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

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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

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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

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

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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

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

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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?

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

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

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

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

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

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

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

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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 )

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Multidimensional Agents Outcomes = {X1,..., Xm} bid =(bid(X1),....,bid(Xm)) b=(bid,..., bid ) B(b) optimal algorithm with fixed tie breaking rule A(bid ) m single-player functions View bid as a virtual coalition C of m single-parameter agents 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

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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 polytime if algorithm is Can we deal with unbounded domains?

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