How Bad is Selfish Routing A survey on existing models for selfish routing Professor John Lui, David Yau and Dah-Ming Qiu presented by Joe W.J. Jiang 2004-05-18

How Bad is Selfish Routing

A survey on existing models for selfish routing

Professor John Lui, David Yau and Dah-Ming Qiu

presented by Joe W.J. Jiang

2004-05-18

Outline of my talk

• Introduction to selfish routing

• Preliminaries and Nash equilibrium

• How bad is selfish routing

• Other models on selfish routing & related work

• Conclusions and problems

Introduction to selfish routing

• Routing in the Internet:• RIP (distance vector routing, Bellman-Ford)• OSPF (link state routing, Dijkstra)• BGP (exterior gateway protocol)

• These routing metrics of the above protocols are generally based on hop counts.

• There is an inherent inefficiency from the users’ perspective: bandwidth, latency, jitter.

• There is an incentive for users to choose routes themselves.

Selfish routing in the Internet

• Source routing : Nimrod -- route information is contained in the header of route request

• Overlay routing: Detour or RON – routing via peer nodes in the overlay network

• Such end-to-end route selection is selfish by nature, optimizing their own performance without considering others’.

Selfishness in the Internet

• Internet: users with a multitude of diverse economic interests

browsersroutersservers

• Selfishness: parties will deviate from their protocol if it is in their interest.

• How to study these problems:Algorithmic Game Theory – algorithms + game theory

Where are you?






Routing Problems

Optimization problem:

• given a network, a traffic rate between each pair of nodes

• latency function of each edge

• objective: the total latency is minimized

Nash Equilibrium

• A Nash Equilibrium is a set of strategies (one status) one for each player, such that no player has incentive to unilaterally change his action.

• Players are in equilibrium if a change in strategies by any one of them would lead that player to earn less than current strategy.

• It is well known that Nash equilibria do not in general optimize social welfare – “Prisoner’s Dilemma”.

Braess’s Paradox

• the price of anarchy:

2/1.5 = 4/3!

x 1

s t

x1

1/2

1/2

average latency= 1+0.5 =1.5

0

average latency= 1+1 =2

1

Some Algorithmic Issues

• Price of AnarchyA measure of degradation of performance caused by lack of cooperation (regulation) – selfishness.

• Mechanism DesignHow to design games so that selfish behaviors would lead to desire outcome.

• Coalitional GamesE.g., how to share costs incurred by a group of users.

Mathematical Models

• A directed graph G=(V, E)• source-sink pairs {si, ti} for i=1,..,k• rate ri 0 of traffic between si and ti for

each i=1,..,k

• set of si-ti paths Pi• P• for each edge e, a latency function le(•)

– nonnegative, differentiable, non-decreasing.

i iP

Mathematical Model – Traffic and Flows

• A flow vector specifies a traffic pattern fp

= amount of flow on si-ti path P• flow of an edge e• A flow f is said to be feasible if for all i,

• We call triple (G, r, l) an instance.• The latency of a path P • cost of all flows C(f) -- total latency

PeP Pe ff:

iPP P rfi

Pe eeP flfl )()(

Ee eee

PPP fflfflfC )()()(

P

Flows and game theory

• Flow represents routes of many noncooperative agents– each agent controlling infinitesimally small a

mount • cars in a highway system • packets in a network

• The cost (total latency) of a flow represents social welfare.

• Agents are selfish in that• minimize personal latency• do not care about social welfare

Flows at Nash equilibrium

• A flow is at Nash equilibrium (or is a Nash flow) if no agent can improve its latency by changing its path.

otherwisef

PPiff

PPiff

f

flflf

PP,...,k}{i

lrGf

P

P

P

P

PPP

~

where,)~

()( have we],,0[

and , ,1 allfor if mequilibriuNash at

is ),,( instancefor feasible flowA :Definition

2

1

21

211

Wardrop’s Principle

).()( ,0 with and

1every for ifonly and if mequilibriuNash

at is instancefor feasible flowA

21121 flflfP,PP

,...,k}{i

(G,r,l)fLemma:

PPPi

k

iii rfLfC(G,r,l)

fLemma

1

)()( then ,

instancefor mequilibriuNash at flow a is If :

In particular, all paths to which f assigns positive amount of flow, have equal latency, say Li(f)

Optimal Flow

• An optimal flow is a flow that minimizes total latency/ average latency.

• Convex programming:

)()( )()( eeeeeEe

ee fflfcfcMinfMinC

PP

P f

Eeff

,...,k}{irf

P

PePpe

iPP

P

i

0

1

:

Optimal Flow (Solution)

• If the objective function ce(fe)=le(fe)fe is convex, global optimal = local optimal

• We expect a flow to be locally optimal if and only if the marginal benefit of decreasing flow along any si-ti path ≤ the marginal cost of increasing flow along any other si-ti path.

