Selfish Load Balancing

1. Selfish Load Balancing Price of Anarchy (PoA) for four Different Load Balancing Games Variants. (Chapter 20) Shenoda Guirguis - CS3510 Spring 08

2. File Download from mirrored sites The web Shenoda Guirguis - CS3510 Spring 08

3. Selfish Load Balancing (Chapter 20) • Given m machines with speeds s1, …, sm and n tasks with weights w1, …, wn • Let [n] = {1, …, n} denote the set of tasks and [m] = {1, …, m} the set of machines. • One seeks for an assignment A: [n]  [m] of the tasks to the machines that is as balanced as possible. The load of machine j [m] under assignment A is defined as • The objective is to minimize the makespan (i.e. max. load over all machines) Shenoda Guirguis - CS3510 Spring 08

4. Agenda • Problem Definition • Load Balancing Games • Summary of the Results • Pure Equilibria for Identical Machines • Proof of tight bound • Convergence • Mixed Equilibria for Identical Machines • Pure Equilibria for Uniformly Related Machines • Proof of tight bound • Algorithms for computing Pure Equilibria • Mixed Equilibria for Uniformly Related Machines Shenoda Guirguis - CS3510 Spring 08

5. Load Balancing Games • Cost of agent i • Social cost of assignment A • Nash Equilibrium • Pure strategies • Load & max load • Mixed strategies (strategy profile) • Expected load, and expected maximum load i Cost(i) = Lj j i Cost(A) Shenoda Guirguis - CS3510 Spring 08

6. Load Balancing Games • Proposition 20.3: Every instance of the load balancing game admits at least one pure Nash equilibrim • Proof: • An assignment A induces a sorted load vector ( ) • If A is not Nash, then there exist an improvement step • Each improvement step results in a sorted load vetor that is lexicographically smaller • Hence a pure Nash equilibrium is reached after a finite number of improvement steps. Shenoda Guirguis - CS3510 Spring 08

7. Load Balancing Games • Illustration of Proposition 20.3’s proof: i  i j k j k Shenoda Guirguis - CS3510 Spring 08

8. Agenda • Problem Definition • Load Balancing Games • Summary of the Results • Pure Equilibria for Identical Machines • Proof of tight bound • Convergence • Mixed Equilibria for Identical Machines • Pure Equilibria for Uniformly Related Machines • Proof of tight bound • Algorithms for computing Pure Equilibria • Mixed Equilibria for Uniformly Related Machines Shenoda Guirguis - CS3510 Spring 08

9. Summary of the Results Shenoda Guirguis - CS3510 Spring 08

10. Agenda • Problem Definition • Load Balancing Games • Summary of the Results • Pure Equilibria for Identical Machines • Proof of tight bound • Convergence • Mixed Equilibria for Identical Machines • Pure Equilibria for Uniformly Related Machines • Proof of tight bound • Algorithms for computing Pure Equilibria • Mixed Equilibria for Uniformly Related Machines Shenoda Guirguis - CS3510 Spring 08

11. Pure Equilibria for Identical Machines: Proof of tight bound • Theorem 20.5: cost(A)  opt(G) • Proof: • j*highest load machine under A (a Nash)  Cost(A) = • i*smallest job on j* • There are at least 2 jobs assigned to j*(o.w. A is OPT) Theorem • Thus i*  0.5 Cost(A) • Machine j, if , then i*moves. • But A is Nash  • Since opt(G) can not be smaller than the average load: Shenoda Guirguis - CS3510 Spring 08

12. Pure Equilibria for Identical Machines: Proof of tight bound • A lower bound instance • Exercise 20.2 generalizes this example for every m, thus the bound is tight 1 2 1 2 Worst Nash; Cost(A) = 4 Opt; Cost(Opt) = 3 PoA = 4 / 3 = 2 – 2/3 Shenoda Guirguis - CS3510 Spring 08

13. Agenda • Problem Definition • Load Balancing Games • Summary of the Results • Pure Equilibria for Identical Machines • Proof of tight bound • Convergence • Mixed Equilibria for Identical Machines • Pure Equilibria for Uniformly Related Machines • Proof of tight bound • Algorithms for computing Pure Equilibria • Mixed Equilibria for Uniformly Related Machines Shenoda Guirguis - CS3510 Spring 08

14. Pure Equilibria for Identical Machines: Convergence • Theorem 20.6: Let A be any assignment of n tasks to m identical machines. Starting from A, the max-weight best response policy reaches a pure Nash after each agent was activated at most once • Proof: • Show that after task i’s best response (satisfying i), i is never upset again due to other task’s improvement step. • Note that task i is satisfied iff if its task is place d on machine with minimum load due to other tasks, and • note that a best response never decreases the minimum load among the machines. Shenoda Guirguis - CS3510 Spring 08

