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

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

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

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

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

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

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) Utilitarian problems: objective is to maximize the social welfare (  i valuation i (X)) BB mechanisms: sum of payments equals the cost of the solution (skip)

8. Cost-Sharing Mechanisms

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

10. Selfish Agents Each customer/agent  has a private valuation v i for the service  declares a (potentially different) valuation b i  pays P i for the service Agents’ goal is to maximize their own utility: u i (b i ) := v i – P i (b i ) Accept iff my utility ≥ 0!

11. 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 M = (A, P) bibi bjbj P1P1 P4P4 P3P3 P2P2

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

13. M’s Group Strategyproofness U Coalition C No one gains At least one looses (ie, u i > u i ) 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 P i (b) = COST(T) Efficiency (NW): maximize NET WORTH(T) := WORTH(T) - COST(T) where WORTH(T):=  i  T v i … (natural requirements) BB and efficiency are mutually exclusive!

15. Mechanism’s Requirements Budget Balance (BB)   i  T P i (b) = COST(T) … (natural “economic” requirements)

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

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

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

19. U Q 1 =U How to build BB, GSP Mechanisms Q3Q3 QkQk … Q2Q2  (Qk,i) (Qk,i)  (Q2,i) (Q2,i)  (Q3,i) (Q3,i) P i =  (Q k,i) … Monotonicity… [Moulin & Shenker ’97] & [PV04] Cost-sharing method  (, )  Mechanism M(  ) … for all Q subsets of U… for all Q (possibly) outputted by M Cross… Self Cross…  (Q1,i) (Q1,i) Prices do not decrease Group Strategyproof Changes

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

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

22. Sequential Algorithms A is sequential if for some bid vectors reaches a chain of sets Q 1, …, Q |U|, Q |U|+1 =Ø Sequential algorithms admits a self cross- monotonic cost-sharing method U Q 1 =U Q3Q3 Q |U| … Q2Q2...... … Q |U|+1 = Ø BB & GSP Mechanisms

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

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

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

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

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

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

29. Selfish Task Scheduling M1M2M4M3M5 t1t1 t2t2 t3t3 t4t4 t5t5 b1b1 b2b2 b3b3 b4b4 b5b5 t i = 1 / s i (ie, the inverse of the speed) Optimal Makespan: min x max i t i (X) Mechanism design: payments  utility = payment - cost no VCG! Allocation X  cost = t i (X) = t i load i (X) Awarded independently from the execution!

30. Verifiable Selfish Agents t i = 1 i underbids 1/2 1 3 i’s release time should be 2 but… … i’s finishing time is 4 i overbids 2 1 1 i can wait 2 time slots delivering the results in the right time IDEA ([Nisan & Ronen, 99]): No payment for underbidding agents Verification is impossible! t i (X) = load i (X) t i i bids from the set {1/2, 1, 2} Verification = observe jobs’ release time

31. Verification Setting Give the payment if the results are given in time  Machine i gets positive load when reporting b i 1. t i  b i  just wait and get the payment 2. t i > b i  no payment (punish agent i)

32. The Power of Verification bibi load i Classical mechanismsMechanisms w/ Verification algorithms Payment functions NO! TRUTHFUL bibi load i NO! TRUTHFUL bibi load i titi P i (b i, b -i )= W max / b i (= W max s i ) Related to max possible supported cost Scaling up for general speeds Unique Not unique [Archer & Tardos, ‘01] [Auletta & al, ‘04]

33. The Power of Verification: Breaking Lower Bounds M1M2M4M3M5 b1b1 b2b2 b3b3 b4b4 b5b5 p1p1 p2p2 p3p3 p4p4 p5p5 p6p6 p7p7 p8p8 p9p9 weight priority t1t1 t2t2 t3t3 t4t4 t5t5 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 Efficient APX truthful mechanisms w/verification: c-APX algorithm A  c(1+  )-APX mechanism

34. Generalizing Verification Setting Give the payment if the results are given “in time” (ie, consistently with b i )  For the outcome computed in b i, ie, X 1. t i (X)  b i (X)  just wait and get the payment 2. t i (X) > b i (X)  no payment (punish agent i) t i ( ) b i ( ) General cost functions (e.g., router latency)

35. (Optimal) Mechanisms with Verification J1J1 JjJj JnJn …… M1M1 MiMi MmMm …… b ij b i1 b in …… agent 1 agent h agent k … … There exists truthful mechanism with verification We don’t if truthful mechanisms without verification do exist polytime Breaking lower bounds for classical mechanisms concerning many natural problems (eg, variants of SPT problem) Goal: minimizing the makespan (althougt not polynomial-time) Given an algorithm c-apx… a c(1+  )-apx an exact bibi b1b1 bmbm

36. Optimal Collusion-Resistant Mechanisms w/ Verification Coalition C GSP do not consider side payments U + – 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

37. Conclusions Cost-Sharing Games  Simple techniques… … lead to polynomial-time cost-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

38. 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)

39. Questions?