Mechanisms with Verification

Carmine Ventre

[Penna & V, 2007]

[V, WINE ‘06]

Routing in Networkss

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7

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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

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VCG Mechanisms

s

M = (A, P)

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37

7

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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’=∞ = 10 + 3 + 1

Ae’=0 = 3 + 1 + 2 + 3 + 1 - 3 = 7

s

d

Utilitye’ = Pe’ – coste’ = 7 – 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: The Trusted Resource

Used Car market: The Kelley Blue Book – the Trusted Resource (www.kbb.com)

The Trusted Resource

Can we engage a trusted resource within a mechanism allowing (somehow) bids verification?

Time is trusted…

… unless a time machine will be created

Time is Trusted 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

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 2.61… for truthful mechanisms w/o verification [KV07]

Summarizing…

General Monotone Cost Functions Optimizing monotone nondecreasing cost

functions always admits a truthful mechanism with verification (for bounded domain) Will show technique for Finite Domains

Breaking several lower bounds for natural problems Variants of the SPT [Bilò&Gualà&Proietti, 06] Scheduling Unrelated Machines [Nisan&Ronen,

99, MS07, CKV07, G07, KV07] Interdomain Routing [MS07, G07]

Task Scheduling [Nisan&Ronen’99]

Allocation X costi(X) + ti,n= ti,j

Selfish

• Optimal Makespan: minx maxi costi(X)

• Verification (observe machine behavior)

no VCG!

J1 Jj Jn

… …

M1 Mi Mm… …

b1 bi bm… …

tasks

machines

t1 ti tm… …types

Mechanism design: payments

utility = payment - cost

Verification

Give the payment if the results are given “in time”

Machine i gets job j when reporting bi,j

1. ti,j bi,j just wait and get the payment

2. ti,j > bi,j no payment (punish agent i)

Setup

Agent i holds a resource of type ti

X1,…, Xk feasible solutions

(how we use resources) costi(X) = ti(X) = time utility = payment – cost Goal: minimize m(X,t)

No payment ifti(X) > bi(X) (verification)

Truthful mechanism running an optimal algorithm

(t1,…,tn)

Existence of the Payments

Truthfulness (single player):

P(a) - a(A(a)) P(b) - a(A(b))

a b

truth-tellingP(b) - b(A(b)) P(a) - b(A(a))

X=A(a)Y=A(b)

a(Y) - a(X)

b(X) - b(Y)

Must be non-negative

(a,b)

(b,a)

P(a) + (a,b) P(b)

P(b) + (b,a) P(a)

A() A(, b-i)

P() P(, b-i)

Algorithm

Existence of the Payments

Truthful mechanism (A, P)

Can satisfy all P(a) + (a,b) P(b)

There is no cycle of negative length

a b kc…

[Malkhov&Vohra’04][MV’05][Saks&Yu’05]

[Bikhchandani&Chatterji&Lavi&Mu'alem&Nisan&Sen’06]……

Why Verification Helps

a bX

a(Y) - a(X)

Some edges may “disappear”

Y

True type is “a” but report “b”:1. a(Y) b(Y) can “simulate b” and get P(b)2. a(Y) > b(Y) no payment (verification helps)P(a) - a(X) - a(Y)

0voluntary participation

0nonnegative costs

a(Y) > b(Y)

P(a) - a(X) P(b) - a(Y)

Why Verification Helps

a bX

a(Y) - a(X)

Only these edges remain:

Ya(Y) b(Y)

Negative cycles may desappear

Optimal Mechanisms

Algorithm OPT:

• Fix lexicographic order X1 X2 … Xk• Return the lexicographically minimal Xj minimizing m(b,Xj)

Optimal Mechanisms

a bX Y

a(Y) b(Y)

m(a(X),b-i(X)) m(a(Y),b-i(Y))

cZ

b(Z) c(Z)

X is OPT(a,b-i)

c(X) a(X)

m(•,b-i(Y)) is non-decreasing

m(b(Z),b-i(Z)) m(c(Z),b-i(Z)) m(b(Y),b-i(Y))

m(c(X),b-i(X)) m(a(X),b-i(X))

Optimal Mechanisms

a bX Y

a(Y) b(Y)

m(a(X),b-i(X)) = m(a(Y),b-i(Y))

cZ

b(Z) c(Z)

c(X) a(X)

= m(b(Z),b-i(Z)) = m(c(Z),b-i(Z))= m(b(Y),b-i(Y))

= m(c(X),b-i(X)) = m(a(X),b-i(X))

Z XX Y X=Y=Z

Finite Domains

Theorem: Truthful OPT mechanism with verification for any finite domain and any

m(X,b)=m(b1(X),…,bm(X))

non decreasing in the agents’ costs bi(X)

All vertices in a cycle lead to the same outcome

Different proof of existence of exact truthful mechanism w/ verification for makespan on unrelated machines [Nisan&Ronen‘99]

Compound Agents

J1 Jj Jn

… …

M1 Mi Mm… …

agent1 agentl agentk… …

t1 ti tm… …types b1 bi bm… …

Each agent declares more than a type

Verification for Compound Agents Punish agent i whenever uncovered lying over one

of its dimensions (e.g., machines) Collusion-Resistant mechanisms w/ verification

w.r.t. known coalitions

aX

a(Y) - a(X)bY

a = (a1, a2) b = (b1, b2)

Edge (a,b) exists iff a1(Y) b1(Y) and a2(Y) b2(Y)

OPT is implementable w/verification

Compound Agents

Collusion-Resistant for known coalitions mechanisms w/ verification for makespan on unrelated machines makespan on related machines

J1 Jj Jn

… …

M1 Mi Mm… …

agent1 agentl agentk… …

b1bi bm… …

Polynomial timec (1+) - APX

Exponential time Exact mechanisms

Truthful Grids?

Auction

Can grid nodes declare a completion time before actually executing Homer’s task?

Doughnuts.exe

Conclusions

Mechanisms with Verification: a more powerful model… … breaking known lower bounds for natural problems … dealing with the strongest notion of agents’

collusion … describing real-life applications

Further Research

What is the real power of verification? Does the revelation principle hold in the

verification setting? Different definitions for the verification

paradigm (e.g., [Nisan&Ronen 99])