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Algorithms for Incentive-Based Computing

Algorithms for Incentive-Based Computing. Carmine Ventre Università degli Studi di Salerno. … or Merging Research of Different Fields. Computer Science. Economics. “ Worst-case equilibria ” by E. Koutsoupias, C. H. Papadimitriou in STACS ‘99. Auctions. 10. A. 7. 6. B. 6.

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Algorithms for Incentive-Based Computing

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  1. Algorithms for Incentive-Based Computing Carmine Ventre Università degli Studi di Salerno

  2. … or Merging Research of Different Fields Computer Science Economics “Worst-caseequilibria” by E. Koutsoupias, C. H. Papadimitriou in STACS ‘99

  3. Auctions 10 A 7 6 B 6 First price sealed bidauction Problems? It is not truthful (e.g., auctioneer can not maximize his own revenue)

  4. Vickrey Auctions 10 A Bid 8 Bid 12 9 11 Utility is 0 in place of 1 (= 10 – 9) B Utility is -1 (= 10 – 11) in place of 0 Second price sealed bid auction This is truthful and generalizes to the concept of mechanism

  5. Mechanisms • Augment an algorithm with a payment function • i.e., design a truthful mechanism • The payment function should incentive in telling the truth 3 10 1 1 s 2 2 1 3 7 7 4 1

  6. M = (A, P) VCG Mechanisms valuation Utility(3) = payment(3) – cost(3) = 3 – 3 = 0 Utility(9) = payment(9) – cost(9) = 9 – 3 = 6 9 3 10 1 1 2 s 2 1 3 7 4 7 Ae=0 = Ae – be 1 Pe = Ae=∞– Ae=0 if e is selected (0 otherwise) Pe= beif e is selected (0 otherwise) M is truthful iff A is optimal Algorithmicmechanism design by N. Nisan and A. Ronen in STOC ’99 (GEB ‘01)

  7. Vickrey Auction (& VCG Mechanism) Weakness (or Cui Prodest?) • It works only for utilitarian problems: i.e., maximizes the social welfare (e.g., it does not maximize seller revenue) • Adaptation to non-utilitarian problems • Verification Model • It is not budget balanced • Cost-Sharing Budget Balance Mechanisms • It is vulnerable to collusion • Cost-Sharing Budget Balance Mechanisms • Verification model • … (not here) (skip) Utilitarian problems: objective is to maximize the social welfare (ivaluationi(X)) BB mechanisms: sum of payments equals the cost of the solution

  8. Cost-Sharing Mechanisms

  9. real worth is 7 Multicast and Cost-Sharing Accept or reject the service? • A service provider s • Selfish customers U • Who is getting the service? • How to share the cost? is worth 5 ( 7) Pi

  10. Selfish Agents • Each customer/agent • has a private valuation vi for the service • declares a (potentially different) valuation bi • pays Pi for the service • Agents’ goal is to maximize their own utility: ui(bi) := vi – Pi(bi) Accept iff my utility ≥ 0!

  11. M = (A, P) Coping with Selfishness: Mechanism Design • Algorithm A • Who gets serviced (Q(b)) • How to reach Q(b) (Construct tree T) • Payment P • How much each user pay P1 bj bi P2 P4 P3

  12. M = (A, P) M’s Truthfulness (or Strategyproofness) vi For all others players’ declarations b-i it holds ui= ui(vi, b-i) ≥ ui(bi, b-i) = ui for all bi (ie, truthtelling is a dominant strategy)

  13. M’s Group Strategyproofness U Coalition C No one gains At least one looses (ie, ui > ui) C is useless Breaks off C Does this definition fit our intuition of collusion-resistant mechanisms?

  14. Mechanism’s Requirements • Budget Balance (BB) • i T Pi(b) = COST(T) • … (natural “economic” requirements)

  15. Cost-Sharing Budget-Balance Mechanisms [Penna & V, WAOA ’04] [Penna & V, SIROCCO ’05] [Penna & V, STACS ’06]

  16. How to build BB, GSP Mechanisms Cost-sharing methods: distribute COST(Q) among users in Q (Q,i)  0 Q U (Q,i) = 0, i  Q (Q,i) = COST(Q) Idea: associate prices to service set

  17. How to build BB, GSP Mechanisms Cost-sharing method (•,•) Mechanism M() U Q (Q,i) Drop i > bi

  18. (Q1,i) Q1=U Q2 Changes (Qk,i) (Q2,i) Prices do not decrease (Q3,i) Q3 Group Strategyproof … U Qk How to build BB, GSP Mechanisms Cost-sharing method (•,•) Mechanism M() … Monotonicity… Pi = (Qk,i) [Moulin & Shenker ’97] & [PV04] Self Cross… Cross… … for all Q subsets of U … for all Q (possibly) outputted by M

  19. Self cross monotonicity: an example COST(Q) 50% 50% s Q s Pay less than before This is not a cross monotonic cost sharing method!

