|| 1 ss EEKnown relations

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)log( ** nSOCC

)log),(min( Anarchy of Price **** nSCOSCSC

We have:

By considering class service costs, we obtain: