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Mechanisms with Verification

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Mechanisms with Verification

Carmine Ventre

TeessideUniversity

M = (A, P)

Principal

Agents

- Mechanisms with verification use the execution of their algorithmic component as a tool to verify agents’ job
- Payments awarded after the execution…
- … and given only if job done “properly”

- (At least) Three different models
- No monitoring […, Penna & V 09, …]
- Full monitoring [Nisan & Ronen 99]
- Type-based verification [Green & Laffont 86]

- No monitoring
- Agents only work only for the time they really need to complete the job

- Full monitoring
- Agents work for the time they declared to the principal

- Incentive-compatibility constraints impose a number of limitations on mechanisms
- Apart from few simple settings, onlyutilitarian problems admit truthful mechanisms
- Mechanisms cannot be resistant to collusions
- Computational complexity: can we approximate OPT in a truthful way?
- Combinatorial Auctions (CAs) is the paradigmatic problem for which OPT is truthful but NP-hard

Without Verification

With verification

Optimal truthful mechanisms for any non-decreasing cost function

Optimal collusion-resistant mechanisms for weakly-utilitarian cost functions

Truthful deterministic polytime CAs with best apx guarantee possible

- “Only” utilitarian problems have truthful mechanisms
- Mechanisms not resistant to collusion
- Approximate truthful mechanisms for CAs

[Penna & V, 08], [V06]

[Penna & V, 09]

[Krysta & V, 10]

Collusion-resistant mechanisms with verification

M = (A, P)

s

Utility = Payment – cost = – true

M truthful if:

d

Utility (true, , .... , ) ≥ Utility (false, , .... , )

for all true, false, and , ...,

M = (A, P)

Pe’ = Ae’=∞ – Ae’=0 = 7

Ae’=∞ = 14

s

e’

3

Ae’=0 = 10 – 3 = 7

10

1

1

2

2

1

3

7

4

7

1

d

A optimalalgorithm

Pe = Ae=∞ – Ae=0

Utilitye’ = Pe’ – coste’ = 7 – 3

Pe = Ae=∞ – Ae=0

Cost of computed solution w/ e = 0

Cost of best solutionw/o e

Mimimum (A is OPT)

Independent of e

h(b–e)

A(true) A(false)

b–e all but e

Costnondecreasing in the agents’ bids

- Accused of bribery
- ~7,000,000 results on Google
- ~6,000 results on Google news

∑ Utility (true, true, , .... , ) ≥ ∑ Utility (false,false, , .... , )

for all true, false, C and , ...,

in C

in C

Coalition C

+

–

e3 reported value

Pe1(true) = 6 – 1 = 5

s

Pe1(false) = 11 – 1 – 1 = 9

3

e3

e1

11

6

“Pe3(false)” = 1

bribe

1

e2

d

h( ) must be a constant

b–e

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

- h is a constant function
- A(true) A(false)

Coalition C

(A, VCG payments) is a CRM

How to ensureit?

“Impossible” forclassicalmechanisms ([GH05]&[S00])

- TCP segmentstartsat time t
- Expected delivery is time t + 1…
- … buttrue delivery time is t + 3

- Itispossible to partiallyverifydeclarations by observing delivery time
- Otherexamples:
- Distance
- Amount of traffic
- Routesavailability

TCP

3

1

- Give the payment if the results are given “in time”
- Agent is selected when reporting false
- truefalse just wait and get the payment
- true>false no payment (punish agent )

Thm.VCGswith verification are collusion-resistant

For any i ti bi

No agent is caught by verification

A(true) = A(true, (t1, …, tn))

A is OPT

A(false, (t1, …, tn))

Costis monotone

A(false, (b1, …, bn))

VCG hypotheses

= A(false)

At least one agent is caught by verification

Usage of the constant h for boundeddomains

Anyvaluebetweenbmin e bmax

Thm.MinMaxobjectivefunctionsadmit a (1+ε)-apx CRM

- Technique can be extended: OptimizeCost + AVCG for anyfunctionCost
- MinMax extensively studied in AMD
- E.g., Interdomain routing and SchedulingUnrelatedMachines
- Manylowerboundsevenfortwoplayers and exponentialrunningtimemechanisms
- E.g., [NR99], [AT01], [GP06], [CKV07], [MS07], [G07], [PSS08], [MPSS09]

