An Optimal Lower Bound for Anonymous Scheduling Mechanisms

Joint Work with Itai Ashlagi (Harvard Business Scool) and Shahar Dobzinski (CS, Hebrew University)

Ron LaviIndustrial Engineering and Management

Technion - Israel Institute of Technology

Job Scheduling (example)

two workers(M1, M2);three tasks (J1, J2, J3):

Need to assign tasks to workers.A possible assignment:

1 2

J3

J1

J2

J1 J2 J3

M1 2 2

M2 1 3 4

Job Scheduling (example)

two workers(M1, M2);three tasks (J1, J2, J3):

Need to assign tasks to workers.A possible assignment:

1 2

J3

J1

J2

J1 J2 J3

M1 2 2

M2 1 3 4

“makespan” = 4

• n tasks (“jobs”) to be assigned to m workers (“machines”)• Each machine, i, needs cij 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 = jSi cij.

– Our goal is then to minimize the maximal load (a.k.a the “makespan” of the schedule).

Job Scheduling (definition)

Scheduling and Mechanism Design

• Nisan and Ronen (GEB, ‘01): 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 cij for executing the job. cij is private information to machine i.

• A machine may get a payment, Pi, to balance its cost,and its total utility is: Pi - li

• A truthful mechanism:– Machines need to report types– Truthful reporting is a dominant strategy.

Question

• Question: design a truthful mechanism that will reach a “close to optimal” makespan.

• Approximation ratio: worst ratio (over all instances) of the mechanism’s makespan to the optimal makespan.

• The “usual” tool: the Vickrey-Clarke-Groves (VCG) method. Fits cases where we wish to maximize the social welfare.

• Basic observation [Nisan-Ronen]: makespan minimization is inherently different than welfare maximization, hence VCG performs poorly (obtains makespan of up to m times the optimum, i.e. has an approximation ratio of m).

Example

two workers(M1, M2);three tasks (J1, J2, J3):

Need to assign tasks to workers.A possible assignment:

1 2

J3

J1

J2

J1 J2 J3

M1 2 2

M2 1 3 4

“makespan” = 4

Welfare = -3 - 3 -1 = -7

Example

A different assignment:

1 2

J3

J1

J2

Makespan = 5

Tot. Welfare = -2 - 3 -1 = -6

two workers(M1, M2);three tasks (J1, J2, J3):

J1 J2 J3

M1 2 2

M2 1 3 4

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 mechanism design?

• The general status of mechanism design for multi-dimensional 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.

Why is this question important? (2)

Case I: related machines [Archer and Tardos (2001)]machine i has speed si, and cij = cj/si

• Optimal truthful mechanism exists (requires exponential computation).

• Many truthful approximations with polynomial computation:– A randomized PTAS (Dhangwatnotai, Dobzinski, Dughmi, and Roughgarden ‘08 )

– Deterministic 3-approximation (Kovacs ‘05)

Case II: two-value jobs [Lavi and Swamy (2007)]

Each processing time is either high or low, in an unrelated way.

• Randomized 3-approx (exponential computation)

• Deterministic 2-approx (polynomial computation, when lows and highs are equal).

• Extension to a “two-range” domain (Yu, 2009)

Current status: special cases

Current status: lower bounds• Nisan and Ronen (1999): Every truthful mechanism obtains

approximation ratio > 2.

• Christodoulou, Koutsoupias, and Vidali (2007): an improved lower bound (about 2.6).

• 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!

Conjecture (Nisan and Ronen): VCG provides the best possible approximation ratio.

(Given the many positive results for the special cases and the very low lower bounds, skepticism is natural)

A bad instance for VCG

J1 … Jm

M1 t1 … t1

M2 t2 … t2

.

.

.

Mm tm … tm

t1+ > tm > … > t2 > t1

Optimal makespan is t1+

VCG gives makespan m·t1

Our ResultTheorem:

Every anonymous and truthful mechanism with a bounded approximation ratio provides the same assignment as VCG in this instance.

Corollary:VCG obtains the best approximation ratio among all truthful and anonymous mechanisms.

J1 … Jm

M1 t1 … t1

M2 t2 … t2

.

.

.Mm tm … tm

t1+ > tm > … > t2 > t1

Anonymity

Anonymity: if two machines with distinct costs switch types, the assigned jobs also switch (i.e. machine names do not matter).

Natural requirement:

• Algorithmic perspective: the classic scheduling algorithms are anonymous.

• Mechanism design perspective: the mechanisms for the special cases are anonymous.

• Game theory perspective: anonymous games are an important and natural class.

Weak monotonicity (W-MON)• DFN (Lavi, Mu’alem, and Nisan ‘03, Bikhchandani et. al. ‘06):

Fix the declarations of the other machines. Suppose machine i receives a set S of jobs when declaring ci, and a set S’ of jobs when declaring c’i, . Then ci(S’) - c’i(S’) > ci(S) - c’i(S)

• Every truthful mechanism satisfies W-MON.• W-MON is necessary for truthfulness if the domain of types is

convex (Saks and Yu, ‘05; Monderer ‘08).

