1. A Combinatorial Algorithm for Strong Implementation of Social Choice Functions Clemens ThielenStephan Westphal 3rd International Workshop on Computational Social Choice 15 September 2010 thielen@mathematik.uni-kl.de

2. Problem Definition Social choice setting with private information: A Combinatorial Algorithm for Strong Implementation of Social Choice Functions

3. Problem Definition (2) We saw (previous talk): A Combinatorial Algorithm for Strong Implementation of Social Choice Functions

4. Our Results A Combinatorial Algorithm for Strong Implementation of Social Choice Functions

5. System of Inequalities Implementation of a social choice function : A Combinatorial Algorithm for Strong Implementation of Social Choice Functions Weak Implementation Strong Implementation

6. Node Potential Interpretation System can be interpreted as finding a node potential: A Combinatorial Algorithm for Strong Implementation of Social Choice Functions constant

7. Characterization (Weak Implementation) Characterization (Weak Implementation) A Combinatorial Algorithm for Strong Implementation of Social Choice Functions Theorem:[Gui et al. 2005]

8. Characterization (Strong Implementation) Characterization (Strong Implementation) A Combinatorial Algorithm for Strong Implementation of Social Choice Functions Theorem:[this paper]

9. The Algorithm The Algorithm A Combinatorial Algorithm for Strong Implementation of Social Choice Functions For weak implementation : For strong implementation : Strict inequalities in the system must be fullfilled.

10. Perturbation of Node Potentials Perturbation of Node Potentials A Combinatorial Algorithm for Strong Implementation of Social Choice Functions

11. Perturbation in Graph G i Perturbation in Graph G i A Combinatorial Algorithm for Strong Implementation of Social Choice Functions Slack of arc ( x, x ´) Slack of incoming arcs becomes positive

12. Finding the Node x A Combinatorial Algorithm for Strong Implementation of Social Choice Functions You shrink it!

13. Summary of Results A Combinatorial Algorithm for Strong Implementation of Social Choice Functions

14. Thank you! A Combinatorial Algorithm for Strong Implementation of Social Choice Functions