)(xcdx

dc ee

Pe

eeP fcfc )()(

. ,0 with and

1every for ifonly and if form theof

programconvex afor optimal is flowA :

21121 (f)c(f)cfP,PP

,...,k}{i (NLP)

fLemma

PPPi

Beckman’s Interpretation

eeeeeeeeeeee fflflfflfcfl )()())(()()(*

. instancefor

mequilibriuNash at isit ifonly and if optimal is

for feasible flow aThen above. as defined function

cost marginal with , edgeeach for function convex a is

in which instancean be Let :

)(G,r,l

(G,r,l)

fl

e

(x)lx(G,r,l)Corollary

*

*

e

Existence of Nash Equilibrium

.~

then m,equilibriuNash at flows

are ~

if Moreover, m.equilibriuNash at flow

feasible a admits functionslatency ingnondecreas

,continuous with instanceAn :

)fC( C(f)

ff,

(G,r,l)Lemma

PP

P f

Eeff

,...,k}{irf

fh

xhdttlxh

P

PePpe

iPP

P

Eeee

e

x

ee

i

0

1

:subject to ,Min

convex is )( ,)(set :Proof

:

0

A good but not optimal upper bound

)C(f

C(f)ρ(G,r,l)

f

f(G,r,l)Definition

*

*

ratioon coordinati the, mequilibriuNash at flow a and

, flow optimalan with instancean For :

),,( , if 0

lrGdttlxlxCorollaryx

ee

)()(

)(

)(

)()( :Proof

***

0

0

*

fCffl

tl

tl

fflfC

Ee

eee

Ee

f

e

Ee

f

e

Eeeee

e

e

A good but not optimal upper bound (cont)

3),,( 2, ,)( ..

1),,(then

reals enonnegativ and integer positive afor

)( if :

2

,

0 ,

lrGpxxlge

plrG

ap

xaxlCorollary

e

ie

p

i

iiee

Where are you?






A simple bad example

pp /1)1(1

pp /1)1( s t

1

x

1

the price of anarchy = 1/ (3/4) =4/3 !

p

pp p

p

when ,0

)1(1 :latency totalthe1

1*1=1

½*1=1/2

½*1/2=1/4

1

Bicriteria Results

. then ,2for feasible is and

for mequilibriuNash at flow a is If :

)C(fC(f)r,l)(G,f

(G,r,l)fTheorem**

if )(

if )()(

ee

eeee fxxl

fxflxl

eeeee fflxlxlx )()()(

)(

)(

)()()()( ******

fC

ffl

flflffCffl

Eeeee

eeeeee

eeee

)(22)(

)()(

)()0()( )()0(***

*

fCrfL

ffLffl

fLlflfLl

iii

i PPPi

PP

iPPiP

i

)()(

)(2)(

)()()(

*

**

***

fCfC

fCffl

fCfCffl

PP

eeee

Bicriteria Results (cont)

. then ,2for feasible is and


)C(fC(f)r,l)(G,f

(G,r,l)fTheorem**

.1

then ,1for feasible is and


)C(fγ

C(f)γ)r,l)(G,(f

(G,r,l)fTheorem

**

Worst-Case Ratio of 4/3 with Linear Latency Functions

eee

e eeee

eeeee

bxaxl

fbfafC

babxa(x) lEe

2)(function cost marginal thedenote

)(

)0,( edgeeach For

*

2

Pee

*ee

Pee

*ee

*Pi

*

Peeee

Peeee

Pi

eee

bfabfa

,fPPP,i

G fb

bfabfa

fPPP,i

Gf

Eebxa(x)l

(G,r,l)Lemma

22

0 with and pair

sink-sourceeach for iff in optimal (globally) if flow a )(

,0 with and pair

sink-sourceeach for iff in mequilibriuNash at is flow a (a)

Then, .each for

functionslatency edge with instancean be Let :

corollary

m.equilibriuNash

at isit iff optimal is for feasible flow a ,vector

rateany for then , form theof is function

latency edgeeach in which network a ba Let :

(G,r,l)r

xa(x)ll

GCorollary

eee

Important results

. respect to with oflatency theequals 2 respect to

with path on flow theincreasing ofcost marginal the(b)

,2for optimal is 2 flow the(a)

thenm,equilibriuNash at flow a is and

functionslatency linear has Suppose :

fPf

P

,l)r (G,f

f

(G,r,l)Lemma

The most important theorem:

Theorem If (G, r, l) has linear latency functions, then ρ(G, r, l)≤4/3

Proof of 4/3 coordination ratio

cost of optimal at rate r

=

cost of optimal at rate r/2

cost of increasing from optimal at rate r/2 to optimal at rate r

+

optimal at r/2 C(f/2)≥1/4 • C(f)

