Selfish Load Balancing Price of Anarchy (PoA) for four Different Load Balancing Games Variants. (Chapter 20) 4/4/2019 Shenoda Guirguis - CS3510 Spring 08

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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) 4/4/2019 Shenoda Guirguis - CS3510 Spring 08

Shenoda Guirguis - CS3510 Spring 08 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 Algorithms for computing Pure Equilibria Mixed Equilibria for Uniformly Related Machines 4/4/2019 Shenoda Guirguis - CS3510 Spring 08

Shenoda Guirguis - CS3510 Spring 08 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) 4/4/2019 Shenoda Guirguis - CS3510 Spring 08

Shenoda Guirguis - CS3510 Spring 08 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. 4/4/2019 Shenoda Guirguis - CS3510 Spring 08

Shenoda Guirguis - CS3510 Spring 08 Load Balancing Games Illustration of Proposition 20.3’s proof: i  i j k j k 4/4/2019 Shenoda Guirguis - CS3510 Spring 08

Shenoda Guirguis - CS3510 Spring 08 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 Algorithms for computing Pure Equilibria Mixed Equilibria for Uniformly Related Machines 4/4/2019 Shenoda Guirguis - CS3510 Spring 08

Shenoda Guirguis - CS3510 Spring 08 Summary of the Results Identical Machines Uniformly Related Machines Pure Equilibria Mixed 4/4/2019 Shenoda Guirguis - CS3510 Spring 08

Shenoda Guirguis - CS3510 Spring 08 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 Algorithms for computing Pure Equilibria Mixed Equilibria for Uniformly Related Machines 4/4/2019 Shenoda Guirguis - CS3510 Spring 08

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: 4/4/2019 Shenoda Guirguis - CS3510 Spring 08

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 4/4/2019 Shenoda Guirguis - CS3510 Spring 08

Shenoda Guirguis - CS3510 Spring 08 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 Algorithms for computing Pure Equilibria Mixed Equilibria for Uniformly Related Machines 4/4/2019 Shenoda Guirguis - CS3510 Spring 08

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. 4/4/2019 Shenoda Guirguis - CS3510 Spring 08

Shenoda Guirguis - CS3510 Spring 08 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 Algorithms for computing Pure Equilibria Mixed Equilibria for Uniformly Related Machines 4/4/2019 Shenoda Guirguis - CS3510 Spring 08

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 4/4/2019 Shenoda Guirguis - CS3510 Spring 08

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. 4/4/2019 Shenoda Guirguis - CS3510 Spring 08

Shenoda Guirguis - CS3510 Spring 08 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 Algorithms for computing Pure Equilibria Mixed Equilibria for Uniformly Related Machines 4/4/2019 Shenoda Guirguis - CS3510 Spring 08

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 4/4/2019 Shenoda Guirguis - CS3510 Spring 08

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 4/4/2019 Shenoda Guirguis - CS3510 Spring 08

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: 4/4/2019 Shenoda Guirguis - CS3510 Spring 08

Shenoda Guirguis - CS3510 Spring 08 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 Algorithms for computing Pure Equilibria Mixed Equilibria for Uniformly Related Machines 4/4/2019 Shenoda Guirguis - CS3510 Spring 08

Shenoda Guirguis - CS3510 Spring 08 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. 4/4/2019 Shenoda Guirguis - CS3510 Spring 08

Shenoda Guirguis - CS3510 Spring 08 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 Algorithms for computing Pure Equilibria Mixed Equilibria for Uniformly Related Machines 4/4/2019 Shenoda Guirguis - CS3510 Spring 08

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; 4/4/2019 Shenoda Guirguis - CS3510 Spring 08

Shenoda Guirguis - CS3510 Spring 08 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 Algorithms for computing Pure Equilibria Mixed Equilibria for Uniformly Related Machines 4/4/2019 Shenoda Guirguis - CS3510 Spring 08