Collusion-Resistant Mechanisms with Verification Yielding Optimal Solutions

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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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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 (link load). No Input Knowledge. 3. 10. 1. 1. 2. Selfishness. - PowerPoint PPT Presentation

Transcript of Collusion-Resistant Mechanisms with Verification Yielding Optimal Solutions

Page 1: Collusion-Resistant Mechanisms with Verification Yielding Optimal Solutions

Collusion-Resistant Mechanisms with Verification Yielding Optimal Solutions

Carmine Ventre (University of Liverpool)

Joint work with: Paolo Penna (University of Salerno)

Page 2: Collusion-Resistant Mechanisms with Verification Yielding Optimal Solutions

Routing in Networkss

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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

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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 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

31TCP

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 bid1. 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-resistants

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e

e’

Truthfulness • e’ has no way to enter the

solution by unilaterally lying• In coalition they can make the

cut really expensive

UtilityC(true)= Pe – 2true

10+Petrue

11+Petrue

truePe’ = 0

UtilityC(bid)=Pe’ – 10bid ≥ 10 + Pe – 10 > UtilityC(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

ths

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2-resistant Algorithms

t=(true, true, , .... , )

ths

b’

ths

t’≥

b’ =

b=(bid, bid, , .... , )

t’ =

in

out

thsb’

thst’

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 bidmin & bidmax

Payment (b) =

h - if outths

b’

h if in

Thm. Algorithm A 2-resistant (A,Payment) is a CRM w/ verification

Proof Idea.

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Proof (continued)

in

out

thsb’

thst’

No agent is caught by verification Each is not worse by truthtelling

btin in

in

in

out

out out

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

thsb’

h - ≥ h -ths

t’ths

b’ h - true ≥ h -ths

b’

true

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Simplifying Resistance Conditiont=(true, true, , .... , )

ths

b’

ths

t’≥

b’ =

b=(bid, bid, , .... , )

t’ =

in

out

thsb’

thst’

b- =(bid , , .... , )

t- =(true , , .... , )

bid ≥ true (Verification doesn’t work)

b=(bid , , .... , ) t=(true , , .... , ) bid ≥ true

b’ = b- t’ = t-

in

out

thsb’

thst’

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 AgentsOutcomes = {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?