On Scheduling Mechanisms: Theory, Practice and Pricing Ahuva Mu’alem SISL, Caltech

Motivation Mechanisms ≈ auctions & reverse- auctions ≈ optimization problems with strategic constraints

Scheduling Problem n jobs to be assigned to m machines t ij = time required to process job j on machine i Goal: Minimize the maximum load (“makespan”) It’s a well-studied NP-hard problem with [1.5, 2] approximability lower and upper bounds [Lenstra, Shmoys, Tardos’87]

Example: 2 jobs, 3 machines The optimal allocation has a makespan of 1 Any other allocation has makespan > 1 Machine m 2 is related to m 3 but not to m 1 rank > 1 is called “multi-dimensional” j1j1 j2j2 m1m m2m2 61 m3m3 122

The Mechanism Design Problem n jobs to be assigned to m strategic machines Machine i has a private cost c i (j) = t ij Goal: Design a scheduling algorithm ALG and a compensation function p (payment) such that the mechanism M(ALG, p) minimizes the makespan in a truthful manner (reporting its true private cost is a dominant strategy for any strategic machine, assuming quasi-linearity)

In their seminal paper [Nisan, Ronen ’99] asked: How well this goal can be approximated in a TRUTH-TELLING manner? The single-dimensional case is solved! A deterministic truthful (1+ε)-approximation mechanism exists in time polynomial(m,n), if all machines are related [Archer, Tardos ’01], [Auletta et al. ’04], [Andelman et al. ’05 + ’07], [Kovacs ’05 + ’07], [Dhangwatnotai, Dobzinski, Dughmi, Roughgarden ‘08], [Christodoulou, Kovacs ’10]

The Multi-Dimensional Case

Deterministic Truthful Mechanism Example: m 1 gets 3 jobs, and is “truthfully” paid 3, resulting in a makespan of 3-3ε; the optimal is 1 Can we do better w.r.t makespan? Job-by-Job Mechanism [NR]: Assign each job to the fastest machine and pay the 2 nd cheapest cost j1j1 j2j2 j3j3 m1m1 1-ε m2m2 111 m3m3 1+δ11

Multi-Parameter2 machinesm machines DeterministicGT-UB is 2 GT-LB is 2  tight GT-UB is m GT-LB is 2.61  A huge gap! [Nisan, Ronen ‘99], [Christodoulou, Koutsoupias, Vidali ‘07], [Koutsoupias, Vidali ‘07] GT –UB = game theoretic upper bound, LB = lower bound

Multi-Parameter2 machinesm machines DeterministicGT-UB is 2 GT-LB is 2  tight GT-UB is m GT-LB is 2.61  A huge gap! [Nisan, Ronen ‘99], [Christodoulou, Koutsoupias, Vidali ‘07], [Koutsoupias, Vidali ‘07] Specifically, the OPTIMAL algorithm w.r.t makespan cannot be truthfully implementable. This justify our focus on APPROXIMATION algorithms

Truthful Randomized Mechanisms Definition: A truthful randomized mechanism is a probability distribution D M over truthful deterministic mechanisms (“with the same D M for every declared cost”) Examples: (1) “Random Dictator”; (2) Run the Job-by-Job mechanism on 2 machines selected uniformly at random

Randomized Lower Bounds Thm [M, Schapira]: Any truthful randomized mechanism for minimizing the makespan cannot achieve approximation ratio better than 2-1/m. The same holds for truthfulness in expectation (using a different proof technique). Remark: very few GT-LBs are known for randomized truthful mechanisms

Proof Idea Yao’s Principle: Find a probability distribution D C over machine costs on which any truthful deterministic mechanism fails to provide the expected approximation of 2-1/m w.r.t makespan Weak-Monotonicity: Theorem [BCRMNS ‘06]: If M(ALG, p) is a truthful mechanism then for every costs c i, d i, c -i it holds that c i (S i ) + d i (T i ) ≤ d i (S i ) + c i (T i ) where ALG(c i, c -i ) = S i and ALG(d i, c -i ) = T i

