St ackelberg Scheduling St rat egies Tim Roughgarden Cornell - - PDF document

▶

Feb 04, 2024 363 likes •553 views

St ackelberg Scheduling St rat egies Tim Roughgarden Cornell Universit y 1 The Model m machines 1,2,...,m A quant it y r of j obs j obs are small (model in a ct s way) For each machine i, a load- dependent lat ency f unct

SLIDE 1

St ackelberg Scheduling St rat egies

Tim Roughgarden Cornell Universit y

SLIDE 2

The Model

m machines 1,2,...,m
A quant it y r of j obs

– j obs are small (model in a ct s way)

For each machine i, a load-

dependent lat ency f unct ion l i(•)

– assume cont inuous, nondecreasing Example: (r=1) l 1(x)=1 l 2(x)=x2 ½ j obs have lat ency 1 j obs have lat ency ¼ ½

SLIDE 3

Equilibria

which j ob assignment s ar e

“st able”? – j obs cont r olled by self ish, noncooper at ive agent s – no j ob want s t o swit ch machines (no j ob should be envious)

vs.

l 1(x)=1 l 2(x)=x2 ½ ½ 1 x2 1

SLIDE 4

vs.

Def : an assignment is at Nash

equilibrium (is a Nash assignment ) if :

all used machines have equal lat ency
unused machines have great er lat ency

Fact : always have exist ence, uniqueness

l 1(x)=1 l 2(x)=x2 ½ ½ 1 x2 1

SLIDE 5

How Good is an Assignment ?

The Cost of an Assignment :

C(x) = cost or t ot al lat ency

exper ienced by assignment x: Σi xi • l i(xi)

our not ion of syst em perf ormance
can opt imize in poly-t ime

Example: cost = ½

1 +½
¼= ?

1 x2 ½ ½

SLIDE 6

How Good are Nash Assignment s?

Goal: prove t hat Nash

assignment s ar e near-opt imal

want a laissez f aire approach t o

regulat ing users

Problem: f alse in general!

Example: (r=1, k large)

vs.

1 xk 1 1 xk

1-? Nash cost s 1 OPT cost s ≈ 0

SLIDE 7

Near-Opt imal Nash Assignment s

Old Approach: weaken model,

compar e Nash vs. OPT

(due t o [Roughgarden/ Tardos 00])

gener al lat ency f ns, weaker OPT

Thm 1: cost of Nash =cost of OPT at rat e 2r

st r onger OPT, linear lat ency f ns

(l i(x)=aix+bi)

Thm 2: cost of Nash = 4/ 3 cost of OP T (at rat e r)

SLIDE 8

Taming Self ishness t hr ough a Manager

New Appr oach:

not all j obs need be cont r olled

by self ish users

– “cent rally cont rolled” vs. “self ishly cont rolled” j obs – behavior of self ish users depends

n assignment of managed j obs

Goal:

assign cent rally cont rolled j obs

t o induce “good” self ish behavior

– see also [Kor lis, Lazar, Or da 97]

SLIDE 9

St ackelberg St rat egies

St ackelberg st r at egy =

assignment of cent rally cont r olled j obs ⇒ yields an induced equilibrium

Basic Quest ions:
what ’s t he best st r at egy?
can we comput e/ charact erize it ?
how inef f icient is t he best

induced equilibr ium?

are we provably near-opt imal?

SLIDE 10

Our Result s - General Lat ency Funct ions

Theorem 1: Can ef f icient ly

comput e a st r at egy inducing an equilibrium wit h cost =(1/ ß) × cost of opt assignment

(ß = f ract ion of cent rally assigned j obs)

Fact :

(1/ ß) × OPT is best possible 1 [x/ (1-ß)] k 1-ß ß

SLIDE 11

Our Result s - Linear Lat ency Funct ions

Theorem 2: Can ef f icient ly

comput e a st r at egy inducing an equilibrium wit h cost = [4/ (3+ß)] × cost of OPT

(ß = f ract ion of cent rally assigned j obs)

Fact :

[4/ (3+ß)] × OPT is best possible 1 x/ (1-ß) 1-ß ß

SLIDE 12

What makes a st rat egy (in)ef f ect ive?

The Scale St rat egy

comput e opt imal assignment x
f all j obs, assign cent r ally

cont r olled j obs via ß • x

OPT I nduced Eq

1 (3/ 2)x

1/ 3

1 (3/ 2)x

2/ 3 1/ 3 2/ 3

Moral: avoid machines t hat self ish users will (over)use anyways

SLIDE 13

The LLF St rat egy

Largest Lat ency First (LLF):

comput e opt , x, f or all j obs
assign x i j obs t o i in or der of

decreasing l i(xi)’s (unt il no managed j obs remain)

l 3=½ (3/ 2)x 1

5/ 12

2x

¼

1/ 3

l 1=1 l 2=½ (3/ 2)x 1

5/ 12

2x

1/ 12

OPT (r=1) LLF (ß=½ )

SLIDE 14

LLF wit h General Lat ency Funct ions

Theor em 1: The LLF st r at egy induces an equilibr ium wit h cost = (1/ ß) × cost of opt assignment . Proof idea: Exploit it er at ive st r uct ur e of LLF t o pr oceed by induct ion on # of machines.

