Mechanisms with verification
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Mechanisms with Verification. Carmine Ventre Teesside University. M = (A, P ). Mechanism design. Principal. Agents. When do you pay?. Do you pay?. Mechanisms with verification.

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Mechanisms with verification

Mechanisms with Verification

Carmine Ventre

TeessideUniversity


Mechanism design

M = (A, P)

Mechanism design

Principal

Agents


When do you pay

When do you pay?


Do you pay

Do you pay?


Mechanisms with verification1

Mechanisms with verification

  • 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 vs full monitoring

No vs. Full monitoring

  • 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


Why verification

Why Verification?

  • 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


Why verification 2

Why Verification? (2)

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

Collusion-resistant mechanisms with verification


Truthful mechanisms

M = (A, P)

Truthful Mechanisms

s

Utility = Payment – cost = – true

M truthful if:

d

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

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


Vcg mechanisms

M = (A, P)

VCG Mechanisms

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


Inside vcg payments

Inside VCG Payments

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


Describing real world collusions

Describing Real World: Collusions

  • Accused of bribery

    • ~7,000,000 results on Google

    • ~6,000 results on Google news


Collusion resistant mechanisms

Collusion-Resistant Mechanisms

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

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

in C

in C

Coalition C

+


Vcgs and collusions

VCGs and Collusions

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)


Constructing collusion resistant mechanisms crms

Constructing Collusion-Resistant Mechanisms (CRMs)

  • h is a constant function

  • A(true)  A(false)

Coalition C

(A, VCG payments) is a CRM

How to ensureit?

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


Describing real world verification

Describing Real World: Verification

  • 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


The verification setting

The Verification Setting

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

    • Agent is selected when reporting false

    • truefalse just wait and get the payment

    • true>false no payment (punish agent )


Exploiting verification optimal crms

Thm.VCGswith verification are collusion-resistant

Exploiting Verification: Optimal CRMs

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


Approximate crms

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

Approximate CRMs

  • 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]


Applications

Applications

* = 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

Truthful mechanisms for monotone cost functions


Abstract set up

No payment if

ti(X)>bi(X)

(verification)

(t1,…,tn)

Abstract setup

  • 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)


Existence of the payments

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

Existence of the Payments

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)


Existence of the payments1

There is no cycle of negative length

a

b

k

c

Existence of the Payments

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]……


Why verification helps

a(Y) > b(Y)

a

b

a(Y) - a(X)

 0

 0

voluntary participation

nonnegative costs

Why Verification Helps

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)


Why verification helps1

a

b

a(Y) - a(X)

Why Verification Helps

Only these edges remain:

X

a(Y)  b(Y)

Y

Negative cycles may disappear


Optimal mechanisms

Optimal Mechanisms

  • Algorithm OPT:

  • Fix lexicographic order

  • X1 X2  …  Xk

  • Return the lexicographically minimal

  • Xj minimizing m(b,Xj)


Optimal mechanisms1

a

b

c

X is OPT(a,b-i)

m(•,b-i(Y)) is non-decreasing

Optimal Mechanisms

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))


Optimal mechanisms2

a

b

c

X=Y=Z

Optimal Mechanisms

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


Finite domains

Finite Domains

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

Type-based verification


Principal agent classical model

Principal-Agent Classical Model

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


Implementation of social choice functions

Implementation of Social choice functions

  • 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


Toy example tall short f

Toy Example: Tall-Short f

f

>180 cm

>

X2

X1


Implementation of tally short f

Implementation of Tally-Short f

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


Principal agent model with partial verification green laffont 86

Principal-Agent Model with Partial Verification [Green&Laffont 86]

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


M implementation of tally short f

M-Implementation of Tally-Short f

<

=

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!


Conclusions

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

  • 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


Further research

FurtherResearch

  • 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]


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