Selfish Load Balancing

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Selfish Load Balancing

• Price of Anarchy (PoA) for four Different Load
  Balancing Games Variants. (Chapter 20)

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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)

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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

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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)
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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.

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Load Balancing Games

• Illustration of Proposition 20.3s proof

i
?
i
j
k
j
k
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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

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Summary of the Results
Identical Machines Uniformly Related Machines
Pure Equilibria
Mixed Equilibria
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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

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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

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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
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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

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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 is best response
  (satisfying i), i is never upset again due to
  other tasks 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.

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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

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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

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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.

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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

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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

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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
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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)?Lk1, then
  A(i) ?Lk. (prove by contradiction)
• Thus, weight assigned to machines in Lk1 under A
  is assigned to machines in Lk under A , thus

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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

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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 ? gt0
• Feldmann et. al. (2003) presented an efficient
  Nashification algorithm for any assignment,
  without increasing makespan.

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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

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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

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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