strong implementation of social choice functions in dominant strategies clemens thielensven o....
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Strong Implementation of Social Choice Functions in Dominant Strategies
Clemens Thielen Sven O. Krumke
3rd International Workshop on Computational Social Choice
15 September 2010
Problem Definition
Social choice setting with private information:
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n jobs j 1; : : : ; j n with processing requirements p1; : : :;pn ¸ 0n jobs j 1; : : : ; j n
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Mechanisms
Types
Bids
Social Choices
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Mechanism:
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strategy α1
strategy αn
g
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Agent 1
Agent n
Utilities and Equilibria
Definition:
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valuation of the output
payment obtained
Utilities and Equilibria
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Definition:
Definition:
Strong Implementation
Definition:
Strong Implementability
Strong Implementability Problem
The Strong Implementability Problem:
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Encoding length:
Augmented Revelation Mechanisms
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Definition:
Augmented Revelation Principle:[Mookherjee, Reichelstein 1990]
„incentive compatibility“
Previous Results
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Previous Results (2)
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Our Results
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Augmented Revelation Principle
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Augmented Revelation Principle:[Mookherjee, Reichelstein 1990]
Augmented Revelation Principle for Dominant Strategies:[this paper]
General Idea (I)
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To obtain an augmented revelation mechanism:•
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Definition:
see definition to follow soon
Selective Elimination
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agent i
SN
¹s1
¹s2
Selective Elimination
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Bad Pairs and Elimination Definition:
Definition:
Definition:
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Two Important Steps
Theorem 2 (selective elimination is necessary):
Theorem 3 (selective elimination is sufficient):
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Structure of the Algorithm
guess
guess
verify
Theorem 3 + close look at the proof
Definition of selective elimination
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The Verification
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General Approach:
Main Observation:
The Payment Polyhedron
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The Payment Polyhedron (I)
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Inequalities encode which bids are dominant bids.
Incentive compatibili
ty&
dominant bids
The Payment Polyhedron (II)
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Inequalities encode conditions of selective elimination
The Payment Polyhedron (II)
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Inequalities encode conditions of selective elimination
Verification Issues
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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.
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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
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NP-complete
!
Thank you!
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