Gibbard-Satterthwaite theorem

• Definition of efficiency
• How can the consequence $x\in X]beefficientunderthepreferencepair[$ \succsim \in \mathscr{D}_I$?
  • $y{\succsim }_{i}x\forall i\in I$
  • $y{\succ }_{j}x\exists j\in I$
• is that there is no consequence $y$ that satisfies
  • In other words, if any one person can improve without making anyone worse off, that is not efficient.
  • social choice function $f:{\mathcal{D}}_i\to X$, if the consequence $f(\succsim )]isefficientunderthepreference[$ \succsim] for all preference pairs [$\in {\mathcal{D}}_I$ $f$ is said to be efficient.

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