Collusion-Resistant Mechanisms with Verification Yielding Optimal Solutions

1. Collusion-Resistant Mechanisms with Verification Yielding Optimal Solutions Paolo Penna and Carmine Ventre

2. Already on these screens... • Concept of mechanisms with verification • Construction of optimal mechanisms w/ verification • A class of social choice functions admitting CRMs w/ verification for any bounded domain • Construction of optimal truthful mechanisms w/ verification for any bounded domain and any cost function of a certain form • Shown the technique only for finite domains collusion-resistant

3. 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

4. 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

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

6. 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

7. 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 + –

8. 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?

9. 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

10. (The general) 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 ) Utility = Payment – cost = – true

11. Comparison with [NR99] verification setting • Two different declarations • Type • Execution time • verification (reported exe time ≥ true one) • Allocation depends on • Payments depend on ,

12. [NR99] verification: agent not caught (i.e., bid≥ cost) Utility = Payment – reported cost = – bid , NR cost = 10 mins bid = 3 hrs e.g., not usable for TCP example “Physical” assumption

13. [NR99] verification: agent caught (i.e., bid< cost) Utility = Payment – true cost = – cost , NR (Easy) Thm. Mechanism truthful (resp. CR) in [NR99] verification model Mechanism truthful (resp. CR) in our verification model

14. CRMs w/verification for single-parameter bounded domains • Agents aka as “binary” (in/out outcomes) • e.g., controls edges • any number between two known constants bidmin & bidmax

15. CRMs w/verification for single-parameter bounded domains: ideas s • Sufficient Properties • Pay all agents(!!!) • Algorithm 2-resistant 3 10 1 1 2 2 true true 11+Pe 10+Pe Truthfulness 1 3 e 7 • e’ has no way to enter the solution by unilaterally lying • In coalition they can make the cut really expensive 2 true Pe’ = 0 7 10 4 1 e’ bid true UtilityC(bid)=Pe’ – 10 ≥ 10 + Pe– 10 > UtilityC(true) true UtilityC(true)= Pe – 2

16. 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]

17. 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

18. 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

19. 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 -

20. 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

21. Applications • Optimal CRMs for: • MST • k-items auctions • Cheaper payments wrt mechanisms of previous “episode” • Optimal truthful mechanisms for multidimensional agents bidding from bounded domains and non-decreasing cost functions of the form Cost(bid , ..., bid )

22. 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 optimal 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

23. 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? • Threshold-monotone vs. utilitarian algorithms