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Optimal Collusion-Resistant Mechanisms with Verification Carmine Ventre Joint work with Paolo Penna

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

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

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

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VCG Mechanisms 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’=∞ = 14 A e’=0 = 10 – 3 = 7 s Utility e’ = P e’ – cost e’ = 7 – 3 d

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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

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Describing Real World: Collusions Accused of bribery ~900,000 results on Google 6,463 results on Google news Are VCGs collusion-resistant mechanisms?

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

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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

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Preventing Collusions is expensive Pay all the agents(!!!) 2 10 e e’ Truthfulness e’ to enter the solution by unilaterally lying must underbid (competition, i.e., non-cooperative behaviour) In coalition they can make the cut really expensive (cooperative behaviour) Utility C (true)= P e – 2 true 10+P e true 11+P e true P e’ = 0 Utility C (false)=P e’ – 10 false ≥ 10 + P e – 10 > Utility C (true) true s 1 2 3 10 2 1 1 4 3 7 7 1 d

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Constructing Collusion-Resistant Mechanisms (CRMs) h is a constant function Pay all the agents A(true) A(false) Coalition C (A, VCG payments) is a CRM How to ensure it?“Impossible” for classical mechanisms ([GH05]&[S00])

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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 31 TCP IDEA ([Nisan & Ronen, 99]): No payment for agents caught by verification

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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 )

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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 VCGs with verification are collusion-resistant Any value between b min e b max

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Approximate CRMs Extending technique above: Optimize MinMax + A VCG 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] MinMax objective functions admit a (1+ε)-apx CRM

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Applications * = FPTAS for a constant number of machines # = PTAS for a constant number of machines

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Conclusions Collusion-Resistant mechanisms with verification for arbitrary bounded domains optimizing generalization of utilitarian (VCG) cost functions Overcome many impossibility results by using a real-world hypothesis (verification) Efficient Mechanisms Mechanism is polytime if algorithm is

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Further Research Frugality of payment scheme? Can we deal with unbounded domains? What is the real power of verification? Explore different definitions for the verification paradigm [Nisan&Ronen, 1999] [Green & Laffont, 1986]...... for which we can also look for untruthful mechanisms Apply verification to CAs

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