Strong Implementation of Social Choice Functions in Dominant Strategies Clemens ThielenSven O. Krumke 3rd International Workshop on Computational Social Choice 15 September 2010 TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: A AAAA A A A A AA A

Problem Definition Social choice setting with private information: Strong Implementability

Mechanisms TypesBidsSocial Choices Mechanism: strategy α 1 strategy α n g Strong Implementability Agent 1 Agent n

Utilities and Equilibria Definition: Strong Implementability valuation of the output payment obtained

Utilities and Equilibria Strong Implementability Definition:

Strong Implementation Definition: Strong Implementability

Strong Implementability Problem Strong Implementability Problem The Strong Implementability Problem: Strong Implementability Encoding length:

Augmented Revelation Mechanisms Augmented Revelation Mechanisms Strong Implementability Definition: Augmented Revelation Principle: [Mookherjee, Reichelstein 1990] „incentive compatibility“

Previous Results Strong Implementability

Previous Results (2) Strong Implementability

Our Results Strong Implementability

Augmented Revelation Principle Strong Implementability Augmented Revelation Principle: [Mookherjee, Reichelstein 1990] Augmented Revelation Principle for Dominant Strategies: [this paper]

General Idea (I) Strong Implementability To obtain an augmented revelation mechanism: Definition: see definition to follow soon

Selective Elimination Strong Implementability agent i

Selective Elimination Strong Implementability

Bad Pairs and Elimination Definition: Strong Implementability

Two Important Steps Theorem 2 (selective elimination is necessary): Theorem 3 (selective elimination is sufficient): Strong Implementability

Structure of the Algorithm guess verify Theorem 3 + close look at the proof Definition of selective elimination Strong Implementability

The Verification The Verification Strong Implementability General Approach: Main Observation:

The Payment Polyhedron The Payment Polyhedron Strong Implementability

The Payment Polyhedron (I) The Payment Polyhedron (I) Strong Implementability Inequalities encode which bids are dominant bids. Incentive compatibility & dominant bids

The Payment Polyhedron (II) The Payment Polyhedron (II) Strong Implementability Inequalities encode conditions of selective elimination

The Payment Polyhedron (II) The Payment Polyhedron (II) Strong Implementability Inequalities encode conditions of selective elimination

Verification Issues Strong Implementability 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. Strong Implementability

Conclusion Strong Implementability in dominant strategies  NP 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 Strong Implementability NP- complete!

Thank you! Strong Implementability