COMP558 Network Games Martin Gairing University of Liverpool, - - PowerPoint PPT Presentation

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 w1, . . . , wn ◮ m parallel machines with speeds s1, . . . , sm

◮ identical machines: s1 = s2 = · · · = sm = 1 ◮ related machines: else

◮ A : [n] → [m] .. assignment of tasks to machines ◮ Load of machine j ∈ [m] under assignment A:

ℓj =

• i∈[n]:A(i)=j

wi sj

◮ 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 ℓA(i) ◮ social cost:

cost(A) = maxj∈[m]{ℓj}

Pure Nash equilibrium Assignment A is a pure Nash equilibrium if for all player i ∈ [n] and all machines j ∈ [m]: ℓA(i) ≤ ℓj + wi sj 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: ℓj := 0 for all j ∈ [m] ◮ Sort task in non-increasing order w1 ≥ w2 ≥ · · · ≥ wn ◮ For i from 1 to n do

◮ A(i) := arg minj∈[m]{ℓj + wi

sj }

◮ ℓA(i) := ℓA(i) +

wi sA(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):

8 7 7 1 5 2 1 6 5 5 3 2 6 6 10 9 6 7 6 6 1 5 5 3 3 5 6 5 2 1 3 2 5 6 5 11

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 Ω(2

√n). [ EVENDAR ET AL., 2003] [ FELDMANN 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 2n. 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 wi makes a best response. Then for all tasks j with wj ≥ wi, 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(m2n).

[ FELDMANN 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 List scheduling 2 − 1

m [Gra66]

LPT

4 3 − 1 3m [Gra69]

1.52 ≤[Fri87] ≤ 5

3

Multifit

13 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

• n 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

• ptimum 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 PoA(G) = max

P∈Nash(G)

cost(P)

• pt(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

◮ 2 identical machines:

◮ s1 = s2 = 1

◮ 4 tasks

◮ w1 = w2 = 2 ◮ w3 = w4 = 1

PoA(G)

◮ (pure) PoA(G) = 4 3 ◮ (mixed) PoA(G) ≥ cost(P) 3

Optimum (a) & worst pure NE (b)

(a) (b)

Mixed NE P

◮ Each task i ∈ [4] chooses

each machine j ∈ [2] with probability pj

i = 1 2. ◮ cost(P) = E[cost(A)] = ???

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 identical machines 2 −

2 m+1

(a) Θ

• log m

log log m

• related machines

Θ

• log m

log log m

• (b)

Θ

• 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 Γ−1(k) = Θ

• log k

log log k

• COMP558 – Network Games

Load Balacing Games · 2.15

Price of Anarchy

Upper Bound Theorem 2.7 (b)

◮ Consider G with

◮ s1 ≥ · · · ≥ sm ◮ w.l.o.g. assume opt(G) = 1 ◮ A : [n] → [m] is NE ◮ Denote c = ⌊ cost(A)

• pt(G) ⌋ = ⌊cost(A)⌋

◮ L = [1, . . . , m] .. list of machines ◮ Lk .. max prefix of L such that ℓj ≥ k for all j ∈ Lk

c−1 c−2 c−3 c Lc−1 Lc−2 Lc−3

We show the recurrence

◮ |Lk| ≥ (k +1)·|Lk+1| for 0 ≤ k ≤ c −2 ◮ |Lc−1| ≥ 1

Solving the recurrence yields

◮ m = |L0|≥(c − 1)! = Γ(c) ◮ And thus c ≤ Γ−1(m).

COMP558 – Network Games Load Balacing Games · 2.16

Price of Anarchy

Lower Bound Theorem 2.7 (b)

4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 2 2 2 2 2 2 2 2 1 1 1 1 8 8 8 8 8 8 8 8 4 4 2 2 1 1 4 3 2 1 3 1 3 * 6 3 * 6 * 4 3 * 6 * 4 * 2

Construction for PoA = 4.

COMP558 – Network Games Load Balacing Games · 2.17

Price of Anarchy

Lower Bound Theorem 2.7 (b)

General construction:

f(1) f(2) f(k−1) f(k)

k−2 k−2

1 1 1 2 2 k−2 k−1 k 1 1 1 2 2 2 2

k−1 k−1 k−2

2 2 2 2 2

k−1 k−1 k−1

2 2 2 2

k−3 k−3

1 2 2

k−2 k−2 k−2 k−3 k−3

For 1 ≤ i ≤ k:

◮ f(i) tasks of weight 2i−1 ◮ f(i) machines with speed 2i−1

Recursive definition of f:

◮ f(k) = k ◮ f(k − 1) = 2(k − 1)2 ◮ f(i) = 2 · i · f(i + 1)

COMP558 – Network Games Load Balacing Games · 2.18