On Scheduling Mechanisms Theory Practice and Pricing Ahuva

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

Motivation • Mechanisms ≈ auctions & reverseauctions ≈ optimization problems with strategic constraints

Scheduling Problem • n jobs to be assigned to m machines • tij = 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]

j 1 j 2 m 1 0. 5 0. 6 m 2 6 1 m 3 12 2 • 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”

The Mechanism Design Problem • n jobs to be assigned to m strategic machines • Machine i has a private cost ci(j) = tij • 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 • Job-by-Job Mechanism [NR]: Assign each job to the fastest machine and pay the 2 nd cheapest cost • Example: m 1 gets 3 jobs, and is “truthfully” paid 3, resulting in a makespan of 3 -3ε; the optimal is 1 • j 1 j 2 j 3 m 1 1 -ε 1 -ε m 2 1 1 1 m 3 1+δ 1 1 Can we do better w. r. t makespan?

Multi-Parameter 2 machines m machines Deterministic GT-UB is 2 GT-UB is m GT-LB is 2. 61 tight A huge gap! [Nisan, Ronen ‘ 99], [Christodoulou, Koutsoupias, Vidali ‘ 07], [Koutsoupias, Vidali ‘ 07] GT –UB = game theoretic upper bound, LB = lower bound

Multi-Parameter 2 machines m machines Deterministic GT-UB is 2 GT-UB is m GT-LB is 2. 61 tight 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 DM over truthful deterministic mechanisms (“with the same DM 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: Weak-Monotonicity: • Find a probability distribution DC over machine costs on which any truthful deterministic mechanism fails to provide the expected approximation of 2 -1/m w. r. t makespan • Theorem [BCRMNS ‘ 06]: If M(ALG, p) is a truthful mechanism then for every costs ci, di, c-i it holds that ci(Si) + di(Ti) ≤ di(Si) + ci (Ti) where ALG(ci, c-i) = Si and ALG(di, c-i) = Ti

Proof Idea Yao’s Principle: Weak-Monotonicity: • Find a probability distribution over inputs on which any truthful deterministic mechanism fails to provide the expected approximation ratio of 21/m w. r. t makespan • Thm[Roberts 79], [Rochet 87]: If M(ALG, p) is a truthful mechanism, then for any costs ci, di, c-i it holds that ci(Ci) + di(Di) ≤ di(Ci) + ci(Di) where the subset of jobs Ci, Di are defined by ALG(ci, c-i) = Ci and ALG(di, c-i) = Di

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

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

Randomized Lower Bound m 1 m 2 j 1 j 2 j 3 4 99/ε 4 4 j 1 j 2 j 3 4 99/ε ε 4+ε j 1 j 2 j 3 ε 99/ε 4+ε 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 weakmonotonicity), the expected ratio then is at least: p 2 · 8 / (4+2ε) + p 3 · 1 = 3/2 - ε’

Randomized Lower Bound m 1 m 2 j 1 j 2 j 3 4 99/ε 4 4 j 1 j 2 j 3 4 99/ε ε 4+ε j 1 j 2 j 3 ε 99/ε 4+ε 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’

Randomized Lower Bound m 1 m 2 j 1 j 2 j 3 4 99/ε 4 4 j 1 j 2 j 3 4 99/ε ε 4+ε j 1 j 2 j 3 ε 99/ε 4+ε 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 Deterministic Randomized 2 machines m machines UB is 2 UB is m GT-LB is 2. 61 UB is 1. 5963 UB is (m+5)/2 GT-LB 1. 5 GT-LB 2 -1/m [Nisan, Ronen ‘ 99] [Christodoulou, Koutsoupias, Vidali ‘ 07] [Koutsoupias, Vidali ‘ 07] 1. 75 → 1. 67 → 1. 5963 [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(Si) - ci(Si) ≥ p(Sk) - ci(Sk) for every 1≤ i, k ≤m, where ALG(c) = (S 1, S 2, …, Sm) [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(Si) - ci(Si) ≥ p(Sk) - ci(Sk) for every 1≤ i, k ≤m, where ALG(c) = (S 1, S 2, …, Sm) • 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 CS-GT-UB is 2 Deterministic Multi-Parameter GT-UB is 3/2 CS-GT-UB is log(m) GT-LB is 3/2 GT-LB is log(m)/loglog(m) Deterministic Single-Parameter Randomized Multi-Parameter CS-GT-UB is 1+ε GT-LB is 1 ? ? [M ’ 09] [Hartline, Ieong, Schapira, Zohar ’ 09] [Cohen, Feldman, Fiat, Kaplan, Olonetsky ’ 10] [M ‘ 09] + preventing simple post-auction resales in quasi-poly time

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 Archive @HUJI

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

Setting (cont. ) The utility of job j is uj = -wj Fj - pj • • • Fj is the flow time: duration from arrival to completion pj > 0 is the payment of job j uj < 0, [Heydenreich, Muller and Uetz ’ 06]. Remark 1: tj and wj are independent Remark 2: to generate wj we used a bimodal distribution

The SRG Model • Honest Arrivals and Runtimes • Big Conservative Group: 90% of the users always declare wj [0. 9 wj , wj ] uniformly at random • Small Aggressive Group: 10% of the users declare wj [0. 1 wj , wj ] 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

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

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

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

• Nash-Equilibrium for 1 st 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 ).

Conclusions (Empirical Part) • 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.

Thank You