Mechanisms with Verification

1. Mechanisms with Verification Carmine Ventre TeessideUniversity

2. M = (A, P) Mechanism design Principal Agents

3. When do you pay?

4. Do you pay?

5. Mechanisms with verification • Mechanisms with verification use the execution of their algorithmic component as a tool to verify agents’ job • Payments awarded after the execution… • … and given only if job done “properly” • (At least) Three different models • No monitoring […, Penna & V 09, …] • Full monitoring [Nisan & Ronen 99] • Type-based verification [Green & Laffont 86]

6. No vs. Full monitoring • No monitoring • Agents only work only for the time they really need to complete the job • Full monitoring • Agents work for the time they declared to the principal

7. Why Verification? • Incentive-compatibility constraints impose a number of limitations on mechanisms • Apart from few simple settings, onlyutilitarian problems admit truthful mechanisms • Mechanisms cannot be resistant to collusions • Computational complexity: can we approximate OPT in a truthful way? • Combinatorial Auctions (CAs) is the paradigmatic problem for which OPT is truthful but NP-hard

8. Why Verification? (2) Without Verification With verification Optimal truthful mechanisms for any non-decreasing cost function Optimal collusion-resistant mechanisms for weakly-utilitarian cost functions Truthful deterministic polytime CAs with best apx guarantee possible • “Only” utilitarian problems have truthful mechanisms • Mechanisms not resistant to collusion • Approximate truthful mechanisms for CAs [Penna & V, 08], [V06] [Penna & V, 09] [Krysta & V, 10]

9. Collusion-resistant mechanisms with verification

10. M = (A, P) Truthful Mechanisms s Utility = Payment – cost = – true M truthful if: d Utility (true, , .... , ) ≥ Utility (false, , .... , ) for all true, false, and , ...,

11. M = (A, P) VCG Mechanisms Pe’ = Ae’=∞ – Ae’=0 = 7 Ae’=∞ = 14 s e’ 3 Ae’=0 = 10 – 3 = 7 10 1 1 2 2 1 3 7 4 7 1 d A optimalalgorithm Pe = Ae=∞ – Ae=0 Utilitye’ = Pe’ – coste’ = 7 – 3

12. Inside VCG Payments Pe = Ae=∞ – Ae=0 Cost of computed solution w/ e = 0 Cost of best solutionw/o e Mimimum (A is OPT) Independent of e h(b–e) A(true)  A(false) b–e all but e Costnondecreasing in the agents’ bids

13. Describing Real World: Collusions • Accused of bribery • ~7,000,000 results on Google • ~6,000 results on Google news

14. Collusion-Resistant Mechanisms ∑ Utility (true, true, , .... , ) ≥ ∑ Utility (false,false, , .... , ) for all true, false, C and , ..., in C in C Coalition C + –

15. VCGs and Collusions e3 reported value Pe1(true) = 6 – 1 = 5 s Pe1(false) = 11 – 1 – 1 = 9 3 e3 e1 11 6 “Pe3(false)” = 1 bribe 1 e2 d h( ) must be a constant b–e “Promise 10% of my new payment” (briber)

16. Constructing Collusion-Resistant Mechanisms (CRMs) • h is a constant function • A(true)  A(false) Coalition C (A, VCG payments) is a CRM How to ensureit? “Impossible” forclassicalmechanisms ([GH05]&[S00])

17. Describing Real World: Verification • TCP segmentstartsat time t • Expected delivery is time t + 1… • … buttrue delivery time is t + 3 • Itispossible to partiallyverifydeclarations by observing delivery time • Otherexamples: • Distance • Amount of traffic • Routesavailability TCP 3 1

18. The Verification Setting • Give the payment if the results are given “in time” • Agent is selected when reporting false • truefalse just wait and get the payment • true>false no payment (punish agent )

19. Thm.VCGswith verification are collusion-resistant Exploiting Verification: Optimal CRMs For any i ti bi No agent is caught by verification A(true) = A(true, (t1, …, tn)) A is OPT  A(false, (t1, …, tn)) Costis monotone  A(false, (b1, …, bn)) VCG hypotheses = A(false) At least one agent is caught by verification Usage of the constant h for boundeddomains Anyvaluebetweenbmin e bmax

20. Thm.MinMaxobjectivefunctionsadmit a (1+ε)-apx CRM Approximate CRMs • Technique can be extended: OptimizeCost + AVCG for anyfunctionCost • MinMax extensively studied in AMD • E.g., Interdomain routing and SchedulingUnrelatedMachines • Manylowerboundsevenfortwoplayers and exponentialrunningtimemechanisms • E.g., [NR99], [AT01], [GP06], [CKV07], [MS07], [G07], [PSS08], [MPSS09]

21. Applications * = FPTAS for a constant number of machines # = PTAS for a constant number of machines † = FPTAS for any number of machines

22. Truthful mechanisms for monotone cost functions

23. No payment if ti(X)>bi(X) (verification) (t1,…,tn) Abstract setup • Agent i holds a resource of typeti • X1,…,Xk feasible solutions (how we use resources) • costi(X) = ti(X) = time • utility = payment – cost • Goal: minimize m(X,t)