* = FPTAS for a constant number of machines

# = PTAS for a constant number of machines

† = FPTAS for any number of machines

Truthful mechanisms for monotone cost functions

No payment if

ti(X)>bi(X)

(verification)

(t1,…,tn)

- Agent i holds a resource of typeti
- X1,…,Xk feasible solutions
(how we use resources)

- costi(X) = ti(X) = time
- utility = payment – cost
- Goal: minimize m(X,t)

truth-telling

a

b

(a,b)

a(Y) - a(X)

X=A(a)

Y=A(b)

Algorithm

b(X) - b(Y)

(b,a)

Must be non-negative

A() A(, b-i)

P() P(, b-i)

Truthfulness (single player):

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

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

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

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

There is no cycle of negative length

a

b

k

c

…

Truthful mechanism (A, P)

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

[Malkhov&Vohra’04][MV’05][Saks&Yu’05]

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

a(Y) > b(Y)

a

b

a(Y) - a(X)

0

0

voluntary participation

nonnegative costs

Some edges may “disappear”

X

Y

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

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

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

a

b

a(Y) - a(X)

Only these edges remain:

X

a(Y) b(Y)

Y

Negative cycles may disappear

- Algorithm OPT:
- Fix lexicographic order
- X1 X2 … Xk
- Return the lexicographically minimal
- Xj minimizing m(b,Xj)

a

b

c

X is OPT(a,b-i)

m(•,b-i(Y)) is non-decreasing

a(Y) b(Y)

b(Z) c(Z)

X

Y

Z

c(X) a(X)

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

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

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

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

a

b

c

X=Y=Z

a(Y) b(Y)

b(Z) c(Z)

X

Y

Z

c(X) a(X)

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

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

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

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

X Y

Z

X

All vertices in a cycle lead to the same outcome

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

m(X,b)

non decreasing in the agents’ costs

*Similar result can be proved for bounded domains with a different technique

Type-based verification

Maximize utility

“Implement” f

No Payment issued

Outcome function g

Declaration domain D

f:D->O social choice function

Observetype t in D

Declare BR(t)

BR(t) is a t’ in D such that utility t(g(t’)) is maximized

Outcome g(BR(t)) is implemented

- g implements f iff
g(BR(t))=f(t)

- g truthfully implements f iff g implements f &
BR(t)=t

Revelation Principle: for all f

f implementable f truthfully implementable

f(t)=g(t)

D

f(t)=x

g(t’)=x

t’

t

There are no alternatives to truthfulness

f

>180 cm

>

X2

X1

D = {t1, t2, t3}

t1=[170-180]

t2=[181-190]

ti(x2) > ti(x1)

t3=[190+]

t1(x1)-t1(x2)<0

t1(x1)-t1(x2)<0

t2(x2)-t2(x2)=0

t2

t1

t3

types

t2(x2)-t2(x1)>0

t3(x2)-t3(x2)=0

g=f

X1

X2

X2

t3(x2)-t3(x1)>0

f is truthfully implementable iff there are no negative-weight edges

f is not truthfully implementable

nor implementable

t1=[170-180]

t2=[181-190]

t3=[190+]

<

<

=

t1

t2

t3

20+ cm

>

=

X1

X2

X2

>

t defines a set of allowed messages M(t)

BR(t) is a t’ in M(t) such that utility t(g(t’)) is maximized

<

=

t1

t2

t3

>

=

X1

X2

X2

f

g

X1

X1

X2

- [GL86] show that Revelation Principle holds only if NRC holds
- Nested Range Condition

holds in uninteresting cases

t

t’

t’’

[Singh&Wittman, 2001]

Yes! There are alternatives to truthfulness!

- 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

- Collusion-Resistant mechanisms with verification for arbitrary bounded domains optimizing generalization of utilitarian (VCG) cost functions
- Mechanism is polytimeif algorithm is

- Optimal truthfulmechanisms for any non-decreasing cost function when agents bid from bounded domains
- Sometimes, computing payments might be unfeasible

- Can we deal with unbounded domains?
- Whatis the realpower of verification?
- Frugality of payment schemes?
- Mechanismswith verificationwithoutmoney? [Koutsoupias11], [Fotakis, Krysta & V, ongoing]
- Explore different definitions for the verification paradigm
- [Nisan&Ronen, 1999]
- [Green & Laffont, 1986]...
- ... for which we can also look for untruthfulmechanisms

- Probabilisticverification [Caragiannis, Elkind, Szegedy & Yu, 2012]
- …