Transcript Truthful Mechanism Design for Multi

Truthful Mechanism Design for
Multi-Dimensional Scheduling via
Cycle Monotonicity
Ron Lavi
Chaitanya Swamy
and
IE&M, The Technion
U. of Waterloo
Job scheduling
• n tasks (“jobs”) to be assigned to m workers (“machines”)
• Each machine, i, needs pij time units to complete job j.
• Our goal: to assign jobs to machines to complete all jobs
as soon as possible. More formally:
– Let Si denote the set of jobs assigned to machine i,
and define the load of a machine: li = jSi pij.
– Our goal is then to minimize the maximal load (a.k.a
the “makespan” of the schedule).
Example
Two machines, three jobs:
Job
p1j
p2j
J1
2
1
J2
2
3
J3
3
4
A possible assignment:
J2
J3
J1
1
2
Example
Two machines, three jobs:
Job
p1j
p2j
J1
2
1
J2
2
3
J3
3
4
A possible assignment:
l1 = 3
l2 = 4
J2
J3
J1
Makespan = 4
1
2
Scheduling and Mechanism Design
• The workers/machines are selfish entities, each one is acting
to maximize her individual utility.
• If job j is assigned to machine i, it will incur a cost pij for
executing the job.
• A machine may get a payment, Pi, and its total utility is: Pi - li
• Question: design a truthful mechanism (in dominant
strategies) that will reach a “close to optimal” makespan.
– First raised by Nisan and Ronen (GEB, 2001).
– Basic observation: makespan minimization is inherently
different than welfare maximization, hence VCG performs
poorly (obtains makespan of up to m times the optimum).
Example
Two machines, three jobs:
Job
p1j
p2j
J1
2
1
J2
2
3
J3
3
4
A possible assignment:
J2
J3
J1
Max. Makespan = 4
Tot. Welfare = -3 - 3 -1 = -7
1
2
Example
Two machines, three jobs:
Job
p1j
p2j
J1
2
1
J2
2
3
J3
3
4
A possible assignment:
J2
J3
J1
Max. Makespan = 5
Tot. Welfare = -2 - 3 -1 = -6
1
2
Why is this question important? (1)
• Significant to several disciplines:
– Computer Science
– Operations Research
• Makespan minimization is similar to a Rawls’ max-min
criteria -- gives a justification from social choice theory.
– The implicit goal: assign tasks to workers in a fair
manner (rather than in a socially efficient manner).
– Can we do it via classic mechanism design?
Why is this question important? (2)
• The general status of mechanism design for multidimensional domains is still unclear.
– What social choice functions can be implemented?
– Few possibilities, few impossibilities, more questions
than answers.
• Scheduling is a multi-dimensional domain, and is
becoming one of the important domains for which we
need to determine the possibilities - impossibilities
border.
Current status (1)
• Nisan and Ronen (1999): a lower bound of 2 for truthful
deterministic approximations (regardless of computational
issues).
– But only give a m-approximation upper bound (VCG) -the gap is very large.
• Christodoulou, Koutsoupias, and Vidali (2007): an
improved lower bound (about 2.4).
• Mu’alem and Schapira (2007): a 2-(1/m) lower bound for
randomized mechanisms and truthfulness in expectation.
• No non-trivial truthful approximation (i.e. o(m)) is known!
Current status (2)
• Archer and Tardos (2001) study the special case of
related machines: each machine has speed si,
and pij = pj/si.
– The optimum is implementable (but NP-hard).
– Many truthful approximations suggested since.
The current-best: a deterministic 3-approximation
by Kovacs (2005).
– Also, a truthful PTAS for a fixed number of
machines, by Andelman, Azar, and Sorani (2004).
• Note: this is a single-dimensional domain, thus it
demonstrates again the contrast between single and
multi dimensionality.
A multi-dimensional special case
• We study a special case of two fixed values: pij  {Lj , Hj}
– Values are fixed and known to the mechanism.
– Still a multi-dimensional domain.
– Generalizes the classic “restricted machines” model
(pij  {pj,
}).
• Result 1: The optimal allocation is not implementable
deterministically. Best possible truthful approximation > 1.14.