24. truth-telling a b (a,b) a(Y) - a(X) X=A(a) Y=A(b) Algorithm b(X) - b(Y) (b,a) Must be non-negative Existence of the Payments A() A(, b-i) P()  P(, b-i) Truthfulness (single player): P(a) - a(A(a))  P(b) - a(A(b)) P(a) + (a,b) P(b) P(b) - b(A(b))  P(a) - b(A(a)) P(b) + (b,a)  P(a)

25. There is no cycle of negative length a b k c … Existence of the Payments Truthful mechanism (A, P) Can satisfy all P(a) + (a,b) P(b) [Malkhov&Vohra’04][MV’05][Saks&Yu’05] [Bikhchandani&Chatterji&Lavi&Mu'alem&Nisan&Sen’06]……

26. a(Y) > b(Y) a b a(Y) - a(X)  0  0 voluntary participation nonnegative costs Why Verification Helps Some edges may “disappear” X Y • True type is “a” but report “b”: • a(Y)  b(Y)can “simulate b” and get P(b) • a(Y) > b(Y)no payment (verification helps) P(a) - a(X)  P(b) - a(Y) P(a) - a(X)  - a(Y)

27. a b a(Y) - a(X) Why Verification Helps Only these edges remain: X a(Y)  b(Y) Y Negative cycles may disappear

28. Optimal Mechanisms • Algorithm OPT: • Fix lexicographic order • X1 X2  …  Xk • Return the lexicographically minimal • Xj minimizing m(b,Xj)

29. a b c X is OPT(a,b-i) m(•,b-i(Y)) is non-decreasing Optimal Mechanisms a(Y)  b(Y) b(Z)  c(Z) X Y Z c(X)  a(X) m(a(X),b-i(X))  m(a(Y),b-i(Y))  m(b(Y),b-i(Y))  m(b(Z),b-i(Z))  m(c(Z),b-i(Z))  m(c(X),b-i(X))  m(a(X),b-i(X))

30. a b c X=Y=Z Optimal Mechanisms a(Y)  b(Y) b(Z)  c(Z) X Y Z c(X)  a(X) m(a(X),b-i(X)) = m(a(Y),b-i(Y)) = m(b(Y),b-i(Y)) = m(b(Z),b-i(Z)) = m(c(Z),b-i(Z)) = m(c(X),b-i(X)) = m(a(X),b-i(X)) X  Y  Z  X

31. Finite Domains All vertices in a cycle lead to the same outcome Theorem: Truthful OPT mechanism with verification for any finite domain* and any m(X,b) non decreasing in the agents’ costs *Similar result can be proved for bounded domains with a different technique

32. Type-based verification

33. Principal-Agent Classical Model Maximize utility “Implement” f No Payment issued Outcome function g Declaration domain D f:D->O social choice function Observetype t in D Declare BR(t) BR(t) is a t’ in D such that utility t(g(t’)) is maximized Outcome g(BR(t)) is implemented

34. Implementation of Social choice functions • g implements f iff g(BR(t))=f(t) • g truthfully implements f iff g implements f & BR(t)=t Revelation Principle: for all f f implementable f truthfully implementable f(t)=g(t) D f(t)=x g(t’)=x t’ t There are no alternatives to truthfulness

35. Toy Example: Tall-Short f f >180 cm > X2 X1

36. Implementation of Tally-Short f D = {t1, t2, t3} t1=[170-180] t2=[181-190] ti(x2) > ti(x1) t3=[190+] t1(x1)-t1(x2)<0 t1(x1)-t1(x2)<0 t2(x2)-t2(x2)=0 t2 t1 t3 types t2(x2)-t2(x1)>0 t3(x2)-t3(x2)=0 g=f X1 X2 X2 t3(x2)-t3(x1)>0 f is truthfully implementable iff there are no negative-weight edges f is not truthfully implementable nor implementable

37. Principal-Agent Model with Partial Verification [Green&Laffont 86] t1=[170-180] t2=[181-190] t3=[190+] < < = t1 t2 t3 20+ cm > = X1 X2 X2 > t defines a set of allowed messages M(t) BR(t) is a t’ in M(t) such that utility t(g(t’)) is maximized

38. M-Implementation of Tally-Short f < = t1 t2 t3 > = X1 X2 X2 f g X1 X1 X2 • [GL86] show that Revelation Principle holds only if NRC holds • Nested Range Condition holds in uninteresting cases t t’ t’’ [Singh&Wittman, 2001] Yes! There are alternatives to truthfulness!

39. Conclusions • Mechanisms with Verification: a more powerful model… • … breaking known lower bounds for natural problems • … dealing with the strongest notion of agents’ collusion • …describing real-life applications • Collusion-Resistant mechanisms with verification for arbitrary bounded domains optimizing generalization of utilitarian (VCG) cost functions • Mechanism is polytimeif algorithm is • Optimal truthfulmechanisms for any non-decreasing cost function when agents bid from bounded domains • Sometimes, computing payments might be unfeasible

40. FurtherResearch • Can we deal with unbounded domains? • Whatis the realpower of verification? • Frugality of payment schemes? • Mechanismswith verificationwithoutmoney? [Koutsoupias11], [Fotakis, Krysta & V, ongoing] • Explore different definitions for the verification paradigm • [Nisan&Ronen, 1999] • [Green & Laffont, 1986]... • ... for which we can also look for untruthfulmechanisms • Probabilisticverification [Caragiannis, Elkind, Szegedy & Yu, 2012] • …