Non-linear objectives in mechanism design

1. Non-linear objectivesin mechanism design Shuchi Chawla University of Wisconsin – Madison FOCS 2012

2. Shuchi Chawla: Non-linear objectives So far today… • Revenue & Social Welfare This talk: • Non-linear functions of type & allocation • Question: how well can we optimize in strategic settings? Do Bayesian assumptions help?

3. Shuchi Chawla: Non-linear objectives Algorithmic mechanism design Three desiderata: • Computational efficiency • Incentive compatibility • Optimize/approximate objective Main theme in AMD: all three not always achievable together What should we give up?

4. Shuchi Chawla: Non-linear objectives AMD tradeoffs Social welfare has gap=1 Standard approximation question Overall OPT OPT-IC OPT-E Black-box OPT-IC+E Social welfare can have large gap, e.g. comb. auctions Bayesian social welfare has small gap

5. Shuchi Chawla: Non-linear objectives AMD tradeoffs Social welfare has gap=1 • Single-parameter: each agent has a single value • Monotone objectives: unilateral increase in an agent’s value causes OPT to allocate more to the agent • IC condition: unilateral increase in an agent’s value results in larger allocation Question 2: Black-box reductions for single-parameter monotone objectives Overall OPT All single-parameter “monotone” objectives have gap=1 OPT-IC OPT-E Black-box Question 1: OPT vs OPT-IC gap for multi-parameter non-linear objectives Prior-free Bayesian OPT-IC+E Bayesian social welfare has small gap (sometimes)

6. Shuchi Chawla: Non-linear objectives Rest of this talk Part I • The makespan objective • Impossibility of black-box reductions for makespan Part II • Bayesian truthful approximations for makespan • Other non-linear objectives; Open problems

7. Shuchi Chawla: Non-linear objectives Part I.1: Minimizing makespan

8. Shuchi Chawla: Non-linear objectives Scheduling to minimize makespan • n jobs, m machines • Jobs have different runtimes on different machines • Makespan= completion time of last job “Unrelated instance” Makespan

9. Shuchi Chawla: Non-linear objectives Scheduling to minimize makespan Strategic setting [Nisan Ronen’99]: • Machines are “selfish workers”; jobs’ runtimes are private • Mechanism = (schedule, payments to machines) • Machines’ objective: maximize payment – work done • Want assignment+payments to induce truthtelling

10. Shuchi Chawla: Non-linear objectives Why makespan? • Important CS problem • Captures the difficulty with non-linear objectives A single agent can disproportionately affect objective • Has received the most attention in AGT

11. Shuchi Chawla: Non-linear objectives Single-parameter makespan • Each machine has a speed; each job has a size • Runtime of job j on machine i = (size of j)/(speed of i) • Monotone objective “Related instance” Makespan

12. Shuchi Chawla: Non-linear objectives A history of prior-free scheduling • Truthful approximations for related machines • Archer-Tardos’01: constantapprox • Dhangwatnotai et al.’08: PTAS • Unrelated machines: upper & lower bounds • Nisan-Ronen’99: mapproximation • Nisan-Ronen’99: lower bound of 2 • Christodoulou et al.’07: 2.41; Koutsoupias-Vidali’07: 2.61 • Mu’alem-Shapira’07: randomized, fractional mechanisms • Ashlagi-Dobzinski-Lavi’09: lower bound of m for anonymous mechanisms Overall OPT OPT-IC OPT-E OPT-IC+E

13. Shuchi Chawla: Non-linear objectives Bayesian model for scheduling • Unrelated setting: Running time of every job on every machine drawn independently from known distribution • Related setting: Speed of every machine drawn independently from known distribution; jobs sizes fixed • Objective: Expected min makespan

14. Shuchi Chawla: Non-linear objectives Part I.2: Black-box transformations

15. Shuchi Chawla: Non-linear objectives Black-box transformations (cf. Nicole’s talk) Algorithm Allocation x Transformation Input v Payment p GOAL: for every algorithm, transformation preserves quality of solution and satisfies incentive compatibility.

16. Shuchi Chawla: Non-linear objectives Black-box transformations • Social welfare: can transform any approx. algorithm into BIC mechanism with “no” loss in expected performance. [Hartline-Lucier’10, Hartline-Kleinberg-Malekian’11, Bei-Huang’11] • Is this possible for other objectives? • Makespan: For any polytime BIC transformation, there is a makespan problem and algorithm such that mech.’s expected makespanis polynomially larger than alg.’s. [C.-Immorlica-Lucier’12] NO!

