Scheduling Games in the Dark Nguyen Kim Thang (joint work with Christoph Durr)

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Scheduling Games in the Dark

Nguyen Kim Thang

(joint work with Christoph Durr)

Aarhus

March 9th, 09

Tuesday, November 16, 2010

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Outline

Scheduling Games

Definition & Motivation Summary of results

Existence of pure Nash equilibrium

Inefficiency of equilibria

Potential argument

Price of anarchy

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Scheduling Games

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Scheduling Games

jobs (players) and machines: a job chooses a machine to execute. The processing time of job on machine is

n m

i j p_ij

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Scheduling Games

n m

i j p_ij

Such a job-machine assignment is a strategy profile. σ The load of machine is _j !_j = !

i:σ(i)=j

p_ij

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Scheduling Games

Each machine specifies a policy how jobs assigned to the machine are to be scheduled (e.g., SPT, LPT, ...).

n m

i j p_ij

The cost of a job is its completion time. c_i i

Such a job-machine assignment is a strategy profile. σ The load of machine is _j !_j = !

i:σ(i)=j

p_ij

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Coordination Mechanism

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Coordination Mechanism

Coordination mechanism aim to design policies such that the self-interested behaviors lead to equilibria with small PoA.

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Coordination Mechanism

Communication is hard or costly (large-scale autonomous systems in the Internet).

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Coordination Mechanism

Policies are designed based on local information.

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Coordination Mechanism

Strongly local policy: a machine looks only at proc. time of jobs assigned to the machine.

σ(i) = j p_ij

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Coordination Mechanism

Strongly local policy: a machine looks only at proc. time of jobs assigned to the machine.

σ(i) = j p_ij

Local policy: depends only on the parameters

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Policies

Strongly local policies:

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Policies

Shortest Processing Time First (SPT)

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Policies

Shortest Processing Time First (SPT) machine 1

machine 2 machine 3

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Policies

machine 2 machine 3

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Policies

machine 2 machine 3

Longest Processing Time First (LPT)

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Policies

machine 2 machine 3

Longest Processing Time First (LPT) MAKESPAN: if

=

σ(i) = j

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Policies

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Policies

Local policies:

Inefficiency-based policy: greedily schedule jobs in increasing order of where ρ_i = p_ij/q_i

q_i = min

j^! p_ij^!

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Policies

Local policies:

Inefficiency-based policy: greedily schedule jobs in increasing order of where ρ_i = p_ij/q_i

q_i = min

j^! p_ij^!

Typically, a policy depends on the processing time of jobs assigned to the machine.

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Non-clairvoyant policies

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Non-clairvoyant policies

What about policies that do not require this knowledge?

Incomplete information games Private information of jobs

Jobs can influence on their completion time by misreporting their processing time

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Non-clairvoyant policies

What about policies that do not require this knowledge?

Incomplete information games Private information of jobs

Jobs can influence on their completion time by misreporting their processing time

Non-clairvoyant policies existence

Nash equilibrium small PoA

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Natural policies

RANDOM: schedules jobs in a random order.

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Natural policies

c_i = p_ij + 1 2

!

i^!:σ(i^!)=j,i^!!=i

p_i^!_j

σ i j

In the strategy profile , is assigned to

= 1

2 (p_ij + !_j)

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Natural policies

A job on machine has an incentive to move to machine iff: ⁱ _j^! ^j

p_ij + !_j > 2p_ij^! + !_j^! c_i = p_ij + 1

2

!

i^!:σ(i^!)=j,i^!!=i

p_i^!_j

σ i j

In the strategy profile , is assigned to

= 1

2 (p_ij + !_j)

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Natural policies

EQUI: schedules jobs in parallel, assigning each job an equal fraction of the processor.

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Natural policies

A B C

C D

p_A = 1 p_B = 1 p_C = 2

p_D = 3

0 4 6 7

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Natural policies

c_i = p_1j + . . . + p_i₋_1,j + (k − i + 1)p_ij

If there are jobs on machine s.t: k j ^p1j ≤ . . . ≤ p_kj

A B C

C D

p_A = 1 p_B = 1 p_C = 2

p_D = 3

0 4 6 7

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Natural policies

c_i = p_1j + . . . + p_i₋_1,j + (k − i + 1)p_ij

A B C

C D

p_A = 1 p_B = 1 p_C = 2

p_D = 3

0 4 6 7

p_1j ≤ . . . ≤ p_ij ≤ . . . ≤ p_kj

sum number

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Models

Def of models:

Def: A job is balanced if i max p_ij/ min p_ij ≤ 2

Identical machines: for some length Uniform machines: for some speed Unrelated machines: arbitrary

p_ij = p_i ∀j p_ij = p_i/s_j

p_ij

s_j p_i

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Existence of equilibrium

The game under EQUI policy is a potential game.