At least (r/2) •L ≥1/2 • C(f)

Lemma

k

ii

**i

*

*

ii

**i

*

)r(fLδ)C(f

δ)r,l)(G,(

δf

ts

) (fLf

(G,r,l)Lemma

1

least at cost has 1 instance problem the

for flow feasible a ,0any for Then, . respect towith

path an on flow increasing ofcost marginal minimum

thebeLet flow. optimalan is for which functions

latencylinear with instancean is Suppose :

)(least at cost would to

from flow of units additional routingthen

in ingnondecreas is each if that note

**

*

fLrts

r

rL

iiii

i

ii

A simple example

x 1

s t

x1

0

1

2

31

2

111)( cost total

122

1flow

2

1 additionalby increaseCost

24))((*)(

*

2/12/1

*

fC

xxxxfLxxi

Proof of lemma

)()()()(

)(function cost theofconvexity *****

2

eeeeeeeeee

eee

flfffflffl

xbxaxlx

k

i PPPPi

k

i PPPPP

Eeeeee

Eeeee

Eeeee

i

i

fffLfC

ffflfC

flffffl

fflfC

1

****

1

****

*****

))(()(

))(()(

)()()(

)()(

Proof of ρ(G, r, l)≤4/3

)(4

3)(

)(4

1

4

1

2

1

4

1)2/(

)(2

1)2/(

2)()2/(

2)2/()2/()(

*

2

2

1

1

**

fCfC

fC

fbfa

fbfafC

fCfC

rfLfC

rfLfCfC

eeeee

eeeee

k

i

ii

k

i

ii

. respect to with oflatency theequals

2 respect to with path on flow the

increasing ofcost marginal the:Recall

fP

fP

Extensions

• Flows at Approximate Nash Equilibrium

• Finitely Many Agents: Splittable Flow

• Finitely Many Agents: Unsplittable Flow

• Central regulation.

Where are you?






Related Papers

• How bad is selfish routing -- Roughgarden & Tardos

• Worst-case Equilibrium -- Koutsoupias & Papadimitriou

• The Price of Selfish Routing -- Mavronicolas & Spirakis

• Realistic Models for Selfish Routing in the Internet -- Akella

KP model (task allocation model)

Main emphasizes on service costMain emphasizes on service cost(routing cost neglected)(routing cost neglected)

Cost = service costCost = service cost

mm servers servers

nn jobs jobs

KP model (cont)

Main emphasizes on service costMain emphasizes on service cost(routing cost neglected)(routing cost neglected)

Scheduling-type problemsScheduling-type problems::Schedule tasks to minimize the execution time (cost)Schedule tasks to minimize the execution time (cost)

Routing in a network consisting of parallel links onlyRouting in a network consisting of parallel links only

KP model (cont)

• simple routing model:– two nodes

–m parallel links with speeds si

– (1 · i · m)

– n jobs with weights wj

– (1 · j · n)

• service cost:– the delay of a connection is proportion

al to load on link

Cost measure

• After each job selects a link:Jobs(j) = jobs assigned to link jCost of jobs assigned to link j:

• Total weight of jobs assigned to link j over the speed of link j

• (Total) cost of a configurationmaxj {Cj}

• Social optimum (minimized cost)min maxj {Cj }

Results• Koutsoupias and Papadimitriou’99

– defined the problem– solved some of most basic cases

• for 2 identical links price of anarchy = 1.5• for 2 links price of anarchy is ¸ ¼ 1.618• for m identical links price of anarchy is

• for m links price of anarchy is

KP’s conjecture

• Koutsoupias-Papadimitriou conjecture – for m identical links– price of anarchy is

– “most natural behavior (random) is worst”– proved by Mavronicolas & Spirakis

Akella’s Model

• Selfish users choose routes that maximize the bandwidth available to the flow.

• Bandwidth available to agent i

• Objective function is total bandwidth used by all users

• The price of anarchy in a network with n flows ban be as large as Ω(n)

iagent of RTT is where,/1

/1i

ii

ie r

r

rb

Where are you?






Conclusions & Problems

• Selfish behaviors would degrade the performance of the network.

• However, some simulation results on Internet show that selfish routing is close to optimal routing. ???

• Other problem: route oscillation (Internet/ overlay network)

• Goal: how to design network or design games (what information should users know? ) so that selfish behavior would

lead to desired outcome?

Thank you for your attention!

The End

Documents

How Bad is Selfish Routing A survey on existing models for selfish routing Professor John Lui, David Yau and Dah-Ming Qiu presented by Joe W.J. Jiang 2004-05-18