17. Shuchi Chawla: Non-linear objectives Single-parameter makespan n jobs,size of job j is xj x4 x1 x2 x3 collection F of feasible assignments v1 • v2 • v3 • v4 • v5 • v6 • v7 • v8 m machines,machine i has speed vi ~ Fi

18. Shuchi Chawla: Non-linear objectives Proof outline • Define makespan instance (feasibility constraints, value distribution). • Find algorithm with low expected makespan. • Use monotonicity condition to show any BIC transformation has high expected makespan. Key issue: Transformation must rely on algorithm to understand/satisfy feasibility constraint That is, transformation must return an allocation that it observes the algorithm return Higher speed ⇒ higher expected load

19. Shuchi Chawla: Non-linear objectives Makespan Instance m1/2 jobs, large size xj = α x1 x2 xk m/2 jobs, small size xj = 1 x1 x2 xm/2 feasibility set F = {at most one job per machine} v1 • v2 • v3 • v4 • v5 • v6 • v7 • v8 m machines, speeds vi ~ Uniform{low = 1, high = α}

20. Shuchi Chawla: Non-linear objectives Approximation Algorithm • If (m/2 ± m3/4) machines report high speed, • assign large jobs to fast machines (at random) • assign small jobs to slow machines (at random) • assign NO job to all remaining machines • Else • assign all jobs randomly

21. Shuchi Chawla: Non-linear objectives Approximation Algorithm high speeds low speeds Note 1: By Chernoff, expected makespan is low. Note 2: Expected allocation is not monotone.

22. Shuchi Chawla: Non-linear objectives Transformation To fix non-monotonicity, must more often: • allocate nothing to low speed machines, or • allocate something to high speed machines.

23. Shuchi Chawla: Non-linear objectives Transformation Input v: 1 • 1 • 1 • 1 • α • α • α • α Each “fast” machine gets large job with probability m-1/2 α • α • α • α • 1 • 1 • 1 • 1 Query v’: pretend some low machines are high and vice versa... then with high probability, makespan is high.

24. Shuchi Chawla: Non-linear objectives Transformation Input v: 1 • 1 • 1 • 1 • α • α • α • α Each machine gets large job with probability m-1/2 1 • 1 • 1 • 1 • 1 • 1 • 1 • 1 Query v’: pretend number of high machines deviates from expectation.. then with high probability, makespan is high.

25. Shuchi Chawla: Non-linear objectives Recap and other results [C.-Immorlica-Lucier’12] • For any BIC transformation, there is an alg. such that the transformation’s makespan is polynomially larger than the algorithm’s • even when the algorithm is a constant approximation • What about other non-linear functions? • Ironing doesn’t work • Gap increases with non-linearity

26. Non-linear objectivesin mechanism design Shuchi Chawla University of Wisconsin – Madison Part II

27. Shuchi Chawla: Non-linear objectives Recap of part I • A representative non-linear objective: makespan • Black-box transformations are essentially impossible for makespan: objective function increases by polynomial factor Overall OPT OPT-IC OPT-E Black-box OPT-IC+E

28. Shuchi Chawla: Non-linear objectives Part II.1: Bayesian approximation for makespan

29. Shuchi Chawla: Non-linear objectives Recall: scheduling to minimize makespan • n jobs, m machines • Jobs’ runtimes drawn from known indep. distributions • Makespan= completion time of last job • Prior-free setting: any anonymous truthful mechanism is at best an m approximation. Makespan

30. Shuchi Chawla: Non-linear objectives A truthful mechanism: MinWork For every job: • Assign the job to the machine that reports the lowest runtime • Pay the machine the job’s running time on its “second best” machine m’ • “Second-price” payments: induce truthtelling • Makespan ≤ sum of best runtimes of all jobs ≤ total work done in optimal schedule ≤ m x optimal makespan ⇒ m-approximation to makespan

31. Shuchi Chawla: Bayesian scheduling Overcoming the lower bound • Ashlagi et al.’s lower bound of m for makespan • Ordered instance: machine i is better than machine i+1 for all jobs • Running times within 1+eps of each other • Any truthful mechanism must allocate all jobs to machine 1 • How do Bayesian assumptions help? • Knowledge of distribution => we can penalize allocations that are always bad for the given instance • A priori identical machines: bad instances have extremely low probability

32. Shuchi Chawla: Non-linear objectives Prior-independent approximation (cf. Tim’s talk) • Unknown Bayesian prior, but belongs to some “nice” family • In particular, the runtime of a job j is identically distributed on every machine. • That is, machines are a priori identical • However, any instantiation of runtimes is an unrelated instance • Result: There exists a truthful prior-independent mechanism that achieves an O(n/m) approximation to expected makespan (*) [C.-Hartline-Malec-Sivan’12]

33. Shuchi Chawla: Non-linear objectives Benchmark • Hindsight OPT • For any instantiation, finds the optimal makespan • OPT1/2 • Discards m/2 machines randomly • For any instantiation, finds optimal makespan over remaining machines • For many distributions, OPT1/2 ~ constant. OPT • Key property: min over 2 draws ~ 2 times a single draw • Includes all “MHR” distributions, e.g. uniform, exponential, normal,…

