An Optimal Lower Bound for Anonymous Scheduling Mechanisms Ron Lavi Industrial Engineering and Management Technion - Israel Institute of Technology Joint Work with Itai Ashlagi (Harvard Business Scool) and Shahar Dobzinski (CS, Hebrew University)

Job Scheduling (example) two workers(M1, M2); three tasks (J1, J2, J3): Need to assign tasks to workers. A possible assignment: J1 J2 J3 M1 2 3 M2 1 4 J2 J3 J1 1 2

Job Scheduling (example) two workers(M1, M2); three tasks (J1, J2, J3): Need to assign tasks to workers. A possible assignment: J1 J2 J3 M1 2 3 M2 1 4 J2 J3 J1 “makespan” = 4 1 2

Job Scheduling (definition) 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 = jSi cij. Our goal is then to minimize the maximal load (a.k.a the “makespan” of the schedule).

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: J1 J2 J3 M1 2 3 M2 1 4 J2 J3 “makespan” = 4 Welfare = -3 - 3 -1 = -7 J1 1 2

Example two workers(M1, M2); three tasks (J1, J2, J3): A different assignment: J1 J2 J3 M1 2 3 M2 1 4 J2 J3 J1 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 mechanism design?

Why is this question important? (2) 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.

Current status: special cases 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: 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 t1+e > tm > … > t2 > t1 J1 … Jm M1 t1 M2 t2 . Mm tm Optimal makespan is t1+e VCG gives makespan m·t1 t1+e > tm > … > t2 > t1

Our Result t1+e > tm > … > t2 > t1 Theorem: 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 M2 t2 . Mm tm t1+e > 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 M2 t2 . Mm tm J1 … Jm M1 t1 - e M2 t2 . Mm tm Which sets S’ satisfy ci(S’) - c’i(S’) > ci(S) - c’i(S) = em ?

Example 1 J1 … Jm M1 t1 M2 t2 . Mm tm J1 … Jm M1 t1 - e M2 t2 . Mm tm Which sets S’ satisfy ci(S’) - c’i(S’) > ci(S) - c’i(S) = em ?

Example 2 J1 J2 J3 M1 x y z M2 ? M3 J1 J2 J3 M1 x - e y - e z + e M2 ? M3 Which sets S’ satisfy ci(S’) - c’i(S’) > ci(S) - c’i(S) = e2 ?

Example 3 J1 J2 J3 M1 1 M2 J1 J2 J3 M1 e 1+e M2 1 Remarks: this almost finishes the proof of the lower bound of 2. Nisan and Ronen use “one hop” arguments, similar to this. The other lower bounds use increasingly longer hops. We use an inductive argument that enables us to identify very long hops that give us the optimal lower bound.

Overview of proof t3j> t2j > t1 >> d Proof is by induction on the number of jobs. In this overview: only 3 machines and 3 jobs. “main lemma”: in the following instance, M1 receives all jobs, where x,y in {t1,}. Proof is by induction on number of ’s. In this overview I will assume correctness for x=y=  and for x=t1 and y= , and will prove the claim for x=y=t1. J1 J2 J3 M1 x y d M2 t21 t22 t23 M3 t31 t32 t33 t3j> t2j > t1 >> d

Induction steps t3> t2 > t1 >> a >> d J1 J2 J3 M1 t1

Step 1 J1 J2 J3 M1 t1 a M2 t2 M3 t3 This induces a mechanism on three machines and two jobs and by the induction assumption the lowest machine must get both jobs.

Induction steps t3> t2 > t1 >> a >> d J1 J2 J3 M1 t1

Step 2 J1 J2 J3 M1 t1 d M2 t2 a M3 t3 J1 J2 J3 M1 d 2d M2 T2 t2 a M3 WMON Towards Contradiction: makespan = a >> 2 = optimal makespan. Thus the mechanism does not provide a finite approx ratio, a contradiction.

