Fast Convergence of Selfish Re-Routing

1. Fast Convergence of Selfish Re-Routing Eyal Even-Dar,Tel-Aviv University Yishay Mansour,Tel-Aviv University

2. Overview • Routing on Parallel links • Model • Coordination Ratio • Migration • Distributed model • Convergence results • Few Types of Equilibrium: • termination, migration, overall.

3. 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.

4. Model: Users and Links m links n users

5. Model: Users and Links job weights m links n users

6. Model: Users and Links m links n users

7. 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

8. Nash Equilibrium m links n users

9. 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!

10. 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

11. 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

12. ESS: Migration m links n users Scheduler

13. ESS: Migration m links n users Scheduler

14. ESS: Migration m links n users Scheduler

15. 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

16. 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

17. Potential games and routing [EKM, ICALP 2003]

18. Example of Perfect Potential • Identical users and links • Potential: • User moving from link i to link j:

19. 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

20. 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)

21. 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

22. Two Links: Balance Policy • Assume n is even • Migration: • From Overloaded to Underloaded with p= d(t)/L1(t) • Expected load: E[Li(t+1)]=n/2 • Theorem: converges in expected O(loglog n) 2 d(t) L1(t) L2(t)

23. 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.

24. 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 200 400 200

25. Two Links: Nash ReRouting • Loads (n=2K): • L1 = K+d • L2 = K-d • Nash ReRouting: • Migration prob: • Diff. Exp. Loads! • Similar Convergence bound 2d L1 L2

26. 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!

27. 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.

28. 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) L1 L4 L2 L3

29. 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

30. Other results • Link Speeds: • Results and analysis carry over. • Weighted Users • lower bound: • Two links: (n) • Exponential weights • High Probability results.

31. Future work • Nash Computation • Nash ReRouting • many links • Sub-Game Perfect Eq. • Two Links • Weighted users: • Algorithms • Two links O(log Wmax) ?