  20. Self cross monotonicity: an example (2) COST(Q) 100% s This is not a cross monotonic cost sharing method! Q This guy pays 0 s M() cannot drop him Pay less than before Idea: some subsets do not “appear”. We need  monotone only for possible subsets generated by M()

  21. Q1=U Q2 Q3 … U … Q|U| Sequential Algorithms • A is sequential if for some bid vectors reaches a chain of sets Q1, …, Q|U|, Q|U|+1=Ø • Sequential algorithms admits a self cross-monotonic cost-sharing method . . . BB & GSP Mechanisms Q|U|+1=Ø

  22. s s s s U MST(Q) MST Q u pay    v v prune u Q > T* Optimal Sequential Algorithm for Steiner Tree Game T + = opt s Q Q u v v T +  +  opt Steiner tree v is the last node added by Prim’s MST algorithm

  23. s s U MST(Q) MST pay prune Q Adding Fairness to Our Mechanisms • Payment is still self cross-monotonic • Is it possible to have no free rider? • No! Unless P=NP opt Steiner tree

  24. M = (A, P) Can we do better without Sequential Algorithms? M is SP, BB, … M for 2 users A is sequential “Natural” GSP Mechanisms A is sequential

  25. Mechanisms with Verification [Ferrante, Parlato, Sorrentino & V, WAOA 2005] [Auletta, De Prisco, Penna, Persiano & V, ICALP 2006] [V, WINE 2006] [Penna & V. , 2007]

  26. Motivating Verification Model Used Car market: The Kelley Blue Book – the Trusted Resource (www.kbb.com)

  27. The Trusted Resource Time is trusted… … unless a time machine will be created Can we engage a trusted resource within a mechanism allowing (somehow) bids verification?

  28. no VCG! Selfish Task Scheduling Awarded independently from the execution! Mechanism design: payments  utility = payment - cost Optimal Makespan: minx maxi ti(X) AllocationX  cost = ti(X) = ti• loadi(X) M1 M2 M3 M4 M5 b1 b2 b3 b4 b5 t1 t2 t3 t4 t5 ti = 1 / si (ie, the inverse of the speed)

  29. Verifiable Selfish Agents Verification = observe jobs’ release time 3 Verification is impossible! ti(X) = loadi(X) •ti i bids from the set {1/2, 1, 2} 1 1/2 i underbids i’s release time should be 2 but… … i’s finishing time is 4 ti= 1 i can wait 2 time slots delivering the results in the right time 1 i overbids 1 2 IDEA ([Nisan & Ronen, 99]): No payment for underbidding agents

  30. The Power of Verification Classical mechanisms Mechanisms w/ Verification algorithms loadi loadi NO! NO! TRUTHFUL TRUTHFUL [Archer & Tardos, ‘01] [Auletta & al, ‘04] bi bi Payment functions Not unique loadi Unique Pi(bi, b-i)=Wmax/ bi (= Wmax •si) Related to max possible supported cost Scaling up for general speeds bi ti [Archer & Tardos, ‘01]

  31. The Power of Verification: Breaking Lower Bounds Efficient APX truthful mechanisms w/verification: c-APX algorithm A c(1+)-APX mechanism weight p2 p3 p4 p5 p6 p7 p8 p9 p1 priority M1 M2 M3 M4 M5 b1 b2 b4 b5 t3 b3 t1 t2 t4 t5 Goal: Design a polytime truthful mechanism optimizing the weighted completion time (ie, weighted sum scheduling) No 1.54-apx truthful mechanism without verification [Archer & Tardos, 2001] (1+)-APX truthful mechanism w/ verification for a constant number of machines

  32. … … Jj Jn J1 bi1 bij bin … … M1 Mi Mm b1 bm bi agentk agent1 … agenth … (Optimal) Mechanisms with Verification Breaking lower bounds for classical mechanisms concerning many natural problems (eg, variants of SPT problem) … … Given an algorithm c-apx… Goal: minimizing the makespan a c(1+)-apx an exact There exists truthful mechanism with verification We don’t if truthful mechanisms without verification do exist polytime (althougt not polynomial-time)

  33. Optimal Collusion-Resistant Mechanisms w/ Verification GSP do not consider side payments U Coalition C Collusion-Resistant mechanisms are impossible unless using posted-price ([Goldberg & Hartline, 2005]) If OPT is truthful via VCG mechanism without verification + Exists a VCG-like payment function such that OPT is collusion-resistant with verification –

  34. Conclusions • Cost-Sharing Games • Simple techniques… • … lead to polynomial-timecost-sharing mechanisms for NP-Hard problem Steiner Tree • … not so unfair (unless P=NP) • … characterize natural class of cost-sharing mechanisms • Mechanisms with Verification • More powerful model… • … breaking known lower bounds for natural problems • … dealing with a strong notion of agents’ collusion

  35. Further Research • Cost-Sharing Mechanisms • Full characterization • What is the power of not “natural” mechanisms? • Price of Fairness • Tradeoff between budget balance and efficiency • Mechanisms with Verification • What is the real power of verification? • Does the revelation principle hold in the verification setting? • Different definitions for the verification paradigm (e.g., Nisan&Ronen 99)

  36. Questions?

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