Theorem:

The game under RANDOM policy is a potential game for 2 unrelated machines with balanced jobs but it is not for more than 3 machines. For uniform machines,

balanced jobs, there always exists equilibrium.

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Inefficiency

Theorem: For unrelated machines, the PoA of policy EQUI is at most 2m – interestingly, that matches the best clairvoyant policy.

PoA is not increased when processing times are unknown to the

machines.

the worst NE

PoA

OPT

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Existence of equilibrium

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Standard definitions

Def: A job is unhappy if it can decrease its cost by

changing the strategy (other players’ strategies are fixed) Def: a best response (best move) of a job is the

strategy which minimizes the cost of the job (while other players’ strategies are fixed)

Def: Best-response dynamic is a process that let an arbitrary unhappy job make a best response.

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RANDOM, uniform machines

Theorem:

The game under RANDOM policy is a potential game for uniform machines with balanced jobs

(balanced speeds).

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RANDOM, uniform machines

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RANDOM, uniform machines

Jobs have length

Machines have speed

p₁ ≤ p₂ ≤ . . . ≤ p_n

s₁ ≥ s₂ ≥ . . . ≥ s_m

p

_ij

= p

_i

/s

_j

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RANDOM, uniform machines

Jobs have length

Machines have speed

p₁ ≤ p₂ ≤ . . . ≤ p_n

s₁ ≥ s₂ ≥ . . . ≥ s_m

p

_ij

= p

_i

/s

_j

Lemma: Consider a job making a best move from to . If there is a new unhappy job with index greater than , then s_a > s_b

i i

a b

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RANDOM, uniform machines

Jobs have length

Machines have speed

p₁ ≤ p₂ ≤ . . . ≤ p_n

s₁ ≥ s₂ ≥ . . . ≥ s_m

p

_ij

= p

_i

/s

_j

i i

a b

Proof: let be a new unhappy job. _i^!

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RANDOM, uniform machines

Jobs have length

Machines have speed

p₁ ≤ p₂ ≤ . . . ≤ p_n

s₁ ≥ s₂ ≥ . . . ≥ s_m

p

_ij

= p

_i

/s

_j

i i

a b

was happy on machine and now has an incentive to move to machine ⁱ

! c i^!

a

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RANDOM, uniform machines

Jobs have length

Machines have speed

p₁ ≤ p₂ ≤ . . . ≤ p_n

s₁ ≥ s₂ ≥ . . . ≥ s_m

p

_ij

= p

_i

/s

_j

i i

a b

! i^!

c b

! c i^!

a

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RANDOM, uniform machines

! c i^!

a

!_c + p_i^!/s_c ≤ !_b + 2p_i^!/s_b

!_c + p_i^!/s_c > (!_a − p_i/s_a) + 2p_i^!/s_a

!_a + p_i/s_a > !_b + 2p_i/s_b

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RANDOM, uniform machines

! c i^!

a

!_c + p_i^!/s_c ≤ !_b + 2p_i^!/s_b

!_c + p_i^!/s_c > (!_a − p_i/s_a) + 2p_i^!/s_a

!_a + p_i/s_a > !_b + 2p_i/s_b

(s_a − s_b)(p_i^! − p_i) > 0 Hence,

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RANDOM, uniform machines

! i^!

c b

(p_i − p_i^!)(2s_b − s_c) > 0

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RANDOM, uniform machines

! i^!

c b

(p_i − p_i^!)(2s_b − s_c) > 0 The lemma follows.

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Potential function

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Potential function

Dynamic: among all unhappy jobs, let the one with the greatest index make a best move.

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Potential function

For any strategy profile , let be the unhappy job with

greatest index. ^σ ^t

f_σ(i) =

!1 if 1 ≤ i ≤ t, 0 otherwise.

1 t 0

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Potential function

For any strategy profile , let be the unhappy job with

greatest index. ^σ ^t

f_σ(i) =

!1 if 1 ≤ i ≤ t, 0 otherwise.

1 t 0

lex. decreasesΦ(σ) = (f_σ(1), s_σ(1), . . . , f_σ(n), s_σ(n))

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Potential function

lex. decreasesΦ(σ) = (f_σ(1), s_σ(1), . . . , f_σ(n), s_σ(n))

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Potential function

lex. decreasesΦ(σ) = (f_σ(1), s_σ(1), . . . , f_σ(n), s_σ(n)) is the unhappy player with greatest index in t^! σ^!

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Potential function

t^! < t

Φ(σ) = (1, s_σ(1), . . . , 1, s_σ(t^!₎, 1, s_σ(t^!₊₁₎, . . . , 1, s_σ(t), . . .) Φ(σ^!) = (1, s_σ(1), . . . , 1, s_σ(t^!₎, 0, s_σ(t^!₊₁₎, . . . , 0, s_σ^!_(t), . . .)

