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Selfish Load Balancing

Selfish Load Balancing. Price of Anarchy (PoA) for four Different Load Balancing Games Variants. (Chapter 20). File Download from mirrored sites. The web. Selfish Load Balancing (Chapter 20). Given m machines with speeds s 1 , …, s m and n tasks with weights w 1 , …, w n

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Selfish Load Balancing

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  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

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