Alternatives to truthfulness are hard to recognize
Download
1 / 15

Alternatives to Truthfulness Are Hard to Recognize - PowerPoint PPT Presentation


  • 79 Views
  • Uploaded on

Alternatives to Truthfulness Are Hard to Recognize. Carmine Ventre (U. of Liverpool) Joint work with: Vincenzo Auletta & Paolo Penna & Giuseppe Persiano (U. of Salerno). Principal-Agent Classical Model. Maximize utility. “Implement” f. Outcome function g. Declaration domain D.

loader
I am the owner, or an agent authorized to act on behalf of the owner, of the copyrighted work described.
capcha
Download Presentation

PowerPoint Slideshow about ' Alternatives to Truthfulness Are Hard to Recognize' - laura-vincent


An Image/Link below is provided (as is) to download presentation

Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.


- - - - - - - - - - - - - - - - - - - - - - - - - - E N D - - - - - - - - - - - - - - - - - - - - - - - - - -
Presentation Transcript
Alternatives to truthfulness are hard to recognize

Alternatives to Truthfulness Are Hard to Recognize

Carmine Ventre (U. of Liverpool)

Joint work with:

Vincenzo Auletta & Paolo Penna & Giuseppe Persiano (U. of Salerno)


Principal agent classical model
Principal-Agent Classical Model

Maximize utility

“Implement” f

Outcome function g

Declaration domain D

f:D->O social choice function

Observe his type 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

Principal awards no payment


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

types

t1

t2

t3

t2(x2)-t2(x1)>0

t3(x2)-t3(x2)=0

g=f

X1

X2

X2

t3(x2)-t3(x1)>0

Tested in time poly in |D|

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

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 [Green&Laffont 86]

<

=

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!


But they are hard to find
But They are Hard to Find [Green&Laffont 86]

  • Reduction from 3SAT for the following problem

    Implementability

    Input: D, O, f, M

    Task: exists g M-implementing f?

  • We start from a formula with clauses C1,…, Cm and variables x1,…, xn


The gadget for the variable xi
The gadget for the variable xi [Green&Laffont 86]

  • ti(F)>ti(T)

  • ui(F)>ui(T)

  • vi(T)>vi(F)

  • wi(T)>wi(F)

?

T

F

?

T

T

T

g(vi)=g(wi)=F unvalid assignment

g(vi)=F “means” xi=FALSE

vi , wi literal nodes of the gadget

g(wi)=F “means” xi=FALSE (ie, xi=TRUE)


The gadget for the clause cj
The gadget for the clause Cj [Green&Laffont 86]

  • cj(F)<cj(T)

  • dj(T)>dj(F)

T

F

F

To the literal nodes in the variable-gadgets


The reduction
The Reduction [Green&Laffont 86]

F

F

T

T

F

T

F

F

x1=TRUE

x2= FALSE

x3=FALSE

x1=TRUE

x2=*

x3=*

  • If formula is sat, then the assignment defines g implementing f

  • If f is implementable, g defines an assignment sat the formula


Easy m s
“Easy” M’s [Green&Laffont 86]

  • Hardness holds even for outcome sets of size 2 and M’s of maximum outdegree 3

  • Implementability is polynomial-time solvable when the M is a collection of path and cycles (ie, maximum outdegree 1)

    • Simple reduction from 2SAT

  • Gap: Maximum outdegree 2?


Quasi linear agents
Quasi-Linear Agents [Green&Laffont 86]

Maximize utility

Outcome function g

“Implement” f

Payment function p

Declaration domain D

f:D->O social choice function

Observe his type t in D

Declare BR(t)

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


Hardness for qlu agent
Hardness for QLU Agent [Green&Laffont 86]

  • Testing if f is M-truthfully implementable is “easy”

    • Check that there are no negative-weight cycle in weighted graph

  • (Even for outcome sets of size 2) testing M-implementability is hard

    • Reduction similar in spirit to the previous one


Conclusions
Conclusions [Green&Laffont 86]

  • TestingM-truthfulimplementabilityis easy in bothcases

  • Hardnessdepends on the freedomofagents in lying

    • 3 ways: hard

    • 1 way: easy

  • Usealternativestotruthfulnesstoimplement social choicefunctions (more interestingthanTally-Shortone) otherwisenotimplementable


ad