15. Agenda • Problem Definition • Load Balancing Games • Summary of the Results • Pure Equilibria for Identical Machines • Proof of tight bound • Convergence • Mixed Equilibria for Identical Machines • Pure Equilibria for Uniformly Related Machines • Proof of tight bound • Algorithms for computing Pure Equilibria • Mixed Equilibria for Uniformly Related Machines Shenoda Guirguis - CS3510 Spring 08

16. Mixed Equilibria for Identical Machines • Fully Mixed Equilibria • P is the only mixed profile, i.e the only Nash • Theorem 20.12: • The proof uses a mapping of the Fully Mixed Nash Equilibrium to that of placing n balls in m bins Shenoda Guirguis - CS3510 Spring 08

17. Mixed Equilibria for Identical Machines • Theorem 20.13: Given an instance G, Let P = (pij),i[n], j [m] denote any Nash equilibrium strategy profile. Then, it holds that • Proof: • Cost(P) = expected makespan = maximum load • We can trivially generalize Pure Nash results to get maximum expected load. • Utilize weighted Chernoff bound to show that no machine can deviate from its expectation by more than a linear factor, the theorem results directly. Shenoda Guirguis - CS3510 Spring 08

18. Agenda • Problem Definition • Load Balancing Games • Summary of the Results • Pure Equilibria for Identical Machines • Proof of tight bound • Convergence • Mixed Equilibria for Identical Machines • Pure Equilibria for Uniformly Related Machines • Proof of tight bound • Algorithms for computing Pure Equilibria • Mixed Equilibria for Uniformly Related Machines Shenoda Guirguis - CS3510 Spring 08

19. Pure Equilibria for Uniformly Related Machines: Proof of tight bound • Theorem 20.7: given an instance G; n tasks, and m machines with speeds s1, … sn Let A be any Nash equilibrium assignment, Then it holds that • Proof: • Define , then • We show that cost(A) / opt(G)  • Assume s1 s2 …  sn Shenoda Guirguis - CS3510 Spring 08

20. Pure Equilibria for Uniformly Related Machines: Proof of tight bound • Let • Define Lk for k  {0, …, c-1} • Show for 0  k  c -2 & • Solving this recurrence yields c-1. opt(G) c-2. opt(G) c-3. opt(G) Lc-1 Lc-2 Lc-3 Shenoda Guirguis - CS3510 Spring 08

21. Pure Equilibria for Uniformly Related Machines: Proof of tight bound Proof of recurrence: • Assume then Lc-1 is empty under Nash Equ. A, then the load of machine 1 is less than (c-1). opt(G) The makespan machine j has load c. opt(G), then moving one task i to machine 1 decreases cost of i to strictly less than (since ) which contradicts that A is Nash. • Now, let A*be optimal assignment. • Lemma 20.8: for any task i, if A(i)Lk+1, then A*(i) Lk. (prove by contradiction) • Thus, weight assigned to machines in Lk+1 under A is assigned to machines in Lk under A* , thus: Shenoda Guirguis - CS3510 Spring 08

22. Agenda • Problem Definition • Load Balancing Games • Summary of the Results • Pure Equilibria for Identical Machines • Proof of tight bound • Convergence • Mixed Equilibria for Identical Machines • Pure Equilibria for Uniformly Related Machines • Proof of tight bound • Algorithms for computing Pure Equilibria • Mixed Equilibria for Uniformly Related Machines Shenoda Guirguis - CS3510 Spring 08

23. Pure Equilibria for Uniformly Related Machines: Algorithms for computing Pure Equilibria • The LPT (Largest Processing time) scheduling algorithm computes a pure Nash equilibrium for load balancing games on uniformly related machines (Theorem 20.10) • Hochbaum and Shomoys (1988) proposed a polynomial time approximation scheme with ratio of (1+ ) for any given  >0 • Feldmann et. al. (2003) presented an efficient Nashification algorithm for any assignment, without increasing makespan. Shenoda Guirguis - CS3510 Spring 08

24. Agenda • Problem Definition • Load Balancing Games • Summary of the Results • Pure Equilibria for Identical Machines • Proof of tight bound • Convergence • Mixed Equilibria for Identical Machines • Pure Equilibria for Uniformly Related Machines • Proof of tight bound • Algorithms for computing Pure Equilibria • Mixed Equilibria for Uniformly Related Machines Shenoda Guirguis - CS3510 Spring 08

25. Mixed Equilibria for Uniformly Related Machines • Using same approach as in case of Mixed Equilibria for identical machine, one can show first the maximum expected makespan to be • Then using Chernoff bound to show that expected maximum load for each job is not much larger • Only a factor of is lost in the last step. • Then the results follows directly; Shenoda Guirguis - CS3510 Spring 08

26. Agenda • Problem Definition • Load Balancing Games • Summary of the Results • Pure Equilibria for Identical Machines • Proof of tight bound • Convergence • Mixed Equilibria for Identical Machines • Pure Equilibria for Uniformly Related Machines • Proof of tight bound • Algorithms for computing Pure Equilibria • Mixed Equilibria for Uniformly Related Machines Shenoda Guirguis - CS3510 Spring 08