Preface COMP558 Network Games Martin Gairing University of Liverpool, Computer Science Dept 2nd Semester 2013/14 COMP558 – Network Games 0.1 ·
Load balancing games Topic 2: Load balancing games Notation Computing pure Nash equilibria LPT Convergence of Best Responses Price of Anarchy COMP558 – Network Games 0.2 ·
Notation 2 Load Balancing Games Ch.20 Makespan scheduling on uniformly related machines ◮ n tasks with weights w 1 , . . . , w n ◮ m parallel machines with speeds s 1 , . . . , s m ◮ identical machines: s 1 = s 2 = · · · = s m = 1 ◮ related machines: else ◮ A : [ n ] �→ [ m ] .. assignment of tasks to machines ◮ Load of machine j ∈ [ m ] under assignment A : w i � ℓ j = s j i ∈ [ n ]: A ( i )= j ◮ Objective: minimize makespan, aka the maximum load over all machines COMP558 – Network Games Load Balacing Games 2.1 ·
Notation Load Balancing Games Load balancing games ◮ task i ∈ [ n ] is managed by player i ◮ pure strategy A ( i ) for each player i ∈ [ n ] yields assignment A : [ n ] �→ [ m ] ◮ Given assignment A ◮ cost of player i is load of chosen machine ℓ A ( i ) ◮ social cost: cost ( A ) = max j ∈ [ m ] { ℓ j } Pure Nash equilibrium Assignment A is a pure Nash equilibrium if for all player i ∈ [ n ] and all machines j ∈ [ m ] : ℓ A ( i ) ≤ ℓ j + w i s j For an assignment A , call a player satisfied if he cannot decrease his cost by unilaterally changing his strategy. COMP558 – Network Games Load Balacing Games 2.2 ·
Computing pure Nash equilibria Topic 2: Load balancing games Notation Computing pure Nash equilibria LPT Convergence of Best Responses Price of Anarchy COMP558 – Network Games Load Balacing Games 2.3 ·
Computing pure Nash equilibria LPT Existence of pure Nash equilibria Theorem 2.1 (Th. 20.10) The LPT algorithm computes a pure Nash equilibrium in polynomial time. LPT algorithm ◮ Start with empty assignment: ℓ j := 0 for all j ∈ [ m ] ◮ Sort task in non-increasing order w 1 ≥ w 2 ≥ · · · ≥ w n ◮ For i from 1 to n do ◮ A ( i ) := arg min j ∈ [ m ] { ℓ j + w i s j } w i ◮ ℓ A ( i ) := ℓ A ( i ) + s A ( i ) ◮ return A Corollary Every instance of the load balancing game admits a pure Nash equilibrium. COMP558 – Network Games Load Balacing Games 2.4 ·
Computing pure Nash equilibria Convergence of Best Responses Best response sequences 1 Improvement step: change to best response Single player moves his task to machine that minimizes his cost. Example (with identical machines): 1 2 1 5 3 3 1 3 2 2 1 2 6 6 6 6 5 5 5 5 5 5 5 3 11 6 5 6 6 10 9 6 7 8 7 7 http://www.csc.liv.ac.uk/˜gairing/BRapplet/applet.html COMP558 – Network Games Load Balacing Games 2.5 ·
Computing pure Nash equilibria Convergence of Best Responses Best response sequences 2 Why bother? ◮ LPT computes a pure NE. However players have to trust some central authority to run the algorithm. ◮ Best response sequences ◮ take the strategic nature of the players into account ◮ model convergence Theorem 2.2 (Prop. 20.3) For every instance of the load balancing game (with related machines) every best response sequence terminates. Remark There are identical machine instances that have sequences of length √ n ) . Ω( 2 [ E VEN D AR ET AL ., 2003] [ F ELDMANN ET AL ., 2003a] COMP558 – Network Games Load Balacing Games 2.6 ·
Computing pure Nash equilibria Convergence of Best Responses Best response sequences 3 Theorem 2.4 For identical machines the length of any sequence of best responses is at most 2 n . Theorem 2.5 Th. 20.6 Let A : [ n ] �→ [ m ] denote any assignment of n tasks to m identical machines. Starting from A , the max-weight best response policy reaches a pure Nash equilibrium after each agent was activated at most once. Both theorems follow more or less directly from Lemma 2.6 Suppose task i of weight w i makes a best response. Then for all tasks j with w j ≥ w i , either j is satisfied after the best response of i or j was already unsatisfied before. COMP558 – Network Games Load Balacing Games 2.7 ·
Computing pure Nash equilibria Convergence of Best Responses Best response sequences 4 Some remarks ◮ On identical machines the max-weight best response transforms any given assignment A into a pure Nash equilibrium A ′ in time O ( n log n ) . ◮ Best responses do not increase social cost. ◮ So cost ( A ′ ) ≤ cost ( A ) . ◮ Algorithms having this property are called Nashification algorithms. ◮ For related machines there is also a Nashification algorithm with running time O ( m 2 n ) . [ F ELDMANN ET AL ., 2003a] ◮ However, this algorithm is not only based on best responses. ◮ Reason: Lemma 2.6 does not hold for related machines. COMP558 – Network Games Load Balacing Games 2.8 ·