Proof Idea Yao’s Principle: Find a probability distribution over inputs on which any truthful deterministic mechanism fails to provide the expected approximation ratio of 2- 1/m w.r.t makespan Weak-Monotonicity: Thm[Roberts79],[Rochet87]: If M(ALG, p) is a truthful mechanism, then for any costs c i, d i, c -i it holds that c i (C i ) + d i (D i ) ≤ d i (C i ) + c i (D i ) where the subset of jobs C i, D i are defined by ALG(c i, c -i ) = C i and ALG(d i, c -i ) = D i

ALG(c 1, c 2 ) Approximations ALG(d 1, c 2 ) ALG(c 1, d 2 )

ALG(c 1, c 2 ) Truthful Approximations ALG(d 1, c 2 ) ALG(c 1, d 2 )

j 1 j2j2 j3j3 m1m1 499/ε 4 m2m2 4 4 Randomized Lower Bound j 1 j2j2 j3j3 m1m1 499/ε 4 m2m2 ε 4+ε j 1 j2j2 j3j3 m1m1 ε99/ε 4+ε m2m2 99/ε 4 4 The probability of each input is: p 1 = ε, p 2 = p 3 = (1-ε)/2. Case 1: If the deterministic mechanism on the first input has a sub-optimal makespan the expected ratio then is at least p 1 ·(99 /ε)/8 > 3/2 Case 2: Otherwise, suppose wlog it allocates j 1 to m 1, and j 2, j 3 to m 2, the best it can do on the third input is a makepan of 8 (without violation of weak- monotonicity), the expected ratio then is at least: p 2 · 8 / (4+2ε) + p 3 · 1 = 3/2 - ε’

j 1 j2j2 j3j3 m1m1 499/ε 4 m2m2 4 4 Randomized Lower Bound j 1 j2j2 j3j3 m1m1 499/ε 4 m2m2 ε 4+ε j 1 j2j2 j3j3 m1m1 ε99/ε 4+ε m2m2 99/ε 4 4 The probability of each input is: p 1 = ε, p 2 = p 3 = (1-ε)/2. Case 1: If the deterministic mechanism on the first input has a sub-optimal makespan, the expected ratio then is at least p 1 · (99 /ε) / 8 > 3/2 Case 2: Otherwise, suppose wlog it allocates j 3 to m 2, the best it can do on the third input is a makepan of 8 (without violation of weak-monotonicity), the expected ratio then is at least: p 2 · 8 / (4+2ε) + p 3 · 1 = 3/2 - ε’

j 1 j2j2 j3j3 m1m1 499/ε 4 m2m2 4 4 Randomized Lower Bound j 1 j2j2 j3j3 m1m1 499/ε 4 m2m2 ε 4+ε j 1 j2j2 j3j3 m1m1 ε99/ε 4+ε m2m2 99/ε 4 4 The probability of each input is: p 1 = ε, p 2 = p 3 = (1-ε)/2. Case 1: If the deterministic mechanism on the first input has a sub-optimal makespan, the expected ratio then is at least p 1 · (99 /ε) / 8 > 3/2 Case 2: Otherwise, suppose wlog it assigns j 3 to m 2, the best it can do on the third input is a makepan of 8 (without violation of weak-monotonicity), the expected ratio then is at least: p 2 · 8 / (4+2ε) + p 3 · 1 = 3/2 - ε’

Multi-Parameter 2 machinesm machines DeterministicUB is 2 GT-LB is 2 UB is m GT-LB is 2.61 [Nisan, Ronen ‘99] [Christodoulou, Koutsoupias, Vidali ‘07] [Koutsoupias, Vidali ‘07] RandomizedUB is GT-LB 1.5 UB is (m+5)/2 GT-LB 2-1/m 1.75 → 1.67 → [Nisan, Ronen ’99] [M, Schapira’07] [Lu, Yu ’08] The upper bound of (m+5)/2 is built on [Christodoulou, Koutsoupias, Kovacs ’07] [Shmoys, Tardos ‘93]

Envy-Free Design M(ALG, p) is an envy-free design if p(S i ) - c i (S i ) ≥ p(S k ) - c i (S k ) for every 1≤ i, k ≤m, where ALG(c) = (S 1, S 2, …, S m ) [M’09] How well the makespan can be approximated in an ENVY-FREE manner (“no agent is willing to exchange his allocated bundle and payment with any other agent”)?

