Optimal Collusion-Resistant Mechanisms with Verification

Carmine Ventre

Joint work with Paolo Penna

Routing in Networkss

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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

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Truthful Mechanisms

M = (A, P)

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Utility (true, , .... , ) ≥ Utility (false, , .... , ) for all true, false, and , ...,

M truthful if:

Utility = Payment – cost = – true

VCG Mechanisms

M = (A, P)

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Pe = Ae=∞ – Ae=0 if e is selected

(0 otherwise)

M is truthful iff A is optimal

Pe’ = Ae’=∞ – Ae’=0 = 7

e’Ae’=∞ = 14

Ae’=0 = 10 – 3 = 7

s

Utilitye’ = Pe’ – coste’ = 7 – 3

d

Inside VCG Payments

Pe = Ae=∞ – Ae=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 in C

VCGs and Collusions

s

3

1

6e1

e2

e3

Pe1(true) = 6 – 1 = 5

e3 reported value

“Promise 10% of my new payment” (briber)

11

Pe1(false) = 11 – 1 – 1 = 9

“Pe3(false)” = 1

bribe

h( ) must be a constantb–e

d

Preventing Collusions is expensive Pay all the agents(!!!)

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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)UtilityC(true)= Pe – 2

true

10+Petrue

11+Petrue

truePe’ = 0

UtilityC(false)=Pe’ – 10false ≥ 10 + Pe – 10 > UtilityC(true)true

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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

31TCP

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 false1. 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, (t1, …, tn))

A(false, (t1, …, tn))

A(false, (b1, …, bn))

= A(false)

A is OPT

For any i ti bi

Cost is monotone

VCG hypotheses

Usage of the constant h for bounded domains

VCGs with verification are collusion-resistant

Any value between bmin e bmax

Approximate CRMs

Extending technique above: Optimize MinMax + AVCG

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