A Combinatorial Algorithm for Strong Implementation of Social
A Combinatorial Algorithm for Strong Implementation of Social Choice Functions Clemens Thielen Stephan Westphal 3 rd International Workshop on Computational Social Choice 15 September 2010 thielen@mathematik. uni-kl. de
Problem Definition Social choice setting with private information : • • • A Combinatorial Algorithm for Strong Implementation of Social Choice Functions
Problem Definition (2) We saw (previous talk) : • • A Combinatorial Algorithm for Strong Implementation of Social Choice Functions
Our Results • • • A Combinatorial Algorithm for Strong Implementation of Social Choice Functions
System of Inequalities Implementation of a social choice function : Weak Implementation Strong Implementation A Combinatorial Algorithm for Strong Implementation of Social Choice Functions
Node Potential Interpretation System can be interpreted as finding a node potential : constant A Combinatorial Algorithm for Strong Implementation of Social Choice Functions
![Characterization (Weak Implementation) • Theorem: [Gui et al. 2005] A Combinatorial Algorithm for Strong](http://slidetodoc.com/presentation_image_h2/5ffb3b7e25615af648bdbe2c5425e109/image-7.jpg "Characterization (Weak Implementation) • Theorem: [Gui et al. 2005] A Combinatorial Algorithm for Strong")
Characterization (Weak Implementation) • Theorem: [Gui et al. 2005] A Combinatorial Algorithm for Strong Implementation of Social Choice Functions
![Characterization (Strong Implementation) Theorem: [this paper] A Combinatorial Algorithm for Strong Implementation of Social](http://slidetodoc.com/presentation_image_h2/5ffb3b7e25615af648bdbe2c5425e109/image-8.jpg "Characterization (Strong Implementation) Theorem: [this paper] A Combinatorial Algorithm for Strong Implementation of Social")
Characterization (Strong Implementation) Theorem: [this paper] A Combinatorial Algorithm for Strong Implementation of Social Choice Functions
The Algorithm For weak implementation : • • For strong implementation : • Strict inequalities in the system must be fullfilled. A Combinatorial Algorithm for Strong Implementation of Social Choice Functions
Perturbation of Node Potentials • • A Combinatorial Algorithm for Strong Implementation of Social Choice Functions
• • • Perturbation in Graph Gi Slack of arc (x , x ´) Slack of incoming arcs becomes positive A Combinatorial Algorithm for Strong Implementation of Social Choice Functions
Finding the Node x • • You shrink it! A Combinatorial Algorithm for Strong Implementation of Social Choice Functions
Summary of Results • • A Combinatorial Algorithm for Strong Implementation of Social Choice Functions
Thank you! A Combinatorial Algorithm for Strong Implementation of Social Choice Functions
- Slides: 14