– Even when Lj = L, Hj = H
– differentiates this case from the related machines case,
another consequence of the multi-dimensionality.
Main Results
• Result 2: a method to convert any c-approximation
algorithm for the two values case to a randomized truthful in
expectation mechanism that obtains a 3c-approximation.
– This is not polynomial time
• Result 3: (when Lj = L, Hj = H) a deterministic, truthful, and
polynomial time, 2-approximation.
• Twist (novelty?) in analysis: we rely on allocation
monotonicity conditions, not on explicit price constructions.
– Common for single-dimensional domains (as initiated by
Myerson), but not for multi-dimensional domains.
The Randomized Construction: Outline
• Description of the monotonicity conditions that are
sufficient to obtain truthfulness.
• A transition to a fractional domain.
• Achieving monotonicity and makespan approximation in
the fractional domain.
• Rounding the fractional solution.
Truthfulness
• Define:
– An “alternatives set”, A. In our case, all possible
assignments of jobs to machines.
– The “type” of a player, ci : A -> R. Here ci is the machine’s
load for the given assignment.
– Let Vi denote the domain of all valid types.
– An algorithm is a function f: V1  . . .  Vn -> A.
• A mechanism is a tuple M = (f, P1 ,  , Pm ),
where Pi : V  R is the payment function for player i.
• Dfn: Truthful Mechanisms.  ci, c-i, c’i :
Pi(ci, c-i) - ci(f(ci, c-i)) > Pi(c’i, c-i) - ci(f(c’i , c-i))
• For a given algorithm, how do we check if such prices exist?
• Can we come up with an equivalent definition that does not
include existential qualifiers, but, rather, only conditions on f.
Weak Monotonicity
Weak monotonicity (W-MON)
• DFN (Lavi, Mu’alem, and Nisan ‘03, Bikhchandani et. al. ‘06):
Suppose f(ci, c-i)=a and f(c’i, c-i)=b. Then
ci(b) - c’i(b) > ci(a) - c’i(a)
• If there exist prices P such that (f,P) is truthful the f must
satisfy weak-monotonicity.
– If there exist such prices we say that f is implementable.
• THM (Saks and Yu, 2005): In convex domains, f is
implementable if and only if it is weakly monotone.
What does w-mon mean in our case?
H
All costs are
for machine 1
allocation = a
c(a) = 2L + 2H
H
H
L
L
L
1
2
What does w-mon mean in our case?
What happens if machine 1
decreases two jobs to L?
allocation = a
c(a) = 2L + 2H
c’(a) = 4L
H
L
H
L
H
L
L
L
1
2
What does w-mon mean in our case?
WMON:
ci(b) - c’i(b) > ci(a) - c’i(a)
= 2(H - L)
• What allocations (b) satisfy
the inequality?
• The only way to get at least
2(H - L) on the left side is to
assign the two the two jobs
that were decreased to
machine 1.
H
L
H
L
H
L
L
L
1
2
What does w-mon mean in our case?
And what if an outside job is
decreased as well?
H
L
H
L
L
L
1
H
L
L
2
What does w-mon mean in our case?
WMON:
ci(b) - c’i(b) > ci(a) - c’i(a)
= 2(H - L)
=> At least two out of the three jobs
that were decreased must be
assigned to 1 (doesn’t matter which
two, only the numbers matter).
H
L
H
L
L
L
1
H
L
L
2
What does w-mon mean in our case?
And if one job is increased from L to H?
H
L
H
L
H L
L
1
H
L
L
2
What does w-mon mean in our case?
WMON:
ci(b) - c’i(b) > ci(a) - c’i(a)
= (H - L)
=> If the increased job remains on 1
then two of the jobs that were
decreased must be assigned to 1, or
alternatively we can “move out” the
increased job and keep only one of
the decreased jobs.
H
L
H
L
H L
L
1
H
L
L
2
What does w-mon mean in our case?
H
L
etc. etc. etc. …
You can now see where the
monotonicity term comes from.
We get an algorithmic
condition instead of the
game-theoretic definition.
H
L
H L
L
1
H
L
L
2
Cycle monotonicity
• W-MON may be insufficient for implementability in nonconvex domains, like our discrete scheduling domain.
