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

Routing in Networks s Internet Change over time (link load) Private Cost No Input Knowledge Selfishness d

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 d

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

VCG Mechanisms M = (A, P) 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

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

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

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

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

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 d

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])

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

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 )

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

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

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

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

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