If

0 0 1

1 t^!

t

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Potential function

t^! < t

If

0 0 1

1 t^!

t

t^! > t

Φ(σ) = (1, s_σ(1), . . . , 1, s_σ(t), . . . , 0, s_σ(t^!₎, . . .) Φ(σ^!) = (1, s_σ(1), . . . , 1, s_σ^!_(t), . . . , 0, s_σ(t^!₎, . . .)

If

0 0 1

1 t^!

t

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Potential function

t^! < t

If

0 0 1

1 t^!

t

t^! > t

Φ(σ) = (1, s_σ(1), . . . , 1, s_σ(t), . . . , 0, s_σ(t^!₎, . . .) Φ(σ^!) = (1, s , . . . , 1, s ^! , . . . , 0, s ^! , . . .)

If

0 0 1

1 t^!

t

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RANDOM, unrel. machines

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RANDOM, unrel. machines

Theorem:

The game under RANDOM policy is a potential

game for 2 unrelated machines with balanced jobs but it is not for more than 3 machines.

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RANDOM, unrel. machines

Theorem:

The game under RANDOM policy is a potential

game for 2 unrelated machines with balanced jobs but it is not for more than 3 machines.

Proof:

Let be the current profile.σ : {1, . . . , n} →{ 1, 2}

The following potential decreases strictly Φ = |!₁ − !₂| + 3 !

i

max{p_iσ(i) − p_iσ(i), 0}

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EQUI

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EQUI

Theorem:

The game under EQUI policy is a strong potential game.

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EQUI

Theorem:

The game under EQUI policy is a strong potential game.

Proof:

Let be the current profile.

The following exact potential decreases strictly σ

Φ = 1 2

!(c_i + p_iσ(i))

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Inefficiency of equilibria

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Inefficiency

Theorem: For unrelated machines, the PoA of policy EQUI is at most 2m.

The knowledge about jobs’ characteristics is not necessarily needed.

the worst NE

PoA

OPT

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Proof (sketch)

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Proof (sketch)

c_i = p_1j + . . . + p_i₋_1,j + (k − i + 1)p_ij

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Proof (sketch)

Proof: ^qⁱ ^{:= min}

j p_ij

Q(i) := arg min

j p_ij

!n

i=1

q_i ≤ m · OP T

Rename jobs such that ^q1 ≤ q₂ ≤ . . . ≤ q_n c_i = p_1j + . . . + p_i₋_1,j + (k − i + 1)p_ij

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Proof (sketch)

j p_ij

Q(i) := arg min

j p_ij

!n

i=1

q_i ≤ m · OP T

Rename jobs such that ^q1 ≤ q₂ ≤ . . . ≤ q_n Lemma: In any NE,

c_i ≤ 2q₁ + . . . + 2q_i₋₁ + (n − i + 1)q_i c_i = p_1j + . . . + p_i₋_1,j + (k − i + 1)p_ij

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Proof (sketch)

j p_ij

Q(i) := arg min

j p_ij

!n

i=1

q_i ≤ m · OP T

Rename jobs such that ^q1 ≤ q₂ ≤ . . . ≤ q_n Lemma: In any NE,

c_i ≤ 2q₁ + . . . + 2q_i₋₁ + (n − i + 1)q_i c_i = p_1j + . . . + p_i₋_1,j + (k − i + 1)p_ij

Proof: c₁ ≤ nq₁

c₂ ≤ q₁ + (n − 1)q₂

= the worst cost on Q(1)

= the worst cost on Q(2)

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Proof (sketch)

By monotonicity of ^(qi)ⁿ_i=1 makespan = max

i c_i

≤ max

i (2q₁ + . . . 2q_i + (n − i + 1)q_i)

≤ 2 !

i

q_i

≤ 2m · OP T P oA ≤ 2m

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Lower bound

Theorem: The strong PoA of EQUI is at least (m+1)/4.

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Lower bound

Proof (idea):

groups of jobs m J₁, . . . , J_m

Jobs in can be scheduled only on machines J_i i, i + 1

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Lower bound

Proof (idea):

Jobs in can be scheduled only on machines J_i i, i + 1 OPT

J₁ J₂

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Lower bound

Proof (idea):

Jobs in can be scheduled only on machines J_i i, i + 1 NE

J₂

OPT

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Conclusion

Knowledge of jobs’ characteristics is not necessarily needed while restricting to strongly local policies.

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Conclusion

Study the existence of equilibrium for RANDOM in two unrelated machines and in uniform machines.

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Conclusion

Designing local policy with PoA = o(log m)

Study the existence of equilibrium for RANDOM in two unrelated machines and in uniform machines.