common lat ency = L ß Base case:

LLF ⇒ Machine 1’s lat ency ≥ L
OPT has ≥ ß j obs on machine 1

⇒ OPT pays ≥ ßL, we pay L

SLIDE 15

LLF wit h Linear Lat ency Funct ions

Theorem 2: The LLF st rat egy

induces an equilibr ium wit h cost = [4/ (3+ß)] × cost of OPT. Main dif f icult y:

previous argument t oo weak f or:
need det ailed st udy of Nash, OPT

when all lat encies are linear

SLIDE 16

Comput ing Opt imal St rat egies

We' ve seen: LLF has t he best

possible worst -case guanant ee

Quest ion: is t he LLF st rat egy

pt imal on all inst ances?

Bad news: no, in f act :

Theorem 3: Comput ing t he

pt imal st r at egy is NP-hard

(even f or linear lat ency f ns).

Compare t o: Opt , Nash assignment s

SLIDE 17

Open Quest ions

Approximat ing t he opt imal st rat egy:

LLF is best possible using OPT

as a lower bound

bet t er guar ant ees f or LLF via a

bet t er lower bound?

mor e sophist icat ed algor it hms?

– Thm 3 is reduct ion f rom Part it ion – exist ence of a (F)P TAS?

St ackelberg Scheduling St rat egies

Tim Roughgarden Cornell Universit y

The Model

– j obs are small (model in a ct s way)

dependent lat ency f unct ion l i(•)

– assume cont inuous, nondecreasing Example: (r=1) l 1(x)=1 l 2(x)=x2 ½ j obs have lat ency 1 j obs have lat ency ¼ ½

Equilibria

“st able”? – j obs cont r olled by self ish, noncooper at ive agent s – no j ob want s t o swit ch machines (no j ob should be envious)

vs.

l 1(x)=1 l 2(x)=x2 ½ ½ 1 x2 1

More on Equilibrium Assignment s

vs.

Def : an assignment is at Nash

equilibrium (is a Nash assignment ) if :

Fact : always have exist ence, uniqueness

l 1(x)=1 l 2(x)=x2 ½ ½ 1 x2 1

How Good is an Assignment ?

The Cost of an Assignment :

exper ienced by assignment x: Σi xi • l i(xi)

Example: cost = ½

1 x2 ½ ½

How Good are Nash Assignment s?

Goal: prove t hat Nash

assignment s ar e near-opt imal

regulat ing users

Problem: f alse in general!

Example: (r=1, k large)

vs.

1 xk 1 1 xk

1-? Nash cost s 1 OPT cost s ≈ 0

Near-Opt imal Nash Assignment s

Old Approach: weaken model,

compar e Nash vs. OPT

(due t o [Roughgarden/ Tardos 00])

Thm 1: cost of Nash =cost of OPT at rat e 2r

(l i(x)=aix+bi)

Thm 2: cost of Nash = 4/ 3 cost of OP T (at rat e r)

Taming Self ishness t hr ough a Manager

New Appr oach:

by self ish users

– “cent rally cont rolled” vs. “self ishly cont rolled” j obs – behavior of self ish users depends

Goal:

t o induce “good” self ish behavior

– see also [Kor lis, Lazar, Or da 97]

St ackelberg St rat egies

assignment of cent rally cont r olled j obs ⇒ yields an induced equilibrium

induced equilibr ium?

Our Result s - General Lat ency Funct ions

Theorem 1: Can ef f icient ly

comput e a st r at egy inducing an equilibrium wit h cost =(1/ ß) × cost of opt assignment

(ß = f ract ion of cent rally assigned j obs)

Fact :

(1/ ß) × OPT is best possible 1 [x/ (1-ß)] k 1-ß ß

Our Result s - Linear Lat ency Funct ions

Theorem 2: Can ef f icient ly

comput e a st r at egy inducing an equilibrium wit h cost = [4/ (3+ß)] × cost of OPT

(ß = f ract ion of cent rally assigned j obs)

Fact :

[4/ (3+ß)] × OPT is best possible 1 x/ (1-ß) 1-ß ß

What makes a st rat egy (in)ef f ect ive?

The Scale St rat egy

cont r olled j obs via ß • x

OPT I nduced Eq

1 (3/ 2)x

1 (3/ 2)x

Moral: avoid machines t hat self ish users will (over)use anyways

The LLF St rat egy

Largest Lat ency First (LLF):

decreasing l i(xi)’s (unt il no managed j obs remain)

l 3=½ (3/ 2)x 1

2x

¼

l 1=1 l 2=½ (3/ 2)x 1

2x

OPT (r=1) LLF (ß=½ )

LLF wit h General Lat ency Funct ions

Theor em 1: The LLF st r at egy induces an equilibr ium wit h cost = (1/ ß) × cost of opt assignment . Proof idea: Exploit it er at ive st r uct ur e of LLF t o pr oceed by induct ion on # of machines.

common lat ency = L ß Base case:

⇒ OPT pays ≥ ßL, we pay L

LLF wit h Linear Lat ency Funct ions