Crm Esa08 1234869680124198 3

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

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

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

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

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 

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

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? 

Describing Real World: Verification TCP datagram starts at time  t Expected delivery is time t +  1… … but true delivery time is t  TCP 1 3 +3 It is possible to partially  verify declarations by observing delivery time Other examples:  Distance  Amount of traffic  Routes availability  IDEA ([Nisan & Ronen, 99]): No payment for agents caught by verification

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 1. true > bid no payment (punish agent ) 2.

CRMs w/verification for single- parameter bounded domains Agents aka as “binary” (in/out outcomes)  e.g., controls edges  Sufficient Properties  Pay all agents(!!!)  s Algorithm 32-resistant  true Truthfulness Pe’ = 0 10 true e 1 1 • e’ has no way to enter the 11+Pe 2 2 solution by unilaterally lying 2 true 1 • In coalition they 7can make the 10+Pe 10 3 e’ cut really expensive 7 4 true 1 Pe – 2 UtilityC(true)= UtilityC(bid)=Pbid – 10 ≥ 10 + Pe – 10 > UtilityC(true) true e’

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

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

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

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

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

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 )

Multidimensional Agents Outcomes = {X1, ..., Xm} View bid as a virtual coalition C bid =(bid(X1), .... ,bid(Xm)) of m single-parameter agents b=(bid , ..., bid ) B(b) optimal algorithm with A(bid ) m single-player fixed tie breaking rule functions P (b) = ∑ payment (bid ) in C Lemma. If every A is m-resistant then (B,P) is truthful Thm. For non-decreasing cost Every A is (B,P) is function of the form m-resistant truthful Cost(bid , ..., bid ) every A is threshold-monotone

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? 

Crm Esa08 1234869680124198 3

by guest27165 on Mar 17, 2009

Accessibility

Categories

Upload Details

Usage Rights

Statistics

Crm Esa08 1234869680124198 3 Presentation Transcript