(Optimal) Collusion-Resistant Mechanisms with Verification Paolo Penna & Carmine Ventre Università degli Studi di Salerno Italy

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

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 d

VCG Mechanisms s 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 = 5 e’ A e’=∞ = A e’=0 = = 9 s d Utility e’ = P e’ – cost e’ = 5 – 3

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  1,030,000 results on Google  1,635 results on Google news Are VCG mechanisms resistant to collusions?

VCGs and Collusions s d 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

Constructing Collusion-Resistant Mechanisms (CRMs) h is a constant function 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

Verification Setting Give the payment if the results are given “in time”  Agent i is selected when reporting b i 1. t i  b i  just wait and get the payment 2. t i > b i  no payment (punish agent i)

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 Problem has a truthful VCG Problem has an optimal CRM Any value between b min e b max

Approximating CRMs Extending technique above: Optimize MinMax + A VCG Example of MinMax objective functions Interdomain routing Scheduling Unrelated Machines MinMax objective functions admit a (1+ε)-apx CRM Lower bound of 2.7… for truthful mechanisms w/o verification

General Monotone Cost Functions Optimizing monotone nondecreasing cost functions always admits a truthful mechanism with verification (for bounded domain) Breaking several lower bounds for natural problems  Variants of the SPT [Gualà&Proietti, 06]  Minimizing weighted sum scheduling [Archer&Tardos, 01]  Scheduling Unrelated Machines [Nisan&Ronen, 99]