Example 1

J1 … Jm

M1 t1 … t1

M2 t2 … t2

.

.

.Mm tm … tm

J1 … Jm

M1 t1 - … t1 -

M2 t2 … t2

.

.

.Mm tm … tm

Which sets S’ satisfy ci(S’) - c’i(S’) > ci(S) - c’i(S) = m ?

Example 1

J1 … Jm

M1 t1 … t1

M2 t2 … t2

.

.

.Mm tm … tm

J1 … Jm

M1 t1 - … t1 -

M2 t2 … t2

.

.

.Mm tm … tm

Which sets S’ satisfy ci(S’) - c’i(S’) > ci(S) - c’i(S) = m ?

Example 2

J1 J2 J3

M1 x y z

M2 ? ? ?

M3 ? ? ?

J1 J2 J3

M1 x - y - z + M2 ? ?

M3 ? ? ?

Which sets S’ satisfy ci(S’) - c’i(S’) > ci(S) - c’i(S) = 2 ?

Example 3

J1 J2 J3

M1 1 1 1

M2 1 1 1

J1 J2 J3

M1 1+ 1+ M2 1 1 1

Remarks:

1. this almost finishes the proof of the lower bound of 2.

2. Nisan and Ronen use “one hop” arguments, similar to this.

3. The other lower bounds use increasingly longer hops.

4. We use an inductive argument that enables us to identify very long hops that give us the optimal lower bound.

Overview of proof

• Proof is by induction on the number of jobs.• In this overview: only 3 machines and 3 jobs.

• “main lemma”: in the followinginstance, M1 receives all jobs,where x,y in {t1,}.

• Proof is by induction on numberof ’s. In this overview I willassume correctness for x=y= and for x=t1 and y= , and willprove the claim for x=y=t1.

J1 J2 J3

M1 x y

M2 t21 t22 t23

M3 t31 t32 t33

t3j> t2j > t1 >>

Induction steps

J1 J2 J3

M1 t1 t1 a

M2 t2 t2 a

M3 t3 t3 a

t3> t2 > t1 >> a >>

J1 J2 J3

M1 t1 t1

M2 t2 t2 a

M3 t3 t3 a

J1 J2 J3

M1 t1 t1

M2 t2 t2 a

M3 t3 t3 t3

J1 J2 J3

M1 t1 t1

M2 t2 t2 tM3 t3 t3 t3

Step 1

J1 J2 J3

M1 t1 t1 a

M2 t2 t2 a

M3 t3 t3 a

This induces a mechanism on three machines and two jobs and by the induction assumption the lowest machine must get both jobs.

Induction steps

J1 J2 J3

M1 t1 t1 a

M2 t2 t2 a

M3 t3 t3 a

t3> t2 > t1 >> a >>

J1 J2 J3

M1 t1 t1

M2 t2 t2 a

M3 t3 t3 a

J1 J2 J3

M1 t1 t1

M2 t2 t2 a

M3 t3 t3 t3

J1 J2 J3

M1 t1 t1

M2 t2 t2 tM3 t3 t3 t3

Step 2

J1 J2 J3

M1 t1 t1

M2 t2 t2 a

M3 t3 t3 a

WMONTowards

Contradiction:

J1 J2 J3

M1

M2 T2 t2 a

M3 t3 t3 a

makespan = a >> 2 = optimal makespan.Thus the mechanism does not providea finite approx ratio, a contradiction.

Induction steps

J1 J2 J3

M1 t1 t1 a

M2 t2 t2 a

M3 t3 t3 a

t3> t2 > t1 >> a >>

J1 J2 J3

M1 t1 t1

M2 t2 t2 a

M3 t3 t3 a

J1 J2 J3

M1 t1 t1

M2 t2 t2 a

M3 t3 t3 t3

J1 J2 J3

M1 t1 t1

M2 t2 t2 tM3 t3 t3 t3

J1 J2 J3

M1 t1 t1

M2 t2 t2 a

M3 t3 t3 t3

Claim 3(a): If M1 receives either J1 or J2 then it must receive J1 and J2.

Proof:

Step 3

J1 J2 J3

M1 t1+

M2 t2 t2 a

M3 t3 t3 t3

Towards a contradiction A contradiction tothe induction hypothesis of

the “main lemma”(M1 should get everything)

Step 3

J1 J2 J3

M1 t1 t1

M2 t2 t2 a

M3 t3 t3 t3

WMONTowards

Contradiction:

J1 J2 J3

M1

M2 t2 t2 a

M3 t3 t3 t3

makespan > a >> 2 = optimal makespan.Thus the mechanism does not providea finite approx ratio, a contradiction.

Claim 3(b): If M1 receives J1 and J2 it must also receive J3.

