Collusion-Resistant Mechanisms with Verification Yielding Optimal Solutions

1. Collusion-Resistant Mechanisms with Verification Yielding Optimal Solutions Carmine Ventre (University of Liverpool) Joint work with: Paolo Penna (University of Salerno)

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

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

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

5. Optimization & Truthful Mechanisms • Objectives in contrast • Many lower bounds (even for two players and exponential running time mechanisms) • Variants of the SPT [Gualà&Proietti, 06] • Minimizing weighted sum scheduling [Archer&Tardos, 01] • Scheduling Unrelated Machines [Nisan&Ronen, 99], [Christodoulou & Koutsoupias & Vidali 07], … • Workload minimization in interdomain routing [Mu’alem & Schapira, 07], [Gamzu, 07] • & a brand new computational lower bound • CPPP [Papadimitriou &Schapira & Singer, 08] • Study of optimal truthful mechanisms

6. Collusion-Resistant Mechanisms ∑ Utility (true, true, , .... , ) ≥ ∑ Utility (bid, bid, , .... , ) • CRMs are “impossible” to achieve • Posted price [Goldberg & Hartline, 05] • Fixed output [Schummer, 02] • Unbounded apx ratios for all true, bid, C and , ..., in C in C Coalition C + –

7. Describing Real World: Collusions • “Accused of bribery” • 1,030,000 results on Google • 1,635 results on Google news • Can we design CRMs using real-world information?

8. Describing Real World: Verification • 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 TCP 3 1 IDEA ([Nisan & Ronen, 99]): No payment for agents caught by verification

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

10. CRMs w/verification for single-parameter bounded domains s • Agents aka as “binary” (in/out outcomes) • e.g., controls edges • Sufficient Properties • Pay all agents(!!!) • Algorithm 2-resistant 3 10 1 1 2 2 true true 10+Pe 11+Pe 1 3 7 Truthfulness true Pe’ = 0 e 7 • e’ has no way to enter the solution by unilaterally lying • In coalition they can make the cut really expensive 2 4 1 10 e’ bid true UtilityC(bid)=Pe’ – 10 ≥ 10 + Pe– 10 > UtilityC(true) true UtilityC(true)= Pe – 2

11. Truthful Mechanisms w/ Verification: the threshold bid < in bid > out (A,P) truthful with verification A(bid, ) ths ths ths in out bid [Auletta&De Prisco&Penna&Persiano,04]

12. 2-resistant Algorithms t=(true, true, , .... , ) t-=(true , , .... , ) b-=(bid , , .... , ) b=(bid, bid, , .... , ) bid ≥ true (Verification doesn’t work) b’ = t’= t’ b’ ≥ t’ b’ in ths ths ths ths out

13. Exploiting Verification: CRMs w/verification h - if out Payment (b) = h if in b’ ths • (A,Payment) is a CRM w/ verification Thm. Algorithm A 2-resistant Proof Idea. At least one agent is caught by verification Usage of the constant h for bounded domains any number between bidmin & bidmax

14. Proof (continued) • Each is not worse by truthtelling • No agent is caught by verification h - if out Payment (b) = t b h if in in in out in in out out in out out t’ t’ b’ b’ b’ t’ true true ths ths ths ths ths ths • h - true = Utility (b) = Utility (b) • h - true ≥ h - b’ Utility (t) = Utility (t) = ths • h - ≥ h - true • h - ≥ h -

15. Simplifying Resistance Condition t- b- b=(bid , , .... , ) b-=(bid , , .... , ) b=(bid, bid, , .... , ) t-=(true , , .... , ) t=(true, true, , .... , ) t=(true , , .... , ) bid ≥ true (Verification doesn’t work) bid ≥ true b’ = b’ = t’= t’= in Optimal CRMs t’ t’ b’ b’ ≥ b’ t’ out in ths ths ths ths ths ths Thm. Optimal threshold-monotone algorithms with fixed tie breaking are n-resistant out

16. Applications • Optimal CRMs for: • MST • k-items auctions • Cheaper payments wrt [Penna&V,08] • Optimal truthful mechanisms for multidimensional agents bidding from bounded domains and non-decreasing cost functions of the form Cost(bid , ..., bid )

17. Multidimensional Agents Outcomes = {X1, ..., Xm} View bid as a virtual coalition C of m single-parameter agents bid =(bid(X1), .... ,bid(Xm)) b=(bid , ..., bid ) B(b) optimal algorithm with fixed tie breaking rule A(bid ) m single-player functions P (b) = ∑ payment (bid ) in C Lemma. If every A is m-resistant then (B,P) is truthful Thm. For non-decreasing cost function of the form Cost(bid , ..., bid ) every A is threshold-monotone Every A is m-resistant (B,P) is truthful

18. Conclusions • Optimal CRMs with verification for single-parameter bounded domains • Optimal truthful mechanisms for multidimensional bounded domains • Construction tight (removing any of the hypothesis we get an impossibility result) • Overcome many impossibility results by using a real-world hypothesis (verification) • For finite domains: Mechanisms polytimeif algorithm is • Can we deal with unbounded domains?