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Strong Implementation of Social Choice Functions in Dominant Strategies

This paper explores the concept of Strong Implementability in the context of social choice functions with private information, focusing on mechanisms and utilities. It discusses the Strong Implementability Problem, Augmented Revelation Mechanisms, Selective Elimination, and the Payment Polyhedron in detail. The study provides insights into the complexities of verification and establishes the NP-completeness of the Strong Implementability problem. The research also presents useful results such as the Augmented Revelation Principle and procedures with polynomially many steps. Thank you for reading!

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Strong Implementation of Social Choice Functions in Dominant Strategies

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  1. Strong Implementation of Social Choice Functions in Dominant Strategies Clemens ThielenSven O. Krumke 3rd International Workshop on ComputationalSocial Choice 15 September 2010 krumke@mathematik.uni-kl.de TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAAAAAAAAA

  2. Problem Definition • Social choice setting with private information: Strong Implementability

  3. Mechanisms Mechanism: Types Bids Social Choices Agent 1 strategy α1 . . . g strategy αn Agent n . . . Strong Implementability

  4. Utilities andEquilibria Definition: valuationoftheoutput paymentobtained Strong Implementability

  5. Utilities andEquilibria Definition: Definition: Strong Implementability

  6. Strong Implementation Definition: Strong Implementability

  7. Strong Implementability Problem The Strong Implementability Problem: Encoding length: Strong Implementability

  8. Augmented Revelation Mechanisms Definition: Augmented Revelation Principle: [Mookherjee, Reichelstein 1990] „incentivecompatibility“ Strong Implementability

  9. PreviousResults Strong Implementability

  10. Previous Results (2) Strong Implementability

  11. OurResults Strong Implementability

  12. Augmented Revelation Principle Augmented Revelation Principle: [Mookherjee, Reichelstein 1990] Augmented Revelation Principlefor Dominant Strategies: [thispaper] Strong Implementability

  13. General Idea (I) • To obtain an augmented revelation mechanism: seedefinitiontofollowsoon Definition: Strong Implementability

  14. Selective Elimination agenti Strong Implementability

  15. Selective Elimination Strong Implementability

  16. Bad Pairs and Elimination Definition: Definition: Definition: Strong Implementability

  17. TwoImportantSteps Theorem 2 (selectiveeliminationisnecessary): Theorem 3 (selectiveeliminationissufficient): Strong Implementability

  18. StructureoftheAlgorithm Theorem 3 + closelookattheproof guess Definitionofselectiveelimination guess verify Strong Implementability

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

  20. The Payment Polyhedron Strong Implementability

  21. The Payment Polyhedron (I) Incentivecompatibility & dominant bids Inequalitiesencodewhichbidsare dominant bids. Strong Implementability

  22. The Payment Polyhedron (II) Inequalitiesencodeconditionsofselectiveelimination Strong Implementability

  23. The Payment Polyhedron (II) Inequalitiesencodeconditionsofselectiveelimination Strong Implementability

  24. VerificationIssues Here I am! Strong Implementability

  25. VerificationIssues Wehaveto handle strictinequalities. To do so, we must find a point in therelative interiorofthepolyhedron. Thiscanbedonebymeansofthe Ellipsoid Method (directly) orbysolving a sequenceof LPs. Byproduct: Paymentsareofpolynomialencodinglength. Strong Implementability

  26. Conclusion NP-complete! • Strong Implementability in dominant strategiesÎNP • Characterizationresultgeneralizesto infinite type spaces • Open: Is theproblem in P? • Useful(?) results: • Augmented Revelation Principle • Selectiveeliminationprocedurewithpolynomiallymanysteps • Paymentsofpolynomialencodingsize Strong Implementability

  27. Thank you! Strong Implementability

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