34. Shuchi Chawla: Non-linear objectives How to design a truthful multi-parameter mechanism? • A simple powerful class: affine maximizers Maximize an appropriate linear a.k.a. affine function • Essentially, an extension of VCG • For example: • Can assign “costs” to some outcomes, and, minimize total (work – cost) • Can forbid certain outcomes by setting cost = ∞ • Can assign more weight to the work of some agents than that of others

35. Shuchi Chawla: Non-linear objectives The MinWorkmechanism again • Essentially VCG: schedule every job on its best machine • Observe: job j’s runtime in MinWork ≤ job j’s runtime in OPT Furthermore, every job goes to a random machine • If jobs were to be distributed uniformly across machines, we would get good makespan • However, balls-in-bins analysis ⇒ some machine has O(log m/log log m) jobs

36. Shuchi Chawla: Non-linear objectives The MinWork(k) mechanism • Find a min-size matching between jobs and machines that assigns at most k jobs to each machine. • Claim: MinWork(k) is truthful • Proof: It is VCG over a restricted domain.

37. Shuchi Chawla: Non-linear objectives The MinWork(k) mechanism • Find a min-size matching between jobs and machines that assigns at most k jobs to each machine. • Claim: MinWork(k) is truthful • Claim: MinWork(10) gets a constant approximation • Obs1: The schedule is almost balanced • Obs2: Every job still goes to roughly its best machine

38. Shuchi Chawla: Non-linear objectives Obs2: the last entry procedure • Fix job j and imagine adding it last in a greedy fashion. Machine full 1 1 2 2 Space available, so j goes here 3 3 4 4 5 5 6 6 MinWork(3) schedule for all but job j MinWork(3) schedule sorted by j’s preferences

39. Shuchi Chawla: Non-linear objectives Obs2: The last entry procedure • Fix job j and imagine adding it last in a greedy fashion. • The probability that j goes to one of its top i machines is at least 1-(1/k)i • MinWork(k) places j in an even better position • Key claim: Placing j on its ith best machine is no worse than placing 5i independent copies of j on their best (of n/2) machines

40. Shuchi Chawla: Non-linear objectives MinWork(10) analysis Job j’s runtime in MinWork(k) ≤ max5^i independent copies j’s runtime in OPT1/2 ≤ 5itimes j’s runtime in OPT1/2 Here i is an exponential random variable; Note: E[5i] = constant. ∴ MinWork(10)’s makespan ≤ 10 E[maxj (j’s runtime in MW)] ≤ constant times OPT1/2 Stochastic dominance

41. Shuchi Chawla: Non-linear objectives Key technical claim • Placing j on its ith best machine is no worse than placing 5i independent copies of j on their best (of n/2) machines Expt. 1 Expt. 2 n/2 copies of j’s runtime n copies of j’s runtime 5i/2 blocks ith min over n copies max over 5imins over n/2 copies

42. Shuchi Chawla: Non-linear objectives Recap and other results [C.-Hartline-Malec-Sivan’12] • Machines a priori identical, “few” jobs • O(1) prior-independent approximation: MinWork(k) ≤ O(1) OPT1/2 • Compare to Bulow-Klemperer’s result for revenue with k items: VCG ≥ O(1) OPTless k agents • Jobs are also a priori identical: multi-stage mechanisms • Prior-ind.O(√log m) approximation to OPT1/2 • Prior-ind.O((log log m)2)approx to OPT for MHR distributions • Hindsight-OPT1/2 (needs regularity)

43. Shuchi Chawla: Non-linear objectives Part II.2: Other objectives & open problems

44. Shuchi Chawla: Non-linear objectives Open problems for makespan • O(1) prior-ind. approximation for non-identical jobs • Bayesian approximation for non-identical machines • Will need to use the knowledge of prior • Even logarithmic approx is non-trivial • A potential approach: charge a prior-dependent amount for placing each additional job on a machine • Approximation for small-support priors • LP based?

45. Shuchi Chawla: Non-linear objectives Other non-linear objectives • Max-min fairness in scheduling a.k.a. load balancing • Prior-free PTAS for related setting [Epstein-van Stee’10] • Unrelated approximation? • Max-min fairness in welfare a.k.a. the Santa Claus problem • Not monotone! • Single-parameter Bayesian approx? Min makespan 10 3 3 2

46. Shuchi Chawla: Non-linear objectives Conclusions • Non-linear objectives in general much harder than social welfare • Mild stochastic assumptions can help us circumvent strong impossibility results • Multi-parameter mechanisms are difficult to understand, but “affine maximizers” is a powerful subclass. • Lots of nice open problems!