1. Mechanisms with Verification Carmine Ventre Teesside University

2. Mechanism design PrincipalAgents M = (A, P)

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 1. Apart from few simple settings, only utilitarian problems admit truthful mechanisms 2. Mechanisms cannot be resistant to collusions 3. 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 “Only” utilitarian problems have truthful mechanisms Mechanisms not resistant to collusion Approximate truthful mechanisms for CAs 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 [Penna & V, 08], [V06][Penna & V, 09][Krysta & V, 10]

9. Collusion-resistant mechanisms with verification

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

11. VCG Mechanisms 1 2 3 10 2 1 1 4 3 7 7 1 P e’ = A e’=∞ – A e’=0 = 7 e’ A e’=∞ = 14 A e’=0 = 10 – 3 = 7 s Utility e’ = P e’ – cost e’ = 7 – 3 M = (A, P) A optimal algorithm P e = A e=∞ – A e=0 d

12. Inside VCG Payments P e = A e=∞ – A e=0 Cost of best solution w/o e Independent of e h(b –e ) Cost of computed solution w/ e = 0 Mimimum (A is OPT) A(true)  A(false) b –e all but eCost nondecreasing 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 Coalition C + – ∑ Utility (true, true,,...., ) ≥ ∑ Utility (false,false,,...., ) for all true, false, C and,..., in C

15. VCGs and Collusions s 3 1 6e1e1 e2e2 e3e3 P e 1 (true) = 6 – 1 = 5 e 3 reported value “Promise 10% of my new payment” (briber) 11 P e 1 (false) = 11 – 1 – 1 = 9 “P e3 (false)” = 1 bribe h( ) must be a constantb –e d

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 ensure it?“Impossible” for classical mechanisms ([GH05]&[S00])

17. Describing Real World: Verification TCP segment starts at time t  Expected delivery is time t + 1…  … but true delivery time is t + 3 It is possible to partially verify declarations by observing delivery time Other examples:  Distance  Amount of traffic  Routes availability 31 TCP

18. The Verification Setting Give the payment if the results are given “in time”  Agent is selected when reporting false 1. true  false  just wait and get the payment 2. true > false  no payment (punish agent )

19. Exploiting Verification: Optimal CRMs No agent is caught by verification At least one agent is caught by verification A(true) = A(true, (t 1, …, t n ))  A(false, (t 1, …, t n ))  A(false, (b 1, …, b n )) = A(false) A is OPT For any i t i  b i Cost is monotone VCG hypotheses Usage of the constant h for bounded domains Thm. VCGs with verification are collusion-resistant Any value between b min e b max

20. Approximate CRMs Technique can be extended: Optimize Cost + A VCG for any function Cost MinMax extensively studied in AMD  E.g., Interdomain routing and Scheduling Unrelated Machines  Many lower bounds even for two players and exponential running time mechanisms E.g., [NR99], [AT01], [GP06], [CKV07], [MS07], [G07], [PSS08], [MPSS09] Thm. MinMax objective functions admit a (1+ε)-apx CRM

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. Abstract setup Agent i holds a resource of type t i X1,…, Xk feasible solutions (how we use resources) cost i (X) = t i (X) = time utility = payment – cost Goal: minimize m(X, t ) No payment if t i (X) > b i (X) (verification) (t 1,…,t n )

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

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

26. Why Verification Helps ab X a(Y) - a(X) Some edges may “disappear” Y True type is “a” but report “b”: 1.a(Y)  b(Y)  can “simulate b” and get P(b) 2.a(Y) > b(Y)  no payment (verification helps) P(a) - a(X)  P(b) - a(Y) P(a) - a(X)  - a(Y)  0 voluntary participation  0 nonnegative costs a(Y) > b(Y)

27. Why Verification Helps ab X a(Y) - a(X) Only these edges remain: Y a(Y)  b(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. Optimal Mechanisms ab XY a(Y)  b(Y) m(a(X),b -i (X))  m(a(Y),b -i (Y)) c Z b(Z)  c(Z) X is OPT(a,b -i ) c(X)  a(X) m(,b -i (Y)) is non-decreasing  m(b(Z),b -i (Z))  m(c(Z),b -i (Z))  m(b(Y),b -i (Y))  m(c(X),b -i (X))  m(a(X),b -i (X))

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

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

32. Type-based verification

33. Principal-Agent Classical Model Outcome function g “Implement” f Maximize utility f:D->O social choice function Declaration domain D Observe type 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 No Payment issued

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)=xg(t’)=x t t’ D There are no alternatives to truthfulness f(t)=g(t)

35. Toy Example: Tall-Short f > 180 cm > X2X1 f

36. Implementation of Tally-Short f t1 D = {t1, t2, t3} X1 X2 g=f types ti(x2) > ti(x1) f is truthfully implementable iff there are no negative-weight edges t1(x1)-t1(x2)<0 t2(x2)-t2(x1)>0 t2=[181-190] t3=[190+] t1=[170-180] t2t3 t2(x2)-t2(x2)=0 t3(x2)-t3(x2)=0 t3(x2)-t3(x1)>0 f is not truthfully implementablenor implementable

37. Principal-Agent Model with Partial Verification [Green&Laffont 86] t1 X1 X2 < t2t3 = = < > > 20+ cm BR(t) is a t’ in M(t) such that utility t(g(t’)) is maximized t defines a set of allowed messages M(t) t2=[181-190] t3=[190+] t1=[170-180]

38. M-Implementation of Tally-Short f [GL86] show that Revelation Principle holds only if NRC holds  Nested Range Condition t1 X1 X2 t2t3 = = < > f X1 X2g Yes! There are alternatives to truthfulness! tt’t’’ holds in uninteresting cases [Singh&Wittman, 2001]

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 polytime if algorithm is Optimal truthful mechanisms for any non-decreasing cost function when agents bid from bounded domains  Sometimes, computing payments might be unfeasible

40. Further Research Can we deal with unbounded domains? What is the real power of verification? Frugality of payment schemes? Mechanisms with verification without money? [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 untruthful mechanisms  Probabilistic verification [Caragiannis, Elkind, Szegedy & Yu, 2012] ……

41. Truthful Grids? Auction Can grid nodes declare a completion time before actually executing Homer’s task? Doughnuts.exe