in mechanism design Shuchi Chawla University of Wisconsin Madison FOCS 2012 So far today Revenue amp Social Welfare This talk Nonlinear functions of type amp allocation Question how well can we optimize in strategic settings ID: 603999
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Slide1
Non-linear objectivesin mechanism design
Shuchi Chawla
University of Wisconsin – Madison
FOCS 2012Slide2
So far today…
Revenue & Social Welfare
This talk:
Non-linear functions of type & allocationQuestion: how well can we optimize in strategic settings? Do Bayesian assumptions help?
Shuchi Chawla: Non-linear objectivesSlide3
Algorithmic mechanism design
Three desiderata:
Computational efficiency
Incentive compatibility
Optimize/approximate objective
Main theme in AMD: all three not always achievable together
What should we give up?
Shuchi Chawla: Non-linear objectivesSlide4
AMD tradeoffs
Shuchi Chawla: Non-linear objectives
Overall OPT
OPT-IC
OPT-IC+E
OPT-E
Black-box
Social welfare has gap=1
Bayesian social welfare has small gap
Standard approximation question
Social welfare can have large gap, e.g. comb. auctionsSlide5
Social welfare has gap=1
Bayesian social welfare has small gap
AMD tradeoffs
Shuchi Chawla: Non-linear objectives
Overall OPT
OPT-IC
OPT-IC+E
OPT-E
Black-box
Question 1: OPT
vs
OPT-IC gap for multi-parameter non-linear objectives
Question 2: Black-box reductions for single-parameter monotone objectives
Single-parameter
: each agent has a single value
Monotone objectives
: unilateral increase in an agent’s value causes OPT to allocate more to the agent
IC condition
: unilateral increase in
an agent’s
value results in larger
allocation
All single-parameter “monotone” objectives have gap=1
Prior-free
Bayesian
(sometimes)Slide6
Rest of this talk
Part I
The
makespan objectiveImpossibility of black-box reductions for makespanPart IIBayesian truthful approximations for makespan
Other non-linear objectives; Open problems
Shuchi Chawla: Non-linear objectivesSlide7
Part I.1: Minimizing makespan
Shuchi Chawla: Non-linear objectivesSlide8
Scheduling to minimize makespan
n jobs, m machines
Jobs have different runtimes on different machines
Makespan = completion time of last job
Shuchi Chawla: Non-linear objectives
J1
J2
J3
J4
M1
M2
M3
Makespan
“Unrelated instance”Slide9
Scheduling to minimize makespan
Strategic setting [Nisan Ronen’99]:
Machines are “selfish workers”; jobs’ runtimes are private
Mechanism = (schedule, payments to machines)Machines’ objective: maximize payment – work doneWant assignment+payments to induce
truthtelling
Shuchi Chawla: Non-linear objectivesSlide10
Why makespan?
Important CS problem
Captures the difficulty with non-linear objectives
A single agent can disproportionately affect objectiveHas received the most attention in AGT
Shuchi Chawla: Non-linear objectivesSlide11
J1
J2
J3
J4
M1
M2
M3
Single-parameter
makespan
Each machine has a speed; each job has a size
Runtime of job j on machine
i
= (size of j)/(speed of
i
)
Monotone objective
Shuchi Chawla: Non-linear objectives
Makespan
“
R
elated instance”Slide12
A history of prior-free scheduling
Truthful approximations for related machines
Archer-Tardos’01
: constant approxDhangwatnotai
et al.’08
:
PTAS
Unrelated machines: upper & lower boundsNisan-Ronen’99: m approximationNisan-Ronen’99: lower bound of 2Christodoulou et al.’07:
2.41
; Koutsoupias-Vidali’07
:
2.61
Mu’alem-Shapira’07
: randomized, fractional mechanisms
Ashlagi-Dobzinski-Lavi’09: lower bound of m for anonymous mechanismsShuchi Chawla: Non-linear objectives
Overall OPT
OPT-IC
OPT-IC+E
OPT-ESlide13
Bayesian model for schedulingUnrelated setting: Running time of every job on every machine drawn independently from known distribution
Related setting: Speed of every machine drawn independently from known distribution; jobs sizes fixed
Objective: Expected min
makespanShuchi Chawla: Non-linear objectivesSlide14
Part I.2: Black-box transformations
Shuchi Chawla: Non-linear objectivesSlide15
Black-box transformationsShuchi Chawla: Non-linear objectives
Transformation
Algorithm
Input v
Allocation x
Payment p
GOAL: for
every
algorithm, transformation
preserves quality of solution
and
satisfies
incentive compatibility
.
(cf. Nicole’s talk)Slide16
Black-box transformations
Social welfare: can transform any approx. algorithm into BIC mechanism with “no” loss in expected performance.
[
Hartline-Lucier’10, Hartline-Kleinberg-Malekian’11, Bei-Huang’11]
Is this possible for other objectives?