Computing pure Nash equilibria Convergence of Best Responses Nashification + approximation algorithms Remark Combining Nashification algorithms with any approximation algorithm yields an algorithm to compute a pure Nash equilibrium with same performance guarantee in polynomial time. Approximation ratios of scheduling algorithms Algorithm identical machines related machines 2 − 1 List scheduling m [Gra66] 4 1 1 . 52 ≤ [Fri87] ≤ 5 LPT 3 − 3 m [Gra69] 3 13 Multifit 11 ≤ [Fri84] ≤ 1 . 2 1 . 341 ≤ [FL83] ≤ 1 . 4 PTAS 1 + ε [HS87] 1 + ε [HS88] COMP558 – Network Games Load Balacing Games 2.9 ·
Computing pure Nash equilibria Convergence of Best Responses Literature for table on previous slide FL83 D.K. Friesen and M.A. Langston. Bounds for multifit scheduling on uniform processors. SIAM Journal on Computing, 12(1):6070, 1983. Fri84 D.K. Friesen. Tighter bounds for the multifit processor scheduling algorithm. SIAM Journal on Computing, 13(1):170181, 1984. Fri87 D.K. Friesen. Tighter bounds for lpt scheduling on uniform processors. SIAM Journal on Computing, 16(3):554560, 1987. Gra66 R.L. Graham. Bounds for certain multiprocessing anomalies. Bell System Tech. J., 45(1):15631581, 1966. Gra69 R.L. Graham. Bounds on multiprocessing timing anomalies. SIAM Journal of Applied Mathematics, 17(2):416429, 1969. HS87 D.S. Hochbaum and D. Shmoys. Using dual approximation algorithms for scheduling problems: Theoretical and practical results. Journal of the ACM, 34(1):144162, 1987. HS88 D.S. Hochbaum and D. Shmoys. A polynomial approximation scheme for scheduling on uniform processors: using the dual approximation approach. SIAM Journal on Computing, 17(3):539551, 1988. COMP558 – Network Games Load Balacing Games 2.10 ·
Price of Anarchy Topic 2: Load balancing games Notation Computing pure Nash equilibria LPT Convergence of Best Responses Price of Anarchy COMP558 – Network Games Load Balacing Games 2.11 ·
Price of Anarchy Price of Anarchy: Definition Price of anarchy The worst case ratio between the social cost in some NE and the optimum social cost. Formally: ◮ G ( m ) .. set of all instances with m machines ◮ For G ∈ G ( m ) let ◮ Nash ( G ) .. the set of all NE for G (pure or mixed) ◮ opt ( G ) .. minimum social cost over all assignments Definition: Price of Anachy cost ( P ) PoA ( G ) = max opt ( G ) P ∈ Nash ( G ) PoA ( m ) = max G ∈G ( m ) PoA ( G ) COMP558 – Network Games Load Balacing Games 2.12 ·
Price of Anarchy Example Load balancing game G Optimum (a) & worst pure NE (b) ◮ 2 identical machines: (a) (b) ◮ s 1 = s 2 = 1 ◮ 4 tasks ◮ w 1 = w 2 = 2 ◮ w 3 = w 4 = 1 Mixed NE P ◮ Each task i ∈ [ 4 ] chooses PoA ( G ) each machine j ∈ [ 2 ] with ◮ (pure) PoA ( G ) = 4 probability p j i = 1 2 . 3 ◮ (mixed) PoA ( G ) ≥ cost ( P ) ◮ cost ( P ) = E [ cost ( A )] = ??? 3 COMP558 – Network Games Load Balacing Games 2.13 ·
Price of Anarchy Bounds on the Price of Anarchy Theorem 2.7: Tight bounds on the price of anarchy pure NE mixed NE � � log m 2 identical machines 2 − (a) Θ m + 1 log log m � � � � log m log m related machines Θ (b) Θ log log m log log log m ◮ For m = 2, the example on previous slide proves the lower bound in (a). ◮ Exercise: Generalise this example to match the bound for all m . COMP558 – Network Games Load Balacing Games 2.14 ·
Price of Anarchy Preliminaries for Theorem 2.7 (b) ◮ Γ .. gamma function ◮ extension of factorial function ◮ Γ( k ) = ( k − 1 )! for every k ∈ N ◮ Γ − 1 .. inverse gamma function Well known fact about Γ − 1 � log k � Γ − 1 ( k ) = Θ log log k COMP558 – Network Games Load Balacing Games 2.15 ·
Price of Anarchy Upper Bound Theorem 2.7 (b) ◮ Consider G with ◮ s 1 ≥ · · · ≥ s m ◮ w.l.o.g. assume opt ( G ) = 1 ◮ A : [ n ] �→ [ m ] is NE ◮ Denote c = ⌊ cost ( A ) opt ( G ) ⌋ = ⌊ cost ( A ) ⌋ ◮ L = [ 1 , . . . , m ] .. list of machines ◮ L k .. max prefix of L such that ℓ j ≥ k for all j ∈ L k c We show the recurrence c − 1 ◮ | L k | ≥ ( k + 1 ) ·| L k + 1 | for 0 ≤ k ≤ c − 2 c − 2 ◮ | L c − 1 | ≥ 1 c − 3 Solving the recurrence yields L c − 1 ◮ m = | L 0 |≥ ( c − 1 )! = Γ( c ) L c − 2 L c − 3 ◮ And thus c ≤ Γ − 1 ( m ) . COMP558 – Network Games Load Balacing Games 2.16 ·
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