1. Mechanisms with Verification Carmine Ventre [Penna & V, 2007] [V, WINE ‘06]

2. Routing in Networks s 1 2 3 10 2 1 1 4 3 7 7 1 d Internet Change over time (link load) Private Cost No Input Knowledge Selfishness

3. Mechanisms: Dealing w/ Selfishness Augment an algorithm with a payment function The payment function should incentive in telling the truth Design a truthful mechanism s 1 2 3 10 2 1 1 4 3 7 7 1 d

4. VCG Mechanisms s M = (A, P) 1 2 3 10 2 1 1 4 3 7 7 1 P e = A e=∞ – A e=0 if e is selected (0 otherwise) M is truthful iff A is optimal P e’ = A e’=∞ – A e’=0 = 7 e’ A e’=∞ = 10 + 3 + 1 A e’=0 = 3 + 1 + 2 + 3 + 1 - 3 = 7 s d Utility e’ = P e’ – cost e’ = 7 – 3

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

6. Describing Real World: Collusions Accused of bribery  1,030,000 results on Google  1,635 results on Google news Are VCG mechanisms resistant to collusions?

7. VCGs and Collusions s d 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

8. 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])

9. VCG weaknesses It is vulnerable to collusion  Collusion-resistant Mechanisms … (stay tuned)

10. Describing Real World: The Trusted Resource Used Car market: The Kelley Blue Book – the Trusted Resource (www.kbb.com)

11. 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

12. Time is Trusted TCP datagram 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 IDEA ([Nisan & Ronen, 99]): No payment for agents caught by verification

13. 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 Problem has a truthful VCG Problem has an optimal CRM Any value between b min e b max

14. Approximating CRMs Extending technique above: Optimize MinMax + A VCG Example of MinMax objective functions Interdomain routing Scheduling Unrelated Machines MinMax objective functions admit a (1+ε)-apx CRM Lower bound of 2.61… for truthful mechanisms w/o verification [KV07]

15. Summarizing…

16. VCG weaknesses It is vulnerable to collusion  Collusion-resistant Mechanisms for VCG problems It works only for utilitarian problems: i.e., minimize the sum of the costs  Mechanisms minimizing any non-decreasing Cost function

17. General Monotone Cost Functions Optimizing monotone nondecreasing cost functions always admits a truthful mechanism with verification (for bounded domain)  Will show technique for Finite Domains Breaking several lower bounds for natural problems  Variants of the SPT [Bilò&Gualà&Proietti, 06]  Scheduling Unrelated Machines [Nisan&Ronen, 99, MS07, CKV07, G07, KV07]  Interdomain Routing [MS07, G07]

18. Task Scheduling [Nisan&Ronen’99] Allocation X  cost i (X) + t i,n = t i,j Selfish Optimal Makespan: min x max i cost i (X) Verification (observe machine behavior) no VCG! J1J1 JjJj JnJn …… M1M1 MiMi MmMm …… b1b1 bibi bmbm …… tasks machines t1t1 titi tmtm …… types Mechanism design: payments  utility = payment - cost

19. Verification Give the payment if the results are given “in time”  Machine i gets job j when reporting b i,j 1. t i,j  b i,j  just wait and get the payment 2. t i,j > b i,j  no payment (punish agent i)

20. 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) Truthful mechanism running an optimal algorithm (t 1,…,t n )

21. 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

22. 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]……

23. 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)  - a(Y)  0 voluntary participation  0 nonnegative costs a(Y) > b(Y) P(a) - a(X)  P(b) - a(Y)

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

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

26. 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))

27. 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

28. Finite Domains Theorem: Truthful OPT mechanism with verification for any finite domain and any m(X,b)=m(b 1 (X),…,b m (X)) non decreasing in the agents’ costs b i (X) All vertices in a cycle lead to the same outcome Different proof of existence of exact truthful mechanism w/ verification for makespan on unrelated machines [Nisan&Ronen‘99]

29. Compound Agents J1J1 JjJj JnJn …… M1M1 MiMi MmMm …… agent 1 agent l agent k … … t1t1 titi tmtm …… types b1b1 bibi bmbm …… Each agent declares more than a type

30. Verification for Compound Agents Punish agent i whenever uncovered lying over one of its dimensions (e.g., machines) Collusion-Resistant mechanisms w/ verification w.r.t. known coalitions a X a(Y) - a(X) b Y a = (a 1, a 2 ) b = (b 1, b 2 ) Edge ( a, b ) exists iff a 1 (Y)  b 1 (Y) and a 2 (Y)  b 2 (Y) OPT is implementable w/verification

31. Compound Agents Collusion-Resistant for known coalitions mechanisms w/ verification for  makespan on unrelated machines  makespan on related machines J1J1 JjJj JnJn …… M1M1 MiMi MmMm …… agent 1 agent l agent k … … b1b1 bibi bmbm …… Polynomial time c (1+  ) - APX Exponential time Exact mechanisms

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

33. 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

34. Further Research 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])