Download - Fast Convergence of Selfish Re-Routing
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
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rsm
links
Model: Users and Linksn
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job weights
Model: Users and Linksn
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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
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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
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Scheduler
ESS: Migrationn
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Scheduler
ESS: Migrationn
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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) ?