Strong Implementation of Social Choice Functions in Dominant Strategies Clemens ThielenSven O....

Strong Implementation of Social Choice Functions in Dominant Strategies

Clemens Thielen Sven O. Krumke

3rd International Workshop on Computational Social Choice

15 September 2010

[email protected]

Problem Definition

Social choice setting with private information:

•

•

•

•

•

•

n jobs j 1; : : : ; j n with processing requirements p1; : : :;pn ¸ 0n jobs j 1; : : : ; j n

Strong Implementability

Mechanisms

Types

Bids

Social Choices

.

.

.

Mechanism:

.

.

.

strategy α1

strategy αn

g


Agent 1

Agent n

Utilities and Equilibria

Definition:


• •

valuation of the output

payment obtained

Utilities and Equilibria


• •

Definition:

Definition:

Strong Implementation

Definition:


Strong Implementability Problem

The Strong Implementability Problem:


Encoding length:

Augmented Revelation Mechanisms


Definition:

Augmented Revelation Principle:[Mookherjee, Reichelstein 1990]

„incentive compatibility“

Previous Results

•

•

•


Previous Results (2)

•

•

•


Our Results

•

•

•


Augmented Revelation Principle


Augmented Revelation Principle:[Mookherjee, Reichelstein 1990]

Augmented Revelation Principle for Dominant Strategies:[this paper]

General Idea (I)


To obtain an augmented revelation mechanism:•

•

Definition:

see definition to follow soon

Selective Elimination


agent i

SN

¹s1

¹s2

Selective Elimination


Bad Pairs and Elimination Definition:

Definition:

Definition:


Two Important Steps

Theorem 2 (selective elimination is necessary):

Theorem 3 (selective elimination is sufficient):


Structure of the Algorithm

guess

guess

verify

Theorem 3 + close look at the proof

Definition of selective elimination


The Verification


General Approach:

Main Observation:

The Payment Polyhedron


The Payment Polyhedron (I)


Inequalities encode which bids are dominant bids.

Incentive compatibili

ty&

dominant bids

The Payment Polyhedron (II)


Inequalities encode conditions of selective elimination

Verification Issues


Here I am!

Verification Issues

We have to handle strict inequalities.To do so, we must find a point in the

relative interior of the polyhedron.This can be done by means of the

Ellipsoid Method (directly) or by solving a sequence of LPs.

Byproduct: Payments are of polynomial encoding length.


Conclusion

Strong Implementability in dominant strategiesÎNP

Characterization result generalizes to infinite type spaces

Open: Is the problem in P?

Useful(?) results:◦ Augmented Revelation Principle◦ Selective elimination procedure with polynomially many

steps◦ Payments of polynomial encoding size


NP-complete

!

Thank you!


Strong Implementation of Social Choice Functions in Dominant Strategies Clemens ThielenSven O....

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