Envy-Free Design M(ALG, p) is an envy-free design if p(S i ) - c i (S i ) ≥ p(S k ) - c i (S k ) for every 1≤ i, k ≤m, where ALG(c) = (S 1, S 2, …, S m ) Motivation: – BI-CRITERIA optimizations with INDIVIDUAL-LEVEL GUARANTEE – Envy-freeness can lead to dominant strategy mechanisms (e.g., Ascending Auctions with Budgets [Aggarwal et al.‘09]) – Study Algorithms & Pricing for multi-parameter problems

2 machines m machines Deterministic Multi-Parameter CS-GT-UB is 2 GT-UB is 3/2 GT-LB is 3/2 CS-GT-UB is log(m) GT-LB is log(m)/loglog(m) [M ’09] [Hartline, Ieong, Schapira, Zohar ’09] [Cohen, Feldman, Fiat, Kaplan, Olonetsky ’10] Deterministic Single-ParameterCS-GT-UB is 1+ε GT-LB is 1 [M ‘09] + preventing simple post-auction resales in quasi-poly time Randomized Multi-Parameter? ?? ??

Commercial Clouds Simulations on Real-Data: measure “average-case” scenarios, also allow us to study several aspects simultaneously

Mechanism Design Challenges Provider’s Goals: Revenue and Quality of Service vs. Users’ Strategic Behavior » On-Line Setting: jobs/tasks arrive over time » Uncertainties about run time Our Approach [Shudler, Amar, Barak, M. ‘10]: Simulation-based analysis performed on real data taken from The Parallel Workload

Homogeneous Cluster with identical machines Each user submits a single job The type of job j is denoted by: ( r j, t j, w j ) – r j > 0 is the release time (“arrival time”) – t j > 0 is the running time (unknown to the user) – w j > 0 is the value per unit time of delay Setting

The utility of job j is u j = -w j F j - p j F j is the flow time: duration from arrival to completion p j > 0 is the payment of job j u j < 0, [Heydenreich, Muller and Uetz ’06]. Remark 1: t j and w j are independent Remark 2: to generate w j we used a bimodal distribution Setting (cont.)

The SRG Model Honest Arrivals and Runtimes Big Conservative Group: 90% of the users always declare w j  [0.9 w j, w j ] uniformly at random Small Aggressive Group: 10% of the users declare w j  [0.1 w j, w j ] uniformly at random. Aggressive users respond to incentives

► Stability Analysis We formulate a simple one-shot game to model the dynamic interaction between the provider and an aggregate consumer playing on behalf of the aggressive users We then look for a Nash-equilibrium in this restricted game

HB Algorithm: Upon any job arrival or termination, preempt all running jobs and run the waiting jobs with the highest declared w j HBNP Algorithm: Upon any job termination run the waiting job with the highest declared w j. WSPT Algorithm: Upon any job termination run the waiting job with the highest declared w j / t j. – Remark: WSPT has informational advantage by knowing t j. Algorithms

Aggregate user’s payoff is the summation of all aggressive user utilities: ∑u j The “β%” Strategy means that w j  [(1- β) w j, w j ] uniformly at random. Every line has a single best response ! (marked above in red) The k-th price best responses are more “truthful” !

The Provider’s payoff is a function of the Total Revenue and the QoS (total weighted flow time): REV* = ∑ p j / ∑ w j F j REV* nicely behaves: Left column always has the highest values and right column has the smallest values. 1st Price: HB is the best w.r.t. QoS, NPHB is the worst.

Nash-Equilibrium for 1st price is [HB, 50%] with a near-optimal REV* (0.976 ≈ 1.000). Nash-Equilibrium for k-th price (ignoring the WSPT) is [NPHB, 25%]: using a non-socially optimal scheduler increases REV* (prices increase when many high value jobs are delayed in the waiting queue. The aggregate consumer is almost truthful in the NE ).

We introduced the SRG Model: a simple behavioral model to study scenarios with inherent uncertainties. We modeled the dynamic on-line interaction between the provider and consumers as a one shot game and showed the existence of (arguably good) unique ‘pure’ symmetric Nash Equilibrium. Future Work: – Non-linear value and utility models. – Strategic impact of budgets (runtime uncertainty causes unpredicted payments). – Competition among providers in a more direct manner. Conclusions (Empirical Part)

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