Induction steps t3> t2 > t1 >> a >> d J1 J2 J3 M1 t1

A contradiction to the induction hypothesis of the “main lemma” Step 3 Claim 3(a): If M1 receives either J1 or J2 then it must receive J1 and J2. Proof: J1 J2 J3 M1 t1 d M2 t2 a M3 t3 J1 J2 J3 M1 d t1+e M2 t2 a M3 t3 A contradiction to the induction hypothesis of the “main lemma” (M1 should get everything) Towards a contradiction

Step 3 Claim 3(b): If M1 receives J1 and J2 it must also receive J3. Proof: (exactly like step 2) J1 J2 J3 M1 t1 d M2 t2 a M3 t3 J1 J2 J3 M1 d 2d M2 t2 a M3 t3 WMON Towards Contradiction: makespan > a >> 2 = optimal makespan. Thus the mechanism does not provide a finite approx ratio, a contradiction.

Step 3 Claim 3(c): M1 receives either J1 or J2. Proof: otherwise find t1 < t’1 < t’’1 < t2 and d < d‘ << d‘‘ < a such that: J1 J2 J3 M1 t’1 d‘ M2 t2 a M3 t3 J1 J2 J3 M1 t’’1 d‘‘ M2 t2 a M3 t3

Step 3 Claim 3(c): M1 receives either J1 or J2. Proof: otherwise find t1 < t’1 < t’’1 < t2 and d < d‘ << d‘‘ < a such that: J1 J2 J3 M1 t’1 d‘ M2 t2 a M3 t3 J1 J2 J3 M1 t’’1 d‘‘ M2 t2 a M3 t3 By WMON, since t2 - t1 > a. J1 J2 J3 M1 t’’1 d‘‘ M2 t’1 d‘ M3 t3

Step 3 Claim 3(c): M1 receives either J1 or J2. Proof: otherwise find t1 < t’1 < t’’1 < t2 and d < d‘ << d‘‘ < a such that: J1 J2 J3 M1 t’1 d‘ M2 t2 a M3 t3 J1 J2 J3 M1 t’’1 d‘‘ M2 t2 a M3 t3 By WMON, since t2 - t1 > a. J1 J2 J3 M1 t’’1 d‘‘ M2 t’1 d‘ M3 t3 J1 J2 J3 M1 t’’1 d‘‘ M2 t’1 d‘ M3 t3 claim 3(a+b)

Step 3 Claim 3(c): M1 receives either J1 or J2. Proof: otherwise find t1 < t’1 < t’’1 < t2 and d < d‘ << d‘‘ < a such that: J1 J2 J3 M1 t’1 d‘ M2 t2 a M3 t3 J1 J2 J3 M1 t’’1 d‘‘ M2 t2 a M3 t3 By WMON, since t2 - t1 > a. J1 J2 J3 M1 t2 a M2 t’1 d‘ M3 t3 J1 J2 J3 M1 t’’1 d‘‘ M2 t’1 d‘ M3 t3 J1 J2 J3 M1 t’’1 d‘‘ M2 t’1 d‘ M3 t3 WMON claim 3(a+b)

Step 3 Claim 3(c): M1 receives either J1 or J2. Proof: otherwise find t1 < t’1 < t’’1 < t2 and d < d‘ << d‘‘ < a such that: J1 J2 J3 M1 t’1 d‘ M2 t2 a M3 t3 J1 J2 J3 M1 t’’1 d‘‘ M2 t2 a M3 t3 Contradiction to anonymity By WMON, since t2 - t1 > a. J1 J2 J3 M1 t2 a M2 t’1 d‘ M3 t3 J1 J2 J3 M1 t’’1 d‘‘ M2 t’1 d‘ M3 t3 J1 J2 J3 M1 t’’1 d‘‘ M2 t’1 d‘ M3 t3 WMON claim 3(a+b)

Induction steps t3> t2 > t1 >> a >> d J1 J2 J3 M1 t1

Final step “main lemma” Proof similar to that of step 3: J1 J2 J3 M1 t1 M2 t2 M3 t3 t1 “main lemma” 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)

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?