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Bottleneck Routing Games on Grids

Costas BuschRajgopal KannanAlfred Samman

Department of Computer ScienceLouisiana State University

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Talk Outline

Introduction

Basic Game

Channel Game

Extensions

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2-d Grid:

Used in:• Multiprocessor architectures• Wireless mesh networks• can be extended to d-dimensions

n

n

nnnodes

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Each player corresponds to a pair of source-destination

EdgeCongestion

3)( 1 eC

2)( 2 eC

Bottleneck Congestion: 3)(max

eCCEe

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A player may selfishly choose an alternativepath with better congestion

ii CC 31

PlayerCongestion

i

3iC

1iC

Player Congestion: Maximum edge congestion along its path

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Routing is a collection of paths, one path for each player

Utility function for player :i

ii Cppc )(

p

congestionof selected path

Social cost for routing :

CpSC )(p

bottleneck congestion

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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Bends :number of dimension changes plus source and destination

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Price of Stability:

Price of Anarchy:

)1(O

)(n

even with constant bends )1(O

Basic congestion games on grids

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Better bounds with bends

Price of anarchy: nO log

Channel games:

Optimal solution uses at most bends

Path segments are separated accordingto length range

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There is a (non-game) routing algorithmwith bends and approximation ratio

nO log nO log

Optimal solution uses arbitrary number of bends

Final price of anarchy: nO 3log

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Solution without channels: Split Gameschannels are implemented implicitly in space

Similar poly-log price of anarchy bounds

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Some related work:

Arbitrary Bottleneck games [INFOCOM’06], [TCS’09]:

Price of AnarchyNP-hardness

Price of Anarchy DefinitionKoutsoupias, Papadimitriou [STACS’99]

Price of Anarchy for sum of congestion utilities [JACM’02]

1O

|| EO

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Talk Outline

Introduction

Basic Game

Channel Game

Extensions

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

number of players with congestion kCi

Stability is proven through a potential functiondefined over routing vectors:

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PlayerCongestion

3iC

1iC

In best response dynamics a player move improves lexicographically the routing vector

)()( pMpM ]0,...,0,0,3,1,0[]0,...,0,0,0,2,2[

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

Before greedy move kCi

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

After greedy move

ii CkkC

)()( pMpM

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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 pSCpSC

• Is a Nash Equilibrium

• Achieves optimal social cost

1)()(Stability of Price *

min pSCpSC

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Price of AnarchyOptimal solution Nash Equilibrium

1* C 2/nC

)(2/* nnCC

Price of anarchy: High!

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Talk Outline

Introduction

Basic Game

Channel Game

Extensions

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Row:

channelsnlog

Channel holds path segments of length in range:

jA]12,2[ 1 jj

0A1A2A3A

]1,1[]3,2[

]7,4[]15,8[

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1eC

2eC

different channels

same channel

Congestion occurs only with path segmentsin same channel

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Path of player

Consider an arbitrary Nash Equilibriump

i

iCmaximum congestionin path

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must have a special 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 Players: Congestion of Edges :ECE

In Nash Equilibrium social cost is: CpSC )(

0 0

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C 1C1C

0 0

Special Edges in optimal paths of 0

First expansion

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C 1C1C

0 01 1

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1

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

ECE

First expansion

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C 1C1C 2C 2C2C2C

0 01 1

Special Edges in optimal paths of 1

Second expansion

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C 1C1C 2C 2C2C

0 01 1

2C

2 2

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2

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

ECE

Second expansion

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In a similar way we can define:

jj

j

E

jCE

edges use that Players:

least at Congestion of Edges Special :

,,,,

,,,,

3210

3210

EEEE

We obtain expansion sequences:

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jj

j

E

jCE

edges use that Players:

12 :rin far ly sufficient are edges and

r channel somein majority thearech whi

least at Congestion of Edges Special :

1-r

Redefine expansion:

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*1

||||

aCE jj

*1

||)(||

CaE

jCE jj

||

)(|| jj

EjC

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*1

||)(||

CaE

jCE jj

If then )log( * nCC

|||| 1 jj EkE

2|| nE Contradiction

constant k

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)log( * nCOC Therefore:

Price of anarchy:)log()log(* nOnO

CC

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Optimal solutionNash Equilibrium1* C)()( 2 nC

)()( 2* nCCPrice of anarchy:

Tightness of Price of Anarchy

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Talk Outline

Introduction

Basic Game

Channel Game

Extensions

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0A

1A

2A3A

Split game

0A

1A

2A3A

Price of anarchy: )log( 2 nO

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d-dimensional grid

Price of anarchy:

nd

O logChannel game

nd

O 22 log

Price of anarchy:Split game