Proof: (exactly like step 2)

Step 3

J1 J2 J3

M1 t’1 t’1 ‘

M2 t2 t2 a

M3 t3 t3 t3

Claim 3(c): M1 receives either J1 or J2.

Proof: otherwise find t1 < t’1 < t’’1 < t2 and < ‘ << ‘‘ < a such that:

J1 J2 J3

M1 t’’1 t’’1 ‘‘

M2 t2 t2 a

M3 t3 t3 t3

Step 3

J1 J2 J3

M1 t’1 t’1 ‘

M2 t2 t2 a

M3 t3 t3 t3

Claim 3(c): M1 receives either J1 or J2.

Proof: otherwise find t1 < t’1 < t’’1 < t2 and < ‘ << ‘‘ < a such that:

J1 J2 J3

M1 t’’1 t’’1 ‘‘

M2 t2 t2 a

M3 t3 t3 t3

By WMON, since t2 - t1 > a.

J1 J2 J3

M1 t’’1 t’’1 ‘‘

M2 t’1 t’1 ‘

M3 t3 t3 t3

Step 3

J1 J2 J3

M1 t’1 t’1 ‘

M2 t2 t2 a

M3 t3 t3 t3

Claim 3(c): M1 receives either J1 or J2.

Proof: otherwise find t1 < t’1 < t’’1 < t2 and < ‘ << ‘‘ < a such that:

J1 J2 J3

M1 t’’1 t’’1 ‘‘

M2 t2 t2 a

M3 t3 t3 t3

By WMON, since t2 - t1 > a.

J1 J2 J3

M1 t’’1 t’’1 ‘‘

M2 t’1 t’1 ‘

M3 t3 t3 t3

claim 3(a+b)

J1 J2 J3

M1 t’’1 t’’1 ‘‘

M2 t’1 t’1 ‘

M3 t3 t3 t3

Step 3

J1 J2 J3

M1 t’1 t’1 ‘

M2 t2 t2 a

M3 t3 t3 t3

Claim 3(c): M1 receives either J1 or J2.

Proof: otherwise find t1 < t’1 < t’’1 < t2 and < ‘ << ‘‘ < a such that:

J1 J2 J3

M1 t’’1 t’’1 ‘‘

M2 t2 t2 a

M3 t3 t3 t3

By WMON, since t2 - t1 > a.

J1 J2 J3

M1 t’’1 t’’1 ‘‘

M2 t’1 t’1 ‘

M3 t3 t3 t3

claim 3(a+b)

J1 J2 J3

M1 t’’1 t’’1 ‘‘

M2 t’1 t’1 ‘

M3 t3 t3 t3

WMON

J1 J2 J3

M1 t2 t2 a

M2 t’1 t’1 ‘

M3 t3 t3 t3

Step 3

J1 J2 J3

M1 t’1 t’1 ‘

M2 t2 t2 a

M3 t3 t3 t3

Claim 3(c): M1 receives either J1 or J2.

Proof: otherwise find t1 < t’1 < t’’1 < t2 and < ‘ << ‘‘ < a such that:

J1 J2 J3

M1 t’’1 t’’1 ‘‘

M2 t2 t2 a

M3 t3 t3 t3

By WMON, since t2 - t1 > a.

J1 J2 J3

M1 t’’1 t’’1 ‘‘

M2 t’1 t’1 ‘

M3 t3 t3 t3

claim 3(a+b)

J1 J2 J3

M1 t’’1 t’’1 ‘‘

M2 t’1 t’1 ‘

M3 t3 t3 t3

WMON

J1 J2 J3

M1 t2 t2 a

M2 t’1 t’1 ‘

M3 t3 t3 t3

Contradiction to anonymity

Induction steps

J1 J2 J3

M1 t1 t1 a

M2 t2 t2 a

M3 t3 t3 a

t3> t2 > t1 >> a >>

J1 J2 J3

M1 t1 t1

M2 t2 t2 a

M3 t3 t3 a

J1 J2 J3

M1 t1 t1

M2 t2 t2 a

M3 t3 t3 t3

J1 J2 J3

M1 t1 t1

M2 t2 t2 tM3 t3 t3 t3

J1 J2 J3

M1 t1 t1

M2 t2 t2 t2

M3 t3 t3 t3

t1

Final step

Proof similar to that of step 3:

Claim 1: If M1 receives at least one job then it receives all jobs.

Claim 2: M1 must receive at least one job (otherwise we construct a contradiction to anonymity)

“main lemma”

Summary• Study scheduling mechanisms to minimize the makespan.

• Result: VCG is the best anonymous and truthful mechanism.– Negative result, since VCG may output a large makespan.

• Technical method: repeatedly applying WMON to create very long contradiction paths.– Instead of proving a characterization result.

• Further directions:– Can non-anonymous mechanisms do better?– Can randomized mechanisms do better?– Perhaps over a discrete domain?– Perhaps using alternative solution concepts?