• Rochet (1987, JME) describes “cycle monotonicity”, which
generalizes W-MON, and is equivalent to implementability
on every domain (with finite alternative space).
– Gui, Muller, and Vohra (2004) derive prices generically
for every cycle-monotone function.
– Thus any cycle monotone algorithm can be
“automatically” converted to a truthful mechanism.
(this can also be done for W-MON algorithms on convex domains).
• That’s our way of analysis in the paper. In the talk, I will
concentrate on W-MON, for the sake of simplicity.
A fractional solution
Fractional allocations
• For the purpose of analysis we consider the case where
jobs may be assigned fractionally:
– xij denotes the fraction of job j assigned to machine i.
– We have i xij = 1 for every j (every job is fully assigned).
– The load of machine i is li = j xij pij
– Machine i’s value is still minus her load, and her utility is
still Pi - li
• Cycle monotonicity is still equivalent to truthfulness and we
will look for for truthful and approximately optimal fractional
mechanisms.
– This is just an intermediate analysis step. We do not
change our actual initial goal.
Fractional conditions for truthfulness
• The following condition implies w-mon (easy proof omitted):
j, x’ij (pij - p’ij) > xij (pij -p’ij)
(where f(pi, p-i) = x and f(p’i, p-i) = x’)
• In words:
– if pij > p’ij then x’ij > xij
– if pij < p’ij then x’ij < xij
– if pij = p’ij then no relation between x’ij and xij is required.
Corollary: suppose that a fractional algorithm, A, satisfies:
 pij = Lj => xij > 1/m
 pij = Hj => xij < 1/m
Then A satisfies w-mon.
A fractional algorithm
ALG: given any integral allocation, x*, convert it to a
fractional allocation x as follows. For every (i,j) s.t. x*ij=1, do:
• If pij = Hj then set xij=1/m for any i,j.
• If pij = Lj then transfer a fraction 1/m of j from i to every
machine i’ with pi’j=Lj.
A fractional algorithm
ALG: given any integral allocation, x*, convert it to a
fractional allocation x as follows. For every (i,j) s.t. x*ij=1, do:
• If pij = Hj then set xij=1/m for any i,j.
• If pij = Lj then transfer a fraction 1/m of j from i to every
machine i’ with pi’j=Lj.
Properties:
– W-MON follows from previous lemma.
– If x* is c-approx then x is 2c-approx: each machine
gets additional load which is at most the total original
load times 1/m, i.e. at most the original makespan.
• This converts any algorithm to a truthful fractional
mechanism.
Back to Integral Mechanisms
“Rounding” a fractional solution
• We will process a fractional assignment xij to a randomized
assignment: Xij = random variable, indicates if i gets j.
• Lavi and Swamy (2005): given a fractional truthful
mechanism, if E[Xij] = xij then there exist prices such that
the randomized mechanism is truthful in expectation.
• Kumar, Marathe, Parthasarathy, and Srinivasan (2005):
given a fractional allocation, one can construct Xij such that
– E[Xij]= xij
– For every i, (w.p. 1), j Xij pij < j xij pij + maxj: 0 < xij < 1 pij.
• In particular, we can take x* to be the optimal allocation.
This will give us a 3-approximation randomized integral
mechanism which is truthful in expectation.
Remarks
• This has two drawbacks:
– It is not polynomial-time
– Truthfulness in expectation is weaker than deterministic
truthfulness (e.g. requires assuming risk-neutrality).
• With all-identical lows and highs: We show a deterministic
truthful mechanism, with a better approximation ratio, 2.
– Constructions and observations again use cycle
monotonicity, but are completely different otherwise.
– Proof: uses graph flows, is longer and less straightforward.
Summary
• A two-fold motivation for multi-dimensional scheduling:
– A natural problem, related to social choice theory, and to
classic CS and OR.
– Explore in general multi-dimensional mechanism design,
and develop new techniques/insights.
• We demonstrate how to use W-MON / Cycle-Mon to obtain
positive results.
• Actual results are for the “two values” case:
– A general method to convert any algorithm to a truthful in
expectation mechanism with almost the same approx.
– A deterministic 2-approx. truthful mechanism.
– OPT is not implementable, best approx > 1.14.
• Open (and seems hard): the case of k>3 fixed values.