Makespan
: For any
polytime BIC transformation, there is a makespan problem and algorithm such that mech.’s expected makespan is polynomially larger than alg.’s.[C.-Immorlica-Lucier’12]
Shuchi Chawla: Non-linear objectives
NO!Slide17
Single-parameter makespan
v
1
v
2
v
3
v
4
v
5
v
6
v
7
v
8
x
1
m
machines,
machine i has speed vi ~ Fi
n jobs,size of job j is xjx2
x3
x4collection F of feasible assignments
Shuchi Chawla: Non-linear objectivesSlide18
Proof outline
Define
makespan
instance (feasibility constraints, value distribution).Find algorithm with low expected makespan.Use monotonicity condition to show any BIC transformation has high expected makespan.
Key
issue: Transformation must rely on algorithm to
understand/satisfy
feasibility constraintThat is, transformation must return an allocation that it observes the algorithm returnShuchi Chawla: Non-linear objectives
Higher speed ⇒ higher expected loadSlide19
Makespan Instance
f
easibility set
F = {at most one job per machine}
v
1
v
2
v
3
v
4
v
5
v
6
v
7
v
8
x
m/2m
machines, speeds vi ~ Uniform{low = 1, high = α}x2
x1
x
k
x
2
x
1
m
/2 jobs, small size
x
j
= 1
m
1/2
jobs, large size
x
j
=
α
Shuchi Chawla: Non-linear objectivesSlide20
Approximation Algorithm
If (m/2 ± m
3
/4) machines report high speed,assign large jobs to fast machines (at random)assign small jobs to slow machines (at random)assign NO job to all remaining machinesElse
assign all jobs randomly
Shuchi Chawla: Non-linear objectivesSlide21
Approximation Algorithm
h
igh speeds
l
ow speeds
Note 1: By
Chernoff
, expected
makespan
is low.
Note 2:
E
xpected allocation is not monotone.
Shuchi Chawla: Non-linear objectivesSlide22
Transformation
To fix non-monotonicity, must more often:
allocate nothing to low speed machines, or
allocate something to high speed machines.Shuchi Chawla: Non-linear objectivesSlide23
Transformation
1
1
1
1
α
α
α
α
Query v’: pretend some low machines are high and vice versa...
Input v:
α
α
α
α
1
1
1
1
Each “fast” machine gets large job with probability m
-1/2
t
hen with high probability,
makespan
is high.
Shuchi Chawla: Non-linear objectivesSlide24
Transformation
1
1
1
1
α
α
α
α
Query v’:
pretend number of high machines deviates from expectation..
Input v:
1
1
1
1
1
1
1
1
Each machine gets large job with
probability m
-
1/
2
t
hen with high probability,
makespan
is high.
Shuchi Chawla: Non-linear objectivesSlide25
Recap and other resultsFor any BIC transformation, there is an alg. such that the transformation’s
makespan
is
polynomially larger than the algorithm’seven when the algorithm is a constant approximationWhat about other non-linear functions?Ironing doesn’t workGap increases with non-linearity
Shuchi Chawla: Non-linear objectives
[C.-Immorlica-Lucier’12]Slide26
Non-linear objectivesin mechanism design
Shuchi Chawla
University of Wisconsin – Madison
Part IISlide27
Recap of part I
A representative non-linear objective:
makespan
Black-box transformations are essentially impossible for makespan: objective function increases by polynomial factorShuchi Chawla: Non-linear objectives
Overall OPT
OPT-IC
OPT-IC+E
OPT-E
Black-boxSlide28
Part II.1: Bayesian approximation for makespan
Shuchi Chawla: Non-linear objectivesSlide29
Recall: scheduling to minimize makespan
n jobs, m machines
Jobs’ runtimes drawn from known
indep. distributionsMakespan = completion time of last jobPrior-free setting: any anonymous truthful mechanism is at best an m approximation.
Shuchi Chawla: Non-linear objectives
J1
J2
J3
J4
M1
M2
M3
MakespanSlide30
A truthful mechanism: M
inWork
For every job:
Assign the job to the machine that reports the lowest runtimePay the machine the job’s running time on its “second best” machine m’
“Second-price” payments: induce
truthtelling
Makespan
≤ sum of best runtimes of all jobs ≤ total work done in optimal schedule ≤ m x optimal makespan⇒ m-approximation to makespanShuchi Chawla: Non-linear objectivesSlide31
Overcoming the lower bound
Ashlagi
et al.’s lower bound of m for
makespanOrdered instance: machine i is better than machine i+1 for all jobsRunning times within 1+eps of each otherAny truthful mechanism must allocate all jobs to machine 1How do Bayesian assumptions help?
Knowledge of distribution => we can penalize allocations that are always bad for the given instance
A priori identical machines: bad instances have extremely low probability
Shuchi Chawla: Bayesian scheduling
31Slide32
Prior-independent approximation
Unknown Bayesian prior, but belongs to some “nice” family
In particular, the runtime of a job j is identically distributed on every machine.
That is, machines are a priori identicalHowever, any instantiation of runtimes is an unrelated instanceResult: There exists a truthful prior-independent mechanism that achieves an O(n/m) approximation to expected makespan
(*)
[C.-Hartline-Malec-Sivan’12]
Shuchi Chawla: Non-linear objectives
(cf. Tim’s talk)Slide33
Benchmark
Hindsight OPT
For any instantiation, finds the optimal
makespanOPT1/2 Discards m/2 machines randomlyFor any instantiation, finds optimal makespan
over remaining machines
For many distributions, OPT
1/2
~ constant. OPTKey property: min over 2 draws ~ 2 times a single drawIncludes all “MHR” distributions, e.g. uniform, exponential, normal,…Shuchi Chawla: Non-linear objectivesSlide34
How to design a truthful multi-parameter mechanism?
A simple powerful class: affine
maximizers
Maximize an appropriate linear a.k.a. affine functionEssentially, an extension of VCGFor example:Can assign “costs” to some outcomes, and, minimize total (work – cost) Can forbid certain outcomes by setting cost = ∞
Can assign more weight to the work of some agents than that of others
Shuchi Chawla: Non-linear objectivesSlide35
The MinWork
mechanism again
Essentially VCG: schedule every job on its best machineObserve: job j’s runtime in MinWork ≤
job j’s runtime in
OPT Furthermore,
every job goes to a random
machineIf jobs were to be distributed uniformly across machines, we would get good makespanHowever, balls-in-bins analysis ⇒ some machine has O(log m/log log m) jobsShuchi Chawla: Non-linear objectivesSlide36
The MinWork(k) mechanism
Find a min-size matching between jobs and machines that assigns at most k jobs to each machine.
Claim:
MinWork(k) is truthfulProof: It is VCG over a restricted domain.
Shuchi Chawla: Non-linear objectivesSlide37
The MinWork(k) mechanism
Find a min-size matching between jobs and machines that assigns at most k jobs to each machine.
Claim:
MinWork(k) is truthfulClaim: MinWork(10) gets a constant approximation
Obs1: The schedule is almost balanced
Obs2: Every job still goes to roughly its best machine
Shuchi Chawla: Non-linear objectivesSlide38
Obs2: the last entry procedure
Fix job j and imagine adding it last in a greedy fashion.
Shuchi Chawla: Non-linear objectives
1
2
3
4
5
6
MinWork
(3) schedule for all but job j
1
2
3
4
5
6
MinWork
(3) schedule sorted by j’s preferences
Machine full
Space available,
so j goes hereSlide39
Obs2: The last entry procedureFix job j and imagine adding it last in a greedy fashion.
The probability that j goes to one of its top
i
machines is at least 1-(1/k)iMinWork(k) places j in an even better positionKey claim: Placing j on its i
th
best machine is no worse than placing 5
i
independent copies of j on their best (of n/2) machinesShuchi Chawla: Non-linear objectivesSlide40
MinWork(10) analysis
Job j’s runtime in
MinWork
(k) ≤ max5^i independent copies j’s runtime in OPT1/2
≤ 5
i
times j’s runtime in OPT1/2 Here i is an exponential random variable; Note: E[5i] = constant.∴ MinWork(10)’s makespan ≤ 10 E[maxj (j’s
runtime in
MW)]
≤ constant times OPT
1/2
Shuchi Chawla: Non-linear objectives
Stochastic dominanceSlide41
Key technical claim
Placing j on its
i
th best machine is no worse than placing 5i independent copies of j on their best (of n/2) machines
Shuchi Chawla: Non-linear objectives
Expt. 1
n copies of j’s runtime
Expt. 2
5
i
/2 blocks
n/2 copies of j’s runtime
i
th
min over n copies
m
ax over 5
i
mins
over n/2 copiesSlide42
Recap and other results
Machines a priori identical, “few” jobs
O(1) prior-independent approximation:
MinWork(k) ≤ O(1) OPT1/2Compare to Bulow-Klemperer’s result for revenue with k items:
VCG ≥ O(1)
OPT
less
k agentsJobs are also a priori identical: multi-stage mechanismsPrior-ind. O(√log m) approximation to OPT1/2Prior-
ind.
O((log log m)
2
)
approx to OPT for MHR distributions
Shuchi Chawla: Non-linear objectives[C.-Hartline-Malec-Sivan’12]
Hindsight-OPT1/2(needs regularity)Slide43
Part II.2: Other objectives & open problems
Shuchi Chawla: Non-linear objectivesSlide44
Open problems for makespan
O(1) prior-
ind.
approximation for non-identical jobsBayesian approximation for non-identical machinesWill need to use the knowledge of priorEven logarithmic approx is non-trivialA potential approach: charge a prior-dependent amount for placing each additional job on a machine
Approximation for small-support priors
LP based?
Shuchi Chawla: Non-linear objectivesSlide45
Other non-linear objectives
Max-min fairness in scheduling a.k.a. load balancing
Prior-free PTAS for related setting
[Epstein-van Stee’10]Unrelated approximation?Max-min fairness in welfare a.k.a. the Santa Claus problem
Not monotone!
Single-parameter Bayesian
approx
?Shuchi Chawla: Non-linear objectives
Min
m
akespan
10
3
3
2Slide46
ConclusionsNon-linear objectives in general much harder than social welfare
Mild stochastic assumptions can help us circumvent strong impossibility results
Multi-parameter mechanisms are difficult to understand, but “affine
maximizers” is a powerful subclass.Lots of nice open problems!Shuchi Chawla: Non-linear objectives