(Optimal) Collusion-Resistant Mechanisms with Verification

Paolo Penna & Carmine Ventre

Università degli Studi di Salerno

Italy

Routing in Networkss

12

3

10

2

1

1

4

37

7

1

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

12

3

10

2

1

1

4

37

7

1

d

VCG Mechanisms

s

M = (A, P)

12

310

2

1

1

4

37

7

1

Pe = Ae=∞ – Ae=0 if e is selected

(0 otherwise)

M is truthful iff A is optimal

Pe’ = Ae’=∞ – Ae’=0 = 5

e’Ae’=∞ = 10 + 3 + 1

Ae’=0 = 3 + 1 + 2 + 3 + 1 - 3 = 9

s

d

Utilitye’ = Pe’ – coste’ = 5 – 3

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 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

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

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

31TCP

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 bi

1. ti bi just wait and get the payment

2. ti > bi 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, (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

Problem has a truthful VCG Problem has an optimal CRM

Any value between bmin e bmax

Approximating CRMs

Extending technique above: Optimize MinMax + AVCG

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]