Fast Convergence of

Selfish Re-Routing

Eyal Even-Dar, Tel-Aviv UniversityYishay Mansour, Tel-Aviv University

Overview

• Routing on Parallel links– Model– Coordination Ratio– Migration

• Distributed model– Convergence results

• Few Types of Equilibrium:– termination, migration, overall.

Routing on parallel links

• Job scheduling• Classic setting:

– Centralize control– Optimize a global objective function

• minimize MAX load

– Full cooperation

• Game theory setting:– Each user optimizes its objective function

• Load of the machine it selects.

Model: Users and Linksn

use

rsm

links

Model: Users and Linksn

use

rsm

links

job weights

Model: Users and Linksn

use

rsm

links

Model: Links and Users

• Routing:– m links & n users

• Link Model: – Link Mi has speed Si

• User Model:– Weighted: User U has a weight w (U)

– Unrelated: user U has a weight wk(U) on Mk

• Load on link Mi at time t:

– Bi(t) = Users routing on Mi at time t

– Li(t) = [Σj in Bi(t) wi (j) ] / Si

Nash Equilibriumn

use

rsm

links

Model: Nash Equilibrium

• No user can move and lower its load.

• For a user U on link Mi

– For any link Mj

– If U moves to link Mj

– Then Li Lj + wj(U)/Sj

• The load after any move is not lower than before!

Coordination Ratio

• A global optimization function– minimize MAX load

• Coordination Ratio Compares:– Optimal value– Worse Nash Value

• Results for job scheduling [KP,MS,CV,AAR]– Identical: 2 or O(log n / log log n)– Related: O(log n / log log n)– Unrelated: unbounded

Convergence to Nash

• How (fast) users reach the Nash Eq.• Main concern:

– Duration• Non-issue:

– Quality of Nash Eq.• Migration models

– Elementary Step Size (ESS)– Distributed

ESS: Migrationn

use

rsm

links

Scheduler

ESS: Migrationn

use

rsm

links

Scheduler

ESS: Migrationn

use

rsm

links

Scheduler

ESS Migration model [ORS]

• Introduced to study routing• User’s aim: minimize its observed load• Elementary step system:

– Only one user moves at a time.– Scheduler:

• arbitrary; • Specific: random; FIFO; Max Weight; Max Load

– User’s move• improvement/best reply

Potential Games [M+S]• Global Potential function• Relates:

– user utility change – global potential change

• Potential functions:– Perfect/Weighted/Ordinal

• Deterministic Nash Eq.• Equivalent to congestion games.

– Exponential reduction

Potential games and routing [EKM, ICALP

2003]Potential type

Users Links

perfect identical identical

weighted weighted related

ordinal unrelated unrelated

Example of Perfect Potential

• Identical users and links• Potential:

• User moving from link i to link j:

m

ii tLtP

1

2 )(2

1)(

)1(

)()1()1()()1( 2222

ij

ijij

LL

LLLLtPtP

Other results [EKM]

• Identical links:– Max weight user scheduler

• No user moves twice.

– Min weight user scheduler• Exponential lower bound

• Related & Unrelated links:– Various schedulers

This work: Distributed model

• Concurrent migration– Randomized policies– no scheduler

• Major difference:– User might be worse off after

migration

• Convergence time– Identical users: O(log log n)

Distributed Model

• Users: – Identical and Anonymous

• Termination Nash Equilibrium: – Balanced load on links

• Policy– Sets a prob. for migration between

links. • Convergence time

– Number of steps until Termination

Two Links: Balance Policy

• Assume n is even• Migration:

– From Overloadedto Underloaded with p= d(t)/L1(t)

• Expected load:E[Li(t+1)]=n/2

• Theorem: converges in expected O(loglog n)

L1(

t)

L2(

t)2

d(t)

Two Links: Balance Policy

Sketch of Proof:• Two phases:

– Switch phases when d(t) 3 ln 1/• First phase:

– simple Chernoff bound– Completes after O(loglog n ) steps

• Second phase:– Each step terminates with prob.

• Setting =1/T.

/1ln)(3)1( tdtd

1/ln 3/1

Two Links: Nash ReRouting

• Balance: p=1/4• Single user:

– Load on 1: 300 – ¾– Load on 2: 300 – ¼– Best response: STAY!

• Nash ReRouting:– Every migration step

is Nash Equilibrium– Myopic users

400

200

200

Two Links: Nash ReRouting

• Loads (n=2K):– L1 = K+d– L2 = K-d

• Nash ReRouting:

• Migration prob:

• Diff. Exp. Loads!• Similar Convergence bound

L1

L2

2d

12

1

1

L

dp

)-p(LL-p))(- (L 111 121

Two Links: Sub-game Perfect

• Cost accumulate – discounted over time

• User optimizes its discounted return.• Existence: Similar to Stochastic games• Convergence:

– Number of steps O(log log n)– Constants depend on the discount factor!

Two Links: Sub-game Perfect

• Proof ideas:– Let A=1/(1-)– Can “guarantee” 0.5 from any state.

– Bound the value of a state |vd| < 0.5 A

– Migration prob. pd= d/(n+d) +/- O(A/n)

– Low probabilities:• Can not be too small O(1/An)

– Termination in one step in low prob.

Multiple Links: Balance policy

• Loads (n=mK):– Li = K+di

– Over = {i:di > 0}– d = i in OVER di

• Migration prob:– Migrate: di/Li

– Destination: |dk|/d

• Exp. Load: E[Li]=K• Theorem:Õ(loglog n + log m)

L2

L3

L4L

1

Multiple Links: Nash ReRouting

• Always exists:– Similar to Symmetric Players

• Computation:– Independent of n (num. of users)– Exponential in m (num. links)– Algorithm:

• For each link guess support.• Linear set of Eq.

• Convergence: Similar to Balance

Other results

• Link Speeds:– Results and analysis carry over.

• Weighted Users– lower bound:– Two links: (n)

• Exponential weights

• High Probability results.

Future work

• Nash Computation– Nash ReRouting

• many links

– Sub-Game Perfect Eq.• Two Links

• Weighted users: – Algorithms